Course Development Team

Head of Programme : Assoc Prof Tan Chong Hui

Course Developer(s) : Assoc Prof Joseph Lim

Technical Writer : Chloe Chong, ETP

Video Production : Danny Chin, ETP

© 2021 Singapore University of Social Sciences. All rights reserved.

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permission in writing from the Educational Technology & Production, Singapore

University of Social Sciences.

ISBN 9789814873949

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How to cite this Study Guide (APA):

Lim, J. (2021). FIN358 Fixed income and derivative securities (study guide). Singapore

University of Social Sciences.

Release V1.0

Build S1.0.5, T1.5.21

Table of Contents

Table of Contents

Course Guide

1. Welcome…………………………………………………………………………………………………… CG-2

2. Course Description and Aims…………………………………………………………………. CG-3

3. Learning Outcomes…………………………………………………………………………………. CG-5

4. Learning Material……………………………………………………………………………………. CG-6

5. Assessment Overview……………………………………………………………………………… CG-7

6. Course Schedule………………………………………………………………………………………. CG-9

7. Learning Mode………………………………………………………………………………………. CG-10

Study Unit 1: Basics of Options, Forwards, Futures, and Swaps

Learning Outcomes……………………………………………………………………………………. SU1-2

Overview……………………………………………………………………………………………………. SU1-3

Chapter 1: Call and Put Options………………………………………………………………… SU1-4

Chapter 2: Simple Option Strategies…………………………………………………………… SU1-7

Chapter 3: Option Pricing Bounds……………………………………………………………. SU1-21

Chapter 4: Forwards, Futures and Swaps…………………………………………………. SU1-28

Formative Assessment……………………………………………………………………………… SU1-39

References………………………………………………………………………………………………… SU1-45

Study Unit 2: Valuation of Derivative Instruments

Learning Outcomes……………………………………………………………………………………. SU2-2

i

Table of Contents

Overview……………………………………………………………………………………………………. SU2-3

Chapter 1: Introduction to Option Valuation……………………………………………… SU2-4

Chapter 2: Valuation of Options Using the Binomial Option Pricing

Model……………………………………………………………………………………………………….. SU2-12

Chapter 3: Black-Scholes Option Pricing Model……………………………………….. SU2-22

Chapter 4: Option Greeks and Volatility Smile…………………………………………. SU2-29

Chapter 5: Valuation of Futures Contracts and Swaps……………………………… SU2-39

Formative Assessment……………………………………………………………………………… SU2-54

References………………………………………………………………………………………………… SU2-62

Study Unit 3: Derivative Hedging and Combination Strategies

Learning Outcomes……………………………………………………………………………………. SU3-2

Overview……………………………………………………………………………………………………. SU3-3

Chapter 1: Hedging Using Derivatives………………………………………………………. SU3-4

Chapter 2: Trading Strategies Involving Options……………………………………… SU3-15

Formative Assessment……………………………………………………………………………… SU3-21

References………………………………………………………………………………………………… SU3-28

Study Unit 4: Fixed Income Elements, TermStructure and Valuation

Learning Outcomes……………………………………………………………………………………. SU4-2

Overview……………………………………………………………………………………………………. SU4-3

Chapter 1: Elements of Fixed Income…………………………………………………………. SU4-4

Chapter 2: Term Structure of Interest Rates……………………………………………… SU4-17

Chapter 3: Introduction To Fixed Income Valuation…………………………………. SU4-24

ii

Table of Contents

Formative Assessment……………………………………………………………………………… SU4-33

References………………………………………………………………………………………………… SU4-41

Study Unit 5: Understanding Fixed Income Risk and Return

Learning Outcomes……………………………………………………………………………………. SU5-2

Overview……………………………………………………………………………………………………. SU5-3

Chapter 1: Yield Measures, Spot Rates, and Forward Rates……………………….. SU5-4

Chapter 2: Fixed Income Risk: A Preview………………………………………………… SU5-12

Chapter 3: The Measurement of Interest Rate Risk…………………………………… SU5-15

Chapter 4: Fixed Income Credit Risk………………………………………………………… SU5-33

Formative Assessment……………………………………………………………………………… SU5-41

References………………………………………………………………………………………………… SU5-49

Study Unit 6: Asset-Backed Securities and Valuation and Analysis

of Bonds With Embedded Options

Learning Outcomes……………………………………………………………………………………. SU6-2

Overview……………………………………………………………………………………………………. SU6-3

Chapter 1: Mortgage-Backed Securities……………………………………………………… SU6-4

Chapter 2: Asset-Backed Securities…………………………………………………………… SU6-16

Chapter 3: Valuing Bonds With Embedded Options………………………………… SU6-23

Formative Assessment……………………………………………………………………………… SU6-32

References………………………………………………………………………………………………… SU6-40

iii

Table of Contents

iv

List of Tables

List of Tables

Table 1.1 Cash flows of long call, today and at future date…………………………… SU1-22

Table 1.2 Cash flows of long stock and short call, today and at future

date……………………………………………………………………………………………………………….. SU1-23

Table 1.3 Cash flows of long call and short stock today……………………………….. SU1-23

Table 1.4 Cash flows of long call with longer expiration date and short call with

shorter expiration date, today and at future date…………………………………………. SU1-24

Table 1.5 Cash flows of long call with lower exercise price and short call with

higher exercise price, today and at future date…………………………………………….. SU1-25

Table 1.6 Cash flows of long call, long bond and short stock, today and at future

date……………………………………………………………………………………………………………….. SU1-26

Table 1.7 Open interest illustration……………………………………………………………….. SU1-33

Table 1.8 Marking-to-market example………………………………………………………….. SU1-34

Table 2.1 Determinants of option value………………………………………………………… SU2-10

Table 2.2 Cash flows from buying 1 call……………………………………………………….. SU2-13

Table 2.3 Cash flows from buying 1 share and borrowing $46.30………………….. SU2-13

Table 2.4 Payoffs to a portfolio of long 1 share and short 3 calls…………………… SU2-14

Table 2.5 Payoffs to a portfolio of long 2 shares and short 3 calls………………….. SU2-17

Table 2.6 Values of u and d as the number of sub-periods increase………………. SU2-19

Table 2.7 Option Greeks and the effect on call and put options of changes in the

value of the Greek………………………………………………………………………………………… SU2-30

v

List of Tables

Table 2.8 Floating rate swap cash flows……………………………………………………….. SU2-51

Table 4.1 Payment schedules for bullet, fully amortised and partially amortised

bonds………………………………………………………………………………………………………………. SU4-7

Table 5.1 Future value of reinvested cash flows……………………………………………… SU5-7

Table 6.1 Scheduled payments on a sequential pay CMO…………………………….. SU6-11

vi

List of Figures

List of Figures

Figure 1.1 Payoff diagrams for long stock and short stock……………………………… SU1-8

Figure 1.2 Profit diagrams for long stock and short stock………………………………. SU1-9

Figure 1.3 Payoff diagrams for long call and short call…………………………………. SU1-10

Figure 1.4 Profit diagrams for long call and short call………………………………….. SU1-11

Figure 1.5 Payoff diagrams for long put and short put…………………………………. SU1-12

Figure 1.6 Profit diagrams for long put and short put………………………………….. SU1-12

Figure 1.7 Profit diagrams for long stock and long put………………………………… SU1-15

Figure 1.8 Profit diagrams for combined long stock and long put………………… SU1-15

Figure 1.9 Profit diagrams for long call and long put…………………………………… SU1-16

Figure 1.10 Profit diagrams for long straddle……………………………………………….. SU1-17

Figure 1.11 Profit diagram for short straddle………………………………………………… SU1-17

Figure 1.12 Payoff diagrams for long stock and long put……………………………… SU1-18

Figure 1.13 Payoff diagrams for combined long stock and long put……………… SU1-19

Figure 1.14 Payoff diagrams for long call and long bond……………………………… SU1-19

Figure 1.15 Payoff diagrams for combined long call and long bond……………… SU1-20

Figure 1.16 Bounds for option price……………………………………………………………… SU1-26

Figure 1.17 Hedging with forward contract………………………………………………….. SU1-30

Figure 1.18 Interest rate swap……………………………………………………………………….. SU1-37

vii

List of Figures

Figure 2.1 Intrinsic value, time value and option price for a call…………………….. SU2-6

Figure 2.2 Two-period binomial trees……………………………………………………………. SU2-16

Figure 2.3 Four-period binomial trees…………………………………………………………… SU2-18

Figure 2.4 Probability distributions for final stock price……………………………….. SU2-20

Figure 2.5 Call option delta curves……………………………………………………………….. SU2-32

Figure 2.6 Call option gamma curves…………………………………………………………… SU2-33

Figure 2.7 Volatility smile……………………………………………………………………………… SU2-36

Figure 4.1 Treasury STRIP…………………………………………………………………………….. SU4-12

Figure 4.2 Yield curve shapes……………………………………………………………………….. SU4-18

Figure 4.3 Term structure of interest rates…………………………………………………….. SU4-23

Figure 5.1 Yield curves showing Z-spread and OAS for a callable bond……….. SU5-10

Figure 5.2 Price-yield relationship………………………………………………………………… SU5-16

Figure 5.3 Price yield curves of callable and non-callable bonds…………………… SU5-18

Figure 5.4 Price-yield curve for bond with embedded put……………………………. SU5-19

Figure 5.5 Duration: Slope of price-yield curve…………………………………………….. SU5-21

Figure 5.6 Duration and convexity: Slope of price-yield curve……………………… SU5-22

Figure 5.7 Reproduction of Figure 5.5: Duration: Slope of price-yield

curve……………………………………………………………………………………………………………… SU5-26

Figure 5.8 Parallel shift in flat yield curve……………………………………………………. SU5-28

Figure 5.9 Non-parallel shift in upward sloping yield curve:……………………….. SU5-29

viii

List of Figures

Figure 5.10 Non-parallel shift in upward sloping yield curve: Non-proportional

change in spot rates………………………………………………………………………………………. SU5-30

Figure 6.1 Pass-through securities………………………………………………………………….. SU6-8

Figure 6.2 An illustration of tranching and the waterfall structure……………….. SU6-10

Figure 6.3 Binomial tree for 6.5% coupon callable bond with maturity of 4

years……………………………………………………………………………………………………………… SU6-26

ix

List of Figures

x

List of Lesson Recordings

List of Lesson Recordings

Derivatives Basics…………………………………………………………………………………………… SU1-3

Option Basics………………………………………………………………………………………………….. SU1-4

Option Bounds……………………………………………………………………………………………… SU1-21

Forwards and Futures…………………………………………………………………………………… SU1-28

Binomial Option Pricing Model……………………………………………………………………. SU2-12

Black-Scholes Option Pricing Model…………………………………………………………….. SU2-22

Option Greeks……………………………………………………………………………………………….. SU2-29

Volatility Smile……………………………………………………………………………………………… SU2-35

Interest Rate Swap………………………………………………………………………………………… SU2-50

Interest Rate Futures………………………………………………………………………………………. SU3-4

Currency Derivatives……………………………………………………………………………………… SU3-9

Credit Derivatives…………………………………………………………………………………………. SU3-12

Stock Index Futures………………………………………………………………………………………. SU3-13

Spreads………………………………………………………………………………………………………….. SU3-16

Bond Types and Characteristics……………………………………………………………………… SU4-4

Term Structure………………………………………………………………………………………………. SU4-17

Yield Measures, Spot Rates and Forward Rates……………………………………………… SU5-4

Asset-Backed Securities…………………………………………………………………………………… SU6-4

xi

List of Lesson Recordings

Valuing Bonds With Embedded Options………………………………………………………. SU6-23

xii

Course

Guide

Fixed Income and Derivative

Securities

FIN358 Course Guide

1. Welcome

Presenter: Assoc Prof Joseph Lim

This streaming video requires Internet connection. Access it via Wi-Fi to

avoid incurring data charges on your personal mobile plan.

Click here to watch the video. i

Click here for the transcript.

Welcome to the course FIN358, Fixed Income and Derivative Securities, a 5 credit unit

(CU) course.

This Study Guide will be your personal learning resource to take you through the course

learning journey. The guide is divided into two main sections – the Course Guide and

Study Units.

The Course Guide describes the structure for the entire course and provides you with an

overview of the Study Units. It serves as a roadmap of the different learning components

within the course. This Course Guide contains important information regarding the

course learning outcomes, learning materials and resources, assessment breakdown and

additional course information.

i

CG-2

FIN358 Course Guide

2. Course Description and Aims

This course provides you with fundamental knowledge of fixed income and derivative

securities. You will learn about the characteristics of the different securities and how those

characteristics affect their risks and returns. The pricing of these instruments will also be

covered. Various pricing models or methods will be discussed.

For fixed income securities, you will learn how to compute risk and return measures as

well as how various risks or characteristics can affect the return.

Derivatives are often used for hedging purposes and this course helps you understand

the process of hedging. Finally fixed income securities with embedded options will be

examined and a demonstration of how the binomial option pricing model can be adapted

to price such bonds.

Course Structure

This course is a 5-credit unit course presented over six weeks.

There are six Study Units in this course. The following provides an overview of each Study

Unit.

Study Unit 1 – Basics of Options, Forwards, Futures and Swaps

This unit helps you understand the basic characteristics of derivative securities. It

discusses simple option strategies and examines the bounds to option prices.

Study Unit 2 – Valuation of Derivative Securities

This unit will focus on the intrinsic and time value of options. It discusses how the

binomial and Black-Scholes option pricing models can be used to value options. The

sensitivity of price of the option to changes in its determinants will be examined. Finally,

futures and swap valuation will be explained.

CG-3

FIN358 Course Guide

Study Unit 3 – 1. Hedging Using Interest Rate, Currency, Credit and Equity

Derivatives; 2. Options and Futures Combination Securities

The aim of this Study Unit is the application of derivatives for hedging purposes.

Study Unit 4 – Fixed Income Elements, Term Structure and Valuation

This unit will cover the basic characteristics of fixed income securities. The term structure

of interest rates will be examined. This is followed by some fixed income valuation

methods.

Study Unit 5 – Understanding Fixed Income Risk and Return

This unit describes the risk and return of fixed income securities. Several measures of fixed

income risk will be developed.

Study Unit 6 – Asset-Backed Securities and Valuation and Analysis of

Bonds with Embedded Options

This unit introduces the characteristics of asset-backed securities. It also combines the

methodology of the binomial tree which is covered in Study Unit 2 to value bonds which

have embedded options.

CG-4

FIN358 Course Guide

3. Learning Outcomes

Knowledge & Understanding (Theory Component)

By the end of this course, you will be able to:

• Understand the characteristics of various fixed income and derivative securities

• Value fixed income and derivative securities using various methods and models

• Apply your understanding of derivative securities for hedging purposes

Key Skills (Practical Component)

By the end of this course, you will be able to:

• Use software such as Excel to compute the value of fixed income and derivative

securities

• Use Excel to implement the binomial tree model for option valuation as well as the

valuation of bonds with embedded options

• Discuss the use of derivatives for hedging risk

CG-5

FIN358 Course Guide

4. Learning Material

To complete the course, you will need the following learning material(s):

Required Textbook(s)

Gottesman, A. (2016). Derivatives essentials: An introduction to forwards, futures, options and

swaps. Wiley.

Adams, J. F., & Smith, D. J. (2019). Fixed income analysis (4th ed.). Wiley.

If you are enrolled into this course, you will be able to access the eTextbooks here:

To launch eTextbook, you need a VitalSource account which can be created via

Canvas (iBookStore), using your SUSS email address. Access to adopted eTextbook is

restricted by enrolment to this course.

CG-6

FIN358 Course Guide

5. Assessment Overview

The overall assessment weighting for this course is as follows:

Assessment Description Weight Allocation

Participation Participation in class/

online

6%

PCOQ Pre-course quiz 6%

TMA Tutor-Marked Assignment 18%

GBA Group-Based Assignment 20%

ECA End-Course Assessment 50%

TOTAL 100%

The following section provides important information regarding Assessments.

Pre-Course Quiz:

Students are to take the Pre-Course Quiz (PCOQ) prior to embarking on the course. This

constitutes 6% of the course assessment.

Continuous Assessment:

There will be continuous assessment in the form of one Tutor-Marked Assignment

(TMAs) and one Group-Based Assignment, which together constitute 38 percent of the

overall assessment for this course. The two assignments are compulsory and are nonsubstitutable. They will test your understanding of both the fundamental and more

advanced concepts and applications that underlie fixed income and derivative securities.

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FIN358 Course Guide

It is imperative that you read through your Assignment questions and submission

instructions before embarking on your Assignment.

In addition, students will be evaluated on their participation whether in-class or online.

This constitutes 6% of the course assessment.

End-Course Assessment:

The End-Course assessment constitutes the other 50 percent of overall student assessment

and will test your ability to apply marketing-related concepts, theories, and strategies to

particular situations commonly faced by fixed income and derivative securities managers.

All topics covered in the course outline will be assessed.

Passing Mark:

To successfully pass the course, you must obtain a minimum passing mark of 40 percent

for the weighted average of the PCOQ, TMA, GBA and Participation components.

You must also obtain a minimum mark of 40 percent for the final exam. For detailed

information on the course grading policy, please refer to the Student Handbook (Award of

Grades section under Assessment and Examination Regulations). The Student Handbook

is available on the Student Portal.

Non-graded Learning Activities:

Each study unit consists of activities for self-directed learning. These learning activities

are meant to help you assess your own understanding and achievement of the learning

outcomes. The activities can be in the form of Formative Assessments, Quizzes, Review

Questions, or Application-Based Questions. You are expected to complete the suggested

activities either independently or collaboratively.

CG-8

FIN358 Course Guide

6. Course Schedule

To pace yourself and monitor your study progress, pay special attention to your

Course Schedule. It contains study-unit-related activities including Assignments, SelfAssessments, and Examinations. Please refer to the Course Timetable on the Student

Portal for the most current Course Schedule.

Note: Always make it a point to check the Student Portal for announcements and

updates.

CG-9

FIN358 Course Guide

7. Learning Mode

The learning approach for this course is structured along the following lines:

a. Self-study guided by the study guide units. Independent study will require at

least 3 hours per week.

b. Working on assignments, either individually or in groups.

c. Classroom Seminars (3 hours each session, 6 sessions in total).

iStudyGuide

You may be viewing the interactive StudyGuide (iStudyGuide), which is the mobilefriendly version of the Study Guide. The iStudyGuide is developed to enhance your

learning experience with interactive learning activities and engaging multimedia. You

will be able to personalise your learning with digital bookmarking, note-taking, and

highlighting of texts if your reader supports these features.

Interaction with Instructor and Fellow Students

Flexible learning—learning at your own pace, space, and time—is a hallmark at SUSS,

and we strongly encourage you to engage your instructor and fellow students in online

discussion forums. Sharing of ideas through meaningful debates will help broaden your

perspective and crystallise your thinking.

Academic Integrity

As a student of SUSS, you are expected to adhere to the academic standards stipulated

in the Student Handbook, which contains important information regarding academic

policies, academic integrity, and course administration. It is your responsibility to read

and understand the information outlined in the Student Handbook prior to embarking on

the course.

CG-10

Study

Unit1

Basics of Options, Forwards,

Futures, and Swaps

FIN358 Basics of Options, Forwards, Futures, and Swaps

Learning Outcomes

By the end of this unit, you should be able to:

1. Describe the basics of call and put options

2. Discuss simple option strategies and the put-call parity relationship

3. Appraise option pricing bounds

4. Describe the basics of forwards, futures, and swaps

SU1-2

FIN358 Basics of Options, Forwards, Futures, and Swaps

Overview

Derivatives are important financial instruments that serve several purposes.

Derivatives are often used to hedge risks. For example, manufacturers use futures

contracts to hedge the risk of price increases in the raw materials they use. Meanwhile,

farmers use futures contracts to lock in the price for their produce. For airlines, fuel is a big

cost. So, airlines often use futures and swaps to hedge against fuel price increases in order

to avoid increasing ticket prices when the oil price increases. In the financial sector, fund

managers may use options and futures to hedge their portfolios’ positions. In addition,

derivatives allow fund managers alternatives when taking and exiting positions in the

financial markets.

In this study unit, we will learn the basic features of options and futures, as well as

illustrate how they can be used for simple hedging. More elaborate examples of hedging

will be discussed in Study Unit 3.

Lesson Recording

Derivatives Basics

SU1-3

FIN358 Basics of Options, Forwards, Futures, and Swaps

Chapter 1: Call and Put Options

Lesson Recording

Option Basics

1.1 Call and Put Options

Call options and put options are derivative instruments – that is, their values are derived

or dependent on the value of some other asset. That asset is known as the underlying, for

short.

A call option is a financial instrument or contract that gives the owner the right, but not

the obligation, to buy an asset at a fixed price, known as the exercise price or strike price,

on or before a given date known as the expiration date.

For a put option, the owner has the right, but not the obligation, to sell an asset at the

exercise price on or before the expiration date.

Note that for brevity, a call option and a put option are just referred to as a call or a put.

The following are terminology associated with the terms in an option contract:

1. Option buyer is the purchaser of the option.

2. Option writer is the seller of the option.

3. Exercise price is the price that the call buyer needs to pay the call writer for an

asset, if he so wishes, or the price which the put buyer will get when he chooses

to sell the asset to the put writer. The exercise price is set by the exchange when

it lists the option for trading. For options that are not listed – that is, the so-called

over-the-counter options – the exercise price is agreed upon by the buyer and

the seller of the option.

SU1-4

FIN358 Basics of Options, Forwards, Futures, and Swaps

4. Exercise is the act of the buyer of the option exercising his right as stated in the

contract. When a call is exercised, the call buyer tenders the call together with

the exercise price and receives the asset. If it is a stock call option, the call buyer

would receive one share of the company from the call option seller.

However, if a put option is exercised, the put buyer tenders the put together

with the underlying asset and receives the exercise price. If the underlying asset

consists of shares of a company, then when the put is exercised, the put owner

would receive the exercise price.

Note that buyers and sellers of options are third parties. Hence for equity options,

the exercise of calls or puts would not involve company at all.

5. American options and European options differ with regard to when the option can

be exercised. American options can be exercised at any time before expiration

while European options can only be exercised on the expiration date.

Note:

For the rest of the chapter, we use stocks as the underlying asset.

1.1.1 When Is Option Exercise Profitable?

Exercising a call is profitable only if the stock price is greater than the exercise price. For

example, suppose the stock price is $6 and the exercise price is $5. Exercising the call would

involve giving the call together with the $5 exercise price to the option seller and receiving

one share in return. The share can then be sold for $6, netting a profit of $1 ($6 – $5). The

difference between the stock price and the exercise price or (S – X) is known as the intrinsic

value of the call. So, exercising a call makes sense only if the intrinsic value is positive.

Suppose, in the example above, the stock price is $3. Would the call be exercised? Of course

not. Exercising the call means paying $5 for a share that is only worth $3. In this case a call

will not be exercised if the stock price is below the exercise price, or if the intrinsic value

is negative, since the call buyer has the choice whether to do so or not.

SU1-5

FIN358 Basics of Options, Forwards, Futures, and Swaps

For a put, exercising it makes sense only if the stock price is lower than the exercise price.

Suppose the stock price is $6 and the exercise price $10. Exercising the put would involve

buying one share at $6 and tendering this together with the put to receive the exercise

price of $10. This nets a profit of $4 ($10 – $6).

The intrinsic value for a put is the difference between the exercise price and the stock price,

or (X – S). As in a call, the put would only be exercised if its intrinsic value is positive. If

the intrinsic value is negative, the put buyer would do nothing.

Read

Read the following sections from the Derivatives Essentials textbook:

• Gottesman, A. (2016). 2.1 Call option characteristics. In Derivatives Essentials: An

introduction to forwards, futures, options, and swaps (pp. 22-25). Wiley.

• Gottesman, A. (2016). 3.1 Put options characteristics. In Derivatives Essentials:

An introduction to forwards, futures, options, and swaps (pp. 44-46). Wiley.

SU1-6

FIN358 Basics of Options, Forwards, Futures, and Swaps

Chapter 2: Simple Option Strategies

In this section we will discuss how options can be used. Simple option strategies will

include combining stocks with options, and an option with other options. It is easier

to understand these strategies graphically and we will use tools like payoff and profit

diagrams.

2.1 Payoff and Profit Diagrams

Payoff and profit diagrams show the long and short positions for an investment. Payoff

diagrams are useful if we want to compare the values of different investments. If two

investments have the same payoffs, then both investments should cost the same. We will

use this idea later, for example, in demonstrating the put-call parity relationship.

The payoff diagram shows the values of a position for different stock prices. For example,

consider two situations regarding an investment. In the first situation you bought 100

shares of a company at $2 each. If the share price is $5, the value of your shares would be

$500. If you were to sell the shares, the payoff would be $500. Does the payoff change if

you had paid $4 per share? No, you will still get $500 if you sell the 100 shares.

The profit diagram brings in the cost of the investment in the situations above. For the first

situation, if you sell the shares at $5 per share, the profit would be $300, as the profit per

share is difference between the price you sold and the price you bought, that is, $5 less $2

or $3 per share multiplied by 100 shares. For the second situation, the profit is only $100

for your investment.

The diagrams below show the payoffs and profit diagrams for long and short positions

in stock, calls and puts. (A long position is one where you bought the asset, while a short

position is one where you sold the asset.)

SU1-7

FIN358 Basics of Options, Forwards, Futures, and Swaps

2.1.1 Stock: Payoff and Profit Diagrams

Figure 1.1 Payoff diagrams for long stock and short stock

If you have a long position in a stock, the value of your position increases with increases

in the price of the stock. Looking at the first graph on the left in the diagram above, the

value of the long stock position (or the payoff) increases as the stock price rises. Notice

that the payoff starts at zero if the stock price is zero, and that it increases dollar for dollar

with an increase in the stock price. Another way to put this is that the payoff graph is at

an angle of 45 degrees to the horizontal axis.

For the short position in stock (see graph on the right), the value of the position starts at

zero when the price of the stock is zero and decreases with increases in the stock price. In

other words, your position decreases in value when the stock price increases, the opposite

effect of the long stock position.

Another way to look at the two diagrams is that the short stock position is the mirror

image of long stock position, if we were to place the mirror along the horizontal axis. This

is the idea of a zero-sum game – what one party gains, the counterparty loses. Note that

this zero-sum concept applies across the other payoff and profit diagrams shown below

as well.

SU1-8

FIN358 Basics of Options, Forwards, Futures, and Swaps

One thing to note regarding the payoff diagrams for the long and short stock positions is

that there is unlimited upside for the long stock position, as the graph extends upwards

in the north-east direction. But for the short stock position, the downside is unlimited.

This means that the investors who take short positions should be aware of the risk they

are undertaking. They need to be vigilant about their position and institute stop-loss

arrangements for their positions.

Figure 1.2 Profit diagrams for long stock and short stock

For the stock profit diagrams, we bring in the cost of initiating the position. If you buy the

stock for $5 and the price remains at $5, you don’t make a single cent – that is, the profit is

zero. You only make a profit if the stock price rises beyond $5. Every dollar rise increases

the profit by a dollar. (Again the graph is at an angle of 45 degrees to the horizontal axis).

Note that the maximum loss is $5 when the stock price goes to zero.

The profit diagram for the long stock position can be obtained by shifting the whole graph

horizontally to the right by $5, the cost of initiating the position. This is tantamount to

subtracting $5 from every point of the payoff diagram.

The profit diagram for the short stock position can be obtained by shifting the short stock

payoff graph horizontally to the right by $5 or taking the mirror image of the long stock

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profit diagram. Reiterating what we discussed regarding the payoffs of a short position,

losses can be potentially unlimited for short stock positions.

2.1.2 Call: Payoff and Profit Diagrams

Figure 1.3 Payoff diagrams for long call and short call

The payoff of a long call is zero if the stock price is less than $5. As discussed in the section

on exercising a call above, it does not make sense to pay $5 to exercise a call and receive

one share that is worth much less. However, for stock prices higher than $5, the payoff is

the difference between the stock price and the exercise price.

The sloping portion of the payoff diagram for the long call looks similar to that of the long

stock. This is no coincidence. We can think of the call as having the upside of a stock with

a limited downside!

The payoff diagram for a short call or the position of the call writer is just the mirror image

of that for the long call.

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Figure 1.4 Profit diagrams for long call and short call

The profit diagrams for the long and short calls look like the respective payoff diagrams

except that the call profit diagram is shifted downwards by the amount of the call price,

while that of the put is shifted upwards by the price of the put.

The call buyer has to pay a price of C for one call. So, he does not make a profit until the

stock price exceeds the exercise price by C. Note that the call buyer will still exercise the

call if the stock price is above the exercise price, even if the result is a net loss. This occurs

when the stock price is between X and (X + C). The reason for exercising is that the fixed

loss of C is offset by the gain of (S – X) from buying a stock at the exercise price X, which

is lower than the stock price S. Unfortunately, when the stock price is between X and (X +

C), the gain of (S – X) is insufficient to fully offset the fixed loss of C.

2.1.3 Put: Payoff and Profit Diagrams

We shall not discuss the payoff and profit diagrams for put positions except to point out

two things: (1) Where the long call position benefits from an increase in the stock price,

the long put position benefits from a decrease in the stock price; and (2) the long put and

short put positions experience opposite effects from changes in the stock price. Further,

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while the short call position can have unlimited losses, the short put position’s maximum

loss is limited to the exercise price less the put price.

Figure 1.5 Payoff diagrams for long put and short put

Figure 1.6 Profit diagrams for long put and short put

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Read

Read the following sections from:

Gottesman, A. (2016). Call options. In Derivatives Essentials: An introduction to forwards,

futures, options, and swaps (pp. 22-43). Wiley.

• Section 2.2: Long call payoff

• Section 2.3: Long call P&L

• Section 2.4: Short call payoff

• Section 2.5: Short call P&L

• Section 2.6: Long call P&L diagram

• Section 2.7: Short call P&L diagram

• Section 2.8: Call options are zero-sum games

Gottesman, A. (2016). Put options. In Derivatives Essentials: An introduction to forwards,

futures, options, and swaps (pp. 44-64). Wiley.

• Section 3.2: Long put payoff

• Section 3.3: Long put P&L

• Section 3.4: Short put payoff

• Section 3.5: Short put P&L

• Section 3.6: Long put P&L diagram

• Section 3.7: Short put P&L diagram

• Section 3.8: Short put options are zero-sum games

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2.2 Simple Option Strategies

2.2.1 Protective Put

Fund managers and other investors are often concerned that a fall in stock prices would

reduce the value of their stock portfolios. To hedge against the potential loss, a protective

put is used. This limits or protects the loss of a long stock position. As a long put benefits

from a fall in the stock price, the gain from this position can be used to offset the loss in

the stock position. This is demonstrated in the profit diagrams below. Starting with a long

stock position, a put is bought. We buy a put with the same exercise price as the purchase

price of the stock.

The combined position of the stock and the put is found by adding vertically the amounts

for each component. The result looks similar to the profit diagram of a call, where the loss

is limited to the call price, but the upside potential is not affected. This is illustrated by the

sloped portion of the graph, which is parallel to the graph of a long stock position. Every

dollar increase in the stock price from (X + P) yields a dollar of profit for the combined

position.

The profits from the stock, the put, and the combined position at various stock prices are

shown in the small table at the bottom right of the diagram. Any stock price less than the

exercise price of $20 shows a loss of $2. The combined position breaks even when the price

is $22. Beyond this price, the profit of the combined position increases by the same amount

as the increase in stock price, essentially mimicking the long stock position beyond $22.

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Figure 1.7 Profit diagrams for long stock and long put

Figure 1.8 Profit diagrams for combined long stock and long put

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2.2.2 Long Straddle

A straddle consists of a long call position and a long put position, both with the same

exercise price. The profit diagrams below show those of the long call, the long put, the

combined call and put position, and the straddle position, respectively.

The profit diagram of the straddle is obtained in the same fashion as that for the protective

put. We add the profits for the call and the put at each stock price. The resulting straddle

shows that the position yields a profit if the stock price is either below (X – C) or above

(X + C). Note that for simplicity we assume that the call and put have the same price. In

practice, both prices may differ by a small amount.

Looking at the straddle we see that it benefits from large swings in the stock price.

An example of how this can be used is in a national election with two political parties

contesting. If one party wins, it is very good news for stocks as the party is very

pro-business. On the other hand, if the other party wins, the stock market would fall

substantially. This is because the party is anti-business. As we do not know the outcome

of the election ahead of time, the straddle yields a profit regardless of which party wins.

Figure 1.9 Profit diagrams for long call and long put

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Figure 1.10 Profit diagrams for long straddle

Here is an interesting observation: Suppose we were to sell a straddle – that is, sell a call

and a put with the same exercise price. The profit diagram would then be a mirror image

of the long straddle position as shown in the diagram below. Notice that this position

benefits from a relatively calm or flat market. As long as the stock price does not go below

(X – C) or above (X + C), this position will be profitable.

Figure 1.11 Profit diagram for short straddle

The short straddle profit diagram illustrates an important point about options. Options

are financial instruments that are not superfluous, as some critics claim. Consider this: If

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you think that the stock market is going to rise, you will buy stocks to benefit from the

rise. Similarly, if you think that the stock market is going to fall, you would short stocks.

But what would you do to benefit from a stock market that is going to remain relatively

flat? Without options, it is not possible to make a profit from either buying or shorting

stocks.

2.2.3 Put-Call Parity

In Section 2.2.1 on the protective put, we noticed that the graph looks like that for a call.

We look into this in greater detail here.

The payoffs for a long stock and a long put position are shown below.

Figure 1.12 Payoff diagrams for long stock and long put

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Figure 1.13 Payoff diagrams for combined long stock and long put

The payoffs for a long call plus long position in riskless bonds with face value of X are:

Figure 1.14 Payoff diagrams for long call and long bond

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Figure 1.15 Payoff diagrams for combined long call and long bond

Notice that an investment consisting of buying (long) a stock plus buying a put position

(Figure 1.13) results in the same payoff as an investment where a call and a bond are

bought (Figure 1.15). Suppose S, C, and P, are the prices of the stock, call, and put,

respectively when we first make the investments. X is the exercise price for both the call,

the put, as well as the face value of the bond.

The costs of the two investments are:

Stock plus put: P + S

Call plus bond: C + Xe-rt, where “r” is the riskless interest rate for the bond and “t” is the

time to expiration of the options. The bond’s face value or principal has a present value

of Xe-rt

.

Since both investments have the same payoff, their costs should be the same. Thus:

P+ S = C + Xe-rt

This equation describes the put-call parity relationship. It is a useful equation as we do

not need to value both the call and the put, since knowing the value of either would allow

us to value the other.

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Chapter 3: Option Pricing Bounds

Lesson Recording

Option Bounds

Before we examine models of option pricing in the next study unit, we can enhance our

understanding of options by looking at certain option pricing relationships. These pricing

relationships are essentially the bounds for option prices. In other words, while we are

not able to establish the value of an option, we can say that the option value should be

above or below certain values.

