Pricing and Hedging Swaps

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PRICING AND HEDGING SWAPS

by Paul Miron and

Philip Swannell

Published by Euromoney Books

Published by Euromoney Publications PLC Nestor House, Playhouse Yard London EC4V 5EX

Copyright ©1991 Euromoney Publications PLC ISBN 1 85564 052 X

All rights reserved. No part of this book may be reproduced in any form or by any means without permission from the publisher No legal responsibility can be accepted by Euromoney for the information which appears in this publication

Edited at Euromoney by Graham Henderson Typeset by the authors and transferred to film by PW Graphics, London Printed in Great Britain by The Blackmore Press, Dorset Reprinted 1995 by Biddies Ltd, Guildford and King’s Lynn

Contents 1 Introduction 1.1 What the book is about 1.2 For whom the book is intended 1.3 Background knowledge required 1.4 An outline of the book

1 2 3 5 6

2 Defining the swap 2.1 What is an interest rate swap? 2.2 Features of a standard interest rate swap 2.3 Non standard interest rate swaps 2.4 Currency swaps 2.4.1 Why have principal exchanges? 2.4.2 Types of currency swaps

9 9 10 15 18 19 21

3 Background to the swap market 3.1 The development of the market 3.2 Market size 3.3 Developments in book running 3.4 The uses of swaps: a few examples

23 23 25 27 28

4 Hedging Instruments 4.1 Government Bonds 4.1.1 Yield to Maturity 4.1.2 Market details 4.1.3 Modified duration and convexity 4.2 Futures Contracts 4.3 Forward Rate Agreements

31 33 33 41 42 44 48

i

CONTENTS

ii

4.4

Loans and Deposits

5 A simple approach to swap pricing 5.1 Basic concepts 5.1.1 The present value of a cashflow 5.1.2 Accrual basis conversion 5.1.3 Annual versus semi-annual 5.1.4 The value of an annuity 5.2 First worked examples 5.3 Comparison swaps 5.4 Worked examples: pricing 5.5 Worked examples: valuation

6

50 51 53 53 55 56 58 60 62 67 78

5.6 Basis swaps

83

5.7 Pricing currency swaps 5.8 Valuing currency swaps 5.9 Remarks

84 86

Zero coupon pricing 6.1 In defence of zero coupon pricing 6.2 Constructing the discount function 6.2.1 Valuing LIBOR cashflows 6.2.2 Stripping the curve 6.2.3 A worked example 6.2.4 A more complicated example 6.3 Interpolation 6.4 Incorporating Futures 6.5 The futures strip 6.6 Integrating the curves 6.7 Other curves

7 Valuing a swap 7.1 The bid-offer spread 7.2 The fixed leg 7.3 The floating leg 7.4 Special features 7.4.1 LIBOR margins 7.4.2 Back-set and compounded LIBOR

89 91 91 93 93 96 100 103 108 111 112 114 . . 117

121 121 122 123 124 124 125

CONTENTS

7.5

7.4.3 Amortising and rollercoaster swaps 7.4.4 Currency swaps Pricing the swap

iii 128 129 131

8 Interest rate exposure 8.1 A simple example 8.2 An experiment 8.3 The nature of the delta vector 8.4 Par swaps and other par rates 8.5 Equivalent positions 8.6 Analytic deltas 8.7 The case of no futures: a preview 8.8 Using no futures: the maths 8.8.1 How F changes as R changes 8.8.2 How R changes as R´ changes 8.8.3 How F changes as R´ changes 8.8.4 Between the grid points 8.8.5 Portfolio deltas 8.8.6 Equivalent positions 8.8.7 Expanded equivalent positions 8.9 Incorporating futures: a discussion 8.10 Futures: the maths 8.10.1 Building the discount function 8.10.2 Calculating 8.10.3 Calculating 8.10.4 The delta vector 8.10.5 Equivalent positions 8.11 The gamma matrix

133 134

9 Hedging and trading swaps 9.1 Why and what to hedge 9.2 Hedging with bonds 9.3 Calculating the swap PVBP 9.4 Trading against the futures strip 9.5 The swap-FRA arbitrage 9.6 Bond futures hedging 9.7 Reinvestment risk

167 167 169 170 171 174 176 177

135 138 139 139 144 145 147 148 149 150 151 153 154 154 155 156 157 159 160 163 163 164

CONTENTS

iv 9.8 Forward FX arbitrage 9.9 Fixed-fixed currency swaps

179 181

10 Interest rate options 10.1 A toy model 10.2 The standard model 10.3 Cap and floor exposures 10.4 Swaptions 10.5 Other models 10.6 Hedging options

185 186 188 197 201 207 209

11 Managing a portfolio

213 213 217 218 219

11.1 LIBOR exposures 11.2 Cross-currency and cash positions 11.2.1 Cash payments on swaps 11.2.2 Cross-currency cashflows 12 Conclusions

223

A Continuous compounding

225

B Example sterling curves

227

C The delta of a par swap

231

D Zero coupon and additive systems D.1 Definitions D.2 Proof

233 233 235

E Answers to questions of chapter 5

237

Glossary of symbols

240

Bibliography

245

List of Figures 2.1 Diagrammatic representation of an interest rate swap . . 14 19 2.2 Diagrammatic representation of a currency swap

35 4.1 Cashflow schedule of a generic bond 4.2 Cashflow schedule of a bond bought between coupon dates. 37 4.3 Graphical representation of the Newton-Raphson equation. 38 5.1 5.2 5.3 5.4

A simple annuity A general annuity Three swaps/bonds A two and a half year bond

58 59 63 64

6.1 6.2 6.3 6.4 6.5 6.6 6.7

LIBOR payment as principal flows LIBOR payments on a two year swap A one year par swap Cashflows on a three year par swap A one year semi-annual par swap A linearly interpolated function Exponential interpolation

96 97 98

109 111

8.1 8.2 8.3 8.4

A picture of A picture of part of A picture of part of An aid to calculating

146 150 161 162

103 105

10.1 Evolution of bond prices in a toy model

186

11.1 LIBOR exposure over time 11.2 Cross-currency position versus time

217 221

v

List of Tables 3.1 Swap volumes for 1989

26

5.1 A dollar yield curve 5.2 Comparison swap for Example 5.7 5.3 A sterling yield curve 5.4 Comparison swaps for Example 5.9 5.5 Cashflows for the ECU bond and swap of Example 5.10 .

69 70 71 73 76

6.1 6.2 6.3 6.4 6.5

Discount factors for a dollar curve A sterling yield curve Discount function for the curve in Table 6.2 Combined swap and futures dates An FRA generated discount function

7.1 Cashflows to replicate LIBOR payments 7.2 Implied sterling forward rates 7.3 The value of a back-set LIBOR deal 7.4 A rollercoaster swap 7.5 A three year sterling swap two years forward

102 106 109 115 119 124 126 127 129 132

8.1 8.2 8.3 8.4 8.5 8.6 8.7

134 A sterling yield curve Change in portfolio value for a change in individual rates 136 The delta vector 137 The delta of a £10,000,000 par five year swap 140 141 Deltas for £10,000,000 par swaps 142 Portfolio equivalent positions 157 The ordering of the grid points

9.1

PVBP for par sterling swaps vii

171

LIST OF TABLES

viii 9.2

Forward dollar/sterling points

10.1 A toy model for option hedging B.1 B.2 B.3 B.4 B.5

181 211

A sterling yield curve 228 Discount function generated from swaps and deposits . . 228 Sterling futures prices 229 Discount function generated from futures 229 Discount function generated from swaps, deposits and 230 futures .

About the authors Paul Miron and Philip Swannell trade swaps and options at UBS Phillips & Drew, the London subsidiary of Union Bank of Switzerland. In addition, they have developed extensive trading and portfolio management systems for interest rate derivatives. Previously, the authors both worked in the swap group at County Nat West. Paul Miron has a degree in Mathematics and Physics from the University of Warwick and a doctorate in Theoretical Particle Physics from Oxford University. He also spent 1987-88 as a Junior Fellow of Merton College, Oxford. Philip Swannell read Mathematics at St. John’s College. Oxford. Not content with the risk of trading swap portfolios, the authors enjoy rock climbing, and are regularly to be found stuck half way up sea cliffs. Philip Swannell is attempting to learn French.

ix

Acknowledgements The authors wish to thank: Our colleagues at UBS Phillips and Drew, especially the following who made comments on the book: Philippa Skinner, Iain Henderson, Kyran McStay, Jo-Anne Benjafield and Philippe Struk. The authors also thank Jo-Anne Benjafield for the constant supply of cappucini. and diverting thousands of phone calls from persistent brokers. Mark Shackleton and Kelley Kirklin for their comments on several chapters. Robert Lustenburger for carefully reading most of the manuscript. Guido Rauch and Philip Williams, our superiors in the swap group at UBS, who overlooked the odd hour or two spent typing rather than trading. Piers Hartland-Swann at UBS, and Bryan de Caires, Graham Henderson and Charmaine Ferris at Euromoney Publications for their encouragement and help in pursuing this project. Adrian Fitch for a plentiful supply of nourishing lunches, and an unprintable suggestion for the title of this book. This book was typeset by the authors using the preparation system.

xi

document

To Lucy and Marie-Véronique

Chapter 1 Introduction Swap market participants enjoy presenting statistics on the growth of the world-wide interest rate and currency swap market. Perhaps this is understandable at a time when their colleagues in related markets see their jobs threatened by declining volumes and profitability. Nevertheless, it is true that the market has grown from nothing to a size of some three trillion dollars outstanding over the past 10 years. It now occupies an absolutely central position in the capital markets. Most bond issues are swap-related, corporates use swaps to manage their interest rate exposure and investors are now using swaps in addition to bonds. The market shows many signs of maturity. The methods used by the most sophisticated participants to analyse the risks they run have shown similarly rapid evolution. This has been far less widely remarked upon, for it makes for less startling headlines and, for competitive reasons, banks have tried to keep their own techniques secret. Such developments have been vital for several reasons. First, banks have far more flexibility in what they are able to price. This has greatly increased the variety of swap structures. For example, they may be used to buy or sell irregular cashflow schedules, effectively a reinvestment contract. Second, many banks are now far quicker in pricing complicated deals. What might have taken a day to price in 1985 can now be priced and executed in minutes. Banks’ clients increasingly expect this. Third, as the techniques have become more widely used, there has been an effective reduction in the bid-offer spread on complex deals. It is now far harder to make money by ex-

1

2

CHAPTER 1. INTRODUCTION

ecuting such deals and closing out the risk with a group of standard swaps. This puts pressure on profits – it also indicates the maturity of the swap market. Thus, market makers have had to adapt to running and monitoring complex risk positions. Fourth, the techniques have increased the influence of swaps on other markets. Asset swaps, where outstanding Eurobond issues are combined with a swap to suit investors’ requirements, have had a profound effect on the price performance of many Eurobond issues. Yet many asset swaps would not be possible but for the ability of banks to execute swaps to odd end dates with off-market coupons. Another example is the long term foreign exchange market. Major banks often run their long term foreign exchange books in conjunction with their swap books. The price of long term forward foreign exchange is bounded between limits where arbitrage against cross currency swaps is possible.

1.1

What the book is about

This book aims to cover in depth the techniques of pricing and explore their implications for risk management. This area is poorly represented in the literature. Other books concentrate on regulatory, legal and accounting aspects and an explanation of the ways that swaps may be used by banks and corporates. Sometimes, this is accompanied by the ubiquitous “plumbing diagrams”. In these, small boxes represent parties to a swap and arrows represent payments of interest. Magically, investors with an appetite for assets in fixed rate deutschemarks are combined with investors in floating rate minor currencies to produce a complicated diagram and a huge sub-LIBOR margin in dollars for a prestigious AAA rated issuer! While many books deal with general option mathematics and strategy [1], there is remarkably little published material specific to caps, floors and swap options. In this book, established option mathematics is combined with results from our analysis of swaps to yield risk measures for options comparable with those given for swaps.

1.2. FOR WHOM THE BOOK IS INTENDED

3

1.2 For whom the book is intended This book has been written with several groups of potential readers in mind, each being likely to have slightly different uses for it. The groups are:

• Swaps traders and sales people. Modern pricing techniques, and the computer systems that banks have installed to implement them, have had a profound effect on the jobs of many of these people. No longer is it economic to run a swap book as a collection of matched trades. A full understanding of swap pricing will allow traders and sales people to: – structure particular deals, for example, unusual Eurobond issues, in the most advantageous way. Indeed, in order to be creative in this area, it is essential to have a good understanding of swap pricing. This is especially so when the yield curve is steeply sloped. – explain to clients why certain deals, such as forward start swaps, have the prices they do. – understand better the output of their swap pricing systems. Many traders are closely involved in the development of inhouse systems, an area where communication between users and developers is often poor. The reader will be able to compare what his system produces with what modern techniques will allow and perhaps to specify improvements in his software. • Swaps brokers. Brokers often want to calculate the fair price of a complex deal without asking a market maker to quote a price for it. For example, the broker may have to indicate a price for a complex swap long before his client is ready to close the deal. Therefore, many of the broking firms have also developed or bought swap systems. Clearly, the broker will be more interested in the fair price for the

4

CHAPTER 1. INTRODUCTION deal that he thinks a bank should quote than an analysis of the associated risk. That risk will be for his clients, not him, to bear. However, an analysis of the risk also indicates whether he should enter the bid or offered side of the market at different points of the yield curve. The system could chose which side of the market to use automatically, but only at the cost of making it, more than ever, a “black box” which the user may not understand.