3.1 Using Arbitrage to Establish Option Pricing Bounds

A useful tool to establish value is arbitrage. Arbitrage is based on the law of one price –

that is, things that are alike should sell for the same price. In the financial arena, arbitrage

involves the simultaneous buying of an asset and the selling of another asset. If the two

assets have the same value, then arbitrage would result in a profit if those two assets have

different prices. Essentially, the arbitrageur would buy the cheaper asset and sell the more

expensive one. However, note that the cash flows resulting from the arbitrage transactions

may not be realised all at once. Arbitrage profit is possible if we establish that the arbitrage

transaction realises a zero or positive cash flow today and non-negative cash flows in the

future.

In the following sub-sections we will discuss some of these bounds or restrictions on

option pricing, as well as explain how to profit from price discrepancies. We will focus

mainly on restrictions for call pricing.

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3.1.1 Call Price Should Be Greater or Equal to Zero: C≥0

This restriction seems obvious. A call should not have a negative price.

We show that this is to be true by proving that the converse cannot hold, otherwise we

can make a profit through arbitrage.

Suppose C < 0. Then, if we buy a call, we will be paid C instead of paying C in the typical

case where the price of an asset is positive. This results in a positive cash flow today.

Subsequently, in the future, if the stock price is higher than the exercise price, we will

exercise the call, receiving a positive cash flow. Should the stock price be lower than the

exercise price, we do nothing. These actions can be shown in the table below:

Table 1.1 Cash flows of long call, today and at future date

Today At Future Date

S < X S > X

Action Buy call Do nothing Exercise

Cash Flow + C 0 (S – X) > 0

3.1.2 Call Price Should Be Less Than Stock Price: C < S

This restriction should be quite easy to justify. Suppose it were not true – that is, C > S.

Then, buy the lower priced asset, the stock, and sell the higher priced asset, the call. This

nets a profit of (C – S), today. If the call is exercised later, give the stock and receive X. If

the call is not exercised at all, you are left with the stock which has some value. The table

below shows the actions and the corresponding cash flows.

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Table 1.2 Cash flows of long stock and short call, today and at future date

Action Today Cash Flow Today At Future Date

S < X S > X

Buy Stock – S S S – X

Sell Call + C 0 – (S- X)

C – S > 0 S 0

3.1.3 Call Price Should Be Greater or Equal to Its Intrinsic Value: C ≥ Max [0,

S – X]

We can show that arbitrage profits are possible if the restriction is violated. For example,

if C < (S – X), buy the call and exercise immediately. This results in an arbitrage profit of

[(S – X) – C] which is positive.

Table 1.3 Cash flows of long call and short stock today

Action Today Cash Flow

Buy call – C

Exercise Call -X

Sell Stock +S

Net result (S – X) – C > 0

3.1.4 For Calls With Same Exercise Price, Price of Call With Longer

Expiration Should Be Greater or Equal to Price of Call With Shorter

Expiration: C(T2) ≥ C(T1) where (T2 > T1)

Suppose the restriction is violated – that is, the call with the shorter expiration has a higher

price than the call with the longer expiration: C(T1) > C(T2). We can make an arbitrage

profit of C(T1) – C(T2) by buying C(T2) and selling C(T1). If the buyer of C(T1) exercises

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the call, then exercise C(T2). Exercising C(T2) requires paying X to receive one share. Then,

sell the one share to the buyer of C(T1) and receive X. In other words, exercising C(T2)

when C(T1) is exercised are offsetting transactions. However, if C(T1) is not exercised we

still have call C(T2). Even if C(T2) has no intrinsic value, it still has time value, since the

option has not expired. The table below illustrates the actions and corresponding cash

flows.

Table 1.4 Cash flows of long call with longer expiration date and short call with shorter expiration date,

today and at future date

Action Today Cash Flow At Future Date

S < X S > X

Buy C(T2) – C(T2) > 0 S – X

Sell C(T1) + C(T1) 0 – (S – X)

C(T1) – C(T2) > 0 > 0 0

3.1.5 Price of Call With Lower Exercise Price Should Be Greater Than Price

of Call With Higher Exercise Price: C(X1) ≥ C(X2) if X2 > X1

If the restriction is violated, we can make arbitrage profits by using a similar method to

the restrictions above. For call C(X1) which was bought, it will not be exercised if the stock

price is less than X1. However, for stock prices greater than X1, exercising the call results

in a profit of (S1 – X1). For call C(X2) which was sold, the buyer will not exercise the call

if the stock price is less than X2. When the stock price is greater than X2, the sold call will

be exercised. In order to deliver the stock to the buyer of call C(X2) when it is exercised,

we exercise the call C(X1) that was bought. This results in a cash flow of (X2 – X1) which

is positive, since X2 is greater than X1. The results are shown in the table below.

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Table 1.5 Cash flows of long call with lower exercise price and short call with higher exercise price, today

and at future date

Action Today Cash Flow At Future Date

S ≤ X X1 < S ≤ X2 S > X

Buy Call

With Lower

Exercise Price

– C(X1) 0 S – X1 S – X1

Sell Call

With Higher

Exercise Price

+ C(X2) 0 0 – (S – X2)

> 0 0 > 0 X2 – X1 > 0

3.1.6 For A Stock That Pays No Dividends, the Call Price Should Be Greater

or Equal to the Difference Between the Stock Price and the Present Value of

the Exercise Price: C ≥ Max [ 0, S – Xe-rt ]

We can make arbitrage profits if the restriction is violated. To take advantage of the

violation of the restriction, we can do the following: Buy a call, invest the present value of

the exercise price (Xe-rt), and sell the stock. The term e-rt is the continuous time equivalent

of 1/(1+r)t

. The results are shown in the table below. (Note: In order to return the stock

we borrowed earlier when we sold short, we have to buy the stock at a price of ST at the

expiration date.)

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Table 1.6 Cash flows of long call, long bond and short stock, today and at future date

Action Today Cash Flow At Future Date

S

T

≤ X S

T

> X

Buy Call – C 0 S

*

– X

Invest X e-rt – Xe-rt X X

Sell Stock S – ST

– ST

(S – Xe-rt) – C > 0 X – ST

> 0 0

3.1.7 Applying Bounds on Option Price

Based on the restrictions shown in 3.1.1, 3.1.2 and 3.1.3, we see that the C > 0, C < S and

C > (S – X) are bounds to the option price. This is shown by the graphs in the diagram

below:

Figure 1.16 Bounds for option price

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Note that if we apply the restriction in 3.1.6, then the lower bound, as denoted by the line

C > (S – X) in the diagram above, would be shifted to the left.

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Chapter 4: Forwards, Futures and Swaps

Lesson Recording

Forwards and Futures

Forwards, futures and swaps are derivatives which are used by financial institutions and

companies to hedge their risks. Forward and futures contracts have a much longer history

of several hundred years compared to swaps, which came into existence about fifty years

ago. In this chapter, we will be looking at the features of these financial instruments.

Subsequent study units will examine how they can be used for hedging purposes.

4.1 Forward Contracts

A forward contract is an agreement, made today, between two parties, known as

counterparties, to buy or sell an asset at an agreed upon price at a future date. The terms

of the contract are tailored to the needs of the parties involved. This makes the forward

contract a very flexible financial instrument compared to the futures contract whose

contract terms are standardised by the futures exchange in which the futures contract

trades.

The terms of the forward contract relate to the characteristics of the underlying asset, the

method and place of delivery, as well as the settlement process. As these contract terms

are highly customised, this makes it difficult for a third party to take over the contract

should one of the counter-parties wish to get out of the contract. In other words, forward

contracts have low liquidity as they are quite unique and there is no secondary market

that trades in such financial instruments.

Besides low liquidity, forward contracts are beset by the risk of non-performance. Should

one counterparty to the forward contract fail to perform according to the terms of the

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contract, the only recourse the other counterparty has is to sue or initiate a legal process

to compel performance by the defaulting party.

4.1.1 Using Forward Contract to Hedge Risk

The forward contract is often used to lock-in a price. For example, a major expense for

airlines is the jet fuel used in their planes. When the price of oil rises, airlines would need

to pass the oil price increase to the passengers in the form of higher ticket prices, if they

want to maintain their profit margin. However, such a move may cause passengers to

switch to other airlines that maintain their ticket prices despite an increase in oil prices.

So, airlines tend to hedge against possible oil price increases.

Here’s an example of how this works. Suppose the current price of oil is $40 per barrel.

Singapore Airlines will make a decent profit if oil prices were maintained at this level.

However, should the oil price rise and the airline wants to maintain its ticket prices,

its profits would be eroded. Hence, to protect its profit margin, the airline engages in a

forward transaction to buy oil at $40 in, say, one year’s time. So, whether the price of oil

rises to $50 or $60, Singapore Airlines (SIA) will still be able to buy its oil at $40.

Figure 1.17 illustrates how the forward contract fixes the oil SIA buys at $40. The spot/

forward curve shows the profit or loss that the airline would experience as the oil price

changes in the spot/forward market. Suppose the oil price is $60 at the expiration date of

the forward contract. The airline would then be making a $20 loss at this higher oil price

as it is $20 higher than the $40 oil price at which it had based its ticket prices. However,

this loss is offset by the profit of $20 from its forward position.

A question arises, however. What if the oil price went down to $30? SIA would lose $10

on its forward position. This offsets SIA’s gain of $10 in the lower oil price of $30. Thus, if

SIA had not hedged, it would not have to give up this gain. So, was SIA foolish to hedge

against oil price increases? On 20-20 hindsight, this would seem to be the case. However,

if SIA were able to forecast accurately how oil prices would change, and there is hardly

anyone in the world who could do that, then it should be in the oil trading business and

not run an airline!

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Figure 1.17 Hedging with forward contract

4.2 Futures Contracts

4.2.1 Futures Contracts Specifications

Futures contracts are contracts that trade on a futures exchange. Each contract specifies

the following:

• Underlying asset – a commodity (e.g., wheat, gold, etc.) or a financial product (e.g.

Japanese yen, stock index, etc.).

• Quantity of asset – known as the contract size, for example 5,000 bushels for corn,

100 ounces for gold, etc.

• Delivery location – applies to physical assets like wheat, crude oil, etc. Financial

assets are not settled by delivery of the asset but by cash.

• Delivery date – could be a specific date or period during a month known as the

delivery month.

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4.2.2 Futures Contracts Counterparty Risk and Liquidity

One of the drawbacks of forward contracts is the issue of counterparty risk discussed in

an earlier section. Futures exchange overcome this risk by setting up an entity known as

the clearinghouse which is well funded. The clearinghouse is inserted into each futures

contract as a seller to the buyer, or a buyer to the seller in the contract. Thus the buyer or the

seller can be confident that the risk of non-performance by the counterparty is minimised.

As the futures contract terms are standardised and counterparty risk minimised, a

secondary market can develop. Traders after initiating their futures contracts can close

out their positions in those contracts by making a transaction opposite to that which they

originally took. For example, a buyer in a futures contract is obligated to take delivery of

the asset at a future date. However, if he goes on the futures exchange to sell a futures

contract whose terms are the same as his existing contract, then he would have transferred

his obligation to the buyer of the contract to whom he just sold. The question is why would

the new buyer be willing to enter into the contract? The reason is that non-performance is

minimised and the contract terms are standardised, making it easier to find the next buyer

should he wish to close out his position. In other words, there is liquidity for the futures

contract.

4.2.3 Features of Futures Trading

Margins

When a trader buys shares on margin, he does not pay the full amount of the value of the

shares. Instead, he may pay 60% of the value and the other 40% represents the amount he

borrows from the broker. However, when the trader short sells a stock, he does not get the

proceeds of the sale. Instead he has to post a margin – that is, put in funds that serve as a

collateral to pay for losses should the price rise above the level he sold the stock.

Unlike the case of buying a stock, the broker does not finance the trade when a futures

contract is bought or sold. In the futures market, margins serve a similar function as those

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used when short selling a stock. The margin is required as collateral in case of losses

incurred in the trade.

Initial margin

This is the minimum amount that a trader needs to put up before a position in the futures

market can be initiated. The amount is typically between 3 to 12% of the notional value of

the contract, with higher percentages being applied to the more volatile assets.

Maintenance margin

This is minimum level that the margin can fall to before a margin call is triggered, and the

trader is asked to top up the margin to the initial margin level.

Open interest

Open interest measures the number of current contracts outstanding. It is established by

tracking either the cumulative buy or sell open positions. Table 1.7 below illustrates how

this is done.

In Period 1, A buys 5 futures contracts which were sold by B. Open interest is 5. In the next

period, C buys 3 futures which were sold by A who now holds 2 contracts. So, the number

of outstanding contracts is still 5, since no new contracts have been created as C essentially

took over 3 of A’s contracts. In Period 3, D buys 2 contracts sold by B. Finally, in Period

3, D buys 2 contracts from B. As both D and B are making fresh investments, their trade

increases the number of outstanding contracts by 2, giving a new open interest of 7. So, in

order to compute open interest, you will need to find the net position of each trader. Open

interest is then the total number of positions of all traders who long the futures contract

or the total number of positions of all traders who short the futures contract.

Open interest of a particular futures contract starts at zero when the contract is first opened

for trading, and ends with zero when the contract expires. In between, the open interest

provides a clue to the hedging or, conversely, the speculative activity of the traders.

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Table 1.7 Open interest illustration

Investor No. of Contracts

Bought

No. of

Contracts Sold

No. of Long

Contracts

Outstanding

Period 1

A 5 5

B 5

Open Interest 5

Period 2

C 3 3

A 3 2

Open Interest 5

Period 3

0 2 2

B 2

A 2

C 3

Open Interest 7

Marking-to-market

Marking-to-market is the process of valuing, at the end of the day, a trader’s position

based on the settlement price of the futures contract. If the position has a value greater

than the initial margin, the trader can opt to withdraw funds from the account to bring

the margin to the level of the initial margin. Conversely, if the position has a value less

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than the maintenance margin, a margin call is triggered. If new funds are not put up, then

the trader’s position would be closed out by the broker.

Table 1.8 illustrates the process. Each corn contract is for 5,000 bushels. Initial margin and

maintenance margin are set at $2,025 and $1,500, respectively. The trader bought a contract

at a price of $3.27 per bushel. At the end of the first trading day, the price had gone down

to $3.25, and the trader had suffered a $0.02 loss per bushel or $100 loss per contract. This

loss is deducted from the initial margin amount of $2,025 to give a new balance of $1,925.

The process continues until Day 4, when the loss of $250 brings the margin balance down

to $1,475 which is below the maintenance margin of $1,500. The trader has to put $550

($2025 – $1,475) into the margin account to bring it to the initial margin level. This infusion

of funds is shown in the next day’s trading account. The final balance for Day 5 reflects a

loss of $150, which is deducted from the balance of $2,025 (after the payment of $550) to

give the final balance of the day of $1,875.

Table 1.8 Marking-to-market example

Day Settlement

Price

(cents)

Contract

Value ($)

Mark to

Market ($)

Margin

Adjustment

($)

Margin

Balance ($)

0 327 16,350 2,025

1 325 16,250 -100 1,925

2 320 16,000 -250 1,675

3 321 16,050 50 1,725

4 316 15,800 -250 1,475

5 313 15,650 -150 550 1,875

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Convergence property

At the expiration of the futures contract, the futures price will be equal to the then spot

price. If this were not so, an arbitrage profit can be made. How this could be done is left

to the reader who can demonstrate it by using the principle of “buy low, sell high.”

Price limits

Futures exchanges often impose the maximum amount that the futures price can change

in a day. This range of prices is known as the daily price limits. If the price goes up by

the maximum allowed, then the price is said to have gone “limit up” and, conversely,

if the price went down by the maximum allowed, it is a “limit down” move. When the

price limit is hit, trading is either suspended temporarily or for the day, depending on

the regulations. Price limits are established to prevent excessive volatility in the futures

price. Putting a halt to trading allows traders more time to assess both the veracity of the

information as well as the implications of the information instead of just “following the

herd,” and to make better informed decisions.

4.3 Swaps

A swap is a derivative contract in which the counter-parties agree to an exchange of

cash flows. The cash flows involved in a swap are based on the prices or rates of

different financial instruments, and use some notional value of the financial instruments

to compute the amount of cash flows to be exchanged. Swaps do not trade on an exchange.

Rather they are private arrangements between two parties – that is, they trade in the

over-the-counter (OTC) market. While unregulated, practically all swap contracts use

the ISDA Master Agreement, a standard document from the industry trade association,

International Swaps and Derivatives Association (ISDA) to set the terms of the contract.

We describe below some of the more common types of swaps:

• Interest rate swaps involve the exchange of payments based on two different

interest rates. A common interest rate swap is one where the payment based on a

fixed interest rate is exchanged for the payment based on a floating interest rate.

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FIN358 Basics of Options, Forwards, Futures, and Swaps

• In currency swaps, the payments exchanged are based on the principal and interest

rate in one currency with the principal and interest rate of another currency.

• Equity swaps involve the exchange of the returns on a stock index with the returns

on another stock index, or a fixed interest rate, or a floating rate based on LIBOR.

• For a commodity swap, the exchange is between the spot price of a commodity

with a fixed price for that commodity.

• Unlike the swaps described above, where the exchange takes place every period,

a credit default swap (CDS) works more like an insurance policy. The buyer of

the CDS makes regular payments, much like the premiums in insurance, and is

reimbursed by the seller for losses, should a default of a debt issue occur.

4.3.1 Interest Rate Swap Example

In this simple example, we illustrate the use of the interest rate swap. Suppose

Counterparty A is a bank that makes fixed-rate loans. Its funds come from deposits which

are paid a floating rate. The spread that the bank makes on its loans is the difference

between what it charges borrowers less what it pays depositors. Counterparty A is

worried that if interest rates were to rise, the spread it makes on its loans would narrow

and may result in losses for the bank, after accounting for other expenses. Hence, to protect

its profits, it decides to hedge its exposure to fluctuating interest rate risk. Therefore, it goes

into the swap market and, through an investment bank, manages to find Counterparty

B whose situation is opposite to what it faces. Counterparty B receives income which is

based on a floating rate but wants to ensure more certainty in the form of a fixed income.

In the diagram below, we show a plain vanilla interest rate swap depicting the exchange of

the cash flows. Notice that the swap fulfils the needs of the respective parties and hedges

the risks of uncertain cash flows.

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FIN358 Basics of Options, Forwards, Futures, and Swaps

Figure 1.18 Interest rate swap

Read

Read the following sections from:

Gottesman, A. (2016). Forwards and futures. In Derivatives Essentials: An introduction

to forwards, futures, options, and swaps (pp. 3-21). Wiley.

• Section 1.1: Forwards and futures characteristics

• Section 1.2: Long forward payoff

• Section 1.3: Long forward P&L

• Section 1.4: Short forward payoff

• Section 1.5: Short forward P&L

• Section 1.6: Long forward P&L diagram

• Section 1.7: Short forward P&L diagram

• Section 1.8: Forwards are zero-sum games

• Section 1.9: Counterparty credit risk

• Section 1.10: Futures contracts

SU1-37

FIN358 Basics of Options, Forwards, Futures, and Swaps

Gottesman, A. (2016). Interest rate swaps. In Derivatives Essentials: An introduction to

forwards, futures, options, and swaps (pp. 243-263). Wiley.

• Section 14.1 Interest rate swap characteristics

Gottesman, A. (2016). Credit default swaps, cross-currency swaps, and other swaps.

In Derivatives Essentials: An introduction to forwards, futures, options, and swaps (pp.

243-263). Wiley.

• Section 15.1 Credit default swap characteristics

• Section 15.5 Other swap varieties

Reflect 1.1

Consider the following two statements:

• A perfect hedge is one that completely eliminates risk.

• There is no return without risk.

What do these statements tell us?

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FIN358 Basics of Options, Forwards, Futures, and Swaps

Formative Assessment

1. In an option contract, the rights of the counterparties are symmetrical.

a. True

b. False

2. An investor who holds a call will make money if the stock price at expiration is greater

than the exercise price.

a. True

b. False

3. Buying a put is less risky than selling a stock short.

a. True

b. False

4. If an investor is short a straddle, he expects the market to be calm.

a. True

b. False

5. Holding an American call is less risky than holding a European call.

a. True

b. False

6. If the underlying asset price is $90, and the strike price is $100, what is the worth of

the call option at maturity?

a. -$10

b. -$5

c. $0

d. $5

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FIN358 Basics of Options, Forwards, Futures, and Swaps

e. $10

7. The risk for an option writer is higher than for an option buyer.

a. This is true for the call but not for the put.

b. This is true for the put but not the call.

c. This is true for both the call and the put.

d. This is false for both the call and the put.

8. Which of the following statements is true?

a. You cannot sell a call if you do not have the underlying stock.

b. Exercising a call is always profitable.

c. Two calls are essentially the same except for the expiration date. The call with

the longer expiration is worth more than the one with the shorter expiration.

d. Call and put writers face unlimited losses.

9. For a futures contract, the contract value is calculated every day. Any gain or loss is

added to the trader’s account. This process is called ________________.

a. settling

b. maintenance margin checking

c. account reconciliation

d. mark-to-market

10. To calculate the open interest, ________________.

a. sum the outstanding number of hedged positions

b. sum the outstanding number of speculative positions

c. sum the outstanding number of long and short positions

d. sum the outstanding number of short positions

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FIN358 Basics of Options, Forwards, Futures, and Swaps

Solutions or Suggested Answers

Formative Assessment

1. In an option contract, the rights of the counterparties are symmetrical.

a. True

Incorrect

b. False

Correct

2. An investor who holds a call will make money if the stock price at expiration is greater

than the exercise price.

a. True

Correct

b. False

Incorrect

3. Buying a put is less risky than selling a stock short.

a. True

Correct

b. False

Incorrect

4. If an investor is short a straddle, he expects the market to be calm.

a. True

Correct

b. False

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FIN358 Basics of Options, Forwards, Futures, and Swaps

Incorrect

5. Holding an American call is less risky than holding a European call.

a. True

Correct

b. False

Incorrect

6. If the underlying asset price is $90, and the strike price is $100, what is the worth of

the call option at maturity?

a. -$10

Incorrect

b. -$5

Incorrect

c. $0

Correct

d. $5

Incorrect

e. $10

Incorrect

7. The risk for an option writer is higher than for an option buyer.

a. This is true for the call but not for the put.

Correct

b. This is true for the put but not the call.

Incorrect

c. This is true for both the call and the put.

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FIN358 Basics of Options, Forwards, Futures, and Swaps

Incorrect

d. This is false for both the call and the put.

Incorrect

8. Which of the following statements is true?

a. You cannot sell a call if you do not have the underlying stock.

Incorrect

b. Exercising a call is always profitable.

Incorrect

c. Two calls are essentially the same except for the expiration date. The call with

the longer expiration is worth more than the one with the shorter expiration.

Correct

d. Call and put writers face unlimited losses.

Incorrect

9. For a futures contract, the contract value is calculated every day. Any gain or loss is

added to the trader’s account. This process is called ________________.

a. settling

Incorrect

b. maintenance margin checking

Incorrect

c. account reconciliation

Incorrect

d. mark-to-market

Correct

10. To calculate the open interest, ________________.

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FIN358 Basics of Options, Forwards, Futures, and Swaps

a. sum the outstanding number of hedged positions

Incorrect

b. sum the outstanding number of speculative positions

Incorrect

c. sum the outstanding number of long and short positions

Incorrect

d. sum the outstanding number of short positions

Correct

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FIN358 Basics of Options, Forwards, Futures, and Swaps

References

Gottesman, A. (2016). Derivatives essentials: An introduction to forwards, futures, options and

swaps. Wiley.

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FIN358 Basics of Options, Forwards, Futures, and Swaps

SU1-46

Study

Unit2

Valuation of Derivative Instruments

FIN358 Valuation of Derivative Instruments

Learning Outcomes

By the end of this unit, you should be able to:

1. Describe and appraise the intrinsic value and time value of an option, as well as

its determinants

2. Value and discuss options using the Binomial Option Pricing Model

3. Value and discuss options using the Black-Scholes Option Pricing Model

4. Appraise the use of option Greeks

5. Value futures contracts and swaps

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FIN358 Valuation of Derivative Instruments

Overview

In this study unit, we will discuss how options are valued. First, we look at the binomial

option pricing model. It provides a relatively simple but powerful method of pricing

options as there are few assumptions about the properties of the underlying assets. Then,

we examine the Black-Scholes (B-S) option pricing model, which is popularly used. The

advantage of the B-S model over the binomial option pricing model is that it is explicit

about the variables that go into pricing the option. Further, the derivatives of the B-S

option pricing model provide useful information about the sensitivity of the option price

to changes in each of the variables.

How futures and swap contracts can be priced will also be discussed in this study unit.

However, we will not go into as much depth as we do options. The topic of hybrid

securities such as fixed income securities which have embedded options will be discussed

in Study Unit 6.

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FIN358 Valuation of Derivative Instruments

Chapter 1: Introduction to Option Valuation

In Study Unit 1, we looked at the restrictions to option values. This provides a first step to

valuation, as we know what option values should or should not be. This chapter continues

with some fundamentals regarding the option value.

There are two components to the value of an option: the intrinsic value and the time value.

Intrinsic Value

This is the value of the option if it is exercised immediately. For a call, the intrinsic value

is shown as Max [0, S – X]. The maximum function (Max) takes the value of the variable

with the highest value in the brackets. For the call, this rules out negative values. In other

words, the holder of the call option will only exercise the option if (S – X) is greater than

zero – that is, only if S is greater than X.

We can illustrate this with a call option that has an exercise price of $10. If the stock price

is $15 and the investor exercises the call, she will get a payoff of $5. However, if the stock

price is $7, (S – X) = ($7 – $10) = – $3. In this case, it does not make sense for the option to

be exercised as the payoff is negative. The maximum function rules out this action as $0

is greater than -$3.

A call with a positive intrinsic value – that is, the stock price is greater than the exercise

price – is described as “in-the-money.” Should the stock price be very much higher than

the exercise price, we say the call is “deep in-the-money.” The price changes of very

deep in-the-money calls are quite close to the price changes of the underlying stock. An

encompassing term for the different situations, when the stock price is different from the

exercise price, is “moneyness.” So, in-the-money, at-the-money, and out-of-the-money are

just different degrees of moneyness.

If the intrinsic value is zero, the call is considered “out-of-the-money.” Where the stock

price is very much lower than the exercise price, the call is considered deep out-of-theSU2-4

FIN358 Valuation of Derivative Instruments

money. Finally, if the intrinsic value is zero – that is, the stock price is equal to the exercise

price – the option is said to be “at-the-money.”

What about the intrinsic value of a put? The put is profitable if the stock price is less than

the exercise price. However, like the call, a put will not be exercised if it is not profitable

to do so – that is, when the stock price is higher than the exercise price, or x-s < 0. Hence,

we can denote the intrinsic value as Max [0, X – S]. Similar to a call, we say a put is “in-themoney” if the stock price is lower than the exercise price, and “deep in-the-money” if the

stock price is very much lower than the exercise price. Puts are out-of-the-money when

the stock price is higher than the exercise price.

Time value

The time value of an option is not to be confused with the time value of money, although

the option’s time value is affected by the time to expiration. All else equal, the longer the

time to expiration, the higher the time value of the option.

The time value is the price of the option less the intrinsic value. If we exercise an option

before its expiration, we lose the time value of the option. This is a consideration when

the investor is pondering over the early exercise of the option. For a stock that does not

pay dividends, it is never optimal to exercise the option early. (It is left to the student to

prove this using the arbitrage framework in Study Unit 1).

Besides the time to expiration, other variables may affect the option’s time value. The

volatility of the underlying stock’s return is a bigger determinant of time value than time

to expiration. The impact of different variables on option value will be discussed in the

next section. Figure 2.1 below helps to explain time value in relation to the moneyness of

the option.

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FIN358 Valuation of Derivative Instruments

Figure 2.1 Intrinsic value, time value and option price for a call

In Figure 2.1, the graph on the top left corner shows the intrinsic value at different stock

prices. It is 0 when the stock price is below the exercise price. When the call option is inthe-money (S > X), the call option value is the amount by which the stock price exceeds

the exercise price. This is depicted by the portion of the graph which has a slope of 450

.

On the top-right corner of the diagram, the time value graph has a value that is almost

zero at very high or very low prices. This happens when the call is very deep out-of-the

money or very deep in-the-money. The reason is simple. An investor would not want

to pay much for a call that is out-of-the-money. Buying such a call is like taking a low

probability gamble that the call can be in-the-money before expiration. However, as the

stock price gets closer to the exercise price, there is an increasing probability of the call

being profitable. From almost zero, time value increases with an increase in the stock price

until it reaches a maximum at the exercise price. So, buying an out-of-the-money option is

similar to betting that the stock price will rise above the exercise price. An investor would

pay more for the bet as the stock price gets nearer to the exercise price.

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FIN358 Valuation of Derivative Instruments

What if the stock price is at the other end of the range? An investor who buys a call limits

his loss to the call price. However, an investor who buys the stock stands to lose the full

amount of his investment, should the stock price go to zero. Having the loss limited to the

price of a call is similar to an insurance policy where the loss for a policy holder is limited

to the insurance premium. If the stock price is very much higher than the exercise price,

the likelihood of the stock dropping to a price below the exercise price is small. Hence, this

“insurance” feature in the call is not valuable, as reflected in the very small time value.

However, when the stock price gets closer to the exercise price, the call’s insurance feature

becomes more valuable. This is shown in the time value rising to a maximum when the

stock price equals the exercise price.

At the bottom-right corner, the graph shows time value declining to zero as the option

gets closer to its expiration. This decline in time value is slow when the option is first

issued. But the decay in the time value accelerates as the expiration date approaches. A

rule of thumb is that two-thirds of the option’s time value is lost in the last one-third of

the option’s life. The implication of this fact is that investors need to be aware that time

value decay would tend to skew option returns negatively, with the effect being more

pronounced near expiration.

At the lower-left corner, the graph combines the intrinsic and time values to give the option

value. The option value graph approaches the intrinsic value graph when the stock price

is very low or very high, as the time value is very low at either end of the stock price range.

Very deep out-of-the-money options or very deep in-the-money options have very low

liquidity, as few investors are willing to take long-shot bets when the stock price is low

or to pay for the insurance feature when the stock price is high. However, at-the-money

options or options that are slightly in-the-money or slightly out-of-the-money are more

liquid. Investors are interested in such options as the price can go either way from being

in-the-money to out-of-the-money, and vice versa.

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FIN358 Valuation of Derivative Instruments

Determinants of option value

Before we go into the models for valuing options, we first look at the determinants of

option value. There are five main determinants.

• Stock price

As the asset price increases, the call becomes more valuable. It becomes either less

out-of-the-money or more in-the-money. For the put, the reverse is the case as the

put increases in value with decreases in the asset price.

• Exercise price

A lower exercise price makes the call more valuable, but makes the put less

valuable. This means that for a call, the stock can potentially be acquired at a lower

price. Another way to look at this is that it is easier for the call to be in-the-money

when the exercise price is lower than when it is higher. A lower exercise price works

against the put, as exercising the put would fetch a lower amount than if the exercise

price were higher.

• Time to expiration

There are two effects associated with time to expiration. The first is the present value

effect. A longer time to expiration would decrease the present value of the exercise

price. The second effect is the optionality value effect. This is the right, but not the

obligation, to exercise an option. The longer the time to expiration, the higher the

chance that an option which is out-of-the-money becomes an in-the-money option.

Note that it does not matter even if the option becomes even deeper out-of-themoney, as the option holder has the choice not to exercise the option if it is not in

her favour.

The present value effect and the optionality effect are both beneficial to the call

holder. For the put holder, an increase in the time to expiration has opposite results

for the two effects. The increase in time to expiration reduces the present value of

the exercise price, which works against the put holder. However, like the call holder,

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FIN358 Valuation of Derivative Instruments

the put holder would benefit from the longer time to expiration from the optionality

value perspective. These two effects make it difficult to judge the net effect of an

increase in the time to expiration for a put holder. Which effect is stronger depends

on two things: (1) whether interest rates are high or not, and (2) whether the option

is near-the-money as opposed to being deep out-of-the-money. All else equal, a

high interest rate would strengthen the present value effect. As the time value of an

option is highest when the stock price is equal to the exercise price, the optionality

effect would be stronger for near-the-money options.

• Risk-free rate

The higher the risk-free rate, the more valuable the call. The reason is that the

present value of the exercise price decreases with increases in the risk-free rate. This

benefits the call option investor as it would cost less, in present value terms, to

acquire the stock should the option be exercised. However, the lower present value

of the exercise price works against the put holder.

• Volatility of return of underlying asset

Higher volatility is positive for both calls and puts. The reason is that an option is

a one-sided bet. If the option is in the investor’s favour – that is, intrinsic value is

positive – the investor has an incentive to exercise it. However, when the option is

out-of-the money, the investor cannot be forced to exercise. This is unlike positions

in stocks, and forward and futures contracts, where potential gains and losses are

symmetrical. This means that for the investor, volatility is a plus. The higher the

volatility, the higher the potential for gain. But higher volatility does not bring a

corresponding potential for losses. The investor’s loss is capped at the price paid

for the option.

The five determinants described above are the variables in the Black-Scholes model which

will be discussed later. However, there is another determinant of option value.

• Dividends

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FIN358 Valuation of Derivative Instruments

The payment of a dividend has a negative effect on the value of a call but a positive

effect of a put. The reason is that when a stock goes ex-dividend, the stock price will

fall. This fall in the stock price does not result in a corresponding adjustment of the

exercise price unless the dividend is more than ten percent). However, adjustments

are made to the exercise price when there are corporate actions like stock splits,

mergers, take-overs, spin-offs and special distributions of cash and/or stock.

The stock price undergoes a single adjustment by the amount of dividend on the

ex-date. However, this adjustment is anticipated by option prices in the weeks and

months before they are announced. Option models adjust for expected dividends.

This will be discussed later.

The holder of a call that is in-the-money has to make a decision about early exercise

when the firm declares a dividend. Essentially, it is the trade-off between the

dividend amount and the time value of the option that will be lost should the

exercise take place. If the dividend amount is higher than the time value of the

option, then the call should be exercised.

Table 2.1 below summarises the effect of an increase in each variable for calls and puts.

Table 2.1 Determinants of option value

Variable Effect of an increase in the variable on

Call Put

Underlying asset current

price

(+) (-)

Exercise price (-) (+)

Time to expiration (+) (+)/(-)

Risk-free rate (+) (-)

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FIN358 Valuation of Derivative Instruments

Volatility of underlying

asset

(+) (+)

Dividends (-) (+)

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FIN358 Valuation of Derivative Instruments

Chapter 2: Valuation of Options Using the Binomial

Option Pricing Model

Lesson Recording

Binomial Option Pricing Model

This chapter discusses how options are priced using the binomial option pricing model

(BOPM). This model, as the name suggests, assumes that at each point in time, the price

of the underlying asset can either move up or down, ending at two possible points called

nodes. The value at each node depends on the values of the preceding nodes. With the

progression of time, each node will branch out to two further nodes, resulting in a latticelike diagram for asset prices which we call a binomial tree. The ultimate endpoints of the

branches give possible values of the underlying asset. We have a similar binomial tree of

option prices for an option whose underlying asset is the asset. The price of the option

at each node depends on the asset prices in the next period. Working backwards through

the tree, a process called backward induction, we arrive at the value of the option at the

starting point. How this is done will be made clear in the following sections.