• Interest rate option traders and brokers. In chapter three we explore the extent to which the techniques of swap analysis can be combined with option pricing theory in order to analyse the risks run by traders in caps, floors and swap options.

• Corporate treasurers active in risk management. The authors have often been asked by corporate clients to explain the calculations leading to a particular price for a complex deal. Most often this happens in the case of fees to be paid or received in return for cancelling an existing deal. In fact, as swap pricing has increased in sophistication it has become harder to explain how we arrived at our own figures. A tersely worded mathematical definition is hardly likely to convince a treasurer that he is being quoted a reasonable price, particularly in a volatile market. It is possible to get a fair approximation to zero coupon pricing of a deal with simpler, if less flexible, methods. With care, the calculations can be done on a financial calculator, as is explained in chapter five. Certainly, all users, corporates included, should be aware of the value of the swaps they have on their books. Some larger corporates are now buying or developing their own systems. A few have done so and may be as sophisticated as many banks.

• Students of finance. Given the importance of swaps, it is surprising how little they are covered in the finance courses of even the best universities. However, in North American universities there is now a movement towards a more technical treatment of all their subject matter.

1.3.

BACKGROUND KNOWLEDGE REQUIRED

5

With this in mind, we believe that this book would provide a useful grounding for those undertaking finance courses at the MBA and postgraduate level who wish to have a rigorous understanding of swap techniques. There remain interesting and relevant problems to be solved in swaps, as well as the more frequently studied area of contingent claims.

• Risk software designers. The book has obvious appeal for both in-house and independent software designers, though there is far more to system design than mathematical correctness. Robustness, ease of use, level of support and many other features play a part. However, the designer will have a difficult time persuading a trader of the merits of a system whose risk analysis is clearly flawed. After all, it’s the trader’s profit and loss.

• Consultants and accountants. There has grown up a small industry of consultants to the swap market, most often within the established auditing firms. In the past there was a gulf between an accounting approach to swap valuation and that which swap traders felt should be used to assess their performance. This problem has been ameliorated by the adoption of present value accounting methods. It is hoped that this book will help further improve the dialogue between swap traders and their accountants.

1.3

Background knowledge required

It is inevitable in a book of this nature that certain sections presuppose some knowledge of mathematics and finance. Our experience in trading rooms has been that many people decide that swap pricing is too difficult the instant they see a mathematical formula. Don’t panic. It is easy to understand swap pricing knowing only some very simple mathematics, as explained in chapter five. Further, having analysed the risk of a swap it is far easier to understand what the results mean than how they were calculated.

6

CHAPTER 1. INTRODUCTION

The harder sections of the book require only a basic knowledge of calculus and algebra. In Britain, the level might be somewhere between “O” and “A” level. The only exception is in the area of interest rate options. Here a little probability theory will come in useful. A basic level of knowledge of finance is assumed – the reader should have some familiarity with bonds, yields and futures markets. No proficiency in interest rate swaps or options is expected.

1.4

An outline of the book

Our first subject is defining the swap itself. Their variety has grown considerably in recent years. In addition to the generic “plain vanilla” swap, there are cross-currency, basis, amortising, asset, and zero coupon swaps. Chapter four examines the instruments generally used to hedge swaps. Commonly these are bonds, futures, forward rate agreements (FRAs) and. occasionally, loans and deposits. The workings of their markets and pricing is explained, and we show that many of them are interrelated. In chapter five, a simple approach to pricing interest rate and currency swaps is explained in detail, and a number of examples worked through. This also provides an idea of the original techniques used for pricing. In chapter six, the construction of the zero coupon discount function from the various possible yield curves is described, and interpolation methods discussed. Chapter seven gets down to the business of valuing a swap. This is basically a matter of generating the correct cashflows for a given structure and examining the discount function at the corresponding dates. However, there exists a multitude of possibilities: margins, currency exchanges, amortising principals, compounded LIBORs and so on. A detailed procedure for incorporating these and other features is provided. Perhaps the most important issue in running a swap portfolio is measuring and controlling interest rate exposures. The next two chapters cover these in turn. Chapter eight demonstrates the principle of deriving exposures from the pricing algorithm. The central point is that the exposures can be deduced in an unambiguous and analytic

1.4. AN OUTLINE OF THE BOOK

7

way from the equations defining the discount function. This principle can be extended, if desired, to methods other than zero coupon pricing. The formulae for first order (delta) and second order (gamma) interest rate exposure are obtained, and re-expressed in terms of equivalent swap and futures positions. Chapter nine then explains the reasons for and methods of hedging the exposure, using the instruments outlined in chapter four. This is the process whereby a book runner can try to “lock in” the value of his portfolio. The standard trading strategies are also discussed. These can be broadly divided into three categories: spread, yield curve and futures strip trading. In addition, the swap-FRA and long dated foreign exchange arbitrages are examined. Chapter 10 applies the techniques developed for swaps to options. We concentrate on caps (calls on forward rates), floors (puts on forward rates) and options on swaps. There is already a considerable number of publications dealing with other interest rate options (bond options, OTC options etc.), and so this area is left aside. An almost identical procedure to that for swaps is followed to derive the interest rate and volatility risks involved in options, and provide methods of hedging them. Finally, chapter 11 presents some thoughts on managing a portfolio of swaps.

Chapter 2 Defining the swap This chapter describes the economic features of interest rate and currency swaps. First their most general features are covered, then the terms used to describe the detailed structure of a particular swap are given. Subsequently, the market standard structures and the variations to these, such as amortising swaps and forward start swaps are explained. These descriptions are by no means exhaustive, for a troublesome feature of the swap market is that there exist a considerable number of conventions for such matters as deciding the exact dates of payments or calculating the precise amount due on an interest payment date. For each currency there are market standard choices from among the available definitions, which are set out in “1987 Interest Rate and Currency Exchange Definitions” published by ISDA, the International Swap Dealers Association [2].

2.1

What is an interest rate swap?

An interest rate swap is an agreement between two parties. Each contracts to make payments to the other on particular dates in the future. One, known as the fixed rate payer, will make so-called fixed payments. These are predetermined at the outset of the swap. The other, known as the floating rate payer will make payments, the size of which will depend on the future course of interest rates. Typically, the floating rate payments will track rates for six month interbank deposits in London 9

10

CHAPTER 2. DEFINING THE SWAP

(LIBOR). Later in the book we will see how one can rationally put a value on the obligation to make these payments, despite their initially unknown size. The details of a particular interest rate swap must unambiguously determine the timing and size of all payments to be made under the terms of the deal. Interest rate swaps are often used to match the payments on some underlying investment or borrowing. Hence it is not surprising that the payments are determined by amounts of principal, by rates of interest and the period over which they are deemed to accrue.

2.2

Features of a standard interest rate swap

• The notional principal: Fixed and floating payments are calculated as if they were payments of interest on an amount of money borrowed or lent. This amount is referred to as the notional principal. Notice that for interest rate swaps the notional principal never changes hands. • The fixed rate: This is the rate applied to the notional principal to calculate the fixed amounts. From day to day. market participants quote the price at which they are prepared to execute a particular swap by quoting the fixed rate. • Dates of payment: Fixed rate payments are usually paid either annually or every six months. For example, they might be paid every first day of February and August from 1 February 1991 until 1 August 1996. This last payment date is known as the termination date or, more commonly, the maturity date. Two other relevant dates are the trade date, on which the parties agree to do the swap and the effective date, when the first fixed and floating payments start to accrue. Note that, in general, no payments take place on either the trade date or the effective date. Thankfully for those who work in them, banks are not open to accept payment on every day of the year. As with other details of swap structure, there exist a plethora of possible procedures for

2.2.

FEATURES OF A STANDARD INTEREST RATE SWAP

11

overcoming this problem. The most common of these is known as the “modified following business day” convention. Under this scheme, if a payment date would fall on a weekend or bank holiday and the next “good” day is in the same calendar month, then payment is made on this day. Otherwise, payment is made on the preceding day on which banks are open. There are a number of less common conventions: “Following” denotes the next good business day, whether or not it is in a different month; “Preceding” denotes the closest previous business day; “End of month” denotes the last good business day in each month; “IMM” denotes the days on which IMM futures contracts settle. The IMM (International Monetary Market) dates are the third Wednesday in the months March. June. September and December. • The fixed rate payments: Collectively, the fixed rate payments on a swap are known as the fixed leg. Each fixed payment is determined by the notional principal, by the fixed rate and by a quantity known as “the fixed rate day count fraction”. according to :

Fixed Amount

Notional Principal

Fixed Rate

Fixed Rate Day Count Fraction

Broadly, the “fixed rate day count fraction” will be equal to the fraction of a year since the previous payment (or since the effective date). Unfortunately, the swap market is unable to settle on one of several available standard definitions. We are going to need four of these and so give them next. Suppose a swap has notional principal P and fixed rate R. A fixed rate payment is due on D2 = (d2, m2, y2)1. The prior fixed rate payment was on D1 = (d2, m2, y2)1. In the United States “Actual/365(Fixed)”2 is known as bond basis. When the fixed 1 2

For example, if D2 were 17 May 1991 then D2 - (17,5,1990). “Fixed” here refers to the fact that 365 is used regardless of leap years.

12

CHAPTER 2. DEFINING THE SWAP rate is paid on this basis: Fixed Rate Day Count Fraction Hence: Fixed Amount In the United States “Actual/360” is known as money market basis3. In this case: Fixed Rate Day Count Fraction Hence: Fixed Amount Here, D2 – D1 means the number of days from, and including, D1 until, but excluding. D2. If the fixed payments are on a “30/360” basis then: Fixed Rate Day Count Fraction Hence: Fixed Amount Here. 360(D1, D2) is meant to denote the number of days from D1 until D2, calculated assuming that all months have 30 days. To make this precise:

3

Take care. In the United Kingdom, “Actual/365(Fixed)”.

money market basis means

2.2.

FEATURES OF A STANDARD INTEREST RATE SWAP

13

where “max” and “min” denote respectively the greater and the lesser of the two arguments in brackets. Lastly, if the payments are on an equal coupon basis then: Fixed Rate Day Count Fraction

if fixed payments are annual if fixed payments are semi-annual if fixed payments are quarterly

Hence: Fixed Amount

if fixed payments are annual if fixed payments are semi-annual if fixed payments are quarterly

Equal coupon swaps are most often traded in connection with Eurobonds, either swapping the bond issuer’s liabilities or. in the form of an asset swap, swapping the income stream owned by a bond investor. For such swaps the first fixed payment is equal in size to the subsequent fixed payments, even when the first fixed period is only a fraction of a year. • The floating rate payments: Known collectively as the floating leg, standard floating rate payments are said to be “set in advance, paid in arrears”. To explain, each floating rate payment has three dates associated with it: DS, the setting date; D1, when interest starts to accrue and D2, the payment date. The setting date DS is usually two business days prior to the previous floating rate payment date. On this day reference is made to one of several publicly available information services as being, for instance, the rate quoted in London for six month interbank deposits in dollars. D1, when interest starts to accrue, is the prior floating rate payment date (for the first floating payment, it is the effective date of the swap). The payment is calculated according to:

Floating Amount

Notional Principal

Floating Rate

Floating Rate Day Count Fraction

CHAPTER 2.

14

DEFINING THE SWAP

Fixed cashflows

Maturity date

Start date Floating cashflows

Figure 2.1: Diagrammatic representation of an interest rate swap

The floating rate day count fraction is equal to either:

or:

according to the currency of the swap.

• Payment netting: Often a fixed amount will be due on the same day as a floating amount. In this case only the net difference is paid.

Example cashflows: As an example of the calculation of cashflow amounts, we give the fixed cashflows for a swap with the following details:

2.3.

NON STANDARD INTEREST RATE SWAPS Principal Fixed rate Fixed rate daycount fraction Trade date Effective date Termination date Fixed rate payer payment dates

15

$25,000,000 9.84% Actual/360 1 February 1991 5 February 1991 5 February 1996 Each 5 February commencing 5 February 1992 up to and including 5 February 1996 or modified following London and New York business day.

The first cashflow is given by: 25,000,000

9.84 100

365 360

2,494,166.67

This and subsequent fixed cashflows are: Payment date 05-Feb-91 05-Feb-92 05-Feb-93 07-Feb-94 06-Feb-95 05-Feb-96

Fixed payment

No. of days

2,494,166.67 2,501,000.00 2,507,833.33 2,487,333.33 2,487,333.33

365 366 367 364 364

Notice how the payment dates are delayed by weekends. When this happens, the payment amount is adjusted to compensate. This is standard practice in the dollar market.

2.3

Non standard Interest rate swaps

The great majority of swaps executed in the market may be regarded as standard. In particular: • The floating and fixed payments are regular, for example every six months.

16

CHAPTER 2. DEFINING THE SWAP • The term of the swap (the time between the effective date and the termination date) is a whole number of years, most often one, two, three, four, five, seven or 10 years. • One party makes fixed rate payments, the other floating rate payments. • The notional principal remains constant throughout the life of the swap. • The floating rates are set as described above. • The fixed rate remains constant throughout the life of the swap.

As emphasized later in the book, when valuing swaps, we regard them as no more than a collection of cashflows. Therefore there is no reason why a swap needs to satisfy the descriptions above. Indeed some nonstandard structures are fairly common, these are considered next. • An example non-standard swap: Imagine a property developer who has recently completed a commercial office building. He has successfully let the office space at a rent fixed for five years. Suppose also that he has borrowed £25,000,000 at floating rates of interest to finance the development. He expects to use the rental income gradually to repay his borrowings over the five years. Clearly, rising interest rates will increase his borrowing costs, perhaps to a level where the development becomes uneconomic. A good strategy would be for him to pay the fixed rate on a swap whose principal starts at £25.000,000 and reduces at each payment date, reaching zero after five years. This “amortisation structure” would be designed to match his reducing bank borrowings. Thereby, the floating rate payments he receives offsets his interest costs – no matter what the future course of interest rates. In return he pays a predetermined amount. In this way he has effectively hedged4 his exposure to interest rate movements. 4

See the start of chapter four for an explanation of the concept of hedging.

2.3.

NON STANDARD INTEREST RATE SWAPS

17

We have outlined an instance when a corporate might use an amortising swap, a common non-standard swap structure. In chapter five reasons for using many other kinds of swap are suggested. For now. the important point is that the standard interest rate swap can be altered in almost any way imaginable. Examples of such alterations are: • Amortising, accreting and rollercoaster swaps: All of these terms are used to describe interest rate swaps in which the notional principal changes over the life of the swap. In an amortising swap the principal decreases over time, in an accreting swap it increases over time. As the name suggests, a rollercoaster swap has a notional principal which both increases and decreases. • Basis swaps: A swap with two floating legs is known as a basis swap. For example, a five year swap of six month dollar LIBOR against the 30 day dollar commercial paper index set by the Federal Reserve. • LIBOR margins: This is one of the commonest variations to the standard swap structure. Each floating rate is adjusted by a given amount before being used to calculate the floating rate payment. The margin is commonly quoted as a number of basis points, which are each equal to 0.01 percent. A positive margin, one added to the LIBOR rate, is referred to as a margin over LIBOR; a negative margin as a margin under LIBOR. The formula for calculating floating amounts now becomes: Floating Amount

Notional Principal

Floating Rate ±LIBOR margin

Floating Rate Day Count Fraction

• Forward start swaps: The effective date of a swap can be several months, even years, after the trade date. Later we will discuss how the fair price for such a swap is determined by swap market rates prevailing on the trade date. • Zero coupon swaps: In a zero coupon swap there is only one fixed payment. This takes place on the maturity date of the

18

CHAPTER 2. DEFINING THE SWAP swap. Usually the price for such a swap is given by explicitly stating the size of the single fixed payment. This avoids any possible confusion concerning compounding conventions. The name “zero coupon” is given by analogy with zero coupon bonds. These bonds repay principal, but pay no interest. Investors obtain their return by buying them at a discount. To pursue this analogy, the single fixed payment on a zero coupon swap equates to the difference between par and the issue price of a zero coupon bond. • Non-standard floating legs: A host of possibilities exist here. For instance, the floating rate used could be the average of the 6 month LIBOR rates prevailing on each of the four Mondays prior to the usual setting date. Alternatively, in back-set LIBOR swaps the rate used to determine each floating rate payment is that prevailing at the end rather than the beginning of the relevant interest period.

2.4

Currency swaps

Currency swaps are swaps for which the two legs of the swap are each denominated in a different currency. For example, one party might pay fixed rate sterling, the other floating rate ECU. Usually, one leg of a currency swap is floating rate dollars. Such swaps can be used as “building blocks” to create swaps between any desired combination of currencies. So. for example, a bank could combine a swap of fixed rate ECU against floating rate dollars with a second swap between floating rate dollars and fixed rate yen. If the floating rate dollar legs are arranged to net out, then the result is a swap of fixed rate ECU against fixed rate yen5. Each leg of a currency swap requires a notional principal in the appropriate currency. There is a further difference between interest rate swaps and currency swaps which concerns us: the notional principals are exchanged, always on the maturity date (the 5

This is true as far as the swap cashflows are concerned. Under current regulations, currency swaps require considerably greater capital backing than interest rate swaps. “Building” currency swaps in this way is therefore likely to be expensive in terms of capital required.

2.4.

CURRENCY SWAPS

Dollar principal

Sterling principal

19

Fixed sterling cashflows

Sterling principal + final coupon

Floating dollar cashflows

Dollar principal + final coupon

Figure 2.2: Diagrammatic representation of a currency swap final exchange) and often on the effective date (the initial exchange). Consider for a moment a swap of fixed sterling against floating dollars. Under the final exchange, the dollar payer pays the dollar notional principal. In return, the sterling payer pays the sterling notional principals. Conversely, if there is an initial exchange, the dollar payer pays the sterling notional principal, the sterling payer the dollar notional principal6.

2.4.1

Why have principal exchanges?

A US corporate who has issued a five year bond in sterling can use a currency swap to transform its liabilities into dollars. Without a final exchange of principal, the associated swap would not achieve this transformation. To see this, note that while the coupon payments it has to make to bond holders match fixed sterling payments on the swap, the final payment of principal to the bond holders is not matched. Thus if sterling appreciates against the dollar, it will cost the corporate a greater dollar amount to redeem its bond. Swaps are exchanges of interest and principal on underlying debt; without the final exchange 6

The amounts exchanged do not always equal the actual relevant notional principals. Swap documentation uses the terms initial exchange amount and final ex-

change amount for clarity.

20

CHAPTER 2. DEFINING THE SWAP

of principal they are only performing half their task7. Imagine a quiet day in the swap market. Swap rates are not moving. Sterling is trading against the dollar at £1 = $1.70. A bank pays fixed sterling on a £10.000,000 five year swap. Against this it receives floating dollars on $17,000,000. The swap has no initial exchange. The sterling leg of the swap effectively constitutes a liability of the bank. It has a negative value of £10,000,000 (under our assumption that swap rates have not moved since the execution of the deal). Conversely, the dollar leg of the swap constitutes an asset with value $17,000,000. The value of the bank’s position is then:

– £10,000,000 + $17,000,000 £17,000,000 = – £10,000,000 1.70

£0

Suppose now that sterling strengthens against the dollar, moving to $1.705. The bank’s position is now worth:

– £10,000,000 +

£17,000,000 = – £29,325.51 1.705

The bank has made a (mark to market) loss despite the fact that swap rates have not moved. To sum up, we have shown that a currency swap with no initial exchange exposes the counterparties to currency risk on the underlying principal amounts. When currency swaps do have an initial exchange it serves to hedge this risk. Nevertheless, many currency swaps do not have an initial exchange. This is often for the simple reason that one of the counterparties does not have the funds available to exchange. Our US corporate, swapping a bond which was issued some time earlier, is likely to have already found a use for the bond proceeds. The lack of an initial exchange is not a problem for a bank which does not wish to bear the associated currency risk; it can substitute a spot foreign exchange transaction. Why then do interest rate swaps not have exchanges of principal? The answer is straightforward: being in the same currency the payments would net out. 7

Currency swaps with no final exchange do exist, however their price will generally be very different from the price for a standard deal.

2.4. CURRENCY SWAPS

2.4.2

21

Types of currency swaps

With each leg in a different currency one can create the whole gamut of swap structures that we have already seen for interest rate swaps: amortising swaps, forward start swaps and so forth. Two particular structures deserve comment: • Fixed-fixed currency swaps: When both legs of a currency swap are at fixed rates it is often known as a fixed-fixed swap. Their most interesting feature is the relationship with the forward foreign exchange market. Suppose a bank receives fixed sterling against paying fixed dollars. Since all the payments are predetermined, the bank could sell forward its sterling receipts for dollars. This will result in a series of net dollar payments. If the present value8 of these cashflows is positive the bank will be making a nearly9 risk-free or arbitrage profit. In an efficient market such risk-free profit will not be available. One would expect the swap and forward exchange markets to move in tandem to prevent arbitrage. This is generally the case. • Cross-currency basis swaps: Both legs of such swaps are at floating rates. They are useful tools in the creation of other forms of currency swaps. Also, without an initial exchange, they are used to speculate on foreign exchange rates.

8

We are getting a little ahead of ourselves here. The concept of the present value of swap cashflows will be an important topic in future chapters. 9 The bank must still manage a small reinvestment risk.

Chapter

3

Background to the swap market 3.1

The development of the market

The origins of the swap market lie back in the 1970’s - a time when many countries imposed restrictions on the cross-border flow of capital. In the United Kingdom, for example, punitive taxes were levied on currency transactions, in order to encourage investment at home by UK funds. To circumvent such regulations, the idea of the parallel, or back-to-back, loan evolved. As an example of a parallel loan, consider a company in the United States which has a subsidiary in the United Kingdom. The parent company has good access to funding in the dollar market, and the subsidiary requires funding in sterling. Simultaneously, a UK company with a US subsidiary has access to sterling funds, while requiring dollar funding for its US entity. A parallel loan would involve the US parent company lending dollars to the US subsidiary of the UK parent, and a parallel loan in sterling by the UK parent to the UK subsidiary of the US parent. Such a structure sidesteps currency restrictions. A back-toback loan has the same structure as a parallel loan, but has provision for certain default events. As currency regulations gradually disappeared, the parallel loan was superseded by the currency swap – a functionally equivalent transac23

24

CHAPTER 3. BACKGROUND TO THE SWAP MARKET

tion, with the added benefit of flexibility and (relative) liquidity. However, the main reason for the sudden explosion of swap activity in the early 1980’s was the opportunity for capital markets arbitrage. This activity relied on the advantages that various types of counterparty could achieve in different markets – namely, the floating and fixed rate debt markets. The classic example of this type of arbitrage involves two borrowers, one of a sufficiently high creditworthiness to issue fixed rate bonds, and a lesser credit with access limited to the floating rate market. The key concept is one of relative advantage: namely that, even though the better credit can borrow cheaper in either market, the lesser credit holds a relative advantage (i.e. a smaller disadvantage) in the floating market, and the better credit, a relative advantage in the fixed rate market. In this situation, the most economic course of action for each borrower is to raise funds in the market in which it holds its relative advantage. Thus, the better credit will often issue fixed rate debt and, via an interest rate swap, convert this into a floating rate liability. He can, of course, also swap into floating rate in a different currency via a currency swap. The lesser credit wishing to fix its borrowing costs will need to do a mirror image swap, and so the chain becomes complete. One other important development in the swap market has been the increase in the volatility of interest rates and currencies. This has provided many corporate end-users with a reason for careful management of their liabilities. As the market has increased in sophistication, they have been able to hedge themselves exactly against interest rate risk on any cashflow structure, as well as manage their debt according to their interest rate and currency views. As the more basic arbitrage opportunities disappeared in the mid1980’s, banks looked for new incentives for borrowers to utilise the derivatives market. Tremendous growth was seen in what previously would have been regarded as highly esoteric instruments, such as zerocoupon, dual-currency, warrant and index-linked bonds. In addition, swap markets started in many of the smaller currencies, the most notable of recent years being ECU (the European Currency Unit). By the late 1980’s, the swap market had reached such a developed stage, that in some currencies swaps were regarded almost as commodities. The secondary swap market became of major importance, with

3.2.

MARKET SIZE

25

banks keen to unwind (cancel) existing swaps, or assign them (pass their obligations under the swap on to a third party), in order to free up important credit lines. In fact, credit risk has played a pivotal role in determining the major swap players of the 1990’s. Many large corporates, supranationals and sovereign entities have become increasingly selective about the creditworthiness of their swap counterparties, preferring those with either a triple-A or double-A credit rating1. With the downgrading of many Japanese and American banks in recent years, this has left only a select group of prime banks – whether the bulk of the swap business moves to these banks remains to be seen.