2.1 A Simple One-Period Binomial Option Pricing Model

Suppose we have a stock which has a price of $100. The stock price could either go up 10%

or down 5%, so the payoffs to the stock are either $110 or $95 in one year’s time. Suppose

we have a call in which the underlying asset is the stock just described. This call has an

exercise price of $110. If the risk-free interest rate is 5% per annum, what is the price of the

call?

Let’s look at the payoffs of two investments at the end of one year. The first investment

involves buying 1 call, while in the second, we buy 1 share and borrow $90.48.

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FIN358 Valuation of Derivative Instruments

If the stock price is $95 the call’s the payoff will be 0, as the call is out-of-the-money (the

stock price of $95 is below the exercise price of $105). If the stock price is $110, the payoff

to the call will be ($110 – $105) = $5. The results are shown in Table 2.2.

Table 2.2 Cash flows from buying 1 call

Action Today Cash Flow At Future Date

ST=$95 ST=$110

Buy 1 Call -C $0 $5

For the portfolio of 1 share and borrowing $90.48, we have the following payoffs. For the

1 share of stock, the payoff is whatever the price of the stock may be – that is, either $95 or

$110. The loan of $90.48 needs to be repaid with interest. This amounts to $90.48 x (1.05) =

$95. If the stock price is $95, the payoff to our portfolio is $0, since the payoff to the stock of

$95 is used to repay the loan. However, if the stock price is $110, the payoff to the portfolio

is the value of the stock at $110 less the $95 loan repayment, giving a net amount of $15.

This is shown in Table 2.3.

Table 2.3 Cash flows from buying 1 share and borrowing $46.30

Action Today Cash Flow At Future Date

ST=$95 ST=$110

Buy 1 share -$100 $95 $110

Borrow $86.54 +$90.48 -$95 -$95

Portfolio -$9.52 $0 $15

Comparing the payoffs in Tables 2.2 and 2.3, we note that payoffs for the portfolio is three

times the payoffs of the call, for each stock price in one year’s time. Hence, the value of the

stock and loan portfolio must be three times the value of the call. The cost of the portfolio

is $9.52. Hence the value of the call is given by:

3C = $9.52, or C = $3.17

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FIN358 Valuation of Derivative Instruments

What the above shows is that we can replicate the payoffs of the option with an investment

where we buy 1 share of the stock and borrow an amount equal to the present value of the

lower of the two possible stock prices. Replication is an important concept used in option

pricing formulae.

There is another way to look at the transactions. Suppose you buy 1 share and sell 3 calls.

Compute the payoffs of this portfolio if the stock price either decreases to $95 or increases

to $110 in one year’s time.

Table 2.4 Payoffs to a portfolio of long 1 share and short 3 calls

Action Today Cash Flow At Future Date

ST=$95 ST=$110

Buy 1 share -$100 $95 $110

Sell 3 calls +3C $0 -$15

Portfolio -$100 + 3C $95 $95

Table 2.4 shows that regardless of the stock price, the portfolio will yield a payoff of $95

in one year’s time. Since the value of the portfolio is the same whatever the stock price,

the portfolio is essentially riskless. As a riskless portfolio should earn the riskless rate of

return this allows us to price the portfolio as the present value of $95 which is

95 / (1.05) = $90.48. The cost of the portfolio is $100 – 3C. For the investor, this represents

at cash flow of –($100 – 3C) or -$100 + 3C as shown in Table 2.4.

$100 – 3C = $90.48 or C = $9.52/3 = $3.17

Using a riskless portfolio, or riskless hedge, to price the call is very useful. There is no

need to know the risk and return, or the beta and expected return, of the stock. As the

final stock price does not affect the payoff, we do not need the values of the stock’s risk

and return.

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FIN358 Valuation of Derivative Instruments

The information that we need in this method is the hedge ratio, or the ratio of the number

of shares to one call. The hedge ratio is found from the ratio of the range of the option’s

values to the range of the stock values. For the example above, the range of the values for

the option is 5 (5 – 0); while for the stock, the range is 15 (110 – 95), giving a hedge ratio

of 5/15 or 1/3.

When the stock price goes up, we denote its value as uS0 and when it goes down as dS0,

with Cu and Cd as the corresponding call values, respectively. The hedge ratio is given

by:

2.2 Refining the One-Period Binomial Option Pricing Model

In the example above there are only two possible stock prices at the end of the year. This is

unrealistic. The number of possible stock prices should be many times more. This can be

done by having more branches in the binomial tree. To do this we can divide the year into

many short intervals. With each interval giving rise to two branches, a large number of

intervals would result in many branches of the tree, enabling a large number of possible

ending stock prices.

We begin the process by dividing the year into two six-month periods. Over a six-month

period, the amount the stock would move would be half that for a year. Now, at each node,

the stock price can either go up 5% or down 2.5%. Starting at the price of 100, the possible

prices in 6 months would be 100 x 1.05 = 105 and 100 x 0.975 = 97.5. From the prices of

105 and 97.5, we continue with the process. From the node with a stock price of 105, the

following two nodes would be 105 x 1.05 = 110.25 and 105 x 0.975 = 102.375. Note that the

stock price of 102.375 can be arrived in a second way. From the node with a stock price of

97.5, the up-branch results in a price of 97.55 x 1.05 = 102.375.

From the same node with price of 97.5, the down-branch gives a value of 95 x 0.95 = 90.25.

This process gives rise to the binomial trees for the stock prices and the corresponding call

prices, as shown below:

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FIN358 Valuation of Derivative Instruments

Figure 2.2 Two-period binomial trees

As discussed above, the introduction of more intervals or sub-periods of time increases the

number of possible ending prices. When we move from one to two periods, the number

of ending prices increases to three. For a 3-period binomial tree, there will be four possible

ending prices. The pattern is obvious: the number of possible ending prices is one more

than the number of periods.

How do we determine the values of Cuu, Cud, and Cdd? With the call exercise price of

105, Cuu is 110.125 -105 = 5.125. Cud and Cdd are both zero, since their associated stock

prices of 102.375 and 90.0625 are below the 105 exercise price.

Moving backwards in the binomial tree, the next call value to be determined is Cu. This

requires finding the hedge ratio, or the number of shares to hedge one call. The hedge

ratio is:

h = (Cuu – Cud)/(uS0 –dS0) = (5.125 – 0)/( 110.125 – 102.375) = 0.661.

In other words, a portfolio that is long 2 shares at $110 and short 3.03 calls is a riskless

investment as shown below. (Note: Due to rounding error, the two amounts are not exactly

the same.)

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Table 2.5 Payoffs to a portfolio of long 2 shares and short 3 calls

Action Today Cash Flow At Future Date

ST=$102.375 ST=$110.125

Buy 2 shares -$210 $204.75 $220.25

Sell 3 calls +3.03 C $0 -$15.53

Portfolio -$210 + 3.03C $204.75 $204.72

The portfolio which is long 2 shares and short 3.03 calls has the same value of $204.75 in

6-months’ time, regardless of the price of the stock. This means the value of the portfolio

is the present value of $204.75. The 6-month interest rate is 5%/2 = 2.5%.

Hence:

$210 – 3.03C = $204.75/(1.025) or 3.03C = $210 – 199.75 = $10.25

C = $10.25/3.03 or C = 3.38

In Figure 2.3, we extend the model to four 3-month periods. The price changes for up

and down price movements are 10%/4 = 2.5%, and 5%/4 = 1.25%, respectively. With four

periods, we get 5 possible ending prices. If the process is continued with the intervals

becoming shorter and shorter, the possible prices that result would more closely resemble

real-world prices.

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Figure 2.3 Four-period binomial trees

2.3 Applying the Binomial Option Pricing Model in the Real

World

The binomial option pricing model resembles a coin toss experiment. In a coin toss either

head or tail will show up. What is the probability of ending with a stock price of 110.38

compared to 106.34?

110.38 is the result of 4 consecutive up movements in the stock price or getting 4 heads in

a row in a coin toss. We know from simple probability that getting 4 heads is much less

probable than getting 3 heads and 1 tail. Hence, it is much less likely to end up with a

stock price of 110.38 than 106.34. Another way to think about this issue is the number of

ways in which we can get a particular price. For the stock price of 110.38, there is only one

path through the binomial tree. However, for the stock price of 106.34, there are 4 possible

paths: uuud, uudu, uduu and duuu. Our conclusion is that the stock prices at the extreme

ends are less likely than the prices in the middle. This conforms to real world observations

of stock price movements that extreme price moves are less frequent than smaller moves.

If we have two stocks with different return volatilities, the stock with the higher return

volatility will end up with a wider range of prices than the stock with the lower return

volatility. This means that we need to incorporate the volatility of return into the values

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of u and d. To do this, we set u = exp(σ√∆t) and d = exp(-σ√∆t). Notice that the values of u

and d depend on the volatility of the stock’s return, denoted by σ; the standard deviation

of the stocks return; as well as ∆t, the length of each sub-period.

What happens when we keep increasing the number of sub-periods in a year? Table 2.6

shows how the values of u and d change as the number of sub-periods increases. In Figure

2.4, we show the probability distributions of the final stock prices corresponding to the

three sets of values for u and d in Table 2.3. From Figure 2.4, we can see that as the number

of sub-periods increases, the probability distribution looks smoother and conforms more

to the regular bell-shaped normal curve.

Table 2.6 Values of u and d as the number of sub-periods increase

Subperiods, n

3 .333 exp(.173)=1.189 exp(-.173)=.841

6 .167 exp(.123)=1.130 exp(-.095)=.885

20 .015 exp(.067)=1.069 exp(-.067)=.935

(Source: Bodie et al., 2014)

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Figure 2.4 Probability distributions for final stock price

(Source: Bodie et al., 2014)

Read

Read the following sections in:

Gottesman, A. (2016). The binomial option pricing model. In Derivatives Essentials: An

introduction to forwards, futures, options, and swaps (pp. 126-144). Wiley.

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• Section 7.1: Modeling discrete points in time

• Section 7.2: Introduction to the one-period binomial option pricing model

• Section 7.3: Option valuation, one-period binomial option pricing model

• Section 7.4: Two-period binomial model, European-style options

• Section 7.5: Two-period binomial model, American-style option

• Section 7.6: Multi-period binomial option pricing models

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Chapter 3: Black-Scholes Option Pricing Model

Lesson Recording

Black-Scholes Option Pricing Model

One of the very important advances in modern Finance is the Black-Scholes (B-S) Option

Pricing Model. Although other option pricing models have been developed, the B-S model

is still popular and a useful model for us to understand what drives option pricing.

3.1 B-S Model Basics

From Section 1.1, we see that a riskless hedge can be used to price an option. The riskless

hedge depends on the correct ratio between the number of shares to 1 call option – that is,

the correct hedge ratio. As the stock price changes, the hedge ratio (h) needs to be adjusted

to ensure that the portfolio of 1 call and h shares remains riskless. In the B-S model, the

hedge ratio is adjusted continuously with changes in the stock price.

3.2 The B-S Model

For this course, we will not go into the derivation of the B-S model. Rather, we show the

Black-Scholes equation and discuss how changes in the variables of the equation affect the

option price. The B-S call price is as follows:

where

C(S0, t) = call option value at time t

S0 = stock price at time 0

X = exercise price

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(T – t) = time to expiration in years, T = expiration date

r = risk-free rate

σ = underlying volatility of the asset’s return

N(d1)= standard normal cumulative distribution function where

N(d2)= standard normal cumulative distribution function where

3.3 Assumptions of the B-S model

The assumptions underlying the Black-Scholes model are:

• Stock price movements follow a process known as the geometric Brownian motion.

• Securities can be transacted without cost. No transaction costs and no taxes are

levied on income or capital gain.

• All securities are perfectly divisible.

• The underlying stock does not pay a dividend.

• Arbitrage opportunities do not exist.

• Trading does not occur discretely but continuously.

• Borrowing and lending are transacted at the risk-free rate.

• The risk-free rate is assumed to be constant.

• The stock’s return volatility is constant.

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3.4 The Key No-Arbitrage Argument

Take the perspective of the writer of a call option (the argument works for put options as

well, but call options are easier to think with).

The option writer is short the call. At expiry, if it is in-the-money, he will have to deliver

the underlying and receive the exercise price.

If he does nothing during the life of the option, he may end up having to deliver the

underlying at the strike price, when the market price has in fact become very high.

On the other hand, the stock price might have collapsed during the life of the option. In

that case, it would make more sense for the option writer to deposit his cash in the bank

to earn the risk-free rate rather than to hold on to the underlying stock for delivery, since

he would be losing money as its price falls.

These two scenarios represent the extreme cases. In general, for some level of underlying

asset price, the option writer will need to hold on to some shares, interpolating between

the 0-share and the 1-unit of share extreme cases. This holding of shares (in the long

position) counters the short option position. The two financial assets offset each other’s

risks. The number of shares to hold in order to exactly square with the risk of the short

option position is the quantity that is solved for in the Black-Scholes model. This quantity

is known as Delta.

The strategy of holding on to Delta shares of the underlying throughout the life of the

option is known as the Delta hedging strategy. It is the key argument underlying the BlackScholes model.

The Black-Scholes model uses a mathematical equation called a partial differential

equation which, when solved, produces theoretical prices for call and put options as well

as sensitivities of the option price to changes in the underlying variables, a topic discussed

later in this chapter.

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3.5 Implied Volatilities

The direct way of applying the Black-Scholes model proceeds as follows:

• estimate the volatility parameter from the stock price historical data and read off

the other parameters (such as strike price, time to maturity, etc.)

• find the price according to the formula

Now, it is natural to ask: Does this price agree with the price observed in the market? The

answer is often no!

Many reasons have been proposed (e.g., the model needs to be refined) which have given

rise to many other models), etc.

One reason could be that the value of the volatility that we input in the B-S formula is

different from the market’s assessment of volatility. After all, participants in the option

market agree about the values of the other variables – namely, the stock price, the exercise

price, the time to expiration, and the risk-free rate. In fact, in a way, the value of an option

depends on each investor’s assessment of what the future volatility of the returns of the

asset would be. So, we can use the Black-Scholes model to back out the market consensus

regarding the value of volatility. The volatility that results is called the implied volatility.

The implied volatility is a well-accepted concept in the option markets. Financial

information systems often display implied volatilities alongside option prices. As there

is a one-to-one relationship between volatility and option prices, there is the intuitive

conception:

Stock Market Uncertainty => High stock volatility => High option prices

As prices of options are closely linked to the underlying asset price volatility, buying or

selling options are somewhat synonymous with buying or selling volatility.

While we can back out the implied volatility for individual stocks, we can also do the

same for index options. One of the most popularly traded index option is the option on

the S&P 500 stock index. Just as for a single stock, the implied volatility of the S&P 500

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stock index represents the market consensus view on what the future volatility of the stock

market as whole would be like. This implied volatility can be traded just like any asset

through a product known as the VIX which is traded on the CBOE. As the VIX reflects

the overall market’s assessment of what volatility or risk would be like in the future, this

index has been popularly dubbed the “fear index.” High values of the VIX would reflect

the market’s anticipation of higher volatility in the market and is hence a good gauge of

fear pervading investors.

3.6 Dividends

The various assumptions stated for the Black-Scholes model can be relaxed to incorporate

realities about the financial world. One assumption is that the stock does not pay a

dividend. However, dividend payment is quite common. How can the model be adjusted

to incorporate dividends?

For short-term options, one method involves subtracting the present value of the expected

dividends that will be paid during the life of the option from the current value of the stock.

For long-term options, estimating the dividends may be more difficult. In this case, we

adjust the risk-free rate by the dividend yield. Suppose the dividend yield is q, then the

call and put option price formulae become:

where

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3.7 Limitations of the B-S model

While the Black-Scholes model is an elegant model, it has some limitations due to the

assumptions used to derive the model.

A key input in pricing options is the volatility of the underlying asset. B-S assumes this

volatility to be constant. Such an assumption may not be very critical for short-dated

options. However, the assumption of constant volatility may not hold for options with a

longer time to expiration. A less critical assumption about the constant value of a variable

pertains to the risk-free rate. In normal times, the risk-free rate may not change much.

However, in times of great uncertainty, the risk-free rate may be subject to quite wide

swings, especially when interest rates are low.

Another limitation is the assumption that prices follow a process known as the Geometric

Brownian Motion which gives rise to the random variable being distributed lognormally. (This means that returns are distributed normally.) A better characterisation

of the distribution of prices is the Stable distribution which is more fat-tailed. This

distribution shows higher probabilities of extreme price movements, compared to the

normal distribution, and is more in line with price movements during a financial crisis.

3.8 A Summary of the Two Option Pricing Models

We summarise below the key points about the two models.

• The binomial model is a discrete version of the Black-Scholes model. As the number

of time steps in the binomial model increases, the results from the model approaches

that of the Black-Scholes model.

• Option prices produced by models (theoretical prices) give investors an

approximation to what real world prices should be. How close that approximation

is depends on how well the assumptions about real world phenomenon and

behaviour embedded in the models reflect reality. In a sense, theoretical models

assume ideal situations (for example, efficient markets and no market frictions),

which may not prevail. Traders tweak the model to factor some of the real-world

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imperfections. However, while the traders may use the basic B-S model, different

traders will adjust the model differently.

• Financial securities need to be valued for mark-to-market purposes. If traded prices

or price quotes are not available, then financial institutions apply theoretical models

to price financial products in a process called marking-to-model.

• Traders and other market participants use theoretical models to gauge empirical

prices. When theoretical models are calibrated to empirical prices, they are used to

forecast or estimate the future.

Read

Read the following sections:

Gottesman, A. (2016). Understanding pricing and valuation. In Derivatives Essentials:

An introduction to forwards, futures, options, and swaps (pp. 105-125). Wiley.

• Section 6.1: Review of payoff, price, and value equations

• Section 6.2: Value as the present value of expected payoffs

• Section 6.3: Risk-neutral valuation

• Section 6.4: Probability and expected value concepts

• Section 6.5: Understanding the Black-Scholes equation for call value

• Section 6.6: Understanding the Black-Scholes equation for put value

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Chapter 4: Option Greeks and Volatility Smile

Lesson Recording

Option Greeks

This chapter first looks at the sensitivities of option prices to changes in the variables

of the Black-Scholes equation. The different measures of the sensitivities are denoted by

letters from the Greek alphabet, hence the name “Greeks.” These measures are used in

risk management. They are functions of:

• current stock price

• strike price K

• time to maturity T

• volatility σ

• risk-free rate r

These are the important Greeks that we will discuss here:

• Delta

• Gamma

• Vega

• Rho

• Theta

4.1 Calculating the Greeks

The Greeks are essentially partial derivatives of the Black-Scholes option pricing equation.

Partial derivatives are used to measure rates of change as there are more than one

independent variable in equation. As the Greeks apply to both calls and puts, we will

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denote the value of an option by the letter “V” for generality. So the Greeks give the

sensitivity of the change in the value of the option to changes in the variable, keeping

all other variables constant. In the table below, we list the commonly used Greeks, their

symbols, as well as the effects of changes for calls and puts when the variable changes.

Table 2.7 Option Greeks and the effect on call and put options of changes in the value of the Greek

Measure Symbol Effect on Call

with increase in

measure

Effect on Put

with increase in

measure

Delta Increase Decrease

Gamma Increases if S < X

Decreases if S > X

Increases if S < X

Decreases if S > X

Vega Increase Increase

Rho Increase Decrease

Theta Increase Increase

4.2 Applying the Greeks

In this section, for simplicity, we will discuss the application of the Greeks to call options.

The Greeks can be calculated for puts, and one can use it in a similar fashion as for calls

except that for the variables of stock price and risk-free rate, the effect of changes in the

variable is the opposite for puts compared to calls.

Delta

Delta measures the responsiveness of the option price to changes in the underlying stock

price. Call option deltas vary between 0 and plus 1 (put deltas vary between 0 and -1).

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• When a call is deep out-of-the-money, its price would very unresponsive to changes

in the stock price. Hence, delta has a value close to 0.

• When a call is at-the-money, delta is close to 0.5.

• When a call is deep in-the-money, the price of the call moves almost as much as the

stock. Hence, delta is close to 1.

The shape of the call delta curve is shown in Figure 2.5, where we also show how the

curve changes as the call’s expiration draws closer. The delta curve gets steeper as the

option nears the expiration date. It approaches the limit of 0 for out-of-the-money calls,

and 1 for in-the-money calls. What the curves show is that there is high gearing and pricevariability for near-dated and at-the-money options.

Delta provides the first indication of the riskiness of an option position. The higher the

delta, the greater the exposure of the position to the market. Delta is often used for hedging

purposes. As we know how much the option price changes in relation to a change in the

stock price, we can hedge the number of calls to 1 share to create what is called a deltaneutral position. For example, if delta is 0.5, we need 2 calls to hedge the risk of changes

to the price of 1 share.

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Figure 2.5 Call option delta curves

Gamma

Gamma is the second partial derivative of the Black-Scholes equation with respect to the

stock price. It measures the amount by which delta changes for a small change in the stock

price. There are two facts regarding gamma.

• Gamma is always positive

◦ The call price will always gain more from an upward move in the stock price

than it loses from a fall in the stock price of the same size, as price gains are

associated with increasing delta.

◦ Gamma is at the maximum when the stock price is equal to the exercise price.

◦ As the stock price moves away from the exercise price, in either direction,

the value of gamma decreases.

• Gamma changes with the passage of time

◦ As the option gets closer to expiration, the gamma curve gets more peaked

(leptokurtic) (see Figure 2.6).

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◦ An at-the-money option close to expiration has very high gamma as delta

can move suddenly from near 0 to close to 1.

◦ This explains why at-the-money options near expiration have high volatility.

Figure 2.6 Call option gamma curves

Vega

Vega measures the sensitivity of the option price to changes in the volatility of the returns

of the underlying asset. Vega is at the maximum value when the stock price is equal to

the exercise price. Further, the longer the time to expiration, the higher the value of vega,

all else equal. Note that vega declines as the option approaches expiration. Vega takes on

more importance in uncertain markets when there could be quick changes in volatility.

Recall from the earlier section on determinants of option value that volatility benefits

holders of both calls and puts. This means that a long calls or put are long vega.

Rho

Rho measures the sensitivity of the option price to changes in the risk-free interest rate.

A long call or a short put is long rho. For a call, increases in the risk-free rate reduce the

present value of the exercise price and, hence, benefits the call. As the risk-free rate changes

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much less than changes in the price of the underlying asset or changes in volatility, much

less attention is paid to this measure.

Theta

Our discussion of time to expiration in the determinants of option pricing in Chapter 1

looked at the result of an increase in the time value on call and put values. We found that

when the time to expiration increases, the value of a call increases while the change in

the value of the put was indeterminate as there were two factors at work, which result in

opposite directions of change in the value of the put, with the present value effect causing

the put value to decrease while the optionality value causes the put value to increase.

Theta measures the sensitivity of the option price as time to expiration decreases. So we

need to adjust the results in the previous paragraph as we are now considering a decrease

in the time to expiration.

With a decrease in time to expiration, there is erosion in the optionality value. This has a

negative impact on both a long call position and a long put position.

For the present value effect, the shortening of the time to expiration would increase the

present value of the exercise price. The impact on a long call position is negative. However,

the impact on a long put position is positive, since the put holder is receiving the exercise

price.

For a long call position, an increase in theta would cause a decrease in the call value.

Whether the increase in theta decreases or increases the value of the put depends on the

relative strengths of the present value effect and the optionality effect.

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4.3 Volatility Smile

Lesson Recording

Volatility Smile

If the Black-Scholes model were thoroughly correct, then we would be able to estimate

the volatility of the stock, substitute that into the formula, and the calculated theoretical

price of the option would match the price of the option in the market. However, this has

not been the case.

One reason is that the volatility of any stock is seldom constant, so one is never quite sure

of the value for volatility to enter into the Black-Scholes formula.

Nevertheless, it has been reported that before the Black Monday incident of 1987, the

Black-Scholes equation was partially correct in the sense that the implied volatility of any

option of a given stock was equal to that of another option for the same stock. Thus, there

is a constant volatility that can reproduce all the option prices observed in the market for

a given underlying, using the Black-Scholes equation.

However, this equality between the model price and the market price unravelled after

Black Monday. Thus, when we look at the implied volatilities for all the options of a given

stock, we find that they are not equal. In fact, a plot of the volatility at different stock

prices did not show a straight line but a curved shape – somewhat like that of a distinctive

smile, smirk, or curved surface. In other words, the implied volatilities for deep out-of-themoney and deep in-the-money options tended to be higher than the implied volatilities

of options nearer the money. We describe below the volatility smiles for different types of

options.

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Figure 2.7 Volatility smile

(Source: Investopedia)

4.3.1 Foreign Currency Options

If we collate FX options of a given maturity over a given underlying, say EURUSD, we will

obtain a curve on the plane whose vertical axis is the implied volatility and the horizontal

axis is the strike price.

The shape of this curve typically looks like a smile if the two currencies are of roughly

equal strength.

If one of the currencies is deemed stronger that the other, then the implied volatility curve

would not look like a smile but more like a smirk – i.e., with one end shifted higher than

the other.

4.3.2 Equity Options

The implied volatility smile for options of a given maturity and a given underlying stock

looks like a smirk, with the higher end on the side of lower strike prices. At times, this is

also described as a volatility smirk.

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If the implied volatility curve is plotted with OTM options, then the left end of the curve

is plotted with put options while the right end is plotted with call options.

The fact that the smirk is on the left end means that the prices of OTM put options are

higher than would be expected if they were theoretically computed with the Black-Scholes

theory and with the OTM call options as benchmarks.

A reason that has been proposed is that after the Black Monday, traders became aware of

the terrible state of affairs that would occur during a market crash. Thus they started to

price in this fear into OTM put options which are used for insurance purpose.

4.3.3 The Volatility Term Structure and Volatility Surfaces

In plotting a volatility smile, we assume that the underlying and the maturity are both

fixed.

While keeping the underlying fixed, if we allow the maturity to change, then we would

have a sequence of implied volatility curves. When put together, these curves form a

surface called the volatility surface.

This is also the term structure of implied volatility (as term structure means the variation

of market quotes over different maturities).

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Read

Read these sections in:

Gottesman, A. (2016). Introduction to the Greeks. In Derivatives Essentials: An

introduction to forwards, futures, options, and swaps (pp. 145-157). Wiley.

• Section 8.1: Definitions of the Greeks

• Section 8.2: Characteristics of the Greeks

• Section 8.3: Equations for the Greeks

• Section 8.4: Calculating the Greeks

• Section 8.5: Interpreting the Greeks

• Section 8.6: The accuracy of the Greeks

Gottesman, A. (2016). Understanding delta and gamma. In Derivatives Essentials: An

introduction to forwards, futures, options, and swaps (pp. 158-170). Wiley.

• Section 9.1: Describing sensitivity using Delta and Gamma

• Section 9.2: Understanding Delta

• Section 9.3: Delta across the underlying asset price

• Section 9.4: Understanding Delta

• Section 9.5: Gamma across the underlying asset price

Gottesman, A. (2016). Understanding vega, rho and theta. In Derivatives Essentials: An

introduction to forwards, futures, options, and swaps (pp. 177-188). Wiley.

• Section 10.1: Describing sensitivity using Vega, Rho, and Theta

• Section 10.2: Understanding Vega

• Section 10.3: Understanding Rho

• Section 10.4: Understanding Theta

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Chapter 5: Valuation of Futures Contracts and Swaps

5.1 Determining Forward and Futures Prices

The forward price is related to the spot price by the following formula:

Forward = Spot + Cost – Benefit

where Cost and Benefit refer to the cost and benefit that are due to the holding of the asset

that is underlying the forward contract.

This relationship is known as the Cost of Carry Relationship or the Spot-Forward

Relationship.

It is a very important relationship that helps us determine the forward price. It can be

applied to forward contracts regardless of the nature of the underlying. For instance, in

the world of foreign exchange, the relationship is called the interest rate parity.

Similar to forward contracts, sometimes the futures price is conveniently estimated using

the same formula. However, note that futures prices are realised at futures exchanges

through a process of trading. They are not determined by calculation.

5.1.1 Short Selling

Theoretical pricing arguments for derivatives such as forward, futures, and options

require that the underlying asset can be shorted.

Commodity options are usually option contracts whose underlying are futures contracts.

Thus, there is no issue concerning whether the underlying can be shorted or not, since

futures contracts can be entered either in the long or short position.

On the other hand, there are significant limits placed on shorting in most stock markets

to prevent excessive volatility. This will place a limit on the applicability of theoretical

pricing arguments for derivatives.

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5.1.2 Assumptions and Notations

Theoretical pricing models make certain assumptions so the models are tractable. These

are the common assumptions:

1. There is no transaction cost in trading.

2. Either there is no tax, or the tax rate is equal for everyone (in terms of trading

profits).

3. There is a single interest rate – the risk-free rate – in the market at which lending

and borrowing are carried out.

4. Agents will make use of arbitrage opportunities whenever they occur.

Do remember that when theoretical prices are applied, these hold insofar as the

assumptions hold.

We will also use the following common notation:

• T: time at delivery (if 0 is the initiation time, this is also the maturity of the contract)

• : underlying asset price at time 0

• : forward or futures price at time 0

• r: risk-free rate

5.1.3 Forward Price for an Investment Asset

For a stock that does not give dividend, the forward price is calculated using the formula:

.

This formula can be justified by the following no-arbitrage argument.

We will consider the two cases: and , and show that they are impossible

to hold.

Suppose .

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An arbitrageur will then do this:

• at time 0:

◦ borrow in cash to buy the stock

◦ short the forward contract

• at time T:

◦ close the forward contract through delivery of asset

◦ return the loan

Upon delivery, the arbitrageur receives . He needs to pay back to the bank . Since

, he makes a positive profit.

This entire operation is known as an arbitrage opportunity because a positive profit is

made at no risk.

Now, suppose .

An arbitrageur will then do this:

• at time 0:

◦ short the stock and lend out the in cash received

◦ long the forward contract

• at time T:

◦ close the loan to receive

◦ close the forward contract and receive asset with the payment of

Upon delivery, the arbitrageur receives . Since , he makes a positive profit.

This entire operation is again an arbitrage opportunity because a positive profit is made

at no risk.

5.1.4 Known Income

In the case when the stock gives dividends whose present value amounts to I, the forward

price is given by the formula:

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since the dividend received may be used to fund the purchase of the stock for hedging the

shorting for the contract.

5.1.5 Known Yield

Sometimes, the dividend is expressed as a dividend yield (similar to interest being

expressed through interest rate). Let q denote the dividend yield of a stock. Then its

forward price is given by the following formula:

5.1.6 Valuing Forward Contracts

When a forward contract is initiated, its exercise price is set at a level so that it costs nothing

for both counterparties to enter into the contract. However, as time progresses, market

conditions change, resulting in the forward contract assuming a certain value which may

or may not be zero.

Let us work out what this value is.

For simplicity, let’s assume that the underlying asset is a stock that does not give a

dividend.

Consider a forward contract that is initiated at present. Call it . Recall from the previous

section that the forward price or strike price is given by

where 0 is the current time and T is the time to maturity, is the current spot price of the

underlying asset and r is the risk-free rate.

Let’s consider another forward contract that was initiated at some earlier time and has the

same underlying as and a strike price of K. Let’s call this forward contract Fb. We’d like

to find its current value f.

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FIN358 Valuation of Derivative Instruments

Consider the position

i.e., long a unit of Fa and short a unit of Fb.

At maturity, the net cash inflow is:

Thus, the present value of the position is:

But the value of the position can also be expressed as:

Hence,

or

.

We have therefore obtained the value of a forward contract that was initiated a while ago.

In general, if the underlying stock provides dividends, then this formula needs to be

adapted to the new situation as is explained in the following.

Recall from the previous section that the presence of dividends can be expressed in two

ways:

• as the present value of all dividends over the period in discussion I,

• as a dividend yield, q

In the first case, the value of the forward contract would be:

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FIN358 Valuation of Derivative Instruments

while in the second case, its value would be:

5.1.7 Are Forward Prices and Futures Prices Equal?

Discussions on the quality of prices must always be understood with respect to the

contextual assumptions: Are we talking about the equality in the real world (i.e.,

empirically) or on theoretical grounds?

When this question – Are forward prices and futures prices equal? – is asked in the real

world, it will require us to compare the existence of two contracts with similar terms: one

from an OTC market and one from an exchange. Such pairs of contracts are not easily

found, since contracts occupy a competitive space. This means that generally, forward

contracts exist where exchange-traded contracts do not.

On the other head, one may approach the question theoretically. An important distinction

between forward and futures contracts is that the latter is settled daily, while the former is

only settled at maturity. The daily settlement has the implication that cash flows between

the position and the bank (i.e., the depository bank for the trader, since we automatically

assume that he deposits additional cash and borrows cash for his trades from there). This

suggests that if the interest rate of borrowing/lending were to fluctuate, it can make a

difference to the values of forward and futures contracts, with all other terms being equal.

It can be theoretically shown that if the risk-free rate does not change stochastically (i.e.,

there is no uncertainty concerning the risk-free rate), then the forward price and the

futures price are equal. And if not, then they may not be equal.

5.1.8 Futures Prices and Stock Indices

The futures price of stock indices may be theoretically computed by regarding the futures

contract as a forward contract. Thus,

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FIN358 Valuation of Derivative Instruments

shows the theoretical futures price relationship, where is the spot index level, r is the

risk-free rate, T is the maturity of the contract, q is the dividend yield on the index, and

is the futures price.

In order to apply this formula, one needs to find all the values from some

database. The values and T are obvious. The risk-free rate needs to be selected from a

range of rates that are publicly available. Usually one picks the most highly quoted interest

rate for a loan period that is about the same length as from T the market, or one picks a

yield from a government money market instrument to compute an interest rate that we

take to be risk-free.

It is harder to find the dividend yield unless one has access to a good database, partly

because dividend yields on stock indices change regularly. Usually, one needs access to a

paid database for the continually updated data.

5.1.9 Forward and Futures Contracts on Currencies

Currencies are traded for one another. Prices are quoted as exchange rates.

Suppose, for instance, we consider EURUSD. If we state that the exchange rate is EURUSD

1.120, it means that EUR 1 is traded for USD 1.120. If we regard USD to be the domestic

currency and EUR to be the foreign currency, then we could interpret EUR as the asset, and

EURUSD 1.120 to be expressing the price of this asset in terms of the domestic currency.

Under this interpretation, we may regard EUR to be equivalent to a stock that gives

dividends. For EUR, the analogue of dividend is interest. The substitute for dividend yield

is interest rate . The risk-free rate is .

Applying the results that we have obtained for forward contracts over stocks with

dividends expressed in terms of dividend yields, we have the fact that the forward

exchange rate is expressed as:

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FIN358 Valuation of Derivative Instruments

and the value of a forward contract is:

Another thing to note concerning forward or futures on currencies is that the status of

long or short position is ambiguous.

A long position in a forward contract on EURUSD will receive EUR upon the payment

of USD at maturity. But this is equivalent to a short position in a forward contract on

USDEUR. Conversely, a short position in a forward contract on EURUSD is equivalent to

a long position in a forward contract on USDEUR.

5.1.10 Futures on Commodities

The conceptual underpinning for futures on commodities remains the same as before.