3.2

Market size

To give an idea of the size of the markets. Table 3.1 lists the ten largest interest rate and currency swap markets. These are figures collated by ISDA [3] for the year 1989 and represent a significant sample of major swap houses – however, the real size of the market is probably much larger. The table raises a number of noteworthy features: • The market in dollar interest rate swaps is almost nine times as large as the next biggest currency (yen). It appears that, since the ISDA survey, the gap has closed a little, with yen, sterling and deutschemarks all becoming more widely used in periods of considerable interest rate volatility. Of course, the dollar swap is unlikely ever to cede its domination of the market. One needs only to consider the Eurobond market where dollar denominated issues far outstrip those of any other currency. • A significant number of currencies have higher volumes for currency swaps than for interest rate swaps – namely, yen, Swiss francs, Australian dollars. Canadian dollars and ECU. The majority of swaps in these currencies originate as capital markets transactions, where dollar LIBOR is the standard benchmark. 1

A triple-A rated institution is one of the highest creditworthiness, a double-A of only slightly lower quality.

26

CHAPTER 3.

BACKGROUND TO THE SWAP MARKET

Interest rate swap volumes Number of Notional principal Currency local currency contracts (millions) 993,746 36,627 Dollars Yen 17,420,529 5,259 60,611 6,361 Sterling 157,529 6,608 Deutschemarks 84,851 9,029 Australian dollars 265,460 4,131 French francs 1,936 Canadian dollars 34,563 Swiss francs 1,987 46,218 1,054 ECU 17,075 446 Dutch guilders 12.556 Currency swap volumes Number of Notional principal Currency contracts local currency (millions) Dollars 12,492 354,166 6,364 Yen 28,041,825 Swiss francs 2,186 106,567 Australian dollars 3,733 82,215 Deutschemarks 1,629 103,106 ECU 1,222 36,923 Sterling 952 20,047 914 Canadian dollars 41,361 Dutch guilders 21,897 315 French francs 281 55,447 Table 3.1: Swap volumes for 1989 Source: ISDA

Notional principal dollar equivalent (millions) 993,746 128,022 100,417 84,620 67,599 42,016 29,169 28,605 18,998 5,979 Notional principal dollar equivalent (millions) 354,166 201,145 64,823 61,768 53,839 39,948 33,466 32,580 10,132 8,435

3.3. DEVELOPMENTS IN BOOK RUNNING

27

• The relative predominance of some currencies is often subject to the whims of fashion. For example, many Australian corporates have found themselves downgraded by the rating agencies, and now find it hard to trade swaps with non-Australian banks. This has significantly reduced the world-wide activity in Australian dollar swaps since 1988. On the other hand, deutschemark swap volume was boosted in 1990 by the economic uncertainty created by the political unification of East and West Germany, and ECU swap volume has increased dramatically following the increase in ECU denominated bonds from both corporates and, more significantly, governments such as Italy.

3.3

Developments in book running

As we have explained, the earliest examples of swap transactions were matched deals: a bank would act as an arranger, finding two counterparties with equal and opposite requirements, who would then either deal with each other directly, paying a fee to the arranger, or who would each deal with the arranger, who takes a margin. Such deals were often related to new bond issues, so that the details could be established several days in advance. With the increase in swap activity related to asset and liability management, the need arose for banks to be prepared to make a market in the more liquid currencies. Whereas arrangers made their money by taking a margin out of matched deals, market makers profit from making a bid-offer spread (in other words, they aim to receive fixed at their offered side, and pay fixed at their bid side). The ability to do this relied greatly on developments in software, which allowed traders to analyse and hedge complex positions, and accurately assess their risk. There are several important advantages that market making brings. As far as the bank is concerned, one of the most relevant is the greater profit potential. Instead of having to pay a premium to another bank to have dates, principal(s) etc. matched, it can afford to take some mismatch and offset the position with the cheapest structure available (normally a par swap). This type of activity greatly increases the liq-

28

CHAPTER 3. BACKGROUND TO THE SWAP MARKET

uidity of the market, and of any instruments used for hedging. For example, a significant percentage of the daily turnover in some government bond markets is due to swap business.

3.4

The uses of swaps: a few examples

Swaps were invented for the purpose of interest rate risk management. They were not intended to serve as a vehicle for speculating on rates, since there exist better ways of achieving such aims (most notably, the futures market). However, given the complexity of many companies’ balance sheets, the flexibility of the swap can be used to considerable advantage. In this section, we present a few easy examples (based on realistic deals) of structuring swaps to suit the clients’ needs. • Example 1: A company borrows funds for five years on a floating rate basis. Imagine it pays interest on its borrowings of LIBOR, paid every six months (we have ignored the fact that the company would probably have to pay a margin over LIBOR for its funds – this makes little difference to the general idea). In order to protect itself against interest rate movements, it executes an interest rate swap whereby it pays a fixed rate to a bank, who pays the company six month LIBOR, matching its interest payments. If the company was due to draw its funds in a month’s time, it might execute a five year swap starting in one month. • Example 2: Imagine that a corporate wishes to issue a five year Eurobond. It has already issued a great deal of dollar denominated paper, and would therefore benefit by using a different market. However, its main income stream is in dollars in which it wishes to pay interest. If it were to issue a sterling bond and enter into a currency swap whereby it pays dollar LIBOR and receives the bond coupon in sterling, it would achieve its requirements. This strategy would enable the corporate to borrow at a comparatively lower yield in sterling. To be more specific, imagine a bank would pay a spread of 70 over the five year gilt2 in sterling, 2

In other words, a rate of 70 basis points over the yield of the current benchmark five year gilt; see the next chapter for an explanation of yields.

3.4.

THE USES OF SWAPS: A FEW EXAMPLES

29

against receiving dollar LIBOR flat. If the corporate can issue the bond at a yield spread of 50 basis points over the same gilt, then it can achieve its dollar funding at approximately dollar LIBOR less 20 basis points. If it were to bring a straight dollar bond, it would probably need to pay a premium due to the amount of its paper still available, and would thus be unlikely to achieve such fine terms. • Example 3: A building society plans a new sterling mortgage product. It will offer a fixed rate of 14% for the first two years, and 13% for the next two years, with interest paid quarterly. It normally receives floating interest payments monthly on standard mortgages. The society could pay the quarterly interest on its fixed rate mortgages to a bank and swap them for a spread to one month LIBOR. One problem with this approach is that if interest rates fall, then the building society’s clients may seek to refinance their mortgages at lower rates. Of course, the interest rate swap would still be in place. In the United Kingdom, most fixed rate mortgages have heavy prepayment penalties to try to prevent this problem occurring. • Example 4: A construction company borrows funds for five years, paying three month LIBOR. It anticipates receiving income on the property annually from year six. It enters into a swap whereby a bank pays it three month LIBOR for five years, and the company pays an annuity (an equal annual amount) from years six to ten inclusive. Since this structure involves one way payments from the bank to the company for the first five years, then the bank must bear a particularly high credit risk under this swap. This should be reflected in the swap’s pricing.

Chapter 4

Hedging Instruments There is one central reason why swaps have become an indispensable tool within the capital markets: they allow the transfer of risk. A corporate wishing to protect itself against movements in the cost of borrowing may enter into an interest rate swap with a bank. The bank, in turn, may not wish to carry this risk itself. It could, of course, pass the risk along the chain and pay fixed at a lower rate to another institution on a deal having the same structure. However, this is often not possible, and so another approach is required. The solution is hedging l. Hedging is the process of offsetting risk using instruments which are. in some way, correlated. Imagine an investor who owns $100 worth of a five year Eurobond. He is worried that five year interest rates are on the way up. and the price of his bonds will fall (see the following section for an explanation of this point). How can he hedge his position? The trivial solution is simply to sell his bond position – let us assume that the investor wishes to hold on to the bonds, perhaps for tax reasons. We shall examine three alternative possibilities: 1. Pork Bellies: It is probably safe to assume that the price of pork bellies has very little to do with the price of five year Eurobonds. The price of both may occasionally rise at the same time, but the percentage increase would be unlikely to be the same. Thus, one would conclude that pork bellies and Eurobonds are effec1

A general discussion of hedging swap positions is given in chapter 11.

31

32

CHAPTER 4. HEDGING INSTRUMENTS tively uncorrelated. There is no general prescription telling us how many pork bellies to buy or sell at a given moment in order to offset the change in value of the bonds. 2. Ten year Treasuries: Presumably, five and ten year interest rates are related in some manner. This, indeed, is generally true. They usually move in the same direction. However, they do not usually move by the same amount. This is due to a number of technical points explained in the next section. Some traders attempt to take advantage of this relative movement by buying, say, the ten year bond and selling the five year bond, and hoping that the spread between the two widens (i.e. the price of the ten year increases by more than the price of the five year bond). Let us make the (unrealistic) assumption that for every “tick” (l/32nd of one percent) movement in the five year price, the ten year moves by two ticks – then the two instruments are (up to a constant) correlated. One can now easily hedge the $100 position in five year bonds by selling $50 of ten year bonds. The subsequent portfolio of bonds is then, under our assumption, fully insulated against movements in price. The ratio of bonds required, one half, is called the hedge ratio. 3. Another five year dollar Eurobond: Assume this bond has been issued by a triple-A borrower and has a yield which is higher than that of our investor’s Eurobond. It is likely that movements in the Eurobond prices will be correlated to some degree. Once again, the real world displays variations in the difference (usually quoted as a “spread” between the yields), but there is some correlation. The next section demonstrates how to work out the hedge ratio for bonds in general.

This chapter examines the instruments commonly used to hedge swaps. Chapter nine extends these ideas to the details involved in hedging arbitrary interest rate exposures. For the moment, we give a simple introduction to the useful instruments and their pricing.

4.1.

4.1

GOVERNMENT BONDS

33

Government Bonds

Governments usually spend more money than they receive. In order to raise the additional funds required to run the country, the more credit worthy of them generally rely on issuing bonds. The single fact that differentiates these bonds from those issued by a corporate entity in the Eurobond market is that they are assumed to be default free2. In other words, the return an investor earns from such a bond represents the minimum the market expects. Thus, corporate bonds will always provide a higher return to compensate the investor for the additional risk he is bearing. By far the largest of the government borrowers is that of the United States. It issues bonds ranging in maturity from two to thirty years, and shorter maturity bonds, known as Treasury Bills. The US Treasury is constantly reissuing bonds in all the “standard” maturities: two, three, four, five, seven. 10 and 30 years. Thus, there is always guaranteed to be a highly liquid market in these “on-the-run” bonds. One must understand how the markets value and price bonds before understanding how to use them as hedging tools. The fundamental concept is that of yield. Our discussion will be kept to a minimum; the reader interested in pursuing the subject further is recommended to read one of the textbooks covering the bond market [4].

4.1.1

Yield to Maturity

The idea of discounted cashflows will be required in valuing a bond. The calculation of the discount factors relevant to the swap market will be discussed in considerably greater detail in future chapters. For the time being, the simple approach used in bond mathematics is followed. It should be clear that $1 received today is worth more than $1 received in one year3. This is because $1 today can immediately be invested at prevailing rates and thus earn interest. Suppose that $1 can be invested for one year at an annual interest rate of R% (expressed as a decimal). Then: 2

The government bonds considered for hedging purposes are issued by the major industrialised nations. 3 In an economy with positive, non-zero interest rates.

34

CHAPTER 4. HEDGING INSTRUMENTS Value of $1 in one year = $(1 + R)

Turning the argument around, $1 received in a year must be worth $1/(1 + R) today, since this can be invested today at a rate R% to give $1 in one year. Assuming that the rate R% is still available in future years, one can invest $1 today to give $(1 + R) in one year, which, in turn, yields $(1 + R) + $R(1 + R) = $(1 + R)2 in two years. Thus: $1 invested annually at R% for n years

$(1 + R)n in n years.

Or putting it another way: $1 in n years

$1/(1 + R)n today.

The value 1/(1 + R)n is called the “discount factor” for cashflows occurring in n years time. Extending this idea to periodic interest rates is easy. Investing $1 for half a year at R%, gives an amount $(1 + R/2) in six months. Reinvesting this for a further six months at R% gives $(1 + R/2) 2 at maturity. Note that this is not the same amount as one would receive investing $1 for one year at R% – this demonstrates the difference between semi-annual and annual compounding. One can derive a rate 5% at which $1 invested semi-annually for one year is equivalent to $1 invested for one year at the annual rate R%. This requires that:

(4.1)

S is known as the semi-annual equivalent of R, R as the annual equivalent of S. Thus, if R were 10%:

is the equivalent semi-annual rate. This extends to an equivalent quarterly rate Q:

(4.2)

4.1.

GOVERNMENT BONDS

35

+110 +10 o

+10

+ 10

+ 10 1

2

3

4

5

t

-90

Figure 4.1: Cashflow schedule of a generic bond.

and so on. Armed with these simple concepts, one can examine the structure of bond cashflows. Our attention will be confined to “straight” bonds i.e. those with no options attached, which redeem at par (100% of the face value), and have a constant coupon with even periods. Taking a concrete example, imagine purchasing a bond for 90, whose first coupon of 10 is received exactly one year later (so the bond was purchased on a coupon date). Subsequent coupons are paid annually up to, and including, five years after purchase, at which point the bond repays a principal of 100. One can represent the cashflow schedule as shown in Figure 4.1. In these types of figures, upward pointing arrows represent cashflows received, downward pointing lines those paid. The time is marked as t = n where n is the number of years from purchase. The size of each cashflow appears next to each arrow. The yield to maturity (YTM) is defined as that rate at which the

36

CHAPTER 4. HEDGING INSTRUMENTS

discounted cashflows value to zero. In other words:

or, rearranging:

where y is the YTM expressed as a percentage. A value of y = 12.831% satisfies, very nearly, this equation (try it). The YTM gives a generalised measure of the return on a coupon bearing security. It should be noted that a bond yielding y% will only realise an actual return of y% if all the coupons can be reinvested at this rate – a dangerous assumption. This is an example of “reinvestment risk” – a concept that will be revisited. Imagine now a bond which is bought between coupon periods. The cashflows can be represented as in Figure 4.1. Here, there are 30 Eurobond basis days4 between t0 and t1, and 360 between the remaining coupons. The coupon, c, is 12%. and the clean price 100. One must pay for the interest earned in the coupon period currently running – the accrued interest - since the buyer receives a full coupon at t1 and the seller needs to be reimbursed for not receiving this. The full price paid, the “dirty” price Pd, is thus the sum of the “clean” price Pc and the accrued interest:

So. the equation which the YTM. y, must satisfy is:

(4.3) The actual answer is found to be 11.97%. The remainder of this section is rather harder than the rest of this chapter. Its purpose is to establish the way in which the price of a bond changes as its yield changes. This is described in Equation 4.15. 4

Eurobonds accrue interest on a 30/360 day basis. See page 12.

4.1.

GOVERNMENT BONDS

37

Figure 4.2: Cashflow schedule of a bond bought between coupon dates. The non-mathematical reader could skip directly to the next section on page 41, and perhaps also skip the material on bond convexity on pages 43 and 44. Analytically solving Equation 4.3 for y is impossible. Instead, the Newton-Raphson method is generally used to iterate to the solution. Suppose that we wish to solve f (x) = a. If x0 is a first approximation to the solution, a better one is given by:

(4.4) In our case, we wish to solve f (y) = Pd, with f (y) being the right hand side of Equation 4.3. Taking as a first approximation the coupon c, a better approximation is:

(4.5)

38

CHAPTER 4.

HEDGING INSTRUMENTS

Figure 4.3: Graphical representation of the Newton-Raphson equation.

4.1.

GOVERNMENT BONDS

39

Setting:

(4.6) so that:

(4.7) Then, the better approximation. Equation 4.5. becomes:

Where Vc is the result of putting y = c in Equation 4.6 i.e. Vc = Calculating y1 for the case c = 12%, Pd = 111 (P c = 100, A = 11), gives y1 = 11.96996%, quite close to the first approximation and the actual answer. Further details of this method can be found in any standard textbook [5]. The general case is not difficult to guess from the above. First, we establish some notation: Pc

is the clean price (i.e. excluding accrued interest) per 100 face value;

Pd

is the dirty price (i.e. including accrued interest) per 100 face value:

A

is the interest accrued at time of purchase:

H

is the number of coupon payments per annum;

yH

is the yield compounded H times per annum, expressed as a percentage:

N

is the number of coupon payments yet to be made:

f

is the time, expressed as a fraction of the coupon period, from settlement to the first coupon date:

40


c

is the coupon rate, expressed as a percentage of face value per annum (so the actual coupon payment is c/H);

R

is the redemption payment, almost always 100.

Define:

(4.8) then:

(4.9) The coupons are assumed to be spaced by an equal number of days. Although this is sometimes not exactly right, the differences these effects introduce are small enough to be ignored. One can utilise the fact that there is a geometric series in Equation 4.9 by defining: (4.10) (4.11) Thus: (4.12) which is differentiated to obtain:

(4.13) but also: (4.14) and hence:

(4.15) which can be substituted into Equation 4.4 to give the next approximation.

4.1.

4.1.2

GOVERNMENT BONDS

41

Market details

This section gives details of the US and UK government bond markets, the two markets most often used for hedging swaps, and the Eurobond market, used in the creation of asset swaps. • US market: Only the Treasury bond sector will be considered. These instruments pay equal semi-annual coupons with interest accrued on an Actual/Actual basis5. Settlement is commonly the business day following the trade. Bonds are often issued with long first coupons, and care should be taken in yield calculations in these cases. Any market participant can sell Treasuries short. • UK market: UK government bonds are known as “gilts”. Most are either redeemed at a specific date, or at any time between two specified dates. Gilts generally pay equal semi-annual coupons with interest accrued on an Actual/365 basis (so that the accrued interest due can be larger than the next coupon). They usually go “ex-dividend” 37 days prior to the notional coupon date, or. if this is a non-business day, then on the next good day6. As its name implies, an ex-dividend gilt is sold without the next coupon. Settlement is the next business day after the trade. Short selling of gilts is only allowed by Bank of England approved gilt market makers. • Eurobond market: There exists a huge variety of bond structures in the Euromarket. However, the majority of Eurobonds pay equal annual coupons with interest accrued on a 30/360 day basis, with the value date being seven days after the trade date. 5

In other words the denominator, rather than being 360, is twice the number of days from the previous coupon date until the next coupon date (even if either of these two dates is not a good business day). 6 There is an exception if the coupon date is the 5th, 6th, 7th or 8th of January, April. July or October. In this case, the ex-dividend date is the first of the previous month, or. if this is a non-business day. the next good day.

42


4.1.3

Modified duration and convexity

Imagine a bond with yield yH and a security S whose price P(y H ) is a function of the yield. In order to ascertain the hedge ratio between these instruments, one needs to know how the value of each varies as yH varies. This derivative is embodied, in the case of a bond, in a function called the “modified duration” (MD). Specifically: (4.16) for a bond with yield yH compounded H times per annum and dirty price Pd. From Equation 4.15:

(4.17) using the notation of the last section. Returning to the example of Equation 4.3 of a bond whose yield was 11.97% and using this value in Equation 4.7 gives a value for dPd/dy = –1.75813. This is then the rate of change of the dirty price for a one percent movement in the yield7. The modified duration is 1.5839. It should now be clear how to derive the hedge ratio between two securities S1 and S2 with modified durations MD1 and MD2 and dirty prices P1 and P2. One needs to solve: (4.18) for changes in price

7

and hedge ratio

The reader can easily check this with a simple calculation. In Equation 4.3, calculate the price Pd for a yield y = 11.97 and again for y = 11.98. The difference should be one one-hundredth of dPd/dy.

4.1.

GOVERNMENT BONDS

43 (4.19) (4.20)

is the ratio of the change in yield for the securities. To where assume that the yields of the two securities move in parallel is equivalent Under this assumption, for another to setting the ratio bond with dirty price 101 and modified duration 1.4, the hedge ratio would be:

In other words, every $111 (face value) of bond 1 requires $80.43 (face value) of bond 2 as a hedge. Although the modified duration is a good indicator of price variation for very small changes in yield, real markets tend to exhibit somewhat larger movements. In order to allow us to cope with such movements, it is helpful to look at higher order terms in the derivatives with respect to y – the next such term is known as the “convexity”8. The convexity of a bond is defined as: (4.21) The second derivative can easily be expressed as:

(4.22) which involves a summation of N terms. Sometimes9 it is more convenient to have a formula which does not involve such a summation. 8

What is really being examining here are the first few terms in the Taylor expansion of P(y) about y = y0, the present yield:

The second term is effectively the modified duration, and the third the convexity. 9 Fbr example, in order to write a spreadsheet to calculate bond convexity.

44


This is a rather large equation as follows:

(4.23) Going back to the earlier example of Equation 4.3, gives = 4.26474, with the previous values of Pd = 111, c = 12% and y = 11.97%. So, for to 11.98%, the dirty price changes a basis point change in yield. by approximately (see footnote 8 on page 43 for the origin of the factor 1/2). This effect is formally incorporated into our hedge ratio calculations by simply amending Equation 4.19 to:

(4.24) with

4.2

as before.

Futures Contracts

Imagine a farmer who knows for certain that, in three months time, he will have a ton of wheat grain to sell. He is concerned about the possibility of grain prices falling over the intervening period and wishes to hedge himself against price movements. One way for him to do this is to enter into a forward contract with, for example, a baker. This commits him to selling his ton of grain for a price, agreed at the time of striking the deal, at a given date in the future – in this case, three months later. Of course, his ability to enter into this contract depends, ultimately, upon the availability of a counterparty happy to take the risk that prices will fall.

4.2. FUTURES CONTRACTS

45

Forward markets exist, in one form or another, in almost all currency, commodity, bond and money markets. However, they are essentially “over-the-counter” markets. In other words, each transaction has to be tailored to suit the client’s needs and, inevitably, there is a cost associated with this. Some of the forward markets – such as the interest rate and government bond markets – are so frequently used for hedging and speculating that specific instruments have been introduced to facilitate liquidity and ease of transaction. These are called futures contracts. Futures are standardised exchange traded instruments. The main exchanges are the Chicago Mercantile Exchange (CME), the Chicago Board of Trade (CBOT), the London International Financial Futures Exchange (LIFFE) and the Marche a Terme International de France (MATIF) in Paris. All trading is by open outcry. The contracts are standardised in several ways: • Expiry dates are specified. There are usually several expiry dates –generally three months apart for financial futures – for any given instrument. For example. Eurodollar futures (futures contracts on three month dollar interest rates) on the CME expire on the third Wednesday of the month in March. June, September and December (the IMM dates), with a maximum of sixteen dates available at any one time. • Contract size is fixed. The short sterling contract on LIFFE has a standard size of £500,000, the Eurodollar contract a size of $1,000,000. • The “tick size”, the minimum change in price, is specified10. The Eurodollar contract has a tick size of $25. • Settlement is on a next business day basis. One of the features that make futures so attractive is the degree of “gearing” available. To translate, this means that to buy. say, a futures contract on £500,000 three months sterling interest rates does not mean having to put up £500,000 to deal. All that is required is 10

With the exception of a few contracts such as Australian bond futures.

46


an “initial margin” to be maintained. This might involve, for the short sterling contract, depositing £500 per contract in an interest bearing account with the futures brokers. As the value of the position rises or falls, margin is available to be withdrawn or required to be added to maintain the £500 balance. These balancing amounts are called “variation margin”. From the above definitions, it should be clear that futures are just a specialised forward contract. As such, they will converge to the spot price of the underlying instrument at the expiry of the contract. Thus, a December future on three month sterling interest rates will, on the third Wednesday of December when it expires, equal the then current three month LIB OR. In fact, interest rate futures, and a few others, are quoted on a “discount basis”. That is, a price of 90.00 for the sterling contract implies an interest rate of 100 – 90.00 = 10%. The contracts in which we shall have most interest are those on the long gilt and three month dollar and sterling interest rates. The specifications of these contracts are:

Three month sterling interest rate future Contract size Expiry months

£500,000 March. June, September and December.

Delivery day

First business day after last trading day.

Last trading day Quotation Tick size/value Delivery

Third Wednesday of expiry month. 100.00 minus rate of interest. 0.01/£12.50 Cash settlement based on 100.00 minus three month sterling LIB OR on last trading day.

Three month Eurodollar interest rate future Contract size Expiry months Delivery day Last trading day Quotation Tick size/value Delivery

$500,000 March. June, September and December. First business day after last trading day. Third Wednesday of expiry month. 100.00 minus rate of interest. 0.01/S12.50 Cash settlement based on 100.00 minus three month dollar LIB OR on last trading day.

4.2. FUTURES CONTRACTS

47

Long gilt future £50,000 of a notional 9% gilt. March. June. September and December. Any business day in delivery month at seller’s choice. Last trading day Two business days prior to last business day in delivery month. Per £100.00 nominal. Quotation £15.625 Tick size/value Delivery can be of any gilt with redemption dates Delivery between those specified by LIFFE (generally 12-20 years) at the LIFFE market price at 11:00 London time on the second business day prior to delivery calculated by the price factor system (see below). Delivery of a gilt against a short position in gilt futures is a complicated affair. There are a variety of gilts available for delivery. For each of the possibilities, the exchange publishes a “price factor”; this is a conversion factor which determines how much of a given gilt is equivalent to the notional 9% contract gilt. The delivery of a gilt will therefore involve purchase at current prices of the stock in an amount determined by the price factor method. It should be clear that this procedure usually implies a unique “cheapest to deliver” stock which the market will choose. It is outside the scope of this book to develop this matter any further – we shall anyway not need to worry about delivery procedures in future chapters. Consider the following set (called a “strip”) of short sterling futures prices, the settlement prices on 30 April 1990: Contract size Expiry months Delivery day

Expiry June 90 September 90 December 90 March 91 June 91 September 91 December 91 March 92

Price 84.61 84.69 84.89 85.32 85.80 86.25 86.55 86.72

48


What do these tell us about market expectations of sterling interest rates? The June 1990 contract implies that the market believes that three month rates on the third Wednesday in June 1990 (the 20th) will be 100 – 84.60 = 15.4%. Similarly, the September 1990 contract says that three month rates on 19 September 1990 will be 15.1% (an example of an inverse yield curve). Thus, the strip embodies the market’s belief as to forward interest rates. Chapter six will show how to use this information to generate a futures-based discount curve.