Let us begin from:

.

This is the formula for the forward price.

Since physical commodities require storage, storage cost is incurred when holding and

storing them.

Let the storage cost per unit be u.

Then the forward/futures price for commodity underlying is of the form:

.

If we look at empirical market prices, this equality may not hold exactly. Theoreticians

introduce a term to ensure that both sides are equal:

.

This term y is called the convenience yield. It is a theoretical idea that holding a physical

asset brings with it benefits that do not exist when one is long the futures contract alone.

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FIN358 Valuation of Derivative Instruments

But notice that it is only introduced to balance the empirical quantities on both sides of

the equation.

Let us rewrite the above as:

.

The exponential function has the approximation of , hence this equation may be

written as:

Notice that corresponds to the cost of holding the underlying asset (i.e.,

borrowing cash to buy the asset, and the need to pay for storage), while corresponds

to the benefit of holding it.

We may thus re-express the spot-forward relationship into the following form:

Forward = Spot + Cost – Benefit.

There is no conceptual different between such an expression of how the spot and the

forward are related, and the multiplicative form that we began with in this section. They

are the additive and multiplicative expressions of a single fundamental relationship.

5.1.11 The Cost of Carry

The cost that is alluded to above is commonly called the cost of carry (i.e., of carrying the

underlying asset).

For commodities, the cost of carry includes interest cost and storage cost. For stocks, the

cost of carry is the interest cost since storage cost is virtually zero in this case.

5.1.12 Delivery Options

When forward and futures contracts expire, the party in the short position needs to deliver

the underlying to his counterparty.

Forward contracts normally stipulate the exact date of delivery.

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FIN358 Valuation of Derivative Instruments

Futures contracts often allow for delivery at any time during a certain period. As prices

fluctuate continually, this uncertainty creates a complication for the pricing of futures.

Let us consider some complications that may arise.

Suppose that the futures price is expressed as:

,

where c is the cost of carry and yis the convenience yield.

If , the futures price is an increasing function with maturity. But more particular

to the present situation is that for a short seller, where if the decision is between holding

cash (which gives the benefit of r) and holding asset (which gives the benefit of y – u), it

is more beneficial to be holding cash. Hence, the short seller is motivated to deliver the

underlying asset as early as possible.

If , then the opposite holds.

5.1.13 Futures Prices and Expected Spot Prices

Consider the following quantities:

• forward price

• futures price

• spot price

• future spot price

• expected future spot price

Disregarding institutional details (e.g., futures contracts involve daily settlement), we may

regard forward and futures price to be equal.

The forward price and spot price are related by the fundamental cost-of-carry relationship.

The future spot price is of course unknown, which is why forward and futures exist in the

very first place.

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Theoretically, we may reason about the relationship between the forward price and the

expected future spot price in the following manner.

Suppose we enter into the long position of a forward contract whose strike price is . At

maturity T, we buy the asset at . Thus at time T, we have:

.

However, at time 0, we do not know exactly the value of . We may only talk about its

average or expected value at time T.

Bringing these values back to the present by present-valuing with the appropriate rates,

we obtain:

.

Here, note that since is cash received at time T, its PV can be obtained by the discount

factor On the other hand, since the asset is not cash, we cannot discount it back to the

present by multiplying by the discount factor . Instead, we must discount it back to the

present with our personal required rate of return k.

Rewriting, the above equation, we obtain:

.

If k = r (i.e., the rate of return from investing in the asset market is equal to the risk-free

rate), then we say that there is no systematic risk in the underlying asset.

If k > r, we say that there is positive systematic risk in the underlying asset.

If k < r, then we say that there is negative systematic risk in the underlying asset.

Do note that consideration of the expected future price is only done on subjectively

intuitive or objectively theoretical grounds. In reality, we hardly know anything about the

future spot price. Any data only serves as evidence that may be interpreted differently by

different people.

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FIN358 Valuation of Derivative Instruments

5.2 Valuation of Swaps

Lesson Recording

Interest Rate Swap

A swap involves the exchange of two sets of cash flows. In the interest rate swap, one

set of cash flows consists of fixed-rate interest payments while the other set of cash flows

consists of floating-rate interest payments.

The swap is priced at its initiation. This means that we set the terms of the swap such that

the values of the two sets of cash flows, or legs, are equal. After the swap has been priced,

subsequent changes in the floating rate, due to changes in the reference rate (e.g., LIBOR),

will result in the value of the swap changing over time.

Here is a simple example of how we can price a two-year interest rate swap with a notional

amount of $100m. Suppose the fixed-rate leg consists of 4 semi-annual payments. The

floating-rate leg consists of 4 semi-annual payments at LIBOR. Let L1, L2, L3, and L4 be the

respective forward LIBOR rate for periods 1 to 4. The cash flows for the floating leg are

shown in TaBLE 2.8.

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Table 2.8 Floating rate swap cash flows

Six-month

Period

Number

Annual

Libor

Forward

Rate

Semiannual

Libor

Forward

Rate

Actual

Floating

Rate

Payment

At End

Period

Floating

Rate

Forward

Discount

Factor

PV of

Floating

Rate

Payment

At End

of Period

1 4.00% 2.000% $2,000,000 0.9804 $1,960,800

2 4.25% 2.125% $2,125,000 0.9600 $2,040,000

3 4.50% 2.250% $2,250,000 0.9389 $2,112,525

4 4.75% 2.375% $2,375,000 0.9171 $2,178,113

Sum of Discount Factors 3.7964

PV of Floating

Rate Payments

$8,291,438

In Table 2.8, the Libor forward rates can be obtained from Bloomberg or other financial

data provider. It can also be calculated from the spot rates in the Libor yield curve,

assuming the pure expectations theory of interest rate holds, a topic which will be

discussed in later Study Units on Fixed Income. The floating rate payments are based on

the forward (or expected) rates at the time of initiation of the swap, and these are what

the investor expects to get. Of course, after the swap is initiated interest rates will change.

How they change will determine the discount factors used to find the present values of

the floating rate payments.

The discount factor for period 1 is 1/(1 + 0.02) = 0.9804. The discount factor for cash flows

received at the end of period is [1/(1 + 0.02)] x {1/(1 + 0.0215)] = 0.96. In other words, the

long-term spot rate is just the geometric average of the forward rates.

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FIN358 Valuation of Derivative Instruments

At initiation, the present values of the cash flows of both legs should be equal. The cash

flows in the fixed leg consists 4 payments of $100 million x R/2 every 6 months, where R

is the annual fixed- rate payable for the fixed leg. This is an annuity, and its present value

is $100 million x R/2 x (sum of discount factors for 4 periods). So:

PV of payments for fixed-rate = PV of payments for floating-rate

$100 million x R/2 x 3.7964 = $8,291,438

Or R = 4.368%

Once the rate is fixed at 4.368%, changes in Libor would change the value of the stream of

fixed payments. An increase in Libor would reduce the value of fixed leg, while a decrease

in Libor would increase the value.

In our example, the floating rate was the Libor. It does not have to be the case and we

could have Libor plus a premium to reflect higher credit risk. Computing the fixed rate

follows the same procedure in the example.

Note that conceptually, we can start with a fixed rate of, say, 7% for the fixed leg. We then

solve for the unknown premium to be added to Libor to ensure that the present values

of both fixed and floating rates are the same. The premium that we computed would be

used as a part of the terms of the swap.

5.2.1 Two Interpretation of Swaps

a. Package of cash market instruments

For the example above, we can think of the investor as buying a package of cash

market instruments. The transactions are as follows:

• buy $100 million at par of a 2-year floating-rate bond that pays 6-month

Libor every six months

• finance the purchase by borrowing $100 million for 2 years on terms

requiring a 5.93% annual interest rate payable every 6 months

b. Package of forward contracts

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FIN358 Valuation of Derivative Instruments

The other way to think about the swap is that the package consists of a series of

forward contracts. Here, the investor undertakes to pay an amount – the fixed

interest payment – every six months for two years and, in return, he would

receive a package consisting of the “commodity”, which is the 6-month Libor.

Read

Read these following sections in:

Gottesman, A. (2016). Interest rate swaps. In Derivatives Essentials: An introduction to

forwards, futures, options, and swaps (pp. 243-263). Wiley

• Section 14.1: Interest rate swap characteristics

• Section 14.2: Interest rate swap cash flows

• Section 14.3: Calculating interest rate swap cash flows

• Section 14.4: How interest rate swap cash flows can transform cash flows

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FIN358 Valuation of Derivative Instruments

Formative Assessment

1. In a one-period binomial model, if the stock prices at time period 1 can be either $100

or $90, and the corresponding call option prices are $10 and $9, then the delta hedging

ratio is _____________.

a. 0.05

b. 0.10

c. 0.15

d. 0.20

2. As the number of time periods increases, the price obtained by the binomial model

______________.

a. becomes closer and closer to 0

b. becomes closer and closer to 1

c. becomes closer and closer to the price calculated by the Black-Scholes model

d. none of the above

3. The volatility of a stock is essentially _______________.

a. the variance of its price

b. the standard deviation of its price

c. the variance of the differences in the logarithm of its price

d. the standard deviation of the differences in the logarithm of its price

4. The delta of a put option is always ____________.

a. zero

b. positive

c. negative

d. none of the above

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FIN358 Valuation of Derivative Instruments

5. The VIX index ______________.

a. is commonly known as the stock market fear index

b. is a weighted average of option prices of the S&P 500 index

c. was introduced by the CBOE in the 1960s

d. all of the above

6. Delta for a plain vanilla call option is a number that is _________________.

a. always positive

b. always negative

c. between -1 and 1

d. between 0 and 1

7. The volatility smile is a phenomenon in the option market that only makes sense

because the Black-Scholes model does not fully describe the reality it is intended to

describe.

a. True

b. False

8. The shape of a volatility smile in the foreign exchange market can range from being

symmetrical to being asymmetrical depending on ______________.

a. the time of the day

b. the relative strength of the two currencies in an exchange rate

c. whether SGD is involved or not

d. none of the above

9. What does the basis risk as a notion capture?

a. The discrepancy between the price of the asset that needs to be hedged and the

price of the underlying asset of the futures contract.

b. The uncertainty of the trade date of the asset.

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FIN358 Valuation of Derivative Instruments

c. The early closing of the hedging futures contract.

d. All of the above

10. The cost of carry of a currency is _____________.

a. its dividend yield

b. the interest rate of the currency

c. the credit spread of the currency

d. none of the above

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FIN358 Valuation of Derivative Instruments

Solutions or Suggested Answers

Formative Assessment

1. In a one-period binomial model, if the stock prices at time period 1 can be either $100

or $90, and the corresponding call option prices are $10 and $9, then the delta hedging

ratio is _____________.

a. 0.05

Incorrect

b. 0.10

Correct

c. 0.15

Incorrect

d. 0.20

Incorrect

2. As the number of time periods increases, the price obtained by the binomial model

______________.

a. becomes closer and closer to 0

Incorrect

b. becomes closer and closer to 1

Incorrect

c. becomes closer and closer to the price calculated by the Black-Scholes model

Correct

d. none of the above

Incorrect

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FIN358 Valuation of Derivative Instruments

3. The volatility of a stock is essentially _______________.

a. the variance of its price

Incorrect

b. the standard deviation of its price

Incorrect

c. the variance of the differences in the logarithm of its price

Incorrect

d. the standard deviation of the differences in the logarithm of its price

Correct

4. The delta of a put option is always ____________.

a. zero

Incorrect

b. positive

Incorrect

c. negative

Correct

d. none of the above

Incorrect

5. The VIX index ______________.

a. is commonly known as the stock market fear index

Correct

b. is a weighted average of option prices of the S&P 500 index

Incorrect

c. was introduced by the CBOE in the 1960s

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FIN358 Valuation of Derivative Instruments

Incorrect

d. all of the above

Incorrect

6. Delta for a plain vanilla call option is a number that is _________________.

a. always positive

Incorrect

b. always negative

Incorrect

c. between -1 and 1

Incorrect

d. between 0 and 1

Correct

7. The volatility smile is a phenomenon in the option market that only makes sense

because the Black-Scholes model does not fully describe the reality it is intended to

describe.

a. True

Correct

b. False

Incorrect

8. The shape of a volatility smile in the foreign exchange market can range from being

symmetrical to being asymmetrical depending on ______________.

a. the time of the day

Incorrect

b. the relative strength of the two currencies in an exchange rate

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FIN358 Valuation of Derivative Instruments

Correct

c. whether SGD is involved or not

Incorrect

d. none of the above

Incorrect

9. What does the basis risk as a notion capture?

a. The discrepancy between the price of the asset that needs to be hedged and

the price of the underlying asset of the futures contract.

Incorrect

b. The uncertainty of the trade date of the asset.

Incorrect

c. The early closing of the hedging futures contract.

Incorrect

d. All of the above

Correct

10. The cost of carry of a currency is _____________.

a. its dividend yield

Incorrect

b. the interest rate of the currency

Correct

c. the credit spread of the currency

Incorrect

d. none of the above

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FIN358 Valuation of Derivative Instruments

Incorrect

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FIN358 Valuation of Derivative Instruments

References

Gottesman, A. (2016). Derivatives essentials: An introduction to forwards, futures, options and

swaps. Wiley.

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Study

Unit3

Derivative Hedging and

Combination Strategies

FIN358 Derivative Hedging and Combination Strategies

Learning Outcomes

By the end of this unit, you should be able to:

1. Discuss how hedging can be done using interest rate, currency, credit, and equity

derivatives.

2. Appraise the effectiveness of hedging using various types of derivatives.

3. Evaluate option and futures combination strategies.

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FIN358 Derivative Hedging and Combination Strategies

Overview

A key contribution of derivatives is the mitigation of risks that they provide. In the modern

world of business and finance, the firm finds itself facing various risks in its business

operations. On the production side, a firm that produces raw materials faces the risk of

decline in the prices of the ores and minerals it extracts, or the price of the produce from

its farms that it brings to market. For a manufacturer, the minerals or produce are the raw

ingredients that go into the manufacturing process. Any rise in the price of raw materials

means its profit margins will diminish if the firm cannot pass on the increase in raw

material costs to its customers. Financial firms or those firms that have borrowed using

floating rate loans face the risk of increasing interest cost when the interest rate rises. Other

firms and institutions may have lent money or bought the bonds from an issuer and want

to protect their investments should the borrower default on its payments. How the risks

described above can be hedged will be the focus of this study unit.

The principle behind hedging is to use an instrument that produces the opposite price

movement to the price movement of the asset whose price we want to lock in. Essentially,

you hedge a long position in an asset by taking short position in the instrument that tracks

the price of the asset. A long position is one which benefits from a rise in the price of the

asset while a short position gains from a fall in the price. Note that remarks regarding long

or short positions apply for forwards and futures but not all options. While a long position

in a call benefits from an increase in the price of the underlying, a long put benefits from

a decrease in the price.

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FIN358 Derivative Hedging and Combination Strategies

Chapter 1: Hedging Using Derivatives

1.1 Interest Rate Derivatives: Interest Rate Futures

Lesson Recording

Interest Rate Futures

In this section we will explore how interest rate risk may be hedged by interest rate futures

and swaps.

The two well-known interest rate futures are:

• US Treasury bond futures

• Eurodollar futures

Bond futures deal with long-term interest rate risk while the Eurodollar futures deal with

short-term interest rate risk.

1.1.1 Day Count and Quotation Conventions

Bond transactions may not necessarily take place on the coupon payment dates. If the

transaction takes place after the coupon payment date, the seller would have earned

interest on the bond from the last coupon payment date up to the settlement date. As the

interest has been earned but not paid, it is known as the accrued interest. Since it is the

buyer who will receive the next coupon, the buyer needs to pay the seller the accrued

interest.

The price of the bond, the dirty price, is computed as the present value of all cash flows

from the bond. This is not the value of the bond to the buyer, the clean price, as he needs

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FIN358 Derivative Hedging and Combination Strategies

to pay the seller the accrued interest. The clean price and the dirty price are related in the

following manner:

Dirty price – Accrued interest = Clean price

Accrued interest is earned over the period from the last coupon date to the settlement

date for the bond. The length of this period as a proportion of the length of period

between coupon payment dates allows us to prorate the interest earned. There are several

conventions, known as day count conventions, that govern how the length of those

periods are computed. Common day count conventions in the United States are:

• ACT/ACT (for US T-bonds)

• 30/360 (for US corporate and municipal bonds)

• ACT/360 (for US T-bills and money market instruments)

These conventions determine how the number of days in a month or a year, respectively,

should consist of. ACT stands for the actual number of days.

US T-bills are quoted in the following manner: For instance, suppose that the price quote

is 8. This means that the interest rate is 8% of the face value per 360 days (refer to the day

count convention for money market in the US above). This works out to be:

for a face value of over a period of 91 days to maturity.

Consequently, the cash price (i.e., the amount that is required for the security) is:

.

Generally, the relationship between the quoted price P and the cash price Y is:

.

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FIN358 Derivative Hedging and Combination Strategies

1.1.2 Treasury Bond Futures

The key features of the T-bond futures contract at the CME are as follows:

• The seller of the futures contract has to deliver a US T-bond at a certain price

calculated from the futures (strike) price

• The US T-bond can be chosen from a basket which comprises bonds that mature

more than 15 years later and are not callable within 15 years

• The price is computed using the formula:

Most Recent Settlement Price x Conversion Factor + Accrued Interest

• Because a basket of bonds constitutes the eligible deliverable bonds, and since the

bonds are not identical, the conversion factor puts them all on the same value

footing

• The Conversion Factor is defined as the clean price of a theoretical bond with 6%

YTM and has all other characteristics similar to the bond in consideration

• Coupons are rounded to the nearest 3 months – i.e. after rounding, the first coupon

to be paid is either 3 months from the present or 6 months from the present

• Because a basket of bonds constitutes the eligible deliverable bonds, the notion of

Cheapest-To-Deliver bond arises.

• This is because the price of each bond subtracted by the price formula above gives

a quantity that is different from bond to bond

• Hence, the short party will attempt to deliver the cheapest from the basket.

1.1.3 Eurodollar Futures

A Eurodollar futures contract is based on the Eurodollar interest rates. Eurodollars are

deposit of US dollars in banks outside the US. Since the deposits are outside the US,

they are not subject to regulation by the US banking authorities. Hence, such deposits

are considered riskier than deposits in US banks and thus pay a slightly higher interest

rate. As Eurodollar futures are based on short-term interest rates, they are often used for

hedging short-term interest rate risk.

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FIN358 Derivative Hedging and Combination Strategies

The Eurodollar futures contract at the CME is based on a contract unit of $2,500 x Contract

IMM Index. The index as calculated as 100 – R where R is the 3-month Libor rate. If the

index is given as 98, then the 3-month Eurodollar deposit rate is 100 – 98 = 2 or 2%. A

change in the interest rate of one basis point (1/100 x 1%) translates to a change in $2,500

x 0.01 = $25 per contract.

There may be a bit of confusion when the student learns that each Eurodollar futures

contract has a notional or principal amount of $1,000,000. How do we end up with a

contract unit of $2,500? The reason is that we want to make it easy to figure out how a

change in interest rates of a magnitude of 1 percentage point translates to an equivalent

change in the value of the contract. The maturity of the instrument is 3 months while

interest rates are quoted on an annual basis. Therefore, a change in the annual rate of

4% – for example, from 9% to 5% – is equivalent to a 1% change for 3 months. Rather

than convert the change in the annual rate into the equivalent 3-month rate, we adjust

the notional principal from $1,000,000 to one-quarter of it, or $250,000. So, a change in the

annual rate of 1% would result in the change in the value of the contract by $250,000 x

0.01 or $2,500. Therefore, the contract unit of $2,500 gives a convenient way to calculate

changes in the contract value directly from the change in the quoted annual rate – just take

the change in percent and multiply it by $2,500. For example, if the Eurodollar deposit

rate changes from 4% to 4.15% – i.e., a 15-basis point change, the contract value changes

by $2,500 x 0.15 = $375.

Suppose an investor needs to borrow $5 million in 3 months’ time. A fall in interest rates

would be beneficial to her as she could borrow at a lower cost. This is the equivalent of

the price increasing for the futures contract. So, she has a long position with regard to the

future loan of $5 million. By the same token, a rise in interest rates would increase her

borrowing cost. In order to avoid the uncertainty of interest rate movements, she takes a

short position in the interest rate futures. The short position benefits from a fall in the price

of the futures contract, which comes from an increase in interest rate. Hence, the investor

has essentially locked in the current 3-month Eurodollar deposit rate. As she is borrowing

$5m, she needs to buy 5 contracts to hedge interest rate risk.

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FIN358 Derivative Hedging and Combination Strategies

1.2 Interest Rate Swaps

A swap is a common interest rate derivative instrument. It involves a sequence of

transactions to borrow/lend over a future period.

1.2.1 Mechanics of Interest Rate Swaps

An interest rate swap involves:

1. 2 counterparties, one in the long position, one in the short position

• the long is also known as the fixed rate payer/floating rate receiver

• the short is also known as the fixed rate receiver/floating rate payer

2. the interest rate swap refers to a notional amount

3. a floating interest rate (e.g., a LIBOR interest rate), and a fixed rate

• the floating interest rate is reset periodically

• the fixed rate which remains constant for the duration of the swap; It may

be interpreted as the “price” of the swap.

4. two series of cash flows are involved:

• The frequencies of the series may be equal or not

• One series of cash flows, called the fixed leg, is based on the fixed rate.

◦ At each payment date, an amount that is calculated with the fixed

rate upon the notional principal is received by the short, and paid

by the long

• The other series of cash flows, called the floating leg, is based on the

floating rate.

◦ At each payment date, an amount that is calculated with the floating

rate (the most recent floating rate that is set) upon the notional

principal is received by the long, and paid by the short

◦ These interchanges of cash flows occur within the maturity of the

swap

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FIN358 Derivative Hedging and Combination Strategies

◦ The notional principal is for referencing purposes in the calculation

of cash flows. The actual amount does not change hands.

An application of the interest rate swap is as follows.

A corporation may feel that its obligation to pay coupons at fixed interest rate to its

bondholders is a great liability in the market environment, given the current market views.

Hence, it may wish to enter into an appropriate interest rate swap arrangement to pay

floating and to receive fixed – in other words, to be in the short position. With the swap

overlaying the issued bond, the corporation effectively pays floating interests. It collects

fixed cash flows that are then passed on to the bondholders.

1.2.2 Day Count Issues

In the real world, swap cash flows are computed with the consideration of day count

fractions.

In this course, you may or may not need to incorporate day count fractions into your

calculations in questions concerning interest rate swaps. It will depend on the information

given in the question (whether that is sufficient for determining the actual day count

fraction or not) or it will be explicitly stated whether there is a need to use day count

fractions or not.

1.3 Currency Derivatives: Currency Options

Lesson Recording

Currency Derivatives

Currency options, like most currency-related products, are mainly traded over-thecounter. Exchange-traded instruments exist but remain in the minority. They are traded

on the NASDAQ OMX in the United States.

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FIN358 Derivative Hedging and Combination Strategies

A currency option has, as its underlying asset, a currency. The price of a currency or its

exchange rate is quoted in terms of a currency pair – for example, EURUSD. The first

currency in the pair, EUR in this case, is known as the base currency or the asset to be

priced. The second currency, USD in this case, is known as the quote or counter currency.

This is the currency used to price one unit of the asset. Thus, an exchange rate of 1.2 for the

EURUSD means that the price of one euro is US$1.20. As most international transactions

are quoted in U.S. dollars – that is the assets, whether physical or financial, are priced in

U.S. dollar terms – the exchange rate, EURUSD, is known as a direct quote. If the quote

is the amount of foreign currency for each U.S. dollar – e.g., USDEUR – we are using an

indirect quote. Note that the value of USDEUR is the reciprocal of EURUSD.

When firms or investors buy a currency, they don’t take possession of the currency notes,

unlike our transactions at the money changers. Rather, the currency that is bought is left as

a deposit at the bank or institution that sold it. This foreign currency deposit earns interest

much like any other deposits at the bank. Hence, we can think of the currency deposit

as similar to a stock that pays a dividend. The interest rate from the deposit would be

equivalent to the dividend yield of the stock. Consequently, we can apply the modified

Black-Scholes theory to price currency options.

When we are buying or selling a currency the transaction can be looked at in two ways.

Let’s focus on the exchange of currencies using the EURUSD currency pair. If I buy EUR, I

will receive EUR and pay for it with USD. This same transaction can be viewed in another

way. As I give USD to the other party and receive EUR in return, it is as though I sold USD

to the other party who paid for it with EUR.

In the context of options, buying a call with EUR as the underlying asset is the same as

selling a put with the USD as the underlying asset for the EUUSD currency pair.

To understand this, note these relationships:

• Buying a call is having the option to buy the underlying asset

• If the underlying asset is EUR, we buy EUR with USD.

• So, we are buying a call option on EURUSD.

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FIN358 Derivative Hedging and Combination Strategies

• Buying EUR with USD is the same as selling USD and receiving EUR

• The option to sell USD for EUR is similar to having a put option with USD as the

underlying asset of USDEUR

• So, a put option on EURUSD is a call option on USDEUR.

1.3.1 Hedging With Currency Options

Hedging a currency exposure with currency options is not much different from hedging

an asset using an option on the asset. The only thing to note is that since we are dealing

with a currency pair, we need to ensure that the notionals are matched with the foreign

currency notionals.

For example, suppose we have to pay EUR1 million in 6 months for equipment purchased.

If the spot rate of EURUSD today is 1.25, and we want to pay just US$1.25 million in 6

months, we can lock-in the rate of 1.25 by buying a call option on EURUSD at an exercise

price of 1.25 based on a notional 1 million units of EUR. If the EURUSD rate rises to 1.5,

we would have to pay US$1.5 million to obtain the EUR1 million. However, the additional

amount of US$250,000 will be offset by the profit from the option position of US$0.25 x 1

million = US$250,000

Suppose we hedge the currency exposure using USD as the notional. So, we buy a put

option on the USD at the exercise price of 1/1.25 or EUR 0.8. (Note: a rise in the price of

EUR is the same as a fall in the price of USD. When the price of EUR rises to US$1.50,

the price of USD falls to EUR 0.666667. The US$1.25 million earmarked to pay the EUR1

million would only buy EUR(1,250,000 x 0.666667) or EUR833,337. For the put position, the

gain per USD is EUR(0.1333333) was due to the drop in the price of EUR to EUR0.666667

(from EUR0.8), resulting in a gain of EUR0.1333333. This translates to a gain of EUR

(1,250,000 x 0.133333) or EUR166,667. EUR833,337 + EUR166,667 = EUR1 million, which

is the amount needed to pay for the equipment.

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FIN358 Derivative Hedging and Combination Strategies

1.4 Credit Derivatives

Lesson Recording

Credit Derivatives

A credit derivative is a derivative whose value depends on credit risk. Credit derivatives

come in the form of credit default swaps.

Credit default swaps were developed in the 1990s and initiated the market for credit

derivatives. One may see this as part of the growth of the derivatives industry that initially

gained impetus in the 1970s with the set-up of CBOE, the foray of CME into currency

futures, and the publication of the Black-Scholes Theory.

Some aspects of the credit default swaps will be discussed in this section.

1.4.1 Hedging Using Credit Default Swaps

A credit default swap (CDS) works this way:

• Two counterparties are involved in the contract

• Underlying the contract is a reference entity (a bond)

• There is a notional principal (which may differ from that of the bond)

• One party will periodically pay his counterparty an amount of money that is

calculated as a fixed percentage of the notional principal

• This percentage is quite small and is usually quoted in tens of basis points

• This number of basis points is in fact the “price” quote of the contract, or the CDS

spread

• This payment will stop if a credit event occurs

• What constitutes a credit event is clearly spelt out in the contractual documentation

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FIN358 Derivative Hedging and Combination Strategies

• A credit event occurs when the issuer (company) of the reference bond defaults on

the payments

• When a credit event occurs, the counterparty, which has been receiving payment

of credit premiums, must now pay a lump sum of money to make up for the loss

in the value of the bond

• In this way, the value of the bond is protected for the investor who bought the CDS

1.5 Equity Derivatives

Lesson Recording

Stock Index Futures

There are various types of equity derivatives. We discussed calls and puts on stocks

in the section on options in Study Unit 1. Besides calls and puts on individual stocks,

we can have options on a basket of stocks and even stock indices. Futures contracts

tend to be associated with commodity. However, financial futures expanded the field to

financial instruments. Thus, we have futures contracts on individual stocks which is not

that common and futures contracts on stock indexes, e.g., the S&P 500 futures contract.

Finally, the equity swap is another type of equity derivative.

1.5.1 Hedging With Equity Derivatives

We will not be going into the details of hedging with equity derivatives.

For options, protective puts can be used to hedge losses from stock positions. This was

discussed in Study Unit 1.

The principles of hedging using futures are similar to those discussed in our earlier

discussion on hedging interest rate risk. The most popular futures contract used for equity

hedging is the S&P 500 index futures. One important point to note is that the beta of the

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FIN358 Derivative Hedging and Combination Strategies

equity portfolio that is being hedged may be different from the beta of the index futures.

As the S&P 500 index futures is based on the market index, it has a beta of 1. Adjusting

for the different risks between the equity portfolio and the index is as follows:

Number of S&P 500 index futures needed

= (Portfolio Beta x Portfolio value) / ($250 x S&P 500 Index)

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FIN358 Derivative Hedging and Combination Strategies

Chapter 2: Trading Strategies Involving Options

Options can be combined with other derivatives, with themselves, or with their

underlying assets into combinations. These combinations give rise to various payoff

profiles that can be used to shape investment or trading strategies.

We will explore these issues here.

2.1 Principal-Protected Notes

A note is essentially a bond, perhaps with additional features. Sometimes, it is a

combination of fixed income and equity assets.

A principal-protected note (PPN) is a structure that involves a note plus a put option. The

purpose of the put option is to protect the value of the principal of the note if it were to

fall. PPNs are generally created and sold by non-exchange financial institutions.

To put a PPN together is to work out the right price of the combination (fixed income

instrument + equity instrument + put option), the fixed part, the risky opportunistic part,

the protected part, and so on.

2.2 Strategies Involving a Single Option and a Stock

Two common strategies that involve a single option and a stock are the covered call and

the protective put.

The covered call involves the writing of a European call together with the holding of

the underlying asset (one unit of asset for one unit of underlying asset referenced by the

covered call).

The protective put involves the buying of a European put together with the stock itself,

again in suitably matching quantities.

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FIN358 Derivative Hedging and Combination Strategies

The motives of these strategies are as follows:

• Covered call:

◦ The selling of the call is for the collection of premium. But selling a call creates

risk. However, this risk is covered by the long position in the underlying

stock. If the call matures OTM, premium is collected for free. If not, the stock

is delivered at the strike price.

• Protective put:

◦ The put protects the stock if it falls in value. At a suitable strike price, the

intrinsic value of the put option kicks in. The rise in value of the put cancels

off the fall in value of the stock. So, that protective put position remains

hedged and (roughly) constant in value.

2.3 Spreads

Lesson Recording

Spreads

A spread is a trading strategy that involves several options of a similar kind. Here are

some examples:

• Bull spreads

◦ This involves a long position in a European call option with a strike price

and a short position of a European call option on the same underlying at a

higher strike price

• Bear spreads

◦ This involves a long position in a European put option with a strike price

and a short position of a European put option on the same underlying at a

lower strike price

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FIN358 Derivative Hedging and Combination Strategies

• Box spreads

◦ This is a combination of a bull spread and a bear spread at the same set of

strike prices

• Butterfly spreads

◦ This involves:

▪ buying a European call option at X1

▪ selling 2 European call options at X2

▪ buying a European call option at X3

▪ with the constraint that X1 < X2 < X3

• Calendar spreads

◦ This involves selling a European call option of a certain maturity and buying

of a European call option of a longer maturity

◦ It can also be created using put options

• Diagonal spreads

◦ In the above, only one factor varies among the options involved in a single

combination (e.g., strike or maturity)

◦ A diagonal spread is a combination of options such that both maturities and

strike prices may be different for the options involved

In each of these cases, it is important to understand the situation via:

1. a qualitative description of what the strategy is about

2. the payoff diagram

3. the payoff table for different scenarios

It is also worthwhile to note that a given payoff diagram may be constructed using

different option combinations. For instance, for the put-call parity, any position involving

call options can be turned into one that involves put options.

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FIN358 Derivative Hedging and Combination Strategies

2.4 Combinations

An option combination is a strategy that involves call and put options with the same

underlying asset.

Here are some examples:

• Straddle

◦ This involves a long position in a European call and a long position in a

European put with identical strike prices and maturities

• Strip and Strap

◦ A strip involves a long position in a European call and a long position in two

European puts with identical strike price and maturity

◦ A strap involves a long position in two European calls and a long position in

a European put with identical strike price and maturity

• Strangle

◦ This involves a long position in a European call and a long position in a

European put with different strike prices but identical maturities

2.5 Other Payoffs

By using suitable combinations of call and option options (of the same maturity), any

payoff diagram can be well approximated. This means that derivative products can be

structured to cater to different needs of investors.

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FIN358 Derivative Hedging and Combination Strategies

Read

Read the following sections in:

Gottesman, A. (2016). Price and volatility trading strategies. In Derivatives Essentials:

An introduction to forwards, futures, options, and swaps (pp. 188-205). Wiley.

• Section 11.1: Price and volatility views

• Section 11.2: Relating price and volatility views to Delta and Vega

• Section 11.3: Using forwards, calls and puts to monetize views

• Section 11.4: Introduction to straddles

• Section 11.5: Delta and Vega characteristics of long and short straddles

• Section 11.6: The ATN DNS strike price

• Section 11.7: Straddle numerical example

• Section 11.8: P&L diagrams for long and short straddles

• Section 11.9: Breakeven points for long and short straddles

• Section 11.10: Introduction to strangles

• Section 11.11: P&L diagrams for long and short strangles

• Section 11.12: Breakeven points for long and short strangles

• Section 11.13: Summary of simple price and volatility trading strategies

Gottesman, A. (2016). Spread trading strategies. In Derivatives Essentials: An

introduction to forwards, futures, options, and swaps (pp. 223-247). Wiley.

• Section 13.1: Bull and bear spreads using calls

• Section 13.2: Bull and bear spreads using puts

• Section 13.3: Risk reversals

• Section 13.4: Butterfly spreads

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FIN358 Derivative Hedging and Combination Strategies

• Section 13.5: Condor spreads

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FIN358 Derivative Hedging and Combination Strategies

Formative Assessment

1. Which of the following is a combination strategy that involves two or more options

of the same type?

a. Straps

b. Strips

c. Butterfly

d. Spreads

2. The portfolio that comprises a short position in a European call option and a long

position in its underlying stock is known as ___________.

a. a butterfly

b. a protective put

c. a covered call

d. a box spread

3. A reverse butterfly is useful when __________.

a. the market is volatile

b. the market is quiet

c. the investor is not sure where the market is heading

d. none of the above

4. Which of the following is not associated to a credit default swap?

a. Credit event

b. Reference entity

c. Notional principal

d. Duration

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FIN358 Derivative Hedging and Combination Strategies

5. Currency options are theoretically priced by the Black-Scholes formula by

_____________.

a. assuming that the foreign currency is an income yielding asset.

b. assuming that the currencies involved do not change in relative value over

time.

c. assuming that the volatility of the exchange rate is less than 5%.

d. None of the above

6. Ms Lim bought 1,000 shares of Coffee Shop Ltd at $5 which is also the current price.

She bought a put which covered 1,000 shares at an exercise price of $4.00 to protect

against losses from a price decline. The put cost $1 per share. At the expiration of

the put, the share price was $4.50. Ms Lim sold all her shares then. What is Ms Lim’s

overall profit from all her transactions?

a. – $2,000

b. – $1,500

c. – $1,000

d. – $500

7. On June 1, Mr Tan sold 5 September S&P500 futures contract at 1,120.50. On August

20, Mr Tan entered into an offsetting trade with the contract now priced at $1,098.03.