4.3

Forward Rate Agreements

As mentioned in the last section, most commonly used financial instruments have a forward market. In the case of short term interest rates, there exists a particularly useful instrument called the Forward Rate Agreement (FRA). The easiest way to define an FRA is to look at an example. Suppose Bank A sells to Bank B £10,000,000 of 3-6’s FRAs at 15.15%. Then, the banks have entered into a contract which obliges Bank A to pay Bank B three month sterling LIBOR minus 15.15% on £10,000,000 (if this amount is negative i.e. LIBOR is less than 15.15%, then Bank B pays Bank A the net amount). The notation “3-6” thus refers to “start date of underlying - end date of underlying”, with the start date being the “delivery” date of the contract. Say that, in three months, three month LIBOR is 16%. Then Bank A pays Bank B the present value of: (4.25) where D1 is the start date of the underlying (now spot), and D2 is three months hence. There is a convention in the FRA market that the settlement amount is paid at time D1, by discounting Equation 4.25 by: (4.26) where the 16 arises because it is now the current three month LIBOR rate, and is therefore the sensible rate to use for discounting a three

4.3. FORWARD RATE AGREEMENTS

49

month cashflow. In the case of a “6-12”s FRA, the settlement amount would be discounted using the prevailing six month LIBOR. More generally, consider an FRA on an index L with settlement date D1 and maturity D2. If it is traded at a price F on a notional principal P, the settlement amount is: (4.27)

where Dy is the day count basis relevant to the currency (D y = 365 for sterling, and 360 for dollars and most other currencies). If the quantity in Equation 4.27 is positive, the seller receives the amount from the buyer, and vice versa if it is negative. The reader may have noticed that an FRA looks similar to a one period swap. In fact, the only difference lies in the settlement con– vention: FRA convention dictates that payments are discounted to the settlement date D1 and paid then, whereas normal swap settlement in– volves paying in arrears (i.e. undiscounted at D 2 ). Considering this, it appears that a strip of FRAs looks much like a swap – for example, a one year swap with interest paid semi-annually is almost the same as a 0-6’s plus a 6-12’s FRA (in fact, a 0-6’s FRA is really just a loan or deposit, since it is equivalent to an FRA which settles when it is dealt). This equivalence is an obvious hedging opportunity, which will be studied in chapter nine. Since FRAs are just over-the-counter futures, one would expect some similarity in the pricing of these two instruments. Indeed, this is so. Consider the particularly simple situation in which the settlement date of a 3-6’s FRA with price F coincides with the expiry date of a fu– tures contract whose price is P. Then one would expect F = 100 – P. This relationship is not exactly observed in the market. One of the reasons for this is that FRAs require capital against counterparty risk, whereas futures require only variation margin. The difference between F and 100 – P is called the futures-FRA spread, and moves around ac– cording to the differing levels of supply and demand in the two markets. Traders often prefer to trade the spread rather than the underlying in– terest rate.

50

4.4


Loans and Deposits

Loans and deposits are not often used as hedging instruments by swaps and options traders. This is because short term interest rate exposures can usually be hedged more effectively by futures and FRAs, which do not require an “on balance sheet” transaction – one in which the nominal value of the security must be physically paid or received in entering into the deal. Nevertheless, it will be useful to describe them here, since they will be used in constructing the discount function in later chapters. Money market rates – the generic term for the price of a loan or deposit – are generally available from overnight out to one year in maturity. They are traded on a money market basis – Actual/365 for sterling, and Actual/360 for dollars and most European currencies. The offered side of the market (the rate at which a bank is prepared to lend money to a prime counterparty) is usually referred to as LIBOR, although this term strictly refers to the rate, set at ll:00am London time, used for reference purposes in settlements. Similarly, the bid side is referred to as LIBID. These instruments pay one coupon only, at the maturity of the contract. Thus, if one borrows £1.000,000 for nine months at the rate of 15.75%, the interest due in nine months time will be:

assuming the actual number of days between the start of the loan and its maturity to be 275.

Chapter 5

A simple approach to swap princing The advent of zero coupon pricing has provided market participants with consistent algorithms for pricing and valuing all their new and existing swaps. Formerly, accounting for swaps was on an accruals basis, while the “front office” pricing and valuation was carried out using an array of techniques borrowed from bond yield mathematics. Strictly, these techniques are wrong in that they generally make the unwarranted assumption of a flat term structure. Despite this, we are devoting an entire chapter to these methods. There are three reasons for doing so: first, a knowledge of these elementary (even ad-hoc) techniques will be helpful in understanding zero coupon pricing. Second, for the majority of swaps, the results obtained are reasonably accurate i.e. they closely concur with results given by zero coupon pricing. Lastly, not all users of swaps have access to zero coupon pricing software. All results of this chapter can in principle be obtained using a hand held calculator1; they certainly lend themselves to the construction of simple spreadsheet-based programs adequate for monitoring the value of a small number of swaps. This chapter has four objectives: 1

It is easy to make mistakes when doing this. We would not recommend quoting a price for the cancellation of a large swap on the strength of a few unchecked calculations.

51

52

CHAPTER 5. A SIMPLE APPROACH TO SWAP PRICING 1. To price a new interest rate swap. 2. To value an existing interest rate swap. 3. To price a new currency swap. 4. To value an existing currency swap.

To value a swap is to answer the question: given prevailing swap rates, how much could we expect to pay or receive in order to cancel all the swap’s future cashflows? Pricing a swap is to answer a closely related question: given prevailing rates, and also given a particular swap structure, what fixed rate should we expect to pay or to receive? Equivalently, at what fixed rate would the swap have zero value? In fact, it is not always the fixed rate which swap market makers are asked to quote. Often this is given and they are asked to quote a margin on the LIBOR leg of the swap. A market maker might even be given all the details of a swap save its start date and asked to quote the start date nearest to today that he would be prepared to use. Examples 5.7 to 5.9 explain how the start date of a swap affects its price. Under zero coupon pricing, a single method of valuation serves to price almost any swap whatsoever. The same is not true of the techniques of this chapter. Instead there are a set of techniques which can be combined to value and price the majority of interest rate and currency swaps. There is an underlying method used throughout this chapter called the comparison method. For an interest rate swap this is a three step process: Step 1 From a given swap, construct a second swap called the comparison swap designed to: a. have zero value. That is, be at a fixed rate that a swap market maker would be prepared to trade, given prevailing rates for standard swaps. b. have an identical floating leg to the original swap, except possibly in two respects: that the comparison swap have zero margin, and that the period used for calculating the

5.1. BASIC CONCEPTS

53

next floating payment on the comparison swap may be a stub period (i.e. shorter than subsequent periods). c. have identical remaining fixed payment dates to those of the original swap. Step 2 Compare the original swap and the comparison swap and calculate the differences between their cashflows – these are called the residuals. Having so created an appropriate comparison swap, all the undetermined LIBOR cashflows on the two swaps will be identical. Hence the residuals are all predetermined cashflows. Step 3 Calculate the present value of the residuals. By construction, the comparison swap has zero value, so the value of the residuals is equal to the value of the original swap.

5.1

Basic concepts

This section runs through the techniques that allow the comparison method to be put into practice. Wherever appropriate, worked examples and exercises are included. Answers are given in Appendix E.

5.1.1

The present value of a cashflow

Section 4.1.1 explained why a dollar to be received (with certainty) in the future is worth less than a dollar today. In this chapter, we shall calculate the present value of (or discount) cashflows in two different ways: Method 1 - By reference to short term rates for money market deposits. For example, suppose that three month dollar LIBOR is 8.375% on an Actual/360 basis. The value today of a cashflow of $1,000,000 to be received in 92 days time is given by:

54

CHAPTER 5. A SIMPLE APPROACH TO SWAP PRICING In general, when R is an Actual/360 rate, the present value of a cashflow C to be received in t days time is:

(5.1) Alternatively, when R is an Actual/365 rate:

(5.2)

Method 2 - By assuming that a particular rate R will be available for several consecutive periods. In the case of an annual interest rate2 R, the value of a cashflow C occurring in n years is:

(5.3) Alternatively, for a semi-annual rate S:

(5.4)

Questions: 5.1. Assume a six month sterling rate of 15.375%. With a value date of 17 May 1990, what is the present value of £1,000,000 to be received on 19 November 1990 ? 5.2. Assume an annual discount rate of 9.42% on an Actual/360 basis. What is the present value of a cashflow of $1,000,000 to be received in exactly five years time? 2

For Method 2, if R is expressed on an Actual/360 basis it should first be converted to an Actual/365 rate by multiplying by 365/360 – simply because a year has 365 days.

5.1. BASIC CONCEPTS

5.1.2

55

Accrual basis conversion

In section 2.2 we calculated the fixed cashflows for a five year swap at 9.84% annual Actual/360 as below: Payment date 05-Feb-91 05-Feb-92

05-Feb-93 07-Feb-94 06-Feb-95

05-Feb-96

Fixed payment

No. of days

2,494,166.67 2,501,000.00 2,507,833.33 2,487,333.33 2,487,333.33

365

366 367 364

364

Imagine that on 1 February 1991 the five year dollar market was trading at 9.84% Actual/360. What would be the fair price for an otherwise identical swap quoted on an Actual/365 basis? Answer: This is because an Actual/360 swap at 9.84%, and an otherwise identical swap quoted at 9.977%3 on an Actual/365 basis, have exactly the same cashflows. What would be the fair price on a 30/360 basis? The answer in this case is slightly less clear, for the cashflows of an Actual/360 swap never exactly replicate the cashflows of a 30/360 swap, no matter what rate is used. In most years, a 30/360 swap will accrue the same amount of interest as an otherwise identical Actual/365 swap. Hence a close answer will be 9.977% as before. However a slightly better approach is to match the average size of the fixed payments. This method, which attempts to take into account the effect of leap years, leads to a rate of 9.982% being the 30/360 equivalent swap rate. In this case, the two estimates differ by a negligible half a basis point. For some swaps, the difference might be three basis points, which is half the common bid-offer spread in the dollar swap market. The results generalise to give the following formulae:

(5.5) 3

Strictly, 9.976666 ....

56

CHAPTERS.

A SIMPLE APPROACH TO SWAP PRICING

(5.6) (5.7)

R365 is the fair price for the swap quoted on an Actual/365 basis; R360 is the fair price for the swap quoted on an Actual/360 basis; R30/360 is the fair price for the swap quoted on a 30/360 basis; DS

is the start (effective) date of the swap;

DM is the maturity date of the swap. Questions: 5.3. A swap is quoted at 10% on an Actual/360 basis. What would be the quote on an Actual/365 basis? 5.4. A swap market maker is prepared to receive a fixed rate of 15% on an annual Actual/365 basis for a one year swap against three month LIBOR. The swap starts on 31 January 1992 and matures on 31 January 1993. What quote should he make for the same swap on a 30/360 basis?

5.1.3

Annual versus semi-annual

Section 4.1 discussed the annual equivalent R of the semi-annual rate S. The idea was as follows: suppose an investor can choose between either: 1. investing $1 for a year at a rate of R; 2. investing $1 for six months at a rate of S, with the obligation to reinvest the proceeds at a guaranteed rate also equal to S.

5.1. BASIC CONCEPTS

57

He will be indifferent between the two if they yield the same return after one year. Expressing S and R as decimals we saw that this happens when:

(5.8) and: (5.9)

The idea generalises so that if Rn and Rm are rates with respectively n and m compounding periods per year then the two are equivalent when: (5.10)

Questions: 5.5. What is the annual equivalent of the semi-annual rate 10% ? 5.6. What is the quarterly equivalent of the semi-annual rate 9.50% ? The idea of equivalent rates is applied to swaps by making the following assumption: if a swap with annual fixed payments is fairly priced at a rate R, then an otherwise identical swap but with semi-annual fixed payments at a rate S will also be fairly priced4. Immediately we have to introduce two provisos and one caution. The provisos are that the swap must be for a whole number of years and that there should be no LIBOR margin on the floating leg of the swap. The caution is that our assumption, even with its provisos, was incorrect. Look above at how S was derived from R. The rate S held both for the first six months and for the second six months. That is. we assumed a flat yield curve. Translating this into the language of swaps, consider two one year swaps, one with an annual fixed payment of R, the other with two semi-annual fixed payments of S. The two are equivalent if the fixed payments on the semi-annual swap can be “rolled up” to equal 4

And equally for swaps with n and m payments per year, at rates Rn and Rm respectively.

58

CHAPTER 5. A SIMPLE APPROACH TO SWAP PRICING

$100

$100 0

1

$100 2

3

Figure 5.1: A simple annuity the fixed payment on the annual swap i.e. the first semi-annual coupon must be reinvested for a further six months. If the rate that can be locked in today for the reinvestment is not S, then the two swaps are not equivalent to one another (and hence cannot have both been fairly priced). In practice the effect is only noticeable5 for short swaps in currencies with steep yield curves. Sterling is often a good example of such a scenario. As will be explained in later chapters, prices for such swaps are often derived from prevailing prices for interest rate futures contracts. Question: 5.7. Five year dollar swaps are quoted at 10.33% annual Actual/360. What rate should be quoted for a five year swap with semi-annual Actual/360 fixed payments? (Strictly, your first step should be to convert 10.33% to an Actual/365 rate.)

5.1.4

The value of an annuity

An annuity is a regular series of equal payments where the time until the first payment is equal to the time between payments. The fixed leg of a swap with an unadjusted bond basis coupon is a good example of an annuity. Figure 5.1 shows a very simple annuity with time marked in years. Assuming a discounting rate of 10.55%. we can value each of 5

In other words it makes more than one or two basis points difference.

5.1.

59

BASIC CONCEPTS

PMT

PMT

PMT

PMT

PMT

PMT

Figure 5.2: A general annuity the three cashflows using Equation 5.36 and add the results: Value of annuity

$100

$100

$100

2

1.10553

1.1055 1.1055 $246.30

More generally, consider an annuity lasting N years with n payments per year each of size C. Figure 5.2 shows such an annuity with n = 2 and N = 3. Many financial calculators have a built in function to value annuities. You can enter three variables, PMT, N and I where: PMT is the size of each payment (what we called C); N

is the total number of payments (i.e. N × n);

I

is the discount rate divided by the number of payments per year (i.e. R/n). Note that the discount rate should be expressed as a semi-annual rate if there are two payments a year, a quarterly rate if there are four etc.

If you then press the PV button the calculator will return the value of the annuity. In fact it will not calculate the value of each cashflow and add them up. Instead it will take a short cut and evaluate a formula equal to the value of the annuity: (5.11) 6

This equation is used to discount cashflows occurring after the one year grid point D1Y.

60


or in our notation:

(5.12)

Question: 5.8. You have decided that an annual Actual/365 rate of 10.25% is appropriate to discount five year dollar annuities. What is worth more: an annuity of five annual payments of $1,000, or an annuity of ten semi-annual payments of $489 ? As yet we have avoided specifying how to choose a discount rate to value an annuity within the context of swap pricing. We shall make a simple choice – namely the swap rate to the same maturity date. This is market practice, at least so far as the methods of this chapter are still used by swap market makers. It is not hard to object to this choice, particularly when strongly sloped yield curves prevail. However, for the swaps examined here, the rate chosen for the comparison swap will have a greater economic effect than the rate at which the residuals are discounted. For swaps where this is not the case, zero-coupon pricing is probably the best approach.

5.2

First worked examples

Remember the comparison method outlined at the start of the chapter. To value a swap, choose a comparison swap in such a way that it is easy to generate all of the residuals. Although we now have all the tools needed to value residuals, we cannot yet create a great number of comparison swaps. A set of them are immediately to hand, namely the market standard swaps executed at prevailing market rates. The following two examples utilise these. Example 5.1: A bank is prepared to pay 9.42% annual Actual/360 against LIBOR flat for a five year dollar interest rate swap. A client requests that the bank should instead pay 10% in return for an up-front fee paid to the bank. If the swap is on a principal of $10,000,000, how much should the bank quote?

5.2.

FIRST WORKED EXAMPLES

61

Solution 5.1: We need to calculate the present value of a $10,000,000 five year swap at 10%. Since the bank was prepared to pay 9.42% for this structure then a swap at 9.42% will serve as the comparison swap, since it values to par. Both swaps have identical floating legs so the residuals are an annuity formed by the 58 basis points difference between the rates. Assuming for simplicity that there are 365 days between each payment date, then the size of each residual is: $10,000,000

0.58 100

365 360

$58,805.56

So setting: N=5 PMT = 58,805.56 I = 9.42 × 365/360 ≈ 9.551% implies an annuity value of approximately $225,500. This is what the bank should quote to its client. Example 5.2: Suppose that traders are prepared to pay 13.07% semi-annual Actual/365 on ten year sterling swaps. A bank is asked to pay the fixed on a £75,000,000 ten year swap. Fixed payments are to be made annually on a 30/360 basis. In addition the bank must pay £375,000 as an up-front fee to its counterparty. Assume that the bank is also prepared to pay 13.07%. What rate should it quote?

Solution: 5.2 First create a comparison swap by annualising 13.07% to give 13.50%. This holds good as a 30/360 rate. The bank must pay less than 13.50% to compensate for the £375,000 fee. To calculate the value to the bank of each basis point on the rate, set: N = 10 PMT = 75,000,000 × 0.01/100 = £7,500 I = 13.50%

62

CHAPTER 5.


to yield an annuity value of £39,896. The bank must therefore quote below 13.50% by: 375,000 ≈ 9.4basis points 39,896 Hence it must quote 13.406%.

5.3

Comparison swaps

Swaps which have been traded some time in the past do not usually mature on an exact anniversary of today. So to value an arbitrary swap requires comparison swaps with whatever maturity date we wish. In other words, for any date D we must be able to answer the question: what is the fair rate for a swap starting today (or starting out of spot) and maturing on the date D? The answer when D is in exactly one, two, three, five, seven or 10 years time is clear, for such swaps are widely quoted in the market. Is it correct to interpolate (linearly) these rates to obtain fair rates for, say, a four and a half year swap? To explain why it is not correct, we argue by setting up an analogy with bonds. Chapter six explains how to represent the unknown LIBOR payments on a swap by two payments of principal. In this way, interest rate swaps are analogous to bonds whose principal is repaid at maturity. Consider the three bonds shown in Figure 5.3. Carrying through our analogy, we find that a swap is fairly priced if the related bond values to 100 (using swap rates to discount its cashflows). So we have transformed our original question into: if bonds 1 and 2 both value to 100. would bond 3 also to value to 100? In fact the price of bond 3 cannot be deduced from the price of the first two. Instead we invent a reasonable hypothesis, which in this case is: to value a bond's cashflows, discount them at 10.25%. Accordingly, the three bonds have values: 10.25 110.25 Bond 1: 100 1.1025 1.10252 110.25 10.25 10.25 Bond 2: 100 2 1.1025 1.10253 1.1025 5.125 110.25 10.25 Bond 3: 100.12 0.5 1.5 1.1025 1.10252.5 1.1025

5.3.

63

COMPARISON SWAPS

110.25 10.25

Bond 1

110.25 10.25

10.25

Bond 2

110.25 Bond 3

5.125

10.25

Figure 5.3: Three swaps/bonds

64

CHAPTER 5.


110.25 Bond 4

5

10.25

Figure 5.4: A two and a half year bond So our reasonable hypothesis implied that the two and a half year bond did not value to 100. Translating back into the language of swaps, the hypothesis implied that 10.25% was not a fair rate for the two and a half year swap. This is a problem, but not one that is difficult to solve. Bear in mind that the two and a half year swap has a six month first period, that the semi-annual equivalent of 10.25% is 10%, and that 10% accruing for half a year gives a cashflow of 5%. Now construct a second two and a half year bond as in Figure 5.4. Its value is: Bond 4:

5 1.10250.5

10.25 1.10251.5

110.25 1.10252.5

100

Again translating back to the language of swaps, the two and a half year swap is fairly priced at 10.25% if the first period is accrued at the semi-annual equivalent of 10.25% (i.e. at 10%). Extending the idea, we can create comparison swaps to any required maturity by interpolating the fixed rate between observed market rates and using a decompounded rate for the first fixed payment. Decompounded means that if the first fixed period is six months then a semiannual rate is used, if three months then a quarterly rate is used, and so on. More generally, if the swap has n payments per year but a first period of only f days then set m — 365/f in Equation 5.10. Example 5.3: You wish to value a swap with annual fixed payments and with maturity in three years and two months. Three and four year

5.3.

65

COMPARISON SWAPS annual rates are 11% and 11.24% respectively. fixed cashflows on the comparison swap?

What are the

Solution 5.3: Interpolating to three years and two months between the three and four year rates gives 11.04%. Using Equation 5.10 and setting m = 6 and n = 1 gives an equivalent rate for a two month period of 10.564%. So the first fixed cashflow of the comparison swap is 10.564% × 2/12 = 1.76%. Schematically the fixed leg of the comparison swap is: Time 2 months 1 year 2 months 2 years 2 months 3 years 2 months

Cashflow 1.76 11.04 11.04 11.04

Example 5.4: Two and three year rates are both 10.25% annual. What is the fair rate for a two and a half year annual paying swap (with the first period not decompounded)? Solution 5.4: Earlier we saw that 10.25% is not the correct answer but that the correct comparison swap has fixed cashflows: Time

Cashflow

1 2 3

5 10.25 10.25

Suppose the fair rate is x. Then the fair swap has fixed cashflows: Time

Cashflow

1 2 3

x x

66

CHAPTER 5.


As always, the comparison swap is fairly priced. The swap at a rate x is also fairly priced if the residuals have zero present value, or equivalently, if the two fixed legs have the same present value (since the floating legs cancel). Discounting at 10.25% gives:

0.5x 1.10250.5

x 1.10251.5

5

x 2.5

1.1025 ⇒ 2.1235x

10.25 0.5

1.1025 = 21.647

1.5

1.1025

10.25 1.10252.5

⇒ x = 10.194 Notice that the fair two and a half year rate is more than five basis points below both the two and the three year rates. So far we have dealt with the fixed leg of comparison swaps with short (or stub) first periods. The floating leg may also have a stub first period. Remember that, on the comparison swap, the LIBOR settings still to be made must have the same dates as on the original swap, which often means using the prevailing LIBOR rate for the appropriate stub period.

Example 5.5: Imagine that 10 months ago, you traded a four year interest rate swap. It now has three years and two months until maturity. You received the fixed rate at 11.5% on an annual Actual/365 basis against paying six month LIBOR flat. The most recent LIBOR setting was 10.5% Actual/365. Given today's rates as below, illustrate the fixed and floating legs of the swap you own and of an appropriate comparison swap. Maturity 2 months 3 years 4 years

Rate 10.5% Act/365 11% Annual Act/365 11.24% Annual Act/365

Solution 5.5: The cashflows of the swap you own are easy to calculate:

5.4.

WORKED EXAMPLES: PRICING Time

1 year 1 year 2 years 2 years 3 years

2 months 8 months 2 months 8 months 2 months 8 months 2 months

Fixed cashflows 11.50

67

Floating cashflows –5.25 ? ?

11.50

?

11.50

?

11.50

?

?

where the question marks stand for as yet undetermined LIBOR payments. The fixed cashflows of the comparison swap were calculated in Example 5.3. For the LIBOR leg, today’s two month rate of 10.5% is used as the first LIBOR setting. Hence the cashflows for the comparison swap are: Time

1 year 1 year 2 years 2 years 3 years

2 months 8 months 2 months 8 months 2 months 8 months 2 months

Fixed cashflows 1.76 11.04 11.04 11.04

Floating cashflows –1.75 ? ? ? ? ? ?

Again the question marks stand for unknown LIBOR settings. It is because they are the same unknown LIBOR settings as in the previous table that the residual cashflows are all known.

5.4

Worked examples: pricing

The tools to price the great majority of interest rate swaps are now in place. We have not. however, discussed how to price amortising and zero coupon swaps. These structures require the techniques of zero coupon pricing explained in chapter six. This section presents a series

68

CHAPTER 5.


of worked examples, each being designed either to illustrate some small pitfall to avoid, or else to reflect a structure common in the market. Example 5.6: Complete the pricing of the swap of Example 5.5. Solution 5.6: Subtracting cashflows yields a table of residuals as below: Residual Time 6.24 2 months 0.46 1 year 2 months 0.46 2 years 2 months 0.46 3 years 2 months The last three residuals form an annuity starting in two months time. Using Equation 5.11 to discount those three cashflows back to the start date of the annuity gives an annuity value of 1.123 (where N = 3, I = 11.04% and PMT = 0.46). This value, and the residual of 6.24, must be discounted back to today. Using the two month rate of 10.5% gives: Swap value

1.123 + 6.24 1 + 2 × 10.5/1200

7.24

That is, the swap is worth 7.24% of its underlying principal. Delayed (or forward) start swaps are those traded with a start date later than either the trade date, or two business days after the trade date, according to currency. The delay can be several years, but more commonly is up to three months. Two techniques are available to price such deals, one appropriate for a delay of less than one year and the other for delays over one year. Examples 5.7 to 5.9 illustrate the method. Example 5.7: Table 5.1 sets out an example dollar yield curve. Rates up to and including six months are Actual/360 cash rates, subsequent rates

5.4.