Mr Tan’s gain or loss is closest to ___________.

a. $27,000 gain

b. $27,000 loss

c. $5,400 gain

d. $5,400 loss

8. The seller of the T-Bond futures contract has the choice regarding which bond in the

market to deliver.

a. True

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FIN358 Derivative Hedging and Combination Strategies

b. False

9. Mr Lee wants to hedge his short USDJPY currency pair using options.

a. He can hedge by buying a call on the USDJPY rate.

b. He can hedge by selling a put on the JPYUSD rate.

c. He can only hedge using currency futures.

d. Both (a) and (b) are correct.

10. In a Credit Default Swap, the buyer of the swap can end up receiving nothing.

a. True

b. False

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FIN358 Derivative Hedging and Combination Strategies

Solutions or Suggested Answers

Formative Assessment

1. Which of the following is a combination strategy that involves two or more options

of the same type?

a. Straps

Incorrect

b. Strips

Incorrect

c. Butterfly

Incorrect

d. Spreads

Correct

2. The portfolio that comprises a short position in a European call option and a long

position in its underlying stock is known as ___________.

a. a butterfly

Incorrect

b. a protective put

Incorrect

c. a covered call

Correct

d. a box spread

Incorrect

3. A reverse butterfly is useful when __________.

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FIN358 Derivative Hedging and Combination Strategies

a. the market is volatile

Correct

b. the market is quiet

Incorrect

c. the investor is not sure where the market is heading

Incorrect

d. none of the above

Incorrect

4. Which of the following is not associated to a credit default swap?

a. Credit event

Incorrect

b. Reference entity

Incorrect

c. Notional principal

Incorrect

d. Duration

Correct

5. Currency options are theoretically priced by the Black-Scholes formula by

_____________.

a. assuming that the foreign currency is an income yielding asset.

Correct

b. assuming that the currencies involved do not change in relative value over

time.

Incorrect

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FIN358 Derivative Hedging and Combination Strategies

c. assuming that the volatility of the exchange rate is less than 5%.

Incorrect

d. None of the above

Incorrect

6. Ms Lim bought 1,000 shares of Coffee Shop Ltd at $5 which is also the current price.

She bought a put which covered 1,000 shares at an exercise price of $4.00 to protect

against losses from a price decline. The put cost $1 per share. At the expiration of

the put, the share price was $4.50. Ms Lim sold all her shares then. What is Ms Lim’s

overall profit from all her transactions?

a. – $2,000

Incorrect

b. – $1,500

Correct

c. – $1,000

Incorrect

d. – $500

Incorrect

7. On June 1, Mr Tan sold 5 September S&P500 futures contract at 1,120.50. On August

20, Mr Tan entered into an offsetting trade with the contract now priced at $1,098.03.

Mr Tan’s gain or loss is closest to ___________.

a. $27,000 gain

Correct

b. $27,000 loss

Incorrect

c. $5,400 gain

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FIN358 Derivative Hedging and Combination Strategies

Incorrect

d. $5,400 loss

Incorrect

8. The seller of the T-Bond futures contract has the choice regarding which bond in the

market to deliver.

a. True

Correct

b. False

Incorrect

9. Mr Lee wants to hedge his short USDJPY currency pair using options.

a. He can hedge by buying a call on the USDJPY rate.

Incorrect

b. He can hedge by selling a put on the JPYUSD rate.

Incorrect

c. He can only hedge using currency futures.

Incorrect

d. Both (a) and (b) are correct.

Correct

10. In a Credit Default Swap, the buyer of the swap can end up receiving nothing.

a. True

Correct

b. False

Incorrect

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FIN358 Derivative Hedging and Combination Strategies

References

Gottesman, A. (2016). Derivatives essentials: An introduction to forwards, futures, options and

swaps. Wiley.

SU3-28

Study

Unit4

Fixed Income Elements,

TermStructure and Valuation

FIN358 Fixed Income Elements, TermStructure and Valuation

Learning Outcomes

By the end of this unit, you should be able to:

1. Appraise the basic features and terminologies of a bond. These include the bond

indenture, maturity, par value, coupon rate, and bond yield.

2. Discuss the characteristics of different types of bonds, namely, Treasury bonds,

municipal bonds, commercial paper, negotiable certificates of deposit, and

bankers’ acceptance.

3. Examine the term structure of interest rates and discuss the implications of the

different yield curve shapes and theories of term structure.

4. Discuss how bonds can be valued and compute the clean price, full price and

accrued interest.

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FIN358 Fixed Income Elements, TermStructure and Valuation

Overview

This study unit introduces the subject of fixed income securities by first looking at the

terms that describe a bond as well as some of their characteristics. A bond is a contract

between the issuer and the bond investors. It is important to understand some of the key

aspects of this contract as they spell out the rights and obligations of each party. The terms

of the contract have a bearing on the risk and return of the bond.

Underlying the investment in bonds are interest rates. Interest rates or yields are the

returns for different types of bonds. An investor would want to make sense of these rates.

To do so, a tool called the yield curve is used. The yield curve plots the yields of bonds

of the same risk against their maturity. The relationship between the different yields is

commonly referred to as the term structure of interest rates.

The last topic in this unit looks at the valuation of bonds. We introduce some ideas of how

bonds can be valued. Other methods of valuation will be introduced in subsequent study

units.

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FIN358 Fixed Income Elements, TermStructure and Valuation

Chapter 1: Elements of Fixed Income

Lesson Recording

Bond Types and Characteristics

1.1 Introduction to Fixed Income Securities

Fixed income securities, or bonds as they are commonly known, are long-term IOUs issued

by corporations, governments or sovereigns, and supranational organisations like the

World Bank. The typical bond pays a fixed interest or coupon amount (hence the name

fixed income) on a periodic basis, typically every six months. Terms associated with a

bond are:

a. Face or par value or redemption value is the principal amount to be repaid at the

end of the loan.

b. Coupon rate or the nominal interest rate is the annual interest rate payable on

the bond. The coupon rate can be a fixed or variable (floating) rate. The coupon

amount is the coupon rate multiplied by the par value.

Floating rate bonds are called floating rate notes (FRNs) or floaters. Unlike fixed

rate bonds, the coupon payable depends on interest rates in the market. This

bond benefits from rises in the interest rate. The coupon rate is reset periodically,

either on a quarterly or semi-annual basis. This rate is equal to a reference rate

plus a margin, for example, (K + 2%). For a long period, the value of K was based

on the London Interbank Offer Rate (LIBOR). However, because of a scandal

involving the manipulation of the LIBOR, it is now superseded by its successor,

the Secured Overnight Financing Rate (SOFR).

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FIN358 Fixed Income Elements, TermStructure and Valuation

With the step-up coupon rate, the coupon rate is increased by fixed amounts

over time. Another type of structure involves non-payment of coupon for the

first few years. Such a bond is known as the deferred coupon bond, which is

issued for infrastructure projects. This arrangement makes sense as these projects

take considerable time to complete. An infrastructure project would only be able

to deliver cash flows to service the loan, upon its completion.

Some bonds pay no interest at all. Such zero coupon bonds pay interest

implicitly, as they are issued at a discount to the principal or face value.

Credit-linked coupon bonds have coupon rates that change, contingent on

changes in the bond’s credit rating. For example, if the credit rating of the bond

improves (that is, its risk is reduced), then investors would be satisfied to receive

a lower yield for holding the bond. So, the coupon rate is adjusted downwards

to reflect the lower risk.

Another coupon structure is payment-in-kind (PIK). Instead of receiving

coupon payments, bond holders are paid interest in the form of additional bonds.

Finally, in index-linked bonds the coupon is not fixed but linked to some index.

An important index-linked bond is the inflation-linked bond – for example, the

U.S. Treasury Inflation Protected Security (TIPS). When inflation increases, the

principal is adjusted upwards to reflect the increase in inflation. As the coupon

payment is the coupon rate which is now applied to a higher principal amount,

the coupon payment will also be higher.

c. Maturity is the contracted period of payments for the bond. At maturity, the face

value is repaid. A bond’s maturity is usually given in years. A perpetual bond

is a bond that has no maturity. Such a bond is similar to stocks in the sense that

whatever the investor paid the company, for the security, is never repaid.

Bonds that have maturities of one year or less are classified as money market

securities and are known as bills. Bonds with maturities longer than a year are

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FIN358 Fixed Income Elements, TermStructure and Valuation

known as capital market securities. Short-term bonds with maturities less than

ten years are known as notes.

d. Bond indenture, sometimes referred to as the trust deed, is the legal contract that

prescribes the obligations of the issuer and the rights of the bond holders. Some

of the items in the indenture include:

i. Collaterals: These are the assets or guarantees that serve as security in

case there is a loan default. Investors whose securities are not backed

by these collaterals have no claim to the proceeds from the sale of the

assets or cash from the guarantors.

In a collateral trust bond, financial assets serve as the collateral,

while for an equipment trust certificate the collateral is physical assets.

For example, the collateral backing equipment trust bonds could be

equipment and machinery.

Mortgages and accounts receivable can also serve as collateral. These

will be discussed in a later study unit.

ii. Repayment of principal arrangements: There are various ways in

which the principal can be repaid.

In a bullet bond, the principal is repaid in full at maturity.

Payments made by a fully amortised bond consist of equal payments

each period such that the principal will be fully repaid by the bond’s

maturity.

In the hybrid case of the partially amortised bond, the annual

payments are higher than the coupon amount but are insufficient to

fully repay the principal amount by maturity.

Table 4.1 illustrates the three different payment methods.

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FIN358 Fixed Income Elements, TermStructure and Valuation

Table 4.1 Payment schedules for bullet, fully amortised and partially amortised bonds

Bullet Bond (Interest = 5%)

Year Investor

Cash Flows

Interest

Payment

Principal

Payment

Outstanding Principal

at the End of the Year

0 -1,000.00 1,000.00

1 50.00 50.00 0.00 1,000.00

2 50.00 50.00 0.00 1,000.00

3 1,050.00 50.00 1,000.00 0.00

Fully Amortized Bond (Interest=5%)

Year Investor

Cash Flow

Interest

Payment

Principal

Payment

Out standing Principal

at the End of the Year

0 -1,000.00

1 367. 21 50.00 317. 21 682.79

2 367. 21 34.14 333.07 349.72

3 367. 21 17.49 349.72 0.00

Partially Amortized Bond (Interest=5%)

Year Investor

Cash Flows

Interest

Payment

Principal

Payment

Out standing Principal

at the End of the Year

0 -1,000.00

1 183.00 50.00 133.00 867.00

2 183.00 43.35 139.65 727.35

3 183.00 36.37 146.63 580.72

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FIN358 Fixed Income Elements, TermStructure and Valuation

iii. Bond provisions: These are the arrangements that affect the possible

early repayment of part or the whole principal before maturity.

With a call provision the issuer has the right to buy back the bond

before its maturity. Issuers would want to do this if interest rates have

fallen since the bond issue. This provision allows the issuer to issue new

bonds at a lower coupon rate to replace the existing bonds which have

a higher coupon rate. Such a provision disadvantages the bondholders

who would demand a higher coupon rate for such bonds compared

to similar ones without the call feature. Further, bondholders are also

concerned that should interest rates fall substantially within a few years

of the bond’s issuance, the bond will be redeemed. To forestall such an

eventuality, bondholders insist that the bond be protected from being

called for a period of time from the date of issue. This period is known

as the call protection period or lock-out period.

Should the bond be called, it can be exercised in different styles or

ways. The American style allows the issuer to call at any time after

the call protection period. In contrast, the bond can only be called at

maturity under the European style. A third style, the Bermuda style,

lies in between the other two styles in terms of flexibility. Under this

style bonds are callable only on dates specified in a call schedule.

Call and put provisions operate in a similar manner to option calls and

puts. A put provision gives the bond holder the right to sell an asset,

in this case the bond, at an exercise price equal to the face value of

the bond, before maturity. This is tantamount to redeeming the bond

from the issuer who is the put seller. Similar to the call provision, there

is a put protection period during which the bond cannot be put or

sold to the issuer. For a one-time put bond, the bond holder is only

given one chance to exercise the put option before maturity unlike the

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multiple put bond which gives the bondholder multiple opportunities

to exercise the put option.

Repayment of a large amount of money is more onerous than if the

amount is smaller. Hence, to reduce the chance that a large sum will

not be repaid, the sinking fund provision requires the issuer to buy

back a portion of the bonds periodically. The result is a reduction in the

number of bonds outstanding at maturity.

Finally, some bonds have a convertible provision that allows the bond

to be converted into shares of the company. The number of shares

each bond can be converted into is known as the conversion ratio. The

conversion price is the price at which each share can be bought by the

bondholders. The conversion ratio is the par value of the bond divided

by the conversion price. Note that we only need to know either the

conversion ratio or the conversion price. Knowing any one of them

allows us to calculate the other easily.

Following the Great Financial Crisis of 2008, banks were asked to shore

up their capital. One way that banks satisfied the requirement was

through issuing contingent convertible bonds, known as “Cocos”.

Should the bank’s equity value fall below a certain threshold, Cocos are

automatically converted into shares, allowing the bank to fulfil its Tier

1 capital adequacy ratio requirement.

iv. Bond covenants state the promises of the issuer with regard to certain

actions regarding the bond.

Affirmative covenants are undertakings or promises made by the

issuer to take certain actions. Non-compliance with the covenants could

adversely affect the value of the bonds. These actions include the

timely payment of the bond’s interest and principal as well as taxes.

As the assets serve as collateral for the bond, there is a need for the

issuer to maintain properties in good condition, for example. Further,

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compliance is monitored by the trustee who can only do so effectively

if periodic reports are submitted by the issuer.

Negative covenants prohibit the issuer from taking certain actions.

These covenants are put in place to prevent the company’s management

from taking too much risk, for example, increasing the debt level

such that it goes above a certain debt ratio, or taking on certain

types of investments. Other restrictions prevent actions that would

disadvantage bondholders. Examples of such actions include the

disposal of assets or payments of large dividends that reduce the value

of the bond’s collateral. Similarly, the company cannot undertake a

merger or acquisition, unless the terms of the old bonds are carried over

to the new entity. A very important negative covenant is the negative

pledge. This restricts the issuance of new debt unless the new debt

ranks below, or are junior to, the existing debt.

e. Currency denominationdesignates the currency in which the coupons and the

principal of the bond are paid.

If an issuer is domiciled or registered in a certain country and issues a bond

denominated in the currency of that country, that bond is known as a domestic

bond. An example is a Japanese company issuing a Japanese yen-denominated

bond in Japan.

However, should an entity decide to issue a bond in a foreign country, and in

their currency, then the bond is known as a foreign bond. An example, is Honda

Motors issuing a Singapore dollar bond in Singapore.

A bond issued in a country, but not in the currency of the country is known as a

euro bond. An example is either DBS or Honda Motors issuing US dollar bonds

in Singapore.

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Bond payments can be made in more than one currency. Dual currency bonds

could use one currency for coupon payments and another currency for the

repayment of principal.

Currency option bonds allow the bondholder the choice of payment of coupon

or principal in different currencies.

f. Legal entity – Bonds are normally issued under the firm’s name. However, a firm

may want to issue bonds backed by certain assets. The firm does this by setting

up a separate entity to issue the new bonds. As such bonds have lower risk than

the overall firm, bondholders are willing to accept a lower bond yield resulting

in interest savings for the firm.

The separate entity set up to issue the bonds is known as a Special Purpose

Vehicle (SPV) or Special Purpose Entity (SPE). As the assets backing the bond

are held by the SPV the firm’s other bondholders have no claim on them. So the

SPV would have a higher credit rating than the firm itself.

g. Bond identification – Bonds can be identified by the name of the issuer, the

coupon rate and the date of issue. For example a General Motors bond would

be identified as such: General Motors 4.5s of 31/5/30, which means a General

Motors bond paying a 4.5% coupon and maturing on 31/5/30.

1.2 Treasury Bonds

Treasury bonds are issued by the U.S. government. They are considered essentially

default-free since they are backed by the full faith and credit of the U.S. government. Given

their risk-free characteristic such securities would have low returns or yields. The most

popular long-term U.S. Treasury bond has a maturity of 10 years. The yield of this bond

often serves as a benchmark yield for long term bonds. It is combined with the yields of

shorter treasury securities to form a graph known as the Treasury yield curve. The yields of

non-Treasury bonds are based on the Treasury yield curve with risk premiums (additional

yields or returns) added to reflect their higher default risks.

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Typical denominations for U.S. treasury bonds are $1,000 or multiples of $1,000. As

treasury bonds are in high demand, they trade in very active secondary markets. The

most liquid securities are those that have been newly issued and are more liquid than

the existing securities. Their yields are used “benchmark” yields. The next most liquid

securities are those most recently issued. These “on-the-run” securities are followed in

terms of liquidity by the older “off-the-run” securities which have been issued even

earlier.

1.3 Treasury Strips

A treasury strip divides the stream of payments from a treasury bond that pays a coupon

into two components. The first component consists of the all the coupon payments and

is known as an interest only (IO) security. The remaining component is the principal

payment and is called the principal only (PO) security. The PO is subject only to interest

rate risk and its yield serves as a measure of the spot rate for that maturity. POs are also

useful in managing interest rate risk that we shall discuss in Study Unit 5. Collectively, IOs

and POs are known as “Separate Trading of Registered Interest and Principal Securities”

or STRIPS. Figure 4.1 below shows the cash flows from a STRIP.

Figure 4.1 Treasury STRIP

1.4 Municipal Bonds

In the U.S., municipal bonds (munis) are bonds issued by state or local governments. The

typical par value on a municipal bond is $5,000. Interest income on munis is not taxed at

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the federal level and is usually exempt from state and local taxes, if the investor lives in

the state in which the muni is issued. Capital gains are taxable. The tax exemption allows

municipal governments to obtain lower cost financing. With the tax exemption, investors

require lower yields on munis than on taxable bonds with equivalent risk.

General obligation (G.O.) bonds are munis whose credit worthiness depend on the

municipality’s taxing power as well as the amount taxes it can collect.

Revenue bonds are backed only by the revenues of a specific project. General tax revenues

cannot be used to make payments on revenue bonds; thus a revenue bond is riskier than

a G.O.

One cannot directly compare municipal bond rates with taxable corporate bond rates

without adjusting one or the other. The formula to adjust for taxes is simple:

ia = ib * (1 ‐ t) where a and b stand for after tax and before tax, respectively.

Hence, a 4.5% muni bond rate cannot be directly compared to a 6% equivalent risk

corporate bond rate. We must put both bond rates on the same tax basis. If we are in a 28%

tax bracket, the after-tax corporate bond rate ia = 6%*(1‐ 0.28) = 4.32%. Now, we can see

that the investor is better off with the muni rather than the corporate bond.

Muni bond offerings are generally underwritten by investment bankers in a firm

commitment offering. In such an offering the underwriter buys the issue from the firm

paying the bid price and reselling the issue at the offer price.

When the investment banker does not anticipate selling out the bond issue, it may only do

the bond offering on a best efforts offerings basis. Under this method, the underwriter

assists in the sale of the securities but does not buy the issue.

In an underwritten issue, the investment bank may be chosen on the basis of a bid process

where the banker that submits the highest bid price will be selected, or on a negotiated

basis. Municipal issuers generally consider multiple investment bankers before choosing

a lead underwriter. G.O. bonds usually must be issued via a competitive bid.

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Sometimes, bonds are privately placed and usually sold to 10 or fewer large investors.

These investors, usually institutional investors, have assets of over $100 million. In this

case, the bonds need not be registered. Smaller to mid-sized municipal and corporate

borrowers typically use private placements. Private placements can now be traded among

large investors, but the market is very thin. Note that the secondary market for publicly

issued munis is also very thin.

1.5 Corporate Fixed Income Securities

1.5.1 Commercial Paper

Commercial paper is a short-term unsecured promissory note issued by creditworthy

corporations and financial institutions. Because the notes are unsecured and are not very

liquid, commercial papers need to be rated by ratings agencies. The rating strongly affects

the cost of financing with commercial paper.

Low quality paper is often secured by bank lines of credit to obtain a better rating. The

maximum maturity is 270 days (most are less) because the SEC requires formal registration

of securities with maturities greater than 270 days.

Commercial paper is a discount instrument and uses discount quotes similar to T-bills.

The commercial paper market has developed to provide corporations with an alternative

to short-term bank loans. Commercial paper is issued in denominations ranging from

$100,000 to $1 million, with the most common maturities in the 20 to 45-day range. A small

percentage of issuers directly market their own paper, but the bulk is sold through brokers

and dealers.

There is no active secondary market for commercial paper, partly because commercial

paper dealers will buy them from the buyer if the buyer needs the money prior to maturity.

1.5.2 Negotiable Certificates of Deposit

A negotiable certificate of deposit (hereafter CD) is a bearer certificate. It indicates that

a time deposit has been made at the issuing bank from which the bearer can redeem

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the deposit at maturity. Large CDs are negotiable instruments. Negotiable CDs have

a minimum denomination of $100,000, but denominations of $1 million are the most

common. Maturities range from 1, 2, 3, 6 months and 1 year.

Negotiable CD rates are add-on rates (single-payment loans) quoted using the 360-day

convention. Large well-known banks, particularly New York banks, can often pay lower

interest rates on their CDs than other lesser well-known institutions.

There is a secondary market in CDs, although it is not very active. CDs are required to

have ‘substantial interest penalties for early withdrawal.’The secondary market eliminates the

problem of the interest penalty and has increased the banks’ ability to tap on funds that

would otherwise be invested in non‐bank money market securities.

1.5.3 Banker’s Acceptance (BAs)

A banker’s acceptance or draft is often used to facilitate international trade in goods and

services. The seller of the goods writes either a time draft or a sight draft payable by the

buyer of the goods or services. A sight draft is a claim that becomes due and payable upon

presentation to the purchaser. A time draft is a claim that becomes due and payable at a

certain future date specified on the draft.

Because the seller normally will not know the creditworthiness of the buyer (and credit

investigation costs can be quite high), the seller may be reluctant to ship the goods unless

payment can be guaranteed by a third party. Banker’s acceptances (BAs) are a certain

kind of time draft where a bank has agreed to pay the seller of the goods the amount owed

if the buyer cannot or will not pay on the date due. The draft is backed by a letter of credit

drawn on the buyer’s bank, ensuring that the bank will “accept” the draft drawn up by

the seller.

Once the seller can prove that the goods have been shipped in accordance with the contract

and the proper paperwork presented to the buyer, the time draft can be sold as a discount

instrument. The seller of the goods can wait until maturity to receive payment, or more

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likely can discount (sell at a discount) the note with the bank and receive the discounted

face amount immediately. The bank can then hold the acceptance or sell it.

BAs are bearer instruments and are fairly actively traded. Maturities range from 30 to 270

days and BAs are bundled into round lots of $100,000 and $500,000. Their interest rates are

very close to T-bill rates as default risk is quite low since the BA is backed by the importer,

a large bank, and the value of the goods. The amount of BAs outstanding is quite small

compared to the other money market instruments (less than 1% of the total money market

securities outstanding).

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Chapter 2: Term Structure of Interest Rates

Lesson Recording

Term Structure

2.1 Introduction

The cash flows from a fixed income security are the periodic coupons (paid semi-annually

or annually) and the face value (usually paid at the end of maturity). The value (market

price) of a fixed income security is the present value of the expected cash flows discounted

at the market rate of interest.

Yield to maturity (YTM) is the required market rate or the rate that makes the discounted

cash flows from a fixed income security equal to the security’s market price. Overall, the

bond yields we observe are influenced by the risk-free interest rate, expected inflation,

interest rate risk, default risk, liquidity risk, special covenant risk, taxability, and maturity

risk.

The term structure depicts the relationship between maturity and yields for bonds

identical in all respects, except maturity. In practice, ‘identical’ means same rating,

liquidity and hopefully the same coupon (or differential tax effects will be present).

The graph of the term structure, known as the yield curve, can take on any shape, but

upward sloping is most common (meaning longer term bonds promising higher nominal

yields). The Treasury yield curve is a plot of yields on Treasury notes and bonds relative

to maturity.

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2.1.1 Yield Curve Shapes

The commonly observed shape of the yield curve is upward sloping where yields increase

with increases in the time to maturity. Other less commonly observed shapes include

downward sloping and flat. A downward sloping yield curve is the inverse of the upward

sloping yield curve as yields become lower with longer maturity. Hence this curve is

also known as the inverted curve. When yields do not change with maturity, we get the

flat yield curve. Finally, the yield curve can be humped-shaped. Yields rise initially with

increase in maturity but then fall as maturity lengthens. Shown in the diagram below are

various yield curve shapes.

Figure 4.2 Yield curve shapes

2.1.2 Theories of Term Structure

There are three theories which attempt to explain the term structure of interest rates and

how it changes.

Pure Expectations Theory

The yield curve is a graph of spot rates for different maturities. How would investors

decide what an appropriate spot rate should be? According to the pure expectations theory

or unbiased expectations theory the most appropriate yield for each spot rate should be the

corresponding forward rate which is the interest rate that a lender will charge for a future

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loan. Consequently, based on the no arbitrage argument, the long-term interest rate is none

other than the geometric average of one year forward rates. Assuming interest rates are

known with certainty under perfect markets, we prove this relationship below:

If the expected one-year rates are 6%, 7% and 8% for the next three years respectively, and

the three-year rate is 5%, how could one make money on this relationship?

The average of the short term one-year rates is 7%, but the three year rate is only 5%.

One could borrow any given amount such as $1000 for the full three years and invest that

money one year at a time and roll over the investment for three years. The borrowing cost

per year is 5% and the average rate of return is 7%. This is a riskless arbitrage under the

given assumptions that would force the three-year rate and the average of the one-year

rates to converge.

Geometric averages are used to account for compounding. For two‐ or three‐year periods

where the rates are similar, the use of arithmetic averages will give almost identical results.

The critical concept to understand is that according to the pure expectations theory, an

investor is indifferent as to how he invests for the long term. For example, for a N-year

investment, he can invest for N years all at once, or invest for 1 year and roll the investment

over each year for N‐1 times. The investor is indifferent because the future value of the

two alternatives is identical and the riskiness of the two investment strategies is identical

under the given assumptions.

Liquidity Preference Theory

If we invest in a bond for 3 years instead of 2 years, we would expect to get a higher annual

rate of return for the 3-year bond. The reason is simple. Locking up our funds for 3 years

means less flexibility than if the funds were locked up for 2 years as the bond principal is

paid only at the end of 3 years. Hence, we will require a premium to invest over a longer

period.

An alternative to investing for a long period of time is to invest for a short term and rolling

over the investment. For example, instead of investing for 3 years at a go, one could invest

year by year. Each year, when the investment matures, reinvest or roll over the investment

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for another year. This implies that the long-term rate cannotbe the average of the expected

short-term rates.

The long-term rate must equal the average of the short term rates plus what is illogically

called a ‘liquidity premium.’ (It is in fact an illiquidity premium.) The rationale for the

shorter maturity preference is that with uncertainty about future rates, it is riskier to lockin the funds and invest for a longer term rather than investing for a shorter time and rolling

the investment over. The reason is that it is harder to forecast rates further into the future.

This is a modification of the pure expectations theory, but it does not invalidate the logic

of the pure expectations theory. It does imply that long term rates are biased forecasters of

expected future short-term rates. We don’t know very much about the size of the liquidity

premiums. They increase with maturity, and probably do not get much over 100 to 200

basis points.

Market Segmentation Theory

Market segmentation or preferred habitat theory claims that there are two or three distinct

maturity segments (the segments are ill-defined) and market participants will not venture

out of their preferred segment, even if favourable rates may be found in a different

maturity. A less extreme version posits that a sufficient interest rate premium may induce

investors to switch maturity segments.

The idea behind segmentation is that institutions naturally have liabilities of a distinct

maturity, e.g., life insurers have long-term liabilities, so they will not invest for the short

term. Hence, there is no or only a very weak relationship between interest rates of different

maturities. Supply and demand for bonds of a given maturity sets the individual interest

rates.

By inference, there is no reason to construct a term structure as there is no relationship

between long-term rates and expected future short-term rates. This is unlikely to strictly

hold because it suggests that opportunities to take advantage of mispricing of securities

will not be exploited. For example, if the 10-year bond rate is much higher than warranted

by expectations, one could buy the 10-year bond and short a 9-year bond. If the rates on

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different maturities get far enough out of line with expectations, some entity will seek to

exploit the profit opportunity. If existing investors will not exploit the opportunity, new

investors will emerge to do so in a capitalist system. On the other hand, daily changes

in supply and demand as well as non-price conditions can certainly cause long-term

rates to diverge from the average of expected future short-term rates. These create profit

opportunities for astute bond traders. If bond markets are reasonably efficient, these profit

opportunities should not last long.

2.1.3 Components of the Yield Curve

The interest rate that is commonly quoted is known as the nominal interest rate. It is

the compensation for lending out money. First, what is the lender is compensated for is

forgoing the use of the funds. The borrower gets to use the money while the lender can

only do so after he is repaid the borrowed amount. The compensation for deferring the

use of the funds is known as the real rate of interest.

A problem in lending money is that when the borrowed amount is finally repaid, the

amount of goods or services that can be bought may be less than what could be obtained

in the past for the same amount of money. The reason is that prices of goods and services

could have increased. Anticipating a change in purchasing power due to inflation, the

lender would want to be compensated for any loss in purchasing power. In other words,

the lender would demand to be compensated at the same rate as the rate of inflation, or

an inflation premium.

The change in the real interest rate affects all interest rates. (The real interest rate changes

when peoples’ preference for consumption over investment changes.) For example, if

people in general prefer to consume today rather than postpone their consumption to next

year, they may be willing to pay more to borrow money to fulfil their consumption desire.

The impact of a change in the real rate is spread across all maturities. Thus, the shape of

the yield curve is not affected by changes in the real rate of interest.

So, what determines the shape of the yield curve? It is simply our expectations about

future inflation. For example, if inflation is expected to decline in the future, then the

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inflation premium will be lower. In fact, this could explain the shape of the downward

sloping yield curve.

Another component of the nominal interest rate is the interest rate risk premium. The

longer the maturity of the bond, the greater its exposure to changes in interest rates As the

bond price is just the present value of it’s cash flows, a longer maturity makes the impact

of discounting greater. So the longer the maturity of the bond, the higher interest rate risk

premium.

Figure 4.3 decomposes the yield curve into its three: namely, the real rate of interest, the

inflation premium, and the interest rate risk premium. It is evident that the shape of the

yield curve is mainly determined by the inflation premium.

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Figure 4.3 Term structure of interest rates

(Source: Ross et al., 2016)

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Chapter 3: Introduction To Fixed Income Valuation

3.1 General Principles of Valuation

The fundamental principle of financial asset valuation is that the value of an asset is equal

to the sum of the present values of all its expected cash flows. This principle applies

regardless of the financial asset. Typically, bond valuation involves three steps:

1. Estimating the expected cash flows

2. Determining the appropriate discount rate(s)

3. Discounting the expected cash flows with appropriate discount rate(s) and

calculating the net present value (NPV) of the expected cash flows

3.1.1 Estimating Expected Cash Flows

In the context of fixed income securities, cash flows may be either periodic interest income

as determined by the coupon rate, or payment of principal, or both. Cash flows of plain

vanilla bonds such as non‐callable treasury securities and option-free bonds are simple to

project. In contrast, those of callable bonds, putable bonds, mortgage-backed securities and assetbacked securitiesare more difficult to estimate. We shall focus only on the valuation of plain

vanilla bonds in this chapter.

3.1.2 Determining the Appropriate Discount Rate(s)

The minimum discount rateis the yield available in the marketplace for a default-free cash

flow, e.g., yield of the ‘on-the-run,’ or most recently auctioned Treasury security with the

same cash flows. For non-treasury securities, a yield premium over the yield available on

a Treasury security will be required to compensate the investor for default risks.

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The relationship between a bond’s coupon rate, required market yield, and price relative

to its par value can be summarised in the following relationship:

• If couple rate = yield required by market, then price = par value

• If couple rate < yield required by market, then price < par value (discount bond)

• If couple rate > yield required by market, then price > par value (premium bond)

3.2 Bond Valuation by Discounting Expected Cash Flows

The general principle of valuation for all financial assets is straightforward: If we know

all the expected cash flows and the appropriate interest rate(s) to be used to discount the

cash flows, the value of the asset is just the sum of the present values of all the expected

cash flows.

Recall from an earlier chapter that the bond’s coupon rate is the annual dollar coupon

divided by the face value. Although the coupon rate is quoted annually, bonds usually

pay interest semi-annually. If a bond has no periodic interest payment, it is a zero coupon

bond. The required rate of return (RRR) is the annual compound rate that investors feel

they should earn on a bond, given the risk level of the investment. The RRR is used as

the discount rate in the bond price formula to calculate the fair present value (FPV) of the

security. The fair present value is then compared to the existing market price to ascertain

whether the security is over, under, or correctly valued.

The valuation of the bond is similar to finding the present value of a series of cash flows

and can be shown as follows:

The following formula is often used to calculate the present value of an annual coupon

bond, where i is the annual discount rate and n is the number of years to maturity.

Present Value =

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Example:

Calculating the fair Present Value of a Bond

For a $90 annual coupon bond with a face value of $1000 and maturity of 6 years, the

current market price is $945. Do you think it is worth buying if the RRR is 10%?

Answer:

Present Value = = $956.45

This bond would be a good buy since the fair present value of the bond is more than the

market price (the bond is undervalued).

3.3 Valuing Semi-annual Coupon Bonds

In the real world, most of the bonds in the market are semi-annual coupon bonds. The

formula used to calculate the present value of a semi-annual coupon bond is very similar

to that of the annual-coupon bonds. To calculate the fair present value of a bond with semiannual coupon payment, just divide the coupon rate and the annual discount rate by two and

multiply the number of years by two.

Present Value =

For the same bond used in the above example, the fair present value of a bond with semiannual compounding would be:

Present Value = = $955.68

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3.4 Valuing Zero-Coupon Bonds

The valuation of zero-coupon bonds is very straightforward as there is only one cash flow:

the maturity value. The formula to be used is:

Present Value =

where i is the annual discount rate and n is the number of years to maturity. The rationale

for using a 6-month period for compounding is to be consistent with the pricing of a semiannual coupon bond.

3.5 Valuing a Bond Between Coupon Payments

For coupon-paying bonds, if the time of purchase falls between two coupon payments,

the valuation of the bond will become more complicated as you need to differentiate

between the interest earned by the seller and the interest earned by the buyer during the period

between two coupon payments. The interest earned by the seller is the interest that has

accrued between the last coupon payment date and the settlement date and is referred

to as accrued interest. Accrued means the interest was earned but not distributed. At the

time of purchase, the buyer must compensate the seller for the accrued interest. The buyer

recovers the accrued interest when he receives the next coupon payment.

We can follow the previous methods to compute the price of a bond using present value

calculation, with just a simple twist in the formula to reflect the actual coupon period. The

price we calculated is based on the accrued interest embodied in it. This is the full price

that the buyer pays the seller. Therefore, the price is referred to as the Full Price or Dirty

Price.

As the Full Price includes the accrued interest, which the buyer pays the seller first but

recovers it later, it would be useful to know the net price of the bond without the accrued

interest. This is known as the Clean Price or Flat Price, or simply the price. The clean price

is the price that is quoted in the market.

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From the Full Price, the accrued interest must be deducted to determine the price of the

bond.

Clean Price = Full Price – Accrued Interest

Accrued interest is calculated using the formula:

Accrued interest = t/T x PMT

where

t = number of days from the last coupon payment to the settlement period

T = number of days in the coupon period

PMT = coupon payment per period

There are various conventions relating to the computation of “t” and “T” in formula.

The common conventions are Actual/Actual and 30/360. The first is based on the actual

number of days in the month, while the second counts each month as having 30 days

regardless of the actual number of days in that month.

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Example:

Suppose the last coupon was paid on 31 March. The settlement date of the bond was 4

May. The semi-annual coupon payment is $40. Then, the accrued interest is calculated

as follows:

Case 1: Actual/Actual (A/A) convention.

t = 30 + 4 = 34 days (since there are 30 days in April)

T = 30 +31 + 30 + 31 + 31 + 30 = 183 days (the number of days in each month starting

from April)

The accrued interest is (34/183) x $40 = $7.43. This amount is subtracted from the full

price or “dirty price” to determine the clean price or quoted price.

Case 2: 30/360 (convention commonly used for corporate bonds).

t = 30 + 4 = 34 days (every month is 30 days by convention)

T = 30 x 6 = 180 days (the number of days in each month is set at 30)

The accrued interest is (34/180) x $40 = $7.56. This amount is subtracted from the full

price or “dirty price” to determine the clean price or quoted price.

In typical finance textbooks, the cash flows come at the end of the month. In practice,

bonds can be issued at any time. This appears to present a complication. However, when

we discount the cash flows to find the price of the bond at the time of purchase, each

regular period of 6 months is reduced by the fraction t/T of the period. Looking at the

equation below, the “power” to which the terms in the brackets in the denominator is not

a whole number but a whole number less a fraction as shown below:

The above formula can be simplified as follows:

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FIN358 Fixed Income Elements, TermStructure and Valuation

] x

3.6 Valuing a Bond Using Spot Rates

To find the bond price, we discount the cash flows by its yield to maturity which is a single

rate. An alternative to using the yield to maturity is the use of spot rates. Each cash flow is

discounted by its corresponding spot rate. (Recall that a spot rate is the yield to maturity

for a zero coupon bond.)

For example, suppose we have a 3-year 6% coupon bond and the following spot rates.

Time to maturity Spot rate

1 year 3%

2 years 4%

3 years 5%

The value of the bond is given by:

3.7 Matrix Pricing

Bonds are often priced in relation to other bonds. For example, bonds with similar

maturity and risks ought to have similar yields. However, as the bond market is not so

liquid, it may be difficult to find comparable bonds to price a bond. In this case we use

a method call matrix pricing. The name of the method comes from its use of a matrix to

form an array of bonds with the same risk, but with maturities that bracket the maturity

of the bond we wish to price. (Note that for a bond with a fixed maturity and coupon, the

price and the yield are related. Knowing the value of either the price or the yield allows

us to find the other value. Hence, the term “pricing a bond” also refers to finding the yield

of the bond.)

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FIN358 Fixed Income Elements, TermStructure and Valuation

Matrix pricing involves interpolating the yield of the bond in question with two other

yields. In the example below, we want to find the yield of Bond E which has no comparable

bond of the same maturity of 5 years. However we have four bonds – A, B, C, and D –

which have the same risk as the bond but different maturities.

Bond A and Bond B have the same maturity of 3 years, but have different coupon rates

resulting in slightly different yields. We take the average of the yields of these bonds –

which is (3.65% + 3.66%)/2 = 3.655% – to represent the average yield on a 3-year bond

From Bonds C and D, we derive an average of 4.255% for the average yield of a 6-year

bond.

The difference in yield between the average 6-year and 3-year bond is 4.255% – 3.655% =

0.6%. Through linear interpolation, Bond E’s yield is computed as: 3.655% + [(5-3)/(6-3) x

0.6%] = 3.655% + 0.4% = 4.055%

Table 4.2 Matrix pricing

4%

Coupon

5%

Coupon

6%

Coupon

7%

Coupon

3 Years A 3.65% B 3.66%

4 Years

5 Years Bond E

6 Years C 4.25% D 4.26%

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FIN358 Fixed Income Elements, TermStructure and Valuation

Read

Read the following sections in from the Fixed income analysis textbook:

Petitt, B. S. (2019). Fixed income securities: Defining elements. In Fixed income analysis

(4th ed., pp. 3-8). Wiley.

Petitt, B. S. (2019). Fixed income markets: Issuance, trading and funding. In Fixed

income analysis (4th ed., pp. 9-14). Wiley.

• Section 2 Overview of global fixed-income markets

• Section 3 Primary and secondary bond markets

• Section 4 Sovereign bonds

• Section 6 Corporate debt

Petitt, B. S. (2019). Introduction to fixed-income valuation. In Fixed income analysis (4th

ed., pp. 15-26). Wiley.

• Section 2 Bond prices and the time value of money

• Section 3 Prices and yields

• Section 4 The maturity structure of interest rates

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FIN358 Fixed Income Elements, TermStructure and Valuation

Formative Assessment

1. When the yield of a bond is higher than the coupon rate, the bond would trade at

_________________.

a. the par value

b. a discount

c. a premium

d. a price that cannot be determined without more information

2. When the market for bonds is quite illiquid, the method commonly used to price

bonds is ________________.

a. the full price method

b. the liquidity premium method

c. the matrix pricing method

d. none of the above

3. Of the following provisions which might be found in a bond indenture, which would

tend to reduce the coupon interest rate?

a. A convertible feature

b. A put feature

c. A call provision

d. (a) and (b)

4. An example of a Eurobond is _____________.

a. Samsung, a Korean company, issuing a Singapore dollar bond in Singapore

b. Toyota, a Japanese company, issuing a yen bond in Japan

c. SIA, a Singaporean company, issuing a U.S. dollar bond in Singapore

d. none of the above

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FIN358 Fixed Income Elements, TermStructure and Valuation

5. US Treasury Inflation Protected Securities (TIPS) are most likely _____________.

a. inflation-indexed bonds

b. capital-indexed bonds

c. interest-indexed bonds

d. bonds where both interest and capital are indexed

6. The clean price is ____________.

a. Dirty price x (1 + r)

b. Dirty price + accrued interest

c. Dirty price – accrued interest

d. PV of bonds cash flows

7. An investor considers the purchase of a 2-year bond with a 5% coupon rate with

interest paid annually. The 1-year and 2-year spot rates are 3% and 4%, respectively.

The price of the bond is closest to ____________.

a. 101.93

b. 102.15

c. 103.37

d. 105.52

8. An example of an affirmative covenant is the requirement ______________.

a. that dividends will not exceed 40% of earnings

b. to insure and perform periodic maintenance on fixed assets

c. that the debt-to-equity ratio will not exceed 0.5 and times interest earned will

not fall below 6.0

d. that assets are not disposed of unless they are replaced by newer assets

9. Which theory states that if future short-term rates are expected to be lower than

current short-term rates, the yield curve will be downward sloping?

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FIN358 Fixed Income Elements, TermStructure and Valuation

a. Liquidity preference theory

b. Market segmentation theory

c. Expectations theory

d. Eclectic theory

10. Which of the following combinations will result in a sharply increasing yield curve?

a. Increasing expected short rates and increasing liquidity premiums

b. Decreasing expected short rates and increasing liquidity premiums

c. Increasing expected short rates and decreasing liquidity premiums

d. Increasing expected short rates and constant liquidity premiums

e. Constant expected short rates and increasing liquidity premiums

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FIN358 Fixed Income Elements, TermStructure and Valuation

Solutions or Suggested Answers

Formative Assessment

1. When the yield of a bond is higher than the coupon rate, the bond would trade at

_________________.

a. the par value

Incorrect

b. a discount

Correct

c. a premium

Incorrect

d. a price that cannot be determined without more information

Incorrect

2. When the market for bonds is quite illiquid, the method commonly used to price

bonds is ________________.

a. the full price method

Incorrect

b. the liquidity premium method

Incorrect

c. the matrix pricing method

Correct

d. none of the above

Incorrect

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FIN358 Fixed Income Elements, TermStructure and Valuation

3. Of the following provisions which might be found in a bond indenture, which would

tend to reduce the coupon interest rate?

a. A convertible feature

Incorrect

b. A put feature

Incorrect

c. A call provision

Incorrect

d. (a) and (b)

Correct

4. An example of a Eurobond is _____________.

a. Samsung, a Korean company, issuing a Singapore dollar bond in Singapore

Incorrect

b. Toyota, a Japanese company, issuing a yen bond in Japan

Incorrect

c. SIA, a Singaporean company, issuing a U.S. dollar bond in Singapore

Correct

d. none of the above

Incorrect

5. US Treasury Inflation Protected Securities (TIPS) are most likely _____________.

a. inflation-indexed bonds

Incorrect

b. capital-indexed bonds

Correct

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FIN358 Fixed Income Elements, TermStructure and Valuation

c. interest-indexed bonds

Incorrect

d. bonds where both interest and capital are indexed

Incorrect

6. The clean price is ____________.

a. Dirty price x (1 + r)

Incorrect

b. Dirty price + accrued interest

Incorrect

c. Dirty price – accrued interest

Correct

d. PV of bonds cash flows

Incorrect

7. An investor considers the purchase of a 2-year bond with a 5% coupon rate with

interest paid annually. The 1-year and 2-year spot rates are 3% and 4%, respectively.

The price of the bond is closest to ____________.

a. 101.93

Correct

b. 102.15

Incorrect

c. 103.37

Incorrect

d. 105.52

Incorrect

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FIN358 Fixed Income Elements, TermStructure and Valuation

8. An example of an affirmative covenant is the requirement ______________.

a. that dividends will not exceed 40% of earnings

Incorrect

b. to insure and perform periodic maintenance on fixed assets

Correct

c. that the debt-to-equity ratio will not exceed 0.5 and times interest earned will

not fall below 6.0

Incorrect

d. that assets are not disposed of unless they are replaced by newer assets

Incorrect

9. Which theory states that if future short-term rates are expected to be lower than

current short-term rates, the yield curve will be downward sloping?

a. Liquidity preference theory

Incorrect

b. Market segmentation theory

Incorrect

c. Expectations theory

Correct

d. Eclectic theory

Incorrect

10. Which of the following combinations will result in a sharply increasing yield curve?

a. Increasing expected short rates and increasing liquidity premiums

Correct

b. Decreasing expected short rates and increasing liquidity premiums

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FIN358 Fixed Income Elements, TermStructure and Valuation

Incorrect

c. Increasing expected short rates and decreasing liquidity premiums

Incorrect

d. Increasing expected short rates and constant liquidity premiums

Incorrect

e. Constant expected short rates and increasing liquidity premiums

Incorrect

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FIN358 Fixed Income Elements, TermStructure and Valuation

References

Adams, J. F., & Smith, D. J. (2019). Fixed income analysis (4th ed.). Wiley.

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FIN358 Fixed Income Elements, TermStructure and Valuation

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Study

Unit5

Understanding Fixed Income Risk

and Return

FIN358 Understanding Fixed Income Risk and Return

Learning Outcomes

By the end of this unit, you should be able to:

1. Compute and discuss various yield measures

2. Evaluate the amount of risk affected by changing interest rates

3. Compute and appraise different measures of duration

4. Compute and interpret yield volatility

5. Analyse and discuss aspects and components of credit risk

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FIN358 Understanding Fixed Income Risk and Return

Overview

This study unit looks at the various measures of return and risk for bonds.

There are various measures of bond return or yield. They range from the traditional

measures such as current yield, yield to maturity, and bond-equivalent yield. We discuss

holding period yield, spot and forward rates, as well as spreads which are added to

benchmark yields to account for credit, and other factors that cause a bond to require

yields higher than the benchmark.

Much of the study unit is devoted to the important area of bond risk. In Chapter 2, we

talked about various risks aside from interest rate risk. These risks pertain to the particular

circumstances of the bonds or are related to the trading of such bonds. In Chapter 3, we

focus on interest rate risk, which is the risk of bond prices changing as a result of changes

in interest rates. This is followed by the credit aspects of bonds in Chapter 4.

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FIN358 Understanding Fixed Income Risk and Return

Chapter 1: Yield Measures, Spot Rates, and Forward

Rates

Lesson Recording

Yield Measures, Spot Rates and Forward Rates

In this chapter, we will look at the various yield measures and yield spread measures

which can be used to estimate the potential return from a fixed income security. As

investors can receive three sources of return ‒ coupon interest payment, capital gain,

and reinvestment income ‒ from a fixed income security, therefore all the yield measures

should consider all three sources of return.

1.1 Traditional Yield Measures

Traditional yield measures include:

• Nominal yield

This is the bond’s stated yield, as denoted by the coupon rate.

• Current Yield

It is also known as flat yield, interest yield, and running yield.

Current yield = Coupon/Price

It gives the portion of the return of the bond in the form of income.

• Yield to Maturity

This is a commonly used measure of a bond’s total return. It is also known as the

redemption yield.

• Yield to Call

A bond with a call feature allows the bond to be redeemed by the issuer before its

maturity date. The yield to call is computed in the same way as the yield to maturity,

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FIN358 Understanding Fixed Income Risk and Return

except the date when the bond can first be called is substituted for the maturity

date. It is also known as the yield to first call.

• Yield to Put

This yield is similar to the yield to call except that in this case, the option to redeem

the bond lies with the bondholder.

• Yield to Sinker

This is the yield to the sinking fund date when the bond with a sinking fund feature

is retired.

• Yield to Worst

As there are many possible call dates, the yield to worst is the lowest of all the yields

to call.

• Cash Flow Yield

This yield is associated with mortgage-backed securities and is also known as

the mortgage yield. A different name from yield to maturity is used because the

cash flows from mortgage-backed securities are less certain, as compared to the

coupon payments from regular bonds. Similar to yield to maturity, it is the monthly

discount rate that equates the bond price with the net present value of all future

cash flows.

1.1.1 Yield to Maturity

The yield to maturity is the interest rate that will make the present value of a bond’s cash

flow equal to its market price (in this sense, it is similar to internal rate of return). The

calculation of the yield to maturity is complicated in theory; however, most of the financial

calculators can solve the answer easily.

Here is a summary of the relationship between the price of bond, coupon rate, current

yield, and yield to maturity.

Bond selling at Relationship

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FIN358 Understanding Fixed Income Risk and Return

Par Coupon rate = current yield = yield to

maturity

Discount Coupon rate < current yield < yield to

maturity

Premium Coupon rate > current yield > yield to

maturity

1.1.2 Holding Period Yield

In computing the yield to maturity, an assumption is made that intermediate coupons are

reinvested at the same rate as the YTM. However, this assumption is unrealistic. The yield

to maturity is the same as the internal rate of return. As the name suggests, the yield of

the bond is internal, or can only be obtained by investing in the bond itself. Unless the

coupon payments are reinvested in the bond, reinvesting the cash flows from the coupons

elsewhere will not fetch such a return. Hence, investors prefer to use the rates of return

they expect will prevail in the future for the reinvestment of intermediate cash flows. The

yield that results is known as the holding period yield.

The holding period yield is calculated by first compounding all intermediate cash flows

at the relevant rates up to the end of the holding period. The sum of all cash flows at the

end of the period becomes the future value (FV) while the price is the present value (PV).

Then, the holding period yield is calculated as [FV/Price]1/n – 1.

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FIN358 Understanding Fixed Income Risk and Return

Here’s an example showing the computation of the holding period yield. Suppose we

have a 3-year, 8% coupon bond, with YTM equal to 10%, and a price of 94.92. Suppose

intermediate cash flows can be invested at 7%. The cash flows are shown below:

Table 5.1 Future value of reinvested cash flows

Cash Flow Years to Maturity FV

-94.92

8 2 9.16

8 1 8.56

8 0 105

Total 115.608

The holding period yield is:

94.92 x (1 + r)3

= 125.72, or r = 9.802%

1.1.3 The Bond-Equivalent Yield Convention

For semi-annual coupon bond, the market convention adopted to annualise the semiannual yield is to simply double it and call that the yield to maturity, also known as the

bond-equivalent yield. For example, if a semi-annual coupon bond has a 3.5% semi-annual

yield to maturity, then its bond-equivalent yield is 7%.

1.1.4 Limitations of Yield to Maturity

The yield to maturity is more accurate than the current yield measure because the latter

only considers the coupon interest and no other source for an investor’s return. The yield

to maturity, on the contrary, considers not only the coupon income, but any capital gain

or loss, as well as the reinvestment income from interim coupon payments. However,

it assumes that the investor will hold the bond until maturity and, in addition, coupon

payments can be reinvested at an interest rate equal to the yield to maturity. The investor

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FIN358 Understanding Fixed Income Risk and Return

will only realise the yield to maturity stated at the time of purchase if both of these two

assumptions hold.

Any factor that affects these two assumptions may lead to lower returns to the investor.

For example, if the bond is not held to maturity, the investor faces the risk that he may

have to sell for less than the purchase price, resulting in a return that is less than the yield

to maturity, known as the interest rate risk. If the future interest rates are less than the

yield to maturity at the time of purchase, the investor will face the reinvestment risk.

1.2 Treasury Spot Rates

The theoretical spot rates for Treasury securities represent the appropriate set of interest

rates that should be used to value default-free cash flows. The default-free theoretical

spot rate curve can be constructed from the observed Treasury yield curve. The most

commonly used approach to derive the spot rate curve is Bootstrapping. The basic

principle underlying the bootstrapping method is that the value of a Treasury coupon

security is equal to the value of the package of zero-coupon Treasury securities that

duplicates the coupon bond’s cash flows. With the Treasury spot rates, we will be able to

calculate the Zero-Volatility Spread and Option-Adjusted Spread.

1.2.1 Nominal Spread Versus Zero-Volatility Spread

The nominal spread is the difference between the yield for a non-Treasury bond and a

Treasury security with the same maturity. It measures the compensation for the additional

credit risk, option risk (the risks associated with any embedded options), and liquidity

risk that an investor is exposed to by investing in a non-Treasury security rather than a

Treasury security with the same maturity. However, this traditional measure of nominal

spread fails to consider the term structure of the spot rates and the fact that, for bonds

with embedded options, future interest rate volatility may alter its cash flow.

Unlike the nominal spread, which is the spread at a particular maturity, the Zero-Volatility

Spread or Z-spread is a measure of the spread over the entire Treasury spot rate curve, if

the bond is held to maturity (see Figure 5.1 below). Since the Z-spread is measured relative

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FIN358 Understanding Fixed Income Risk and Return

to the Treasury spot rate curve, it represents a spread to compensate for the non-Treasury

security’s credit risk, liquidity risk, and any option risk.

For bullet bonds, unless the yield curve is very steep, the nominal spread will not differ

significantly from the Z-spread. However, for securities where the principal is paid over a

period of time rather than just at maturity, there can be a significant difference, especially

when there is a steep yield curve.

1.2.2 Option-Adjusted Spread

Changes in interest rates and embedded options in a security can change the future cash

flows. However, the Z-spread does not take this into consideration as it assumes that

interest rate volatility is zero.

The Option-Adjusted Spread (OAS) converts the cheapness or richness of a bond into a

spread over the future possible spot rate curves. It is option-adjusted because it allows for

future interest rate volatility to affect the cash flow. The cost of the embedded option, for

a call, can be measured as the difference between the OAS and the Z-spread:

Z-Spread = OAS – option cost (for a callable bond)

For a callable bond, the option cost is the additional spread that the issuer needs to pay

for the call feature. Hence, if the Z-spread is 3%, the OAS 3.75%, the option cost is 0.75%.

This relationship between the Z-spread and the OAS is shown in Figure 5.1 below.

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FIN358 Understanding Fixed Income Risk and Return

Figure 5.1 Yield curves showing Z-spread and OAS for a callable bond

1.3 Forward Rates

A forward rate can be computed from the existing term structure based on the defaultfree theoretical spot rates. The term structure of interest rates is the relationship between the

Treasury spot rates and maturity, also known as the Treasury spot rate curve. Using arbitrage

arguments, given a set of long-term spot rates, one can find the set of individual one-year

forward rates.

If the unbiased expectation theory strictly holds, then forward rates are an unbiased

estimate of the market’s expectation of future interest rates. If there are liquidity premiums,

one should subtract the liquidity premium from the forward rate, before using it as an

estimate of the expected future spot rate. If segmentation strictly holds, the forward rate

has no economic meaning. Since a spot rate is simply a package of short-term forward

rates, it will not make any difference whether we discount cash flows using spot rates or

forward rates.

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FIN358 Understanding Fixed Income Risk and Return

Read

Read the following sections in:

Petitt, B. S. (2019). Introduction to fixed income valuation. In Fixed income analysis (4th

ed., pp. 15-26). Wiley.

• Section 1: Introduction

• Section 2: Bond prices and the time value of money

• Section 3: Prices and yields: Conventions for quotes and calculations

• Section 4: The maturity structure of interest rates

• Section 5: Yield spreads

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FIN358 Understanding Fixed Income Risk and Return

Chapter 2: Fixed Income Risk: A Preview

2.1 Liquidity Risk

An asset is liquid if it can be sold off quickly at or close to its fair value. Liquidity risk

refers to the risk that the bond could only be sold off quickly if its price were marked down

substantially. One indication of this risk is the bid-ask spread. The wider the spread, the

less liquid the asset.

If the bond is held to maturity, there is no issue of liquidity risk. However, if the value

of the bond needs to be marked to the market, then a wide bid-ask spread means that its

value would be much lower than its fair value, assuming fair value is the average of the

bid and ask prices.

The marking-to-market process depends on the availability of prices. There are 3 levels of

liquidity, depending on which price is available. Listed below are the 3 levels in order of

decreasing availability of prices:

• Level 1 – mark the position to market prices

• Level 2 – mark the position to bid-ask quotes

• Level 3 – mark the position to model values

2.2 Exchange Rate Risk

A foreign-currency bond means that changes in the exchange rate will affect the value of

the bond. This is exchange rate risk.

2.3 Inflation Risk

In Study Unit 4 we discussed the components of nominal interest rate. One of the

components is the inflation premium which compensates the investor for loss of

purchasing power. Inflation risk is the risk that the expected risk premium will change.

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FIN358 Understanding Fixed Income Risk and Return

Bonds are typically not protected against inflation, except inflation-indexed bonds like the

Treasury Inflation Protection Security or TIPS.

2.4 Volatility Risk

A bond without any embedded option will not be affected by a change in the expected

yield volatility. Volatility risk arises because changes in volatility will change the value

of a bond’s embedded option. How the price changes depends on whether the bond is

callable or putable. As volatility increases the value of the option, an increase in volatility

will decrease a callable bond’s price but increase a putable bond’s price. For a bond with

an embedded option, its price, as compared to that of a straight bond, will be affected by

the option as follows:

Callable bond price = straight bond price – embedded option price

Putable bond price = straight bond price + embedded option price

2.5 Event Risk

This is the risk that the issuer is unable to pay due to events like:

• Natural disasters: This is the risk of loss to a company not covered by insurance,

due to the effects of the natural disaster.

• Takeover or restructuring: A takeover or restructuring may affect the price of a

company’s bonds if their terms are not protected from changes due to the event.

However, the event can also have systematic effects. For example, a consequence of

the RJR Nabisco LBO was the increase in yields for the whole sector.

• Regulatory changes: The portfolios of institutions subject to regulation can be

affected by regulatory rulings regarding the shares they can hold.

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FIN358 Understanding Fixed Income Risk and Return

2.6 Sovereign Risk

This is the risk that arises from holding sovereign debt. Such debt can default in payments

due to a country’s economic difficulties. Any restructuring of the debt could result in a

substantial haircut.

If a government repudiates its debt, legal means to enforce payments is difficult. The

reasons are the difficulty in suing a government and enforcing a successful verdict.

This does not mean that an investor should shun sovereign debt. Countries have an

incentive to treat investors fairly in order to have good reception for their subsequent debt

issues. Further, an investor should know where the sovereign bonds were issued, as the

governing laws depend on the laws of the country of issue. A foreign sovereign bond

issued in the U.S. would be governed by U.S. laws, which may disallow arbitrary changes

to the terms of the bond.

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FIN358 Understanding Fixed Income Risk and Return

Chapter 3: The Measurement of Interest Rate Risk

The value of a bond moves in the opposite direction to a change in interest rates. As

the interest rate increases, the price of a bond falls. To better understand and control the

interest rate risks, it is therefore necessary to measure and quantify the risks we face when

buying or holding a bond. We will discuss the two main approaches to measuring interest

rate risk: the full valuation approach and the duration/convexity approach.

3.1 The Full Valuation Approach

The full valuation approach measures the interest rate risk exposure of a bond position

or a portfolio by revaluing it whenever there is a change in interest rate. For example, an

investor can measure the interest rate exposure to a 50 basis point, 100 basis point, or 200

basis point instantaneous change in the interest rate. This is also referred to as scenario

analysis as it involves assessing the exposure to interest rate scenarios.

The advantage of the full valuation approach is obvious: it is straightforward and accurate.

If one has a good valuation model, interest rate risks can be measured accurately by

assessing how the value of the portfolio or individual bond change for any given interest

rate scenario. A common issue for using the full valuation approach is which scenarios

should be evaluated to assess interest rate risk exposure. For some regulated entities, there

are specified scenarios established by regulators. On the other hand, risk managers and

highly leveraged investors such as hedge funds tend to look at extreme scenarios to assess

exposure to interest rate changes, which is known as Stress Testing.

Although the full valuation approach seems straightforward and accurate, the

disadvantage of this approach is also obvious: the valuation process can be very time

consuming. This is particularly true if the portfolio has a large number of bonds and the

investor would have to revalue each bond for each scenario. Therefore, managers would

prefer a simple measure that is accurate and can easily be carried out rather than having

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FIN358 Understanding Fixed Income Risk and Return

to revalue an entire portfolio. In the next section, we will discuss such a measure ‒ the

duration/convexity approach.

3.2 The Duration/Convexity Approach

In previous chapters, we have learned that the price volatility of bond is determined by

three characteristics: (1) maturity, (2) coupon rate, and (3) presence of embedded options.

The duration/convexity approach is based on understanding these characteristics of

bonds.

3.2.1 Convexity

To understand the concept of convexity, let us look at option-free bonds first. A

fundamental characteristic of an option-free bond is that the price of the bond changes in

opposite direction to a change in the bond’s yield. When the price/yield relationship for

any option-free bond is graphed, its shape is not linear but, instead, curving towards the

original point, as shown below. This relationship is known as convexity.

Figure 5.2 Price-yield relationship

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FIN358 Understanding Fixed Income Risk and Return

The price sensitivity of a bond to changes in the yield can be measured in terms of the

dollar price change, or the percentage price change. Four properties can be summarised

based on this convex price/yield relationship:

Property 1: Although the price moves in the opposite direction from the change in the

yield, the percentage price change is not the same for all bonds.

Property 2: For small changes in the yield, the percentage price change for a given bond

is roughly the same, whether the yield increases or decreases.

Property 3: For large changes in yield, the percentage price change is not the same for an

increase in yield as it is for a decrease in yield.

Property 4: For a given large change in yield, the percentage price increase is greater than

the percentage price decrease.

All option-free bonds exhibit positive convexity, which means that for a large change in

the interest rates in both directions, the amount of the price appreciation is greater than

the amount of price depreciation.

Now let’s look at the price volatility of bonds with embedded options. In general, the price

of a bond with embedded options is equal to the value of an option-free bond plus or

minus the value of embedded option.

The two most common types of embedded options are call options and put options. A

callable bond exhibits positive convexity at high yield levels and negative convexity at

low yield levels. “Negative convexity” means that for a large change in interest rates, the

amount of the price appreciation is less than the amount of the price depreciation. The

negative and positive convexity of a callable bond can be shown in the following graph:

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FIN358 Understanding Fixed Income Risk and Return

Figure 5.3 Price yield curves of callable and non-callable bonds

As shown from the above graph, when a bond exhibits negative convexity, the bond

compresses in price as yield declines. The reason for the negative convexity is that at

low yield levels, it is very likely that the issuer will call the bond. Thus, the value of

the embedded call option is more valuable to the issuer, and this reduces the bond price

relative to a comparable option-free bond.

The value of a putable bond is equal to the value of an option-free bond plus the value of

the put option. At low yield levels, the price of the putable bond is basically the same as

the price of a comparable option-free bond because the value of the put option is small.

As market yield increases, the price of the putable bond declines, but the decline is less

than that for an option-free bond (see diagram below). The reason is that the put option

cushions the decline in the price of the bond. Remember: the price of the bond is the sum of

the price of an option-free bond and the put option. So, while the option-free component

of the bond falls in value, when the interest rate increases, the fall in price makes the put

option more valuable as its intrinsic value has increased.

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Figure 5.4 Price-yield curve for bond with embedded put

3.3 Duration

Now that you’ve developed an understanding of the price volatility of bond, let’s look

at the duration/convexity approach. Duration is a measure of the approximate price

sensitivity of a bond’s price to interest rate changes. In other words, the duration of a bond

is essentially an elasticity measure, i.e.:

Duration = % change in price / % change in yield

There are various measures of duration. In general, there are two types: yield duration

and curve duration. In yield duration, the change in the bond price is measured in relation

to a change in the bond’s yield to maturity. For curve duration, the change in bond price

comes from a change in the benchmark yield curve. Curve duration is used for bonds

with embedded options as changes to the yield to maturity is not appropriate, since the

embedded option may cause the bond to be redeemed before its maturity.

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The different yield duration measures include Macaulay duration, modified duration,

money duration, and the price value of a basis point (PVBP). Effective duration is a

measure of curve duration.

3.3.1 Macaulay Duration

The Macaulay duration is calculated as follows:

where CFt

is the bond’s cash flow at time t.

From the equation, we see that the Macaulay duration weights each time to maturity “t”

by the proportion of its cash flow’s present value to the price of the bond. If we consider

each cash flow at time t as a mini zero coupon bond, then the price of the bond is nothing

other than the price of a portfolio of zero-coupon bonds. The maturity of this “portfolio”

is nothing other than the weighted average of the maturity of each zero-coupon bond.

For example, consider a 3-year coupon bond. There are four mini zero-coupon bonds,

namely, the three coupon payments and the face value that form this portfolio. The

weighted average maturity of this “portfolio” or its duration is:

Duration does a good job of estimating the percentage price change for a small change in

interest rates. For day-to-day fluctuations, duration works quite well; but when interest

rates move significantly, such as when the Fed makes an announcement of a rate change,

the predicted pricing errors can become significant. The prediction errors arise because

bond prices are not linear with respect to interest rates.

At lower yield rates, bond prices are more sensitive to interest rate changes than at higher

initial promised yields. A given percentage change in interest rates will result in a larger

bond price change for a low yield bond than for a high yield bond. Thus, a graph of bond

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prices versus interest rates would be convex to the origin. The diagram below illustrates

this:

Figure 5.5 Duration: Slope of price-yield curve

Duration does not quite capture this change in sensitivity (or convexity) of bond prices

to interest rates. To improve the approximation provided by duration, an adjustment

for “convexity” can be made. Hence, the duration/convexity approach uses duration

combined with convexity to estimate the percentage price change of a bond caused by

changes in interest rates.

It is noteworthy that duration predicts that the price changes of bonds are symmetric

with respect to an interest rate increase and an interest rate decrease. An examination of

the diagram above indicates that this is not a true assertion. As mentioned above, the

bond’s price with respect to interest rates is convex to the origin. The duration is the first

derivative or slope of the tangent line in the diagram below.

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Figure 5.6 Duration and convexity: Slope of price-yield curve

Hence, the error in the bond price prediction is due to the curvature of the bond price –

bond yield graph. The degree of curvature is called the convexity. Greater convexity leads

to greater pricing prediction errors. The errors can be quite economically significant for

large portfolios and for big interest rate changes.

Note that convexity works in the investor’s favour. While duration over-predicts the price

drop that follows from an interest rate increase, and under‐predicts the price increase that

results from a yield decline, convexity moderates the effect, so investors desire convexity

in their bonds.

The greater the interest rate change, the greater the error in predicted prices and rates of

return from ignoring convexity. All fixed income securities that have cash flows prior to

maturity exhibit convexity.

3.3.2 Modified Duration

The Macaulay duration measures the percentage change in price in relation to a percentage

change in the bond’s yield to maturity. Modified duration extends on this concept. Instead

of the percentage change in price in modified duration, it gives us the change in the price

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itself. Specifically, it gives the approximate percentage change in price for a 100 basis point

change in interest rates. Dividing the Macaulay duration by (1+ yield to maturity), we

get the modified duration measure. An alternative way of calculation, which gives us the

approximate modified duration, is:

Approx. Modified Duration

=(P-

– P+

)/2P0Δy

where

Δy = change in yield in decimal

P0 = initial price of the bond

P

–

= new price of bond if yields decline by Δy

P

+

= new price of bond if yields increase by Δy

3.3.3 Effective Duration

When bonds have embedded options, duration and modified duration cannot be used to

measure the change in the bond price. The reason is that the embedded option in the bond

could change the cash flow pattern of the bond as it could be redeemed prior to maturity.

Some valuation models for bonds with embedded options take into account how changes

in yield will affect the expected cash flows. Thus, when calculating the changes in bond’s

price, those models will take into account not only the discounting at different interest

rates, but also how the expected cash flows may change. This method of calculating

duration is referred to as the effective duration or option-adjusted duration.

The difference between modified duration and effective duration for bonds with

embedded options can be very large. For example, a callable bond could have a modified

duration of 8, but an effective duration of 4. Thus, using modified duration as a measure

of the price sensitivity for a security with embedded option to changes in yield would be

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misleading. Modified duration is appropriate for option-free bonds. Effective duration is

a more appropriate measure for any bond with an embedded option.

Effective Duration

=(P-

– P+

)/2P0x Δcurve

where

Δcurve = change in yield in decimal of the benchmark curve

P0 = initial price of the bond

P

–

= new price of bond if yields decline by Δcurve

P

+

= new price of bond if yields increase by Δcurve

3.3.4 Money Duration and Price Value of a Basis Point (PVBP)

Money duration, as the name suggests, measures the change in the price of the bond in

terms of the dollar amount that results. Multiplying the modified duration by the bond

price gives the money duration measure, which is the dollar amount of change in the price

of the bond in relation to a 100 basis point change in the yield. In other words:

Money duration = Modified duration x Price

If we measure the dollar amount of change due to a 1 basis point change in the interest rate,

we get the Price Value of a Basis Point (PVBP), which is also known as the Dollar Value

of 1 basis point change (DV01). Using the same notation in the computation of modified

duration, PVBP is computed as:

PVBP = (P-

– P+

)/2P0

3.3.5 Portfolio Duration

A portfolio’s duration can be obtained by calculating the weighted average of the

durations of the bonds in the portfolio. The weight of each bond is the ratio of the

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bond’s value to the total value of all bonds. In applying portfolio duration to estimate the

sensitivity of a portfolio to changes in interest rates, it is assumed that the yield for all

bonds in the portfolio changes by the same amount.

The formula for portfolio duration is as follows:

Portfolio Duration = w1D1 + w2D2 + w3D3 + w4D4 + ……+ wnDn

where

wi = market value of bond i/ market value of the portfolio

Di = duration of bond i

n = number of bonds in the portfolio

3.3.6 Summary of Duration Measures

The different duration measures may seem confusing. Let’s see their connections.

First, we start with the bond price equation and differentiate it with respect to (1+r).

Differentiation allows us to find change in one variable in relation to the change in another

variable. Thus, given a change “r” in the interest rate, the change in the bond price is as

follows:

P0 = C/(1+r) + C/(1+r)2

……(C +F)/(1+r)n

dP0/d(1+r) = 1/d(1+r) [C/(1+r) + C/(1+r)2

……(C +F)/(1+r)n

]

After incorporating the price and some manipulation of the equation, we end up with

From the above equation,

Percentage change in price is given by:

∆P0 / P0 = – D/(1+r) x ∆r using Duration, D

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∆P0 / P0 = – MD x ∆r using Modified Duration, MD = D/(1 +r)

∆P0 = – MD x P0 x ∆r

∆P0 = – MoneyD x ∆r using Money Duration, MoneyD = MD x P0

3.3.7 Notes on Modified Duration, Convexity and Yield Curve Assumption

Modified Duration

How did we arrive at this formula? It’s simple. First, note that duration is the first

derivative of the bond pricing equation. This suggests that the slope of the price-yield is

a measure of duration. How is the slope measured? Figure 5.5 (reproduced below) gives

us a clue.

Figure 5.7 Reproduction of Figure 5.5: Duration: Slope of price-yield curve

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The slope of the tangent to the curve at P is computed by taking the average of the slopes

of the lines P1P and PP2.

Slope of line P1P = (P1 – P)/(Y1 – Y) = (P1 – P)/∆r

Slope of line PP2 = (P – P2)/(Y2 – Y) = (P – P2)/∆r

Average of the slopes = [(P1 – P)/∆r + (P – P2)/∆r)]/2

= (P1 – P2)/2∆r

= (P-

– P+)/2∆r 2

where P-

= P1 and P+ = P

As Modified Duration = Duration / P0

Then MD = [(P-

– P+)/2∆r]/ P0 or

Convexity

Convexity measures the degree of curvature of the price-yield curve. Again, we can

approximate convexity by looking at the slope of two lines P1P and PP2 in Figure 5.5.

We start off with a straight line consisting of two segments. In this case the slopes of

both segments are the same. If we swivel the left segment of the line to the right, the two

segments would then form an angle.

For the price-yield curve, we can approximate the curvature by considering the difference

in the slopes of the two lines – P1P and PP2 – in Figure 5.5. Also, note that the greater the

difference in slopes, the more acute is the angle, which connotes higher convexity.

Difference in the slopes = [(P1 – P)/∆r – (P – P2)/∆r)]

= (P1 – P2) – 2P0 /2∆r

= (P-

– P+) – 2P0/2∆r

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where P-

= P1,

and P+ = P2

To find convexity we have to find the second derivative of the bond-price equation with

respect to the change in interest rate, or the first derivative of duration.

Note that the difference in the slopes measures the rate of change in the slopes, or the rate

of change in duration. Dividing this rate of change by the change in interest rate would

give us an approximate convexity measure as shown below:

Yield Curve Assumption

When we compute the Macaulay duration we differentiated the bond price equation with

respect to the interest rate. Notice that the discount rate in the bond price equation is the

same regardless of the maturity of the cash flow. Implicitly, we are making the assumption

that the yield curve is flat. Further, as the interest rate for each maturity changes by the

same amount, we are assuming a parallel shift in the yield curve.

Figure 5.8 Parallel shift in flat yield curve

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There are other measures of duration where the assumptions of a flat yield curve and equal

amount of interest rate change for all maturities are modified to account for an upward

sloping yield curve as well as non-parallel shift in the yield curve.

An alternative measure that accommodates a non-flat yield curve assumes that the

proportional change in the t-period spot rate is the same as the proportional change in the

one-period spot rate(see Figure 5.8), that is:

d(1 + rt

)/(1 + rt

) = d(1 + r1)/(1 + r1)

where r1 < r2 .. < rt or r1 > r2 .. > rt

The result is a duration measure we term “Macaulay’s second duration measure” or D2:

Figure 5.9 Non-parallel shift in upward sloping yield curve:

Proportional change in spot rates

There is yet another measure that is similar to the second Macaulay duration measure but

makes the additional assumptions that the proportional change in the t-period spot rate

is the same as the proportional change in the one-period spot rate multiplied by a factor

Kt

(see Figure 5.10). The result is “Macaulay’s third duration measure” or D3:

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Figure 5.10 Non-parallel shift in upward sloping yield curve: Non-proportional

change in spot rates

Key Rate Durations

If the yield curve does not shift in a parallel manner, yet another alternative is the use of

key rate durations. Key rates could be 3-month, 1-year, 3-year, 5-year, 10-year, etc.

Key rate durations are computed as follows:

• All yields are held constant except for the yield for a particular maturity of the yield

curve which changes by 1 basis point.

• Compute the change in the bond or portfolio value.

• The key rate duration for that maturity is the sensitivity of change in the bond or

portfolio value to a change in yield.

The Ho-Lee key rate duration model, uses 11 key maturities of a Treasury spot rate curve

to derive the key rate durations.

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Summary of Duration Measures

We have discussed a few duration measures. Which are the ones that traders and risk

managers use? In general, modified duration, PVBP or DV01, and Key Rate Duration are

the models used. Using modified duration and DV01 is fairly straightforward, whereas

there are some technical issues when using Key Rate Duration.

Note that with the passage of time, even if interest rates do not change much, the

duration value changes. This means that for effective hedging purposes, adjustments to

the hedging program adopted earlier needs to be made. To do the adjustments, bond

portfolio managers can buy or sell securities as well as change the weightages of the

securities in the portfolio. Alternatively, they can use US Treasury futures or options to

target the key rate durations.

Duration as a Hedging Tool

Given the wide coverage of this course which combines the two subjects of Derivatives

and Fixed Income, we do not have room to discuss bond portfolio management.

Lest the student wonders about the point of learning the various risk measures, we discuss

briefly the use of duration in managing risk.

Many companies (for example, pension funds and insurance companies) have future

liabilities to meet. Pension funds need to make pay-outs to the retired employees while

insurance companies make payments for endowment policies that mature, or life policies

which the policy holder may want to cash out when she retires. As such, these companies

need to invest the cash flows in the form of pension contributions and insurance

premiums.

Long-term bonds are commonly bought for the portfolios as the coupon payments, as well

as the face value at maturity, are certain cash flows that can be used to meet those needs.

A simple method that a portfolio is hedged or immunised against the risk of interest rate

changes is the use of duration matching. This involves matching the duration of the assets

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to the duration of the liabilities. Other methods of immunisation, some of which involve

dynamic rather than passive hedging, are beyond the scope of the course.

Read

Read the following sections in:

Petitt, B. S. (2019). Understanding fixed income risk and return. In Fixed income analysis

(4th ed., pp. 43-54). Wiley.

• Section 1: Introduction

• Section 2: Sources of return

• Section 3: Interest rate risk on fixed-rate bonds

• Section 4: Interest rate risk and yield horizon

• Section 5: Credit and liquidity risk

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Chapter 4: Fixed Income Credit Risk

Credit problems in a bond portfolio are the primary cause of non-payment of bond

obligations for holders of fixed income securities. Thus, the analysis of credit risk is of

primary importance to holders of such securities. An understanding of credit risk and

credit analysis is useful, regardless of whether one works in the credit department of a

rating agency; or if one is involved in presenting loan applications on behalf of a corporate

borrower/issuer; or even for someone who just wishes to increase his chances of success

when seeking a personal loan. Methods of credit evaluation differ for various types of

bonds and by the size of the borrower. In this unit, we explore the primary concerns of

credit analysis for corporate bonds.

4.1 Credit Ratings

Most bonds are rated in terms of default risk by at least one of the major ratings agencies,

typically Moody’s, Standard and Poor’s and Fitch. Many institutions can hold only limited

amounts of unrated or low-rated debt. So, a favourable rating lowers the interest yield

required and increases the number of potential buyers. Junk bonds are bonds rated below

Baa by Moody’s or BBB by S&P. Higher ratings are termed investment gradebonds. In

general, ratings agencies evaluate the industry strength, the firm’s position in the industry,

liquidity, profitability, debt capacity, and (since Sarbanes-Oxley) corporate governance.

Each specific rated issue is also examined for protection provided to investors and the

firm’s ability to pay.

4.2 Traditional Credit Analysis

The purpose of credit analysis is to generate profitable bond portfolios that do not expose

bond holders to excessive amounts of risk. Corporate bonds are essentially loans to

corporations. The credit analysis process will normally be quite detailed and steps in

the process may include meeting up with the corporation’s customers and suppliers

(particularly if there is one or only a few major buyers or suppliers of the product or

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service), procuring a credit history such as a Dun and Bradstreet report, and application

of the four C’s of Credit:

• Character: Quality of management and corporate governance rank at the top of

our list. The capability of management means that the businesses and projects

the company embarks on have been well-thought out with proper due diligence

undertaken. The quality of management is shown in how the company deals with

setbacks and the way it overcomes challenges. Good management enables the

company to deliver cash flows and ensure the fulfilment of their debt obligation.

The other aspect of management is integrity. While corporate governance strictures

provide some checks and balances on the way management does its work,

ultimately, it is the character of people in management that determine whether

the company cuts corners and engages in business in the grey areas. Integrity of

management also means that the stakeholders’ interests are taken into account in

the decisions management makes.

• Capacity: Cash flow to service the loan and adequacy of the corporation’s capital

to prevent insolvency. The stability of cash flows depends on the type of business

the company is in, as well as the adequacy of risk management practices. Further,

judicious use of leverage will also affect the company’s ability to service the debts.

• Collateral: Involves security or asset pledged to add safety to the loan as an available

source of repayment. In the event of default, the collateral pledged is disposed to

make good any outstanding payments of interest and/or capital. However, note

that the absolute priority rule rarely holds during corporate reorganisation.

• Covenants: Covenants are important in both constraining the activities of the

company (negative covenants), as well as mandating action on the part of

management (positive covenants). Stringent covenants may constrain the actions of

the company too much while loose covenants may invite trouble for the creditors,

as it is unrealistic to appeal to the noble nature of humans to “do the right thing.”

The above may be assessed by asking the following questions about the bond issuer on

the given categories.

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4.2.1 Production of Marketable Output (Capacity)

• Are there stable supplies and costs of inputs? For example, for lumber mills

dependent on logging of federal lands, stability of supply has been lacking in recent

years. What are union wage demands and the probability of costly strikes, if any?

• What is the issuer’s competitive advantage? Is it a differentiated product with high

mark-up? The issuer’s sales can be sensitive to the economy and technological

and taste changes. Firms producing commodities rely on supply; or distribution

advantages; or service, volume, and economies of scale to maintain profits.

• Is the issuer in a growing industry, or is the only way to generate sales growth

via capturing market share from competitors? Has the issuer allocated sufficient

resources to marketing the firm’s product? Is the issuer at risk from losing

distribution channels to competitors? Is the issuer dependent on sales to one or only

a few customers?

4.2.2 Management (Character)

• Is management of the issuer trustworthy? Have they been excessively ‘cooking the

books’? Has management cooperated with the credit analysts’ scrutiny? What is the

employee morale and turnover rate? How credible is the business plan? (Does the

issuer have a plan at all?) Is the issuer likely to require additional financing over

the loan period? Are they aware of this? Has the issuer planned for financing of

additional working capital, if needed? To what extent does the business rely on one

or only a few key people who could not be easily replaced?

4.2.3 Capital (Capital and Collateral)

• Does the issuer have sufficient equity? Do the issuer’s managers have significant

equity holdings to ensure they are concerned with firm performance? What is the

issuer’s debt capacity? How specific are the assets to the given issuer? Are they

useful only to firms in the same industry?

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4.2.4 Cash Flow Analysis

The statement of cash flowsis a primary tool in assessing the debt servicing capacity of an

issuer. The cash flow statement breaks down sources and uses of cash into cash flow from

operations, from investment activity, and from financing sources. The credit analyst will

wish to assess that cash flow from operations is sufficient to service the potential loan. A

prudent analyst will also demand a schedule of actual cash disbursements and receipts.

The statement of cash flows does not provide this information!

4.2.5 Ratio Analysis

Various profitability and debt & coverage ratios may be analysed to evaluate the ability

of the corporate issuer to meet its debt obligations. The numbers are calculated from the

data above.

Liquidity ratios

• Current ratio = Current Assets / Current Liabilities

• Quick ratio (acid test) = (Cash + Cash equivalents + Receivables) / Current

Liabilities

These ratios measure the issuer’s ability to pay its debts in the short run. If the issuer sells big

ticket items or has a large number of days’ sales in inventory, the quick ratio may be a

better measure.

Asset management ratios

• Days sales in receivables = (Receivables × 365) / Credit Sales

• Days in inventory = (Inventory × 365) / Cost of Goods Sold

• Sales to fixed assets = Sales / Fixed Assets

• Asset turnover = Sales / Total Assets

Asset management ratios assess management’s ability to manage a given amount of assets

such as inventory or receivables. High ratios may suggest problems in credit management

or inventory management, or simply low sales. The turnover ratios measure the number

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of dollars of revenue generated per dollar invested in the given asset category. Recall that

the asset turnover ratio is a major input into ROE.

Debt (long – term solvency) ratios

• Debt-to-asset or total debt ratio = Total Liabilities / Total Assets

• Times‐Interest‐Earned: EBIT / Interest Expense

• Cash flow-to-debt ratio = (EBIT + Depreciation) / Debt

EBIT is earnings before interest and taxes, EAT is earnings after taxes. These ratios measure

the issuer’s ability to pay off its debts in the long run. These need to be assessed in light of the

earnings variability of the issuer and industry norms. Fixed charges may include interest, lease

and sinking fund payments. The minimum acceptable fixed charge ratio will be higher

with greater cash flow variability. At a minimum, the cash flow-to-debt ratio should also be

greater than the interest rate on the debt.

Profitability ratios

• Gross margin = Gross profit / Sales

• Operating profit margin = Operating profit / Sales

• Net profit margin = EAT / Sales

• Return on assets = EAT / Average total assets

• Return on equity = EAT / Total equity

• Dividend pay-out = Dividends / EAT

These ratios measure the bond issuer’s ability to generate profits (gross, operating, or net)

per dollar of revenue, or per dollar of assets or equity. The first three ratios measure the

management’s ability to control given expense categories. The ROA and ROE ratios are

guides to the firm’s rate of return on invested dollars. Note that market ratios are often

unavailable for firms of this size. Other commonly used tools include the use of common

size and indexed statements to focus on comparisons with competitors or over time.

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4.3 Credit Scoring Models

Ratio analysis has many limitations. First, a ratio by itself tells very little; some benchmark

of comparison is needed. For many smaller corporate borrowers, benchmark data may

not be available. On the other hand, large firm corporate borrowers are often involved in

multiple lines of business, and this can make ratio comparisons difficult as well because

there is no single industry to use as a benchmark. Differences in accounting treatments can

lead to substantial differences in ratios as well, even if the borrower’s management is not

purposefully intending to mislead creditors such as at Enron, WorldCom, Adelphia and

others. Ratings agencies such as Moody’s and Standard and Poor’s provide information

about the credit risk of the borrower. Market analysts such as Value Line, Hoovers and a

plethora of Wall Street analysts provide current (albeit biased) forecasts of future earnings

and growth prospects. Sophisticated credit scoring models have been developed for these

firms such as the Altman’s “Z-Score model”:

Z = 1.2X1 +1.4X2 + 3.3X3 + 0.6X4 +1.0X5

where

X1 = Working capital / Total assets

X2 = Retained earnings / Total assets

X3 = EBIT / Total assets

X4 = Market value of equity / Book value of long term debt

X5 = Sales / Total assets

The higher the Z‐score, the lower the probability of bond issuer default. An issuer with a Z‐score

less than 1.81 is considered to have high default risk. A Z score of 2.99 or more indicates

low default risk, and a Z–score between 1.81 and 2.99 indicates that the bond issuer’s

default risk is indeterminate.

Credit scores are an analytical tool that can determine some of the differences between

issuers that previously defaulted, versus borrowers who did not. They can provide the

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credit analyst with valuable information about whether a potential corporate borrower

closely resembles the profile of issuers who have defaulted on prior corporate debt. This is

especially useful in assessing the financial aspects of credit analysis, but it is weaker in assessing

character and management strength. Thus, it would be an inappropriate purpose of a credit

scoring model to make a large credit analysis decision based entirely on the credit score.

This would ignore too many other relevant factors.

4.4 Special Considerations for High Yield Corporate Bonds

Credit analysis of high-yield corporate bonds entails the same process as described above,

but there are some additional complicating factors that should be considered. These are,

namely: analyses of (a) debt structure, (b) corporate structure and (c) covenants.

The typical debt structure of high-yield issuers includes: (a) bank debt, (b) brokers’ loans,

(c) reset notes, (d) senior debt, (e) senior subordinated debt and payment in kind bonds. It is

important to realise that high-yield issuers rely extensively on bank loans due to lack

of alternative financial sources. Bank debts are usually short term, with floating interest

rates. Thus, banks must be repaid in the near future and a rise in short-term interest rates

may impose severe cash flow problems for an issuer heavily financed by bank debt. The

implication is that the analyst must carefully assess the timing and amount of maturing

debt and sources of repayment, which may be from operating cash flow, refinancing, and/

or sale of assets. In this context, sale of assets may adversely affect future cash flows and

refinancing may result in higher funding costs. Alternatively, in the absence of bank loans,

high-yield issuers may be forced to turn to instruments such as reset notes, payment in

kind bonds or deferred coupon bonds, all of which may adversely impact future cash flows.

High-yield issuers usually have a holding company corporate structure. The assets to pay

creditors of the holding company will come from the operating subsidiaries. The key point

is that the credit analyst must understand the corporate structure in order to assess how

cash will be passed between subsidiaries and the parent company, and among subsidiaries

when evaluating the ability of the parent company to meet its obligations to creditors. Just

looking at financial ratios of the entire holding company structure would not be adequate.

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Read

Read the following sections in:

Petitt, B. S. (2019). Fundamentals of credit analysis. In Fixed income analysis (4th ed.,

pp. 43-54). Wiley.

• Section 1: Introduction

• Section 2: Credit risk

• Section 3: Capital structure, seniority ranking, and recovery rates

• Section 4: Rating agencies, credit ratings, and their role in the debt markets

• Section 5: Traditional credit analysis: corporate debt securities

• Section 6: Credit risk vs return: yields and spreads

• Section 7: Special considerations of high-yield, sovereign, and non-sovereign

credit analysis

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FIN358 Understanding Fixed Income Risk and Return

Formative Assessment

1. The yield spread of a specific bond over the swap rate of the same maturity is:

a. G-spread

b. I-spread

c. Z-spread

d. cannot be determined as swap rates are not interest rates

2. When an investor holds a bond with a Macaulay duration that is greater than his

investment horizon, ______________.

a. the investor is hedged against interest rate risk

b. reinvestment risk dominates, and the investor is at risk of lower rate

c. market price risk dominates, and the investor is at risk of higher rate

d. none of the above

3. As a measure of interest rate changes, the Macaulay duration is most accurate when

the yield curve is ___________.

a. upward sloping

b. downward sloping

c. flat

d. the shape of the yield curve is irrelevant

4. Mr Lim bought a 7% coupon bond at the par value of 100. It had a maturity of 9 year.

Right after the bond was purchased, interest rates rose to 8% and remained at that

rate until maturity. The bond is sold 5 years later. If all the coupons are reinvested

over the holding period, Mr Lim’s 5-year holding period yield or horizon yield would

be closest to _________.

a. 5.6%

b. 6.0%

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FIN358 Understanding Fixed Income Risk and Return

c. 6.6%

d. 7.2%

5. Mr Tan bought a 3 year bond. It paid a 5% coupon annually. He paid 105.66 for the

bond at a YTM of 3%. If the yield changes by 5bps, which of the following is closest

to the bond’s modified duration?

a. 2.78

b. 2.83

c. 3.88

d. 5.34

6. Which of the following statements about duration is correct? A bond’s ________.

a. effective duration is a measure of yield duration

b. modified duration is a measure of curve duration

c. Macaulay duration is a measure that is meaningful for all bonds

d. modified duration cannot be larger than its Macaulay duration

7. Assuming no change in the credit risk of a bond, the presence of an embedded put

option ______________.

a. reduces the effective duration

b. increases the effective duration of the bond

c. does not change the effective duration of the bond

d. decreases the convexity of the bond

8. An option-adjusted spread (OAS) on a callable bond is the Z-spread ___________.

a. over the benchmark spot curve

b. minus the standard swap rate in that currency of the same tenor

c. minus the value of the embedded call option expressed in bps per year

d. none of the above

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FIN358 Understanding Fixed Income Risk and Return

9. A bond portfolio consists of the following 3 fixed-rate bonds. Assume annual coupon

payments and no accrued interest of the bonds. Prices are per 100 of par value.

Bond Maturity Market

Value

Price Coupon YTM Mod.

Dur

A 6 years 170,000 85 2.0% 4.95% 5.42

B 10 years 120,000 80 2.4% 4.99% 8.44

C 15 years 100,000 100 5.0% 5.00% 10.38

390,000

The bond portfolio’s modified duration is closest to ____________.

a. 7.62

b. 7.88

c. 8.08

d. 8.20

10. A limitation of calculating a bond portfolio’s duration as the weighted average of

the yield durations of the individual bonds that compose the portfolio is that it

______________.

a. is less accurate when the yield curve is less steeply sloped

b. is not applicable to portfolios that have bonds with embedded options

c. assumes a parallel shift to the yield curve

d. none of the above

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FIN358 Understanding Fixed Income Risk and Return

Solutions or Suggested Answers

Formative Assessment

1. The yield spread of a specific bond over the swap rate of the same maturity is:

a. G-spread

Incorrect

b. I-spread

Correct

c. Z-spread

Incorrect

d. cannot be determined as swap rates are not interest rates

Incorrect

2. When an investor holds a bond with a Macaulay duration that is greater than his

investment horizon, ______________.

a. the investor is hedged against interest rate risk

Incorrect

b. reinvestment risk dominates, and the investor is at risk of lower rate

Incorrect

c. market price risk dominates, and the investor is at risk of higher rate

Correct

d. none of the above

Incorrect

3. As a measure of interest rate changes, the Macaulay duration is most accurate when

the yield curve is ___________.

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FIN358 Understanding Fixed Income Risk and Return

a. upward sloping

Incorrect

b. downward sloping

Incorrect

c. flat

Correct

d. the shape of the yield curve is irrelevant

Incorrect

4. Mr Lim bought a 7% coupon bond at the par value of 100. It had a maturity of 9 year.

Right after the bond was purchased, interest rates rose to 8% and remained at that

rate until maturity. The bond is sold 5 years later. If all the coupons are reinvested

over the holding period, Mr Lim’s 5-year holding period yield or horizon yield would

be closest to _________.

a. 5.6%

Incorrect

b. 6.0%

Incorrect

c. 6.6%

Correct

d. 7.2%

Incorrect

5. Mr Tan bought a 3 year bond. It paid a 5% coupon annually. He paid 105.66 for the

bond at a YTM of 3%. If the yield changes by 5bps, which of the following is closest

to the bond’s modified duration?

a. 2.78

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FIN358 Understanding Fixed Income Risk and Return

Correct

b. 2.83

Incorrect

c. 3.88

Incorrect

d. 5.34

Incorrect

6. Which of the following statements about duration is correct? A bond’s ________.

a. effective duration is a measure of yield duration

Incorrect

b. modified duration is a measure of curve duration

Incorrect

c. Macaulay duration is a measure that is meaningful for all bonds

Incorrect

d. modified duration cannot be larger than its Macaulay duration

Correct

7. Assuming no change in the credit risk of a bond, the presence of an embedded put

option ______________.

a. reduces the effective duration

Correct

b. increases the effective duration of the bond

Incorrect

c. does not change the effective duration of the bond

Incorrect

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FIN358 Understanding Fixed Income Risk and Return

d. decreases the convexity of the bond

Incorrect

8. An option-adjusted spread (OAS) on a callable bond is the Z-spread ___________.

a. over the benchmark spot curve

Incorrect

b. minus the standard swap rate in that currency of the same tenor

Incorrect

c. minus the value of the embedded call option expressed in bps per year

Correct

d. none of the above

Incorrect

9. A bond portfolio consists of the following 3 fixed-rate bonds. Assume annual coupon

payments and no accrued interest of the bonds. Prices are per 100 of par value.

Bond Maturity Market

Value

Price Coupon YTM Mod.

Dur

A 6 years 170,000 85 2.0% 4.95% 5.42

B 10 years 120,000 80 2.4% 4.99% 8.44

C 15 years 100,000 100 5.0% 5.00% 10.38

390,000

The bond portfolio’s modified duration is closest to ____________.

a. 7.62

Correct

b. 7.88

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FIN358 Understanding Fixed Income Risk and Return

Incorrect

c. 8.08

Incorrect

d. 8.20

Incorrect

10. A limitation of calculating a bond portfolio’s duration as the weighted average of

the yield durations of the individual bonds that compose the portfolio is that it

______________.

a. is less accurate when the yield curve is less steeply sloped

Incorrect

b. is not applicable to portfolios that have bonds with embedded options

Incorrect

c. assumes a parallel shift to the yield curve

Correct

d. none of the above

Incorrect

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FIN358 Understanding Fixed Income Risk and Return

References

Adams, J. F., & Smith, D. J. (2019). Fixed income analysis (4th ed.). Wiley.

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FIN358 Understanding Fixed Income Risk and Return

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Study

Unit6

Asset-Backed Securities and

Valuation and Analysis of Bonds

With Embedded Options

FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

Learning Outcomes

By the end of this unit, you should be able to:

1. Appraise the investment characteristics and risks of various asset-backed

securities

2. Discuss pre-payment risks and tranches in asset-backed securities

3. Discuss the arbitrage-free valuation framework

4. Illustrate the backward induction valuation methodology within the binomial

interest rate tree framework

5. Compute the value of a bond with embedded options using an interest rate tree

6. Explain the effect of volatility on the arbitrage-free value of an option

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

Overview

This last study unit is about asset-backed securities – for example, mortgage-backed

securities. We look at the structures and features of various types of asset-backed

securities.

Such securities are similar to bonds but with features that resemble the optionality of

callable bonds. They can be valued using a binomial method, which is similar to the one

used for valuing options.

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

Chapter 1: Mortgage-Backed Securities

Lesson Recording

Asset-Backed Securities

In this chapter, we will focus on the mortgage-backed securities, which include residential

mortgage-backed securities and commercial mortgage-backed securities. In the United

States, the residential mortgage-backed securities can be divided into two types: (1) those

issued by federal agencies, and (2) those issued by private entities. The former are called

agency mortgage-backed securities, and the latter are called non-agency mortgage-backed

securities.

1.1 Residential Mortgage Loans

A mortgage is a loan secured by the collateral of some specified real estate property which

obliges the borrower to make a pre-determined series of payment. The mortgage gives

the lender the right to “foreclose” on the loan if the borrower defaults and to seize the

property in order to ensure that the debt is paid off. The interest rate on the mortgage

loan is called the mortgage rate or the contract rate. There are many types of mortgage

designs used throughout the world. A mortgage design is a specification of the interest

rate, term of the mortgage, and the manner in which the borrowed funds are repaid. The

most common mortgage design is the fixed rate, level payment, fully amortised mortgage,

which has the following features:

• The mortgage rate is fixed for the life of the mortgage loan

• The dollar amount of each monthly payment is the same for the life of the mortgage

load (i.e., there is “level payment”)

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

• When the last scheduled monthly mortgage payment is made, the remaining

mortgage balance is zero (i.e., the loan is fully amortised)

In the above mortgage design, it is assumed that the homeowners do not pay off any

portion of the mortgage balance prior to the scheduled due date. However, in reality, many

homeowners pay off all or part of their mortgage balance prior to the maturity date, which

is referred to as the pre-payment. The effect of pre-payment is that the amount and timing

of the cash flow from a mortgage loan are not known with certainty. This risk is referred

to as pre-payment risk. We will discuss in more detail about the pre-payment risks later

in the next sections.

1.2 Mortgage Pass-Through Security

A mortgage pass-through security is a security created when one or more holders of

mortgages form a pool of mortgages and sell shares or participation certificates in the

pool. A pool may consist of several thousand or only a few mortgages. When a mortgage

is included in a pool of mortgages that is used as collateral for a mortgage pass-through

security, the mortgage is said to be securitised.

Rather than engaging in hedging activities to limit risk, financial institutions may manage

their risks via loan securitisation, which involves transforming portfolios or pools of loans

into marketable securities.In securitising loans, financial institutions are passing on the

risk of asset transformation to others and choosing to act as asset brokers instead. Because

the broker function is generally less risky than the asset transformation (or financing)

function, securitisation may reduce the rate of return to the selling financial institutions,

unless a sufficient additional volume of transactions can be generated. Nevertheless,

securitisation allows additional alternatives to tailor the risk-return combination financial

institutions choose to bear. In theory, loan securitisation may also reduce the government’s

deposit insurance liability. As mentioned earlier, loan securitisation is the conversion

of loans into marketable securities. This is usually accomplished by placing the loans

in a trust and issuing (selling) marketable securities using the loans as collateral. Loan

securitisation can improve a financial institution’s risk-return trade-off by reducing credit

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

risk, improving the liquidity of the balance sheet, and reducing the regulatory burden

imposed on traditional lending and deposit activities.

Although securitisation has largely been limited to mortgage loans, other loan types are

now being securitised and other institutions are entering the securitisation business. After

origination, mortgages may be placed in a pool held by a trustee and mortgage passthrough securities may then be sold to the public. The pool organiser passes through

all mortgage payments (less a servicing fee) made by the homeowners, including prepayments, to the holders of the pass-through securities on a pro-rata basis. Payments are

thus monthly and are variable, based on how many homeowners pay off their mortgages

early. The pool organiser and/or the government usually provide insurance for the

mortgages in the pool.

The cash flows of mortgage pass-through securities depend on the cash flow of the

underlying pool of mortgages. The cash flows consists of monthly mortgage payments

representing interest, the scheduled repayment of principal, and any pre-payments. The

monthly cash flow for a pass-through is less than the monthly cash flow of the underlying

pool of mortgage by the amount equal to servicing and other fees.

Default risk is generally not a worry for government-backed pass-through securities.

However, these securities carry substantial pre-payment risk. Pass-throughs created

without government or quasi-government involvement may have default risk as well as

pre-payment risk. A limited number of private mortgage pass-through issuers securitise

non-conforming mortgages that do not qualify for government insurance or have

appropriate loan-to-value ratios.

1.3 Advantages of Securitisation

Securitisation is a process of issuing securities whose collateral is backed by a pool of assets

or other securities. Besides a pool of assets consisting of home mortgages, other types of

assets include loans and receivables.

There are many benefits to securitisation such as:

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

• freeing up their funds by selling the firm’s assets. The assets are sold to special

purpose entities which issue securities against the pool of assets held by them.

• providing direct exposure to an asset risk. (This may sound strange. Why would

anyone want to be exposed to risk? The reason is risk goes along with return. To

secure the return from a particular asset, the investor has to accept the risk of that

asset). Suppose an investor wants exposure to the property sector. He could buy

shares in a property development company. However, besides being exposed to

the risk of the properties the developer is building, the investor is also exposed to

other risks such as the risk that stems from the amount of leverage, and how well

the company is run. By contrast, when investing in a Real Estate Investment Trust

(REIT), the risk is focused on the income-producing properties.

• A bank can issue new housing loans if it receives external funds, or if the existing

housing loans are paid up. As housing loans take many years before they are fully

paid back, the bank is somewhat constrained to issue new housing loans. However,

if the bank could securitise the loans, perhaps by selling to an entity which sells

shares to raise funds to buy the loans, the mortgages in the bank’s book can be taken

off. New home loans can be made from the receipt of the existing loans that have

been sold.

• Fixed assets such as houses are rather “lumpy.” We can either buy the whole house

or none of it. Buying a fraction of the house is not feasible. However, if the house is

part of a pool of houses that serve as collateral for securities issued, those securities

would certainly have more liquidity than their underlying assets, which are houses.

• Instead of waiting for the accounts receivable to be paid, securitisation provides

another avenue for funding.

In short, securitisation can reduce re-pricing and funding gap problems (i.e., interest rate

risk) because a long-term fixed rate asset is removed from the balance sheet. The process

can also be used to reduce liquidity risk. The reduction in liquidity risk occurs because the

loans are now marketable, and if an institution wishes to invest in the real estate markets,

mortgage-backed securities that are also liquid can replace the loan investments.

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

For a simple pass-through security, the pool organiser simply passes through all mortgage

payments made by the homeowners, including pre-payments, to the holders of the passthrough securities on a pro-rata basis (see Figure 6.1 below).

Figure 6.1 Pass-through securities

1.4 Pre-Payment Risk

Most mortgages are set up as fully amortised loans,where there will be no remaining

balance at the end of the 30-year or 15-year maturity. However, most mortgages are also

prepaid. Pre-payments increase in falling interest rate environments and leave the passthrough holder with a shorter maturity instrument than expected. The duration is reduced

by pre-payments so the price gains anticipated from falling rates are not as large as

predicted. But the loss in reinvestment income from having to reinvest at lower rates does

occur. This can substantially reduce the investor’s realised rate of return over time. The

dollars of interest earned each month can decline fairly rapidly as interest rates drop and

pre-payments increase. The reduction in total expected cash flows over the life of the passthrough dampens the increase in price associated with the interest rate drop. The future

value of the reinvestment income declines due to the lower reinvestment rate and lower

interest that will be received.

There are three factors that affect the pre-payments. They are:

1. The prevailing mortgage rate

2. Normal housing turnover

3. Characteristics of the underlying mortgage pool

In describing pre-payments, market participants refer to the pre-payment rate or prepayment speed. The pre-payment rate can be measured in several ways:

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

1. Single Monthly Mortality Rate

2. Conditional Pre-Payment Rate

3. PSA Pre-Payment Benchmark

1.5 Collateralised Mortgage Obligation (CMO)

The Collateralised Mortgage Obligation (CMO) was created in 1983 as a means of

repackaging pre-payment risk. CMOs are created by repackaging mortgage payment

streams – or, more typically, by repackaging payments on pass-throughs. In either case, the

innovation of the CMO is to offer different classes or ‘tranches’ with different degrees of

pre-payment protection. The simplest form of CMO is a sequential pay CMO (see below).

This is a profitable activity for CMO backers. Because the CMO investor has a better idea

of the prepayment risk they face; consequently, they are willing to pay more for a CMO

than a pass‐through, ceteris paribus.

CMOs are a hybrid between a pass-through and a bond. With a sequential pay CMO,

separate classes are created with different levels of prepayment protection. Suppose a

sequential pay CMO with a total pool value of $150 million has three classes, A, B, and C

with principal amounts of $50 million per class. The Class A CMO holder would receive all

the initial principal payments (on the entire pool), including all pre-payments whenever

they occur. Until all the Class A holders have been paid off, Class B holders would receive

no principal payments. Likewise, Class C is not affected by any pre-payments until Class

B holders have been paid off. This system of paying off one class before the next class is

paid is known as the waterfall structure, where cash flows cascade from one level to the

next. The multiple classes allow investors to better choose the level of pre-payment risk

desired (see Figure 6.2 below).

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

Figure 6.2 An illustration of tranching and the waterfall structure

Example of Payments on a Sequential Pay CMO

Suppose that the mortgages in the pool have a 9% interest rate, and further

suppose the CMO makes monthly payments. It could make quarterly or semi-annual

payments as well. The mortgage holders make their scheduled monthly payments. If

there are defaults, the pool organiser will make the scheduled payment:

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

Table 6.1 Scheduled payments on a sequential pay CMO

Only Class A holders are initially affected by pre-payments or any principal payments.

Once the Class A principal is totally retired, Class B holders will begin to receive all

principal payments, including pre-payments. CMOs sometimes have six to nine classes.

Class A may have an expected maturity of 2 to 3 years, Class B may have an expected

maturity of 5 to 7 years, and Class C may have a typical expected maturity of 8 to 10 years

or more. Buyers of Class A bonds are seeking short duration mortgage investments and

such securities are typically purchased by FIs with shorter time horizons including thrifts,

banks and P&C insurers. Class B bonds have some pre-payment protection; these appeal

to longer term investors such as banks, pension funds and life insurance companies.

Class C securities are generally desirable investments for institutions seeking long-term

investments and are primarily held by pension funds and life insurers. Each class is

usually quoted at a mark-up over the appropriate maturity Treasury rate. Thus, mortgage

investments offer a higher promised rate than comparable maturity Treasuries with little

or no additional default risk. Actual promised rates depend upon expected pre-payment

patterns. Higher pre-payments result in shorter maturities. Pool organisers can often sell

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

CMO claims for more total value than similar pass-throughs because some investors are

usually willing to pay for the additional pre-payment protection.

The Z class

CMOs usually have a Z classwhich is different from the other classes. The Z class is a

residual claim on the mortgage pool. Initially, the Z class receives no payments, but its

face value increases at a stated coupon rate. Once the principals on all other classes have

been fully paid off, Z class investors begin to receive both interest and principal payments.

Z class investments are long duration and are thus risky. Faster pre-payments will start

payments to the Z class investor sooner. An investor is not sure when payments will begin,

nor how many payments will be received. Typical investors are institutions such as hedge

funds with an appetite for risk that are looking for long-term investments with potentially

higher rates of return.

1.6 Interest-Only and Principal-Only CMOs

Mortgage pools can be used to create mortgage pass-through strips such as interest-only and

principal-only securities. The interest-only (IO) and principal-only (PO) strips are special

types of CMOs. The IO provides the holder a pro-rata claim to all interest payments made

on the pool of mortgages. The PO provides the holder a pro-rata claim on all principal

payments made on the pool. An IO can exhibit negative convexity because as interest rates

fall, pre-payments rise, and the total amount of interest accruing to these securities falls.

Lower rates raise the present value of the cash flows, but the lower overall cash flows can

result in a decline in the value of the IO when rates fall. The converse also holds. If the prepayment effect dominates, the IO will exhibit negative convexity; if the present value effect

dominates, it will not, so the net result depends on changes in prepayment behaviour as

interest rates move. The pre-payment effect, however, is more likely to dominate when

rates are below the coupon rate. IOs that are expected to exhibit negative convexity can

be used by FIs desiring to hedge against rising interest rates. Since IOs are investments

that provide cash flows, these may be more acceptable to managers than caps or futures

positions.

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

POs exhibit greater volatility than an equivalent maturity bond. As interest rates fall and

pre-payments increase, the present value of the cash flows will rise, increasing the value

of the PO and cash payments to the PO holder are accelerated, further increasing the value

of the PO. The converse is also true. In this case, the present value and pre-payment effects

both work in the same direction to increase the PO’s volatility relative to a standard bond.

POs may appeal to investors who want a potentially higher rate of return than that offered

by a standard bond and without facing additional default risk. Certain institutions are

limited in their ability to invest in below investment grade debt, for instance. POs may

also be useful for FIs which wish to increase the interest rate sensitivity of their assets. For

instance, institutions with a negative duration gap or positive re‐pricing gaps may find

POs useful hedging investments. Both IOs and POs can be used to hedge balance sheet

interest rate exposures.

1.7 Planned Amortisation Class CMO

An alternative to a sequential pay CMO is the planned amortisation class (PAC) CMO.

The basic PAC CMO has two classes:

• The PAC or Planned Amortisation Class

For a range of Prepayment Speeds, say 80% of standard to 250% of standard,

the maturity of the PAC and the principal and interest payments received will

not change. That is, payments will be made according to the original planned

amortisation schedule. PAC investors have a higher degree of certainty of the cash

flows they will receive and the maturity of their investments than investors in passthroughs.

• The Support Tranche

The support tranche receives all pre-payments as long as pre-payment speed stays

within the given range. If the pre-payment speed moves outside the standard range,

pre-payments are shared between the PAC and support tranche.

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

1.8 Mortgage-Backed Bonds (MBB)

The MBB is different from a pass-through and a CMO in that the MBB does not remove the

mortgages from the balance sheet. MBBs are also standard bonds that have mortgages as

collateral. There is no ‘passing through’ of mortgage payments (transformed or not) with

MBBs. Thus, MBBs do not offer financial institutions the main advantages of securitisation

discussed above that resulted from removing the mortgages from the balance sheet. They

also leave the issuing financial institutions with substantial pre-payment risk because the

bonds require fixed coupon payments to be paid regardless of the level of pre-payments.

The financial institutions will receive the promised principal, but they must reinvest the

principal at lower interest rates while still paying the higher promised bond interest rate.

As a result, most MBBs have to be over-collateralised in order to receive a high quality

credit rating.

MBBs are advantageous to investors in that the bondholders have no pre-payment risk

(unless the bonds are callable). MBBs may be advantageous to the financial institution

because the bond issue can be used to fund the mortgages and the bond issue will have

a similar maturity to the mortgages’ expected maturity, thus reducing the re-pricing and

duration gap problems. This advantage to the financial institutions comes at increased

risks to the FDIC because the pledged assets backing the mortgage bonds may not be

available to insured depositors in the event of financial institution failure. The MBB

may also actually reduce the financial institution’s liquidity because now the pledged

mortgages cannot be sold; moreover, the financial institution must pledge more mortgages

than bonds issued so that overall liquidity can be reduced. Due to these disadvantages,

MBBs are the least used form of securitisation. With a pass-through security, the investor

bears all of the pre-payment risk. With a non-callable mortgage-backed bond, the issuer

bears all of the pre-payment risk. With a CMO, the pre-payment risk is shared between the

issuer and the investors and the investors can choose their desired level of pre-payment

protection by choosing between the different CMO classes.

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

Read

Read the following sections in:

Petitt, B. S. (2019). Introduction to asset-backed securities. In Fixed income analysis (4th

ed., pp. 27-34). Wiley.

• Section 1: Introduction

• Section 2: Benefits of securitization for economics and financial markets

• Section 3: How securitization works

• Section 4: Residential mortgage loans

• Section 5: Residential mortgage-backed securities

• Section 6: Commercial mortgage-backed securities

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

Chapter 2: Asset-Backed Securities

While the mortgage-backed securities were the first type of asset-backed securities

(ABS), securities backed by other types of assets (e.g., consumer and business loans and

receivables) have been issued throughout the world. Some examples include securities

backed by automobile loans, credit card receivables, home equity loans, manufactured

housing loans, student loans, Small Business Administration loans, corporate loans, and

bonds.

There are three reasons that securitisation occurred first for mortgage loans. First,

mortgage loans are highly standardised. Second, U.S. government mortgage insurance

has limited the need for buyers of mortgage-backed securities to engage in individual

credit risk investigations. Moreover, a secondary market exists for the collateral, so

collateral values can be easily updated and homes generally maintain their collateral

value. Third, mortgage loans are usually for the long term and, therefore, the costs of

securitisation can be efficiently spread through time.

A lack of these characteristics would seem to be the major limiting factor in securitisation

of other loan types. However, standardisation can be achieved at the origination level to

some extent, and pooling can lead to additional standardisation. Third-party participants

may be willing to insure against default risk, or securities buyers may be willing to bear

the default risk. In some cases, such as for credit card loans, the pooling of large numbers of

borrowers and the ability to charge sufficiently high interest rates to offset higher loss rates

can overcome the default risk problem. As familiarity grows and the costs to securitise

fall, shorter-term loans may be securitised if sufficient economies of scale can be achieved,

perhaps by high volume. Nevertheless, the terms and risks of many loans are unique.

Ultimately, it is the ability to accurately assess and measure the level of risk of the pool that

may limit securitisation. The more heterogeneous the loans of a given type and the more

uncertain the collateral values are, the more difficult it will be to successfully securitise a

given loan type.

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

2.1 The Securitisation Process and Features of Asset-backed

Securities

The issuance of an asset-backed security is more complicated than the issuance of a

corporate bond. It evolves a “seller,” a “buyer,” and a special purpose vehicle (SPV), as

well as third-party entities such as attorneys, trustees, and rating agencies, and even a

guarantor.

An entity that wants to raise funds via a securitisation will sell assets to a special purpose

vehicle (SPV); it is referred to as the ‘‘seller’’ in the transaction. The “buyer” of assets in a

securitisation is a special purpose vehicle and this entity, referred to as the issuer or trust,

raises funds to buy the assets via the sale of securities (the asset-backed securities).

There will be an entity in a securitisation that will be responsible for servicing the

loans or receivables. Third-party entities in the securitisation are attorneys, independent

accountants, trustee, rating agencies, servicer, and possibly a guarantor.

2.1.1 Transaction Structure

In a securitisation structure, the bond classes issued can consist of a senior bond that is

tranched so as to redistribute pre-payment risk and one or more subordinate bonds; the

creation of the subordinate bonds provides credit tranching for the structure.

The collateral for an asset-backed security can be either amortising assets (e.g., auto loans

and closed-end home equity loans) or non-amortising assets (e.g., credit card receivables).

For amortising assets, projection of the cash flow requires projecting pre-payments. For

non-amortising assets, pre-payments by an individual borrower do not apply since there

is no schedule of principal repayments.

When the collateral is amortising assets, typically the principal repayments are distributed

to the security holders. When the collateral consists of non-amortising assets, typically

there is a lockout period, or a period where principal repayments are reinvested in

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new assets; after the lockout period, principal repayments are distributed to the security

holders.

The type of collateral ‒ amortising or non-amortising ‒ has an impact on the structure of

the transaction. Typically, when amortising assets are securitised, there is no change in

the composition of the collateral over the life of the securities except for loans that have

been removed due to defaults and full principal repayment due to prepayments or full

amortisation.

In an ABS transaction backed by non-amortising assets, the composition of the collateral

changes. The funds available to pay the security holders are principal repayments

and interest. The interest is distributed to the security holders. However, the principal

repayments can be either: (1) paid out to security holders, or (2) reinvested by purchasing

additional loans.

2.1.2 Credit Enhancements

Asset-backed securities are credit enhanced – that is, there must be support from

somewhere to absorb a certain amount of defaults. Credit enhancement levels are

determined relative to a specific rating desired for a security. There are two general types

of credit enhancement structures: external and internal.

External credit enhancements come in the form of third-party guarantees that provide

for first loss protection against losses up to a specified level. External credit enhancement

includes insurance by a monoline insurer, a guarantee by the seller of the assets, and

a letter of credit. The most common forms of internal credit enhancements are reserve

funds and senior/subordinate structures. The senior/subordinated structure is the most

widely used internal credit support structure with a typical structure having a senior

tranche and one or more non-senior tranches.

For mortgage-related asset-backed securities and non-agency mortgage-backed securities,

there is a concern that pre-payments will erode the protection afforded by the non-senior

(i.e., subordinated) tranches after the deal closes. A shifting interest structure is used to

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protect against deterioration in the senior tranche’s credit protection due to pre-payments

by redistributing the pre-payments disproportionately from the non-senior tranches to the

senior tranche according to a specified schedule.

2.2 Home Equity Loans

A home equity loan (HEL) is a loan backed by residential property. The collateral for a

home equity loan is typically a first lien on residential property and where the loan fails

to satisfy the underwriting standards for inclusion in a loan pool of Ginnie Mae, Fannie

Mae, or Freddie Mac because of the borrower’s impaired credit history or an extremely

high payment-to-income ratio.

Typically, a home equity loan is used by a borrower to consolidate consumer debt using

the current home as collateral rather than to obtain funds to purchase a new home. Home

equity loans can be either closed-ended (i.e., structured the same way as a fully amortising

residential mortgage loan) or open-ended (i.e., homeowner given a credit line).

The monthly cash flow for a home equity loan-backed security backed by closed-end

HELs consists of: (1) net interest, (2) regularly scheduled principal payments, and (3) prepayments. Several studies by Wall Street firms have found that the key difference between

the pre-payment behaviour of HELs and traditional residential mortgages is the important

role played by the credit characteristics of the borrower. Studies strongly suggest that

borrower credit quality is the most important determinant of pre-payments, with the

sensitivity of refinancing to interest rates being greater the higher the borrower’s credit

quality.

2.3 Auto Loan-backed Securities

Auto loan-backed securities are issued by the financial subsidiaries of auto manufacturers,

commercial banks, and independent finance companies and small financial institutions

specialising in auto loans.

The cash flow for auto loan-backed securities consists of regularly scheduled monthly

loan payments (interest and scheduled principal repayments), and any pre-payments. PreSU6-19

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payments on auto loans are not sensitive to interest rates. Pre-payments on auto loanbacked securities are measured in terms of the absolute pre-payment speed (denoted ABS)

which measures monthly prepayments relative to the original collateral amount.

2.4 Student Loan-Backed Securities (SLABS)

SLABS are securities backed by student loans. Student loans cover college cost

(undergraduate, graduate, and professional programmes such as medical and law school)

and tuition for a wide range of vocational and trade schools.

Securities backed by student loans, popularly referred to as SLABS, have similar structural

features as the other asset-backed securities. Student loans involve three periods with

respect to the borrower’s payments ‒ deferment period, grace period, and loan repayment

period. Pre-payments typically occur due to defaults or a loan consolidation (i.e., a loan

to consolidate loans over several years into a single loan).

Issuers of SLABS include the Student Loan Marketing Association (Sallie Mae), traditional

corporate entities, and non-profit organisations. Student loan-backed securities offer a

floating rate. For some issues, the reference rate is the 3-month Treasury bill rate; but for

most issues, the reference rate is SOFR.

2.5 Credit Card Receivable-Backed Securities

Credit card receivable-backed securities are backed by credit card receivables for credit

cards issued by banks, retailers, and travel and entertainment companies. Credit card

deals are structured as a master trust.

For a pool of credit card receivables, the cash flow consists of finance charges collected,

fees, and principal. The principal repayment of a credit card receivable-backed security is

not amortised; instead, during the lockout period, the principal payments made by credit

card borrowers are retained by the trustee and reinvested in additional receivables and

after the lockout period (the principal amortisation period), the principal received by the

trustee is no longer reinvested but paid to investors.

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There are provisions in credit card receivable-backed securities that require early

amortisation of the principal if certain events occur. Since the concept of prepayments

does not apply to credit card receivable-backed securities, participants look at the monthly

payment rate (MPR) which expresses the monthly payment (which includes interest,

finance charges, and any principal) of a credit card receivable portfolio as a percentage of

debt outstanding in the previous month.

2.6 Collateralised Debt Obligation (CDO)

A collateralised debt obligation (CDO) is a security backed by a diversified pool of one or

more of the following types of debt obligations:

• U.S. domestic high-yield corporate bonds

• structured financial products (i.e., mortgage-backed and asset-backed securities)

• emerging market bonds

• bank loans

• special situation loans and distressed debt

A collateralised bond obligation is a CDO in which the underlying pool of debt obligations

consists of bond-type instruments (high-yield corporate and emerging market bonds). In

a CDO, there is an asset manager responsible for managing the portfolio of assets.

The tranches in a CDO include senior tranches, mezzanine tranches, and subordinate/

equity tranche. The senior and mezzanine tranches are rated and the subordinate/equity

tranche is unrated.

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Read

Read the following sections in:

Petitt, B. S. (2019). Introduction to asset-backed securities. In Fixed income analysis (4th

ed., pp. 27-34). Wiley.

• Section 7: Non-mortgage asset-backed securities

• Section 8: Collateralized debt obligations

• Section 6: Commercial mortgage-backed securities

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

Chapter 3: Valuing Bonds With Embedded Options

Lesson Recording

Valuing Bonds With Embedded Options

The presence of an embedded option in a bond structure makes the valuation of such

bonds more complicated. In order to understand how to value a bond with embedded

options, we will need to review some of the fundamental concepts.

3.1 Call Options and Put Options

The two most common types of embedded options are call options and put options.

As interest rates decline, the issuer of a call option bond may call or redeem its debt

obligation. This makes a callable bond less valuable to an investor in a declining interest

rate environment when interest rates decline below the bond’s coupon rate. Thus,

unlike the option-free bond discussed in the previous section, there is very little price

appreciation for the callable bond as interest rates decline beyond a certain critical yield

level. When the callable bond enters this region, the bond is said to exhibit “pricecompression.” In stark contrast to option-free bonds, a callable bond shows much less

price appreciation. This feature of callable bonds is called negative convexity (see Figure

5.3 in Section 3.2.1 of Study Unit 5). When the prevailing interest rates are higher than the

coupon rates, however, the callable bond will have a similar price/yield relationship to

that of an option-free bond.

In the case of bonds with put options or putable bonds, when the prevailing interest rates

are lower than the coupon rates, putable bonds will have a similar price/yield relationship

to that of an option-free bond. As interest rates rise, however, the price of the putable bond

declines, but the price decline is less than that of an option-free bond (see Figure 5.3 in

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

Section 3.2.1 of Study Unit 5). Typically, the put price is par value. Hence, the advantage

to the investor is that if yields rise excessively such that the bonds value falls below the

par value, the investor will exercise his put option and redeem his bond investment at par

value.

Only a few practitioners will develop their own model to value bonds with embedded

options. In practice, portfolio managers and analysts use a proprietary model developed

by either a dealer or a vendor of analytical systems. There are various arbitrage-free

models that have been developed to value a bond with embedded options. They all follow

the same principle: they generate an interest rate tree based on some interest rate volatility

assumption, they require rules for determining when any of the embedded options will

be exercised, and they employ the backward induction methodology.

3.2 Binomial Interest Rate Tree

In this section, we will use the binomial model to demonstrate the issues and assumptions

associated with valuing a bond with embedded options. This model is available with

Bloomberg and other commercial vendors. In practice, these models share all of the

principles described in this binomial model but differ with respect to certain assumptions

that can produce quite different values. The reasons for these differences in valuation must

be understood by a portfolio manager.

Valuation of bonds with embedded options starts with the binomial interest rate tree.

We looked at the valuation of options using the binomial tree in Study Unit 2. For fixed

income, the binomial tree approach is quite similar.

The binomial interest rate tree is basically a network of one-year forward rates derived

in the same way that spot rates were obtained using the bootstrapping technique based

on arbitrage arguments. At each node, there are two possible forward interest rates which

can be derived knowing the coupon rate for the on-the-run issue, assumed interest rate

volatility, an assumed interest rate model and the interest rate at the root of the tree.

It is not essential for you to know how to derive the binomial interest rate tree. Rather, you

should understand how to value a bond with an embedded option given the rates on the

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

tree. The fundamental principle is that when a tree is used to value an on-the-run issue

for the benchmark (e.g., Treasury market), the resulting value should be arbitrage free –

i.e., the tree should generate a value for an on-the-run issue equal to the observed market

value.

To obtain the value of the bond with an embedded option, we repeatedly determine the

value of the bond at each node of binomial interest rate tree (using backward induction)

starting out from the rightmost node. The value of the bond at any node of the interest

rate tree is:

Value at a node = ½ {[(VH +C)/(1+r*)] + [(VL +C)/(1+r*)]}

where

C = coupon payment

VH = bond’s value for the higher 1-year forward rate

VL = bond’s value for the lower 1-year forward rate

r* = the appropriate 1-year forward discount rate at the node

What takes place at each node is either a random event or a decision. The change in interest

rates represents a random event. However, in the presence of an embedded option, there

will be a decision at each node. Specifically, the decision will be whether or not the issuer

or bond holders will exercise the option. For example, in the case of a callable bond where

the call option is exercised by the issuer, the bond value at a node must be changed to

reflect the lesser of its values if it is not called (i.e., the value obtained by applying the

backward induction method) and the call price. Figure 6.2 illustrates the process for a 6.5%

coupon bond with a 4-year maturity and 10% interest rate volatility. The bond is callable

at prices of 102 and 101, in Years 1 and 2 respectively. The fastest way to come to grips with

this valuation technique is to work through the examples given in Section 3 of Chapter 8

in Petitt, B. S. (2019). Fixed income analysis (4th ed.). Wiley.

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

Figure 6.3 Binomial tree for 6.5% coupon callable bond with maturity of 4

years

In Figure 6.3, we show the timeline at top of the figure, where each number depicts the

end of the year corresponding to the number. For example, Time 1 marks the end of Year

1. The end of Year 0, Time 0, corresponds to the beginning of Year 1. The cash flows are

assumed to come at the end of the year.

For each cell, there are four lines of data. The first line gives the interest rate applicable

to the period indicated by two specific times. For example, “r12u” shows the rate for the

end of Year 1 to the end of Year 2. The letter(s) after the two specific times indicate(s) the

path taken to arrive at the node. For example, “r23uu” shows that the node is the result

of two upward moves in the interest rates. The second and fourth lines show the interest

applicable for the next period and the coupon amount. The third line shows the price for

bond, given the two possible values at the end of the next period.

Let’s look at how the bond price is calculated for node shown as “r34udd”. The price of

the bond is the weighted average of present values of the two possible value for the bond

in the next period. For this node,

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

Bond price = 0.5 x (100 + 6.5)/(1 +0.061660) + 0.5 x (100 + 6.5)/(1 +0.061660)

= 100.3146

However, the price calculated is higher than the call price of 100, so the price is set to 100.

The process of calculating the bond price at node r34udd is repeated for all nodes at time

3. This process is repeated for all nodes at Time 2 and continues in a backward recursive

fashion until we arrive at the bond price at Time 0.

3.3 Valuing Mortgage-Backed and Asset-Backed Securities

In this section, we are going to discuss how to value mortgage-backed and asset-backed

securities. To understand the valuation process, we should first look at the cash flow yield

analysis of these securities.

3.3.1 Cash Flow Yield Analysis

The yield on any financial instrument is the interest rate that makes the present value of

the expected cash flow equal to its market price plus accrued interest. When applied to

mortgage-backed and asset-backed securities, this yield is called a cash flow yield.

The cash flow yield is the interest rate that makes the present value of the projected

cash flow for a mortgage-backed or asset-backed security equal to its market price plus

accrued interest. The convention is to compare the yield on mortgage-backed and assetbacked securities to that of a Treasury coupon security by calculating the security’s bondequivalent yield. This measure is found by computing an effective semi-annual rate and

doubling it.

The cash flow yield is based on three assumptions that thereby limit its use as a measure

of relative value:

1. A pre-payment assumption and default/recovery assumption,

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

2. An assumption that the cash flows will be reinvested at the computed cash flow

yield, and

3. An assumption that the investor will hold the security until the last loan in the

pool is paid of

All yield measures suffer from problems that limit their use in assessing a security’s

potential return. The yield to maturity has two major shortcomings as a measure of a

bond’s potential return. To realise the stated yield to maturity, the investor must:

1. reinvest the coupon payments at a rate equal to the yield to maturity, and

2. hold the bond to the maturity date.

Reinvestment risk is the risk of having to reinvest the interest payments at less than the

computed yield. Interest rate risk is the risk associated with having to sell the security

before its maturity date at a price less than the purchase price. These shortcomings are

equally applicable to the cash flow yield measure.

3.3.2 Valuation Approaches for Different Types of ABS

The valuation approach adopted for structured fixed income instruments depends

primarily on the characteristics of the underlying – i.e., the type of receivables or loans

backing the instrument. There are generally instruments with:

ABS With No Pre-Payment

Structured fixed income instruments such as Asset-Backed Security (ABS) backed by

credit card receivables, which have no pre-payment options are valued by discounting

its cash flows using spot interest rates (extracted from a theoretical spot rate curve with

respect to a benchmark such as Treasury market, a specific issuer, or bond sector with a

given credit rating) plus a spread called the Z-spread or zero volatility spread. The Zspread is the spread that will make the present value of cash flows from the ABS (when

discounted at the Treasury spot rate plus this spread) equal to the price of the security. A

trial and error procedure (or search algorithm) is required to determine the zero‐volatility

spread.

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

A Z-spread measured relative to the Treasury spot rate curve represents a spread to

compensate for the non‐Treasury security’s credit risk and liquidity risk.

ABS With Pre-Payment Option and Remote Pre-Payment Probability

Structured fixed income instruments such as ABS backed by automobile loans, which have

pre-payment options that are typically not exercised are valued in the same manner as

those without pre-payment options described above. In this case, the Z-spread measured

relative to the Treasury spot rate curve represents a spread to compensate for the non‐

Treasury security’s credit risk, liquidity risk, and option risk.

ABS With Pre-Payment Option and High Pre-Payment Probability

Structured fixed income instruments such as ABS, which have pre-payment options that

are typically exercised, are valued using either backward induction with a binomial model

(described in Study Unit 4) or Monte Carlo Simulation, depending on whether the cash

flow is interest rate path-dependent.

• Non-Interest Rate Path-Dependent Securities

In the case of callable agency debentures and corporate bonds, the decision to

exercise a call option is not dependent on how interest rates evolved over time. The

decision of the issuer to call a bond will depend only on the level of interest rate, not

the path that the interest rates took to get to that rate. Such structured fixed income

instruments securities are valued using backward induction with a binomial model.

• Interest Rate Path-Dependent Securities

In contrast, there are structured fixed income securities such as mortgage‐backed

securities and CMOs, where pre-payments are interest rate path-dependent

because the current month’s pre-payment rate depends primarily on whether there

have been prior opportunities to refinance. This means that the cash flow received

in one period is determined not only by the current interest rate level, but also by

the path that interest rate took to get to the current level. Monte Carlo Simulation is

used to value such securities. In this technique, an interest rate model and the term

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

structure of interest rates are first used to generate many scenarios of future interest

rate paths, assuming a certain level of interest rate volatility. The structured fixed

income security is then valued on each interest rate path and averaged.

An interest rate model is a model that assumes how interest rates will change over

time. The volatility assumption determines the dispersion of future interest rates,

which are adjusted arbitrarily so that the model will produce arbitrage free values.

(In contrast, note that there is no such adjustment necessary in a binomial model

because the tree is automatically built to be arbitrage-free). Thus, in the case of

a newly issued mortgage pass‐through security with a maturity of 360 months,

numerous adjusted simulated interest path scenarios are generated. Each scenario

(i.e., interest rate path) consists of a path of 360 simulated 1‐month future interest

rates. For each month of the scenario, a monthly interest rate and a mortgage

refinancing rate are generated. The monthly interest rates are used to discount the

projected cash flows in the scenario.

Given the mortgage refinancing rate and a pre-payment model, cash flows on each

interest rate path can be generated. Given these cash flows, the present value of cash

flows for each month within an interest rate path can be calculated. The discount

rate for determining the present value is the simulated spot rate for each month

on the interest rate path plus an appropriate spread. This spread reflects the risks

that the investor feels are associated with realising the cash flows and is called the

option-adjusted spread (OAS). The spot rate on a path can be determined from the

simulated future monthly rates. The present value for each interest rate path is the

sum of the present value of cash flows for each month on that path. In essence, it is

the theoretical value of the pass-through security, if that path was actually realised.

The value of a structured fixed income security is then determined by calculating

the average of the theoretical values from all such simulated interest rate paths.

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Read

Read the following sections in:

Petitt, B. S. (2019). Valuation and analysis of bonds with embedded options. In Fixed

income analysis (4th ed., pp. 77-90). Wiley.

• Section 1: Introduction

• Section 2: Overview of embedded options

• Section 3: Valuable and analysis of callable and putable bonds

• Section 4: Interest rate risk of bonds with embedded options

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

Formative Assessment

1. Securitisation is beneficial for banks because it __________.

a. repackages bank loans into simpler structures

b. increases the funds available for banks to lend

c. allows banks to maintain ownership of their securitised assets

d. results in higher profits than if the bank were to hold on to the loans

2. In a securitisation structure, time tranching provides investors with the ability to

choose between __________.

a. extension and contraction risk

b. fully amortising loans and partially amortising loans

c. call protection and no-call protection

d. senior bonds and subordinated bonds

3. For a mortgage pass-through security, which of the following risks would most likely

increase as interest rates decline?

a. Balloon

b. Extension

c. Contraction

d. Liquidity

4. Which of the following statements concerning the role of a support tranche in a

planned amortisation class collateralised mortgage obligation (PAC CMO) is least

accurate?

a. The purpose of a support tranche is to provide pre-payment protection for one

or more PAC tranches.

b. The support tranches are exposed to extremely high levels of credit risk.

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

c. If pre-payments are too low to maintain the PAC schedule, the shortfall is

provided by the support tranche.

d. None of the above

5. An investor who expects interest rates to fall would likely pick which of the following

tranche of a CMO?

a. Fixed-rate tranche

b. An inverse floating rate tranche

c. A PAC tranche

d. Support tranche

6. The monthly cash flows of a mortgage pass-through security most likely _________.

a. remain constant

b. change when interest rates decline

c. are equal to the cash flows of the underlying pool of mortgages

d. none of the above choices is correct

7. A pre-payment rate of 90 PSA means that investors can expect _______.

a. 90% of borrowers whose mortgages are included in the collateral backing the

mortgage pass-through security to prepay their mortgages

b. 90% of the par value of the mortgage pass-through security to be repaid prior

to the security’s maturity

c. the pre-payment rate of the mortgages included in the collateral backing the

mortgage pass-through security to be 90% of the monthly prepayment rates

forecasted by the PSA model

d. to receive 90% of the payments forecasted by the PSA model

8. In a collateralised mortgage obligation, ________.

a. pre-payment risk is redistributed among different bond classes

b. pre-payment risk is eliminated

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

c. conforming loans form the underlying structure

d. none of the above

9. Commercial mortgage-backed securities (CMBS) mortgages are structured as

nonrecourse loans, which means __________.

a. the borrower cannot be forced into bankruptcy

b. the lender cannot litigate the loan.

c. the lender can only look to the collateral as a means to repay a delinquent loan.

d. all of the above listed options.

10. During the lock-out period, credit card receivable asset-backed securities owners

receive ________.

a. no cash flow

b. only principal payments collected

c. only finance charges collected and fees

d. principal payments less finance charges and fees

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

Solutions or Suggested Answers

Formative Assessment

1. Securitisation is beneficial for banks because it __________.

a. repackages bank loans into simpler structures

Incorrect

b. increases the funds available for banks to lend

Correct

c. allows banks to maintain ownership of their securitised assets

Incorrect

d. results in higher profits than if the bank were to hold on to the loans

Incorrect

2. In a securitisation structure, time tranching provides investors with the ability to

choose between __________.

a. extension and contraction risk

Correct

b. fully amortising loans and partially amortising loans

Incorrect

c. call protection and no-call protection

Incorrect

d. senior bonds and subordinated bonds

Incorrect

3. For a mortgage pass-through security, which of the following risks would most likely

increase as interest rates decline?

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

a. Balloon

Incorrect

b. Extension

Incorrect

c. Contraction

Correct

d. Liquidity

Incorrect

4. Which of the following statements concerning the role of a support tranche in a

planned amortisation class collateralised mortgage obligation (PAC CMO) is least

accurate?

a. The purpose of a support tranche is to provide pre-payment protection for

one or more PAC tranches.

Correct

b. The support tranches are exposed to extremely high levels of credit risk.

Incorrect

c. If pre-payments are too low to maintain the PAC schedule, the shortfall is

provided by the support tranche.

Incorrect

d. None of the above

Incorrect

5. An investor who expects interest rates to fall would likely pick which of the following

tranche of a CMO?

a. Fixed-rate tranche

Incorrect

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

b. An inverse floating rate tranche

Correct

c. A PAC tranche

Incorrect

d. Support tranche

Incorrect

6. The monthly cash flows of a mortgage pass-through security most likely _________.

a. remain constant

Incorrect

b. change when interest rates decline

Correct

c. are equal to the cash flows of the underlying pool of mortgages

Incorrect

d. none of the above choices is correct

Incorrect

7. A pre-payment rate of 90 PSA means that investors can expect _______.

a. 90% of borrowers whose mortgages are included in the collateral backing the

mortgage pass-through security to prepay their mortgages

Incorrect

b. 90% of the par value of the mortgage pass-through security to be repaid prior

to the security’s maturity

Incorrect

c. the pre-payment rate of the mortgages included in the collateral backing the

mortgage pass-through security to be 90% of the monthly prepayment rates

forecasted by the PSA model

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

Correct

d. to receive 90% of the payments forecasted by the PSA model

Incorrect

8. In a collateralised mortgage obligation, ________.

a. pre-payment risk is redistributed among different bond classes

Correct

b. pre-payment risk is eliminated

Incorrect

c. conforming loans form the underlying structure

Incorrect

d. none of the above

Incorrect

9. Commercial mortgage-backed securities (CMBS) mortgages are structured as

nonrecourse loans, which means __________.

a. the borrower cannot be forced into bankruptcy

Incorrect

b. the lender cannot litigate the loan.

Incorrect

c. the lender can only look to the collateral as a means to repay a delinquent

loan.

Incorrect

d. all of the above listed options.

Correct

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

10. During the lock-out period, credit card receivable asset-backed securities owners

receive ________.

a. no cash flow

Incorrect

b. only principal payments collected

Incorrect

c. only finance charges collected and fees

Correct

d. principal payments less finance charges and fees

Incorrect

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FIN358 Asset-Backed Securities and Valuation and Analysis of Bonds With Embedded Optio…

References

Adams, J. F., & Smith, D. J. (2019). Fixed income analysis (4th ed.). Wiley.

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