WORKED EXAMPLES: PRICING

69

Value date: 17th May 1990

Maturity O/N 1W 1M 2M 3M 6M 1Y 2Y 3Y 4Y 5Y

Par rate 8.3125 8.25 8.3125 8.375 8.375 8.5 8.75

9.08 9.24 9.34 9.42

Table 5.1: A dollar yield curve are annual Actual/360 swap rates. Assume that a swap trader is prepared to receive fixed at the four and five year rates shown. What rate should he be prepared to receive for a swap starting one month forward and running for four years? Solution 5.7: A comparison swap (see Table 5.2) is constructed as follows: 9.34 + (9.42 – 9.34) = 9.3466% Fixed rate: Decompounded first rate: 8.9637% Actual/360 First fixed payment: 0.7573 Subsequent fixed payments: 9.4764% First floating rate: 8.3125% Actual/360 First floating payment: 0.7023 0.055 Net first payment: The solution swap is fairly priced when the residuals have zero present value. If each of the last four residuals are of size x then an annuity of four payments of x must have a value at the one month point of –0.055. To solve for x use Equation 5.11 (or a

70

CHAPTER 5.

Time


Comparison swap Fixed Floating 1 month 0.7573 0.7023 1 year 1 month 9.4765 2 years 1 month 9.4765 3 years 1 month 9.4765 4 years 1 month 9.4765

Solution swap Floating Fixed

0 9.4765 9.4765 9.4765 9.4765

0

– – – –

x x x x

Residual 0.055

x x x x

Table 5.2: Comparison swap for Example 5.7 financial calculator) with: N=4 PMT = 1 I = 9.3466 × 365/360 = 9.4765% to give an annuity value of 3.2061. Now x is found by taking the ratio: x = –0.055/3.2061 = –0.0171 So the fixed payments on the solution swap are of size 9.4765 + 0.0171 — 9.4936%. The fixed rate for the solution swap is quoted on an Actual/360 basis so: swap rate = 9.4935 × 360/365 = 9.363% By comparison, the four year rate was 9.34%. Swap traders would say that a one month delay on a four year swap is worth 2.3 basis points. Delayed start swaps are very common and the next example shows a quick and simple calculation of the delayed swap rate. The calculation is the four step process implicit in the previous example. Step 1 Interpolate a rate for the comparison swap. Step 2 Decompound the first fixed period.

5.4.


71

Value date: 17th May 1990

Maturity O/N 1W 1M 2M 3M 6M 1Y 2Y 3Y 4Y 5Y 7Y 10Y

Par rate 14.25 14.75 15 15.125 15.15625 15.375 15.55 14.25 13.95 13.85 13.75 13.55 13.10

Table 5.3: A sterling yield curve Step 3 Compare the decompounded rate with the appropriate LIBOR rate to calculate the first payment on a comparison swap. Step 4 Amortise the result of step three and add to the rate calcu– lated in step one. Example 5.8: Table 5.3 shows a sterling yield curve. All the rates shown are on an Actual/365 basis. The one year rate is annual, rates less than one year are cash rates, and rates over one year are semi– annual. Calculate the implied rate for a five year swap three months forward. Solution 5.8: Step 1 Interpolating to five and a quarter years between the five and seven year rates gives 13.725%.

72

CHAPTER 5. A SIMPLE APPROACH TO SWAP PRICING Step 2 Decompounding gives a quarterly equivalent rate of 13.497%. Step 3 The three month LIBOR is 15.15625% hence the net first cashflow on the comparison swap is (13.497 – 15.15625)/4 = -0.4148. Step 4 Setting PV = –0.4148, N = 10, I = 13.725/2 = 6.8625% we find PMT = –0.0587. Hence the solution swap has cashflows of size 13.725/2 – 0.0587 = 6.8038 and is at a rate 6.8038 X 2 = 13.608%. In this case traders would say that a three month delay on a five year sterling swap “costs” 14.2 basis points. Questions:

5.9. Using the sterling yield curve of Table 5.3 calculate the semi– annual price of a seven year swap six months forward. 5.10. Using the dollar yield curve of Table 5.1 price a three year semi– annual 30/360 swap with a delay of three months. Suppose a swap trader receives fixed on a seven year swap. At the same time he pays fixed on a two year swap (of the same currency, size and roll cycle). For the first two years the two floating legs cancel each other, as do the two fixed legs – except to the extent that the two fixed rates are different. So the trader has effectively manufactured a five year swap two years forward. This idea underlies the pricing of forward start swaps with delays of more than one year and is illustrated in the next example. Example 5.9: Assume that a trader is prepared to receive fixed on a seven year sterling swap at 13.55% and pay fixed on a two year sterling swap at 14.25%. What price should he or she quote to receive fixed on a five year swap two years forward? Solution 5.9: Recall that the standard swap structure in sterling is semi-annual Actual/365. Table 5.4 shows the two comparison swaps, one a long position in a seven year swap, the other a short position

5.4.

73


Time

0.5Y 1Y 1.5Y 2Y 2.5Y 3Y 3.5Y 4Y 4.5Y 5Y 5.5Y 6Y 6.5Y 7Y

Comparison swap 1 Fixed leg 6.775 6.775 6.775 6.775 6.775 6.775 6.775 6.775 6.775 6.775 6.775 6.775 6.775 6.775

Comparison swap 2 Fixed leg –7.125 –7.125 –7.125 –7.125

Solution swap

6.775 – 6.775 – 6.775 – 6.775 – 6.775 – 6.775 – 6.775 – 6.775 – 6.775 – 6.775 –

x x x x x x x x x x

Residual

–0.35 –0.35 –0.35 –0.35 x x x x x x x x x x

Table 5.4: Comparison swaps for Example 5.9

in a two year swap. Taken together, the two comparison swaps have exactly the same floating leg as the solution swap, so the floating legs need not be considered. To solve for the forward swap rate requires an x such that the residuals shown in the right hand column of Table 5.4 have zero present value. The first four cashflows have present value given by: N=4 PMT = – 0.35 I = 6.775% Annuity value = – 1.1916 To solve for x, first suppose that x = 1 and subtract the value of a two year annuity of x from the value of a seven year annuity of x.

74

CHAPTER 5.


N=4

PMT= 1 I = 6.775% ⇒ Annuity value = 3.4045 N = 14 PMT = 1 I = 6.775% ⇒ Annuity value = 8.8647 So. were x = 1, the stream of 10 x’s would have value 8.8647 – 3.4045 = 5.4602; but the stream must have value 1.1916 to cancel the value of the first four residuals. A simple ratio gives:

x

=

1.1916/5.4629 = 0.2181

Forward swap rate

=

2 × (6.775 – x) ≈ 13.11%

Example 5.10:

One of the driving forces behind the growth of the swap market has been the use of swaps to convert the liabilities of Eurobond issuers into floating rate. In fact, during 1989 and 1990 there was a marked reduction in the percentage of Eurobond issues that were swapped. Reliable figures are not published but a fair guess is that two thirds (by volume) of issues in 1988 were swapped compared with only one third of issues in 1990. Nevertheless, new issue swaps still form an important part of the swap market. For a full description of the Eurobond market the reader is referred to one of the standard references [6]. To price a swap related to a bond issue requires knowledge only of the cashflows that the bond issuer pays to the holders of his bonds. A typical bond issue might have had the following details:

5.4.

WORKED EXAMPLES: PRICING Amount Launch date Payment date Maturity date Coupon Issue price Fees Issue price less fees

75

ECU 100,000,000 15 May 1990 17 June 1990 17 June 1993 10 100 1 98

The issue is announced to the market on 15 May 1990. One month later the issuer receives from the banks involved in the issue an amount of 98 % ECU100,000,000 = ECU98,125,000. On 17 June 1991 and 1992 the issuer pays bond holders coupons of exactly ECU 10,625,000. On the maturity date the issuer pays the principal amount plus the final coupon i.e. ECU 110,625,000. Note that the term “fees” is a misnomer. The banks buy ECU 100,000,000 of bonds from the issuer and sell them on to investors. The banks buy the bonds at the issue price less fees, that is at 98 % of face value. Such is the competition to obtain Eurobond business that banks are likely to sell bonds on to investors at or near the same price, implying a near zero profit. Assume that the issuer wants to swap the bond cashflows, so transforming his liabilities from fixed into floating rate ECU. On the launch date he must receive a fixed rate on a forward start swap against paying floating rate ECU plus or minus a margin. The swap is designed so that the fixed leg exactly matches the bond coupons. The floating cashflows under the swap are calcu– lated on a notional principal of ECU 100,000,000. To allow for fair comparisons to be made between raising floating rate funds via a swapped bond issue and other alternatives, the amount raised must equal the amount on which LIBOR is calculated. The bond issue proceeds are only ECU 98,250,000. Therefore the swap also includes an up-front payment of ECU 1,875,000 which takes the proceeds back up to ECU 100,000,000. The cashflows of the bond and swap are illustrated in Table 5.5, where brackets round a number indicate a negative cashflow.

76

CHAPTER 5. Date

17-Jun-90 17-Dec-90 17-Jun-91 17-Dec-91 17-Jun-92 17-Dec-92 17-Jun-93


Bond cashflows 98,125,000 (10,625,000) (10,625,000) (110,625,000)

Swap cashflows Fixed leg Floating leg 1,875,000 (LIBOR + xbp) 10,625,000 (LIBOR + xbp) (LIBOR + xbp) 10,625,000 (LIBOR + xbp) (LIBOR + xbp) 10,625,000 (LIBOR + xbp)

Net 100,000,000 (LIBOR + xbp) (LIBOR + xbp) (LIBOR + xbp) (LIBOR + xbp) (LIBOR + xbp) (LIBOR + xbp) & (100,000,000)

Table 5.5: Cashflows for the ECU bond and swap of Example 5.10 Solution 5.10: To price the swap, assume that on 15 May 1990 one month ECU LIBOR was 10.15625% (Actual/360) and that the swap counterparty is prepared to pay 10.87% (annual 30/360) for both three and four year swaps. The method of Example 5.8 gives a one month forward start swap rate7 of 10.872%. The forward start swap serves as a comparison swap in which case there are three elements to the comparison swap. With their present values as at the swap start date these are: (a) The up-front payment of 1,875,000 Present value = 1,875,000 (b) The difference in the fixed coupons i.e. three payments of: (10.625 – 10.872)/100 × 100,000,000 = –247,000 Present value (I = 10.87%) = –605,038 (c) The six margin cashflows, each of size:

Present value (I = 5.435%) = –25,102x 7

Because the ECU yield curve is almost flat, the forward start rate is almost equal to the spot start rate.

5.4.


77

Solving for x: 1,875,000 – 605,038 – 25,102x = 0 x = 50.6 So the bond issuer has effectively borrowed floating rate ECUs at six month ECU LIBOR plus 50.6 basis points. Example 5.11: – A shortcut for pricing new issue swaps: In the previous example we mechanically applied the comparison method to price a new issue swap. In practice swap traders use a simple shortcut. Recall from section 4.1.1 that the yield of a bond is that discount rate at which the bond cashflows value to zero. The fair margin on a new issue swap is equal to the difference between: (a) the Eurobond yield at the issue price less fees and: (b) the forward start swap rate for a swap against LIBOR flat. As a further refinement, if the swap is against six month LIBOR then the two yields above should be expressed on a semi-annual basis. Applying the shortcut to the new issue swap of Example 5.8 gives: Bond yield at a price of 98

= = Forward start swap rate = = Margin = =

11.398% annual 11.090% semi-annual 10.872% annual 10.592% semi-annual 11.090% – 10.592% 0.498% = 49.8 basis points

The shortcut method implicitly uses the bond yield in place of the forward swap rate as a discounting rate. This accounts for the small difference of 0.8 basis points between the answers given by the two techniques.

78


5.5

Worked examples: valuation

This section contains two worked examples of the valuation of realistic interest rate swaps, one from each of the sterling and dollar swap markets. As long as the idea underlying the comparison method is kept in mind, most interest rate swaps can be valued using the formats set out. However, care is needed to avoid mistakes. Very simple calculations can check whether an answer is of the correct order of magnitude. Example 5.12: The swap described below was a seven year deal when traded on 23 February 1989. Using the yield curve of Table 5.3 and a value date of 17 May 1990, calculate its worth to the fixed rate receiver. Amount Start date Maturity date Fixed rate Existing LIBOR rate

£10,000,000 23-Feb-89 23-Feb-96 11.05% semiannual Act/365 15.125% Act/365

Solution 5.12: First- work out further details of the existing swap. Notice the sign convention that cashflows paid are shown as negative. Last payment date Next payment date Next fixed payment Subsequent fixed payments Next floating payment

Next create a comparison swap:

23-Feb-90 23-Aug-90 £547,958.90 £552,500.00 (£750,034.25)

5.5.

79

WORKED EXAMPLES: VALUATION Amount Start date Maturity date Term Fixed rate

Decompounded first rate

First LIBOR rate Next payment date Next fixed payment Subsequent fixed payments Next floating payment

£10,000,000 17 May 1990 23-Feb-96 5 years 282 days 13.673% Act/365 interpolated between 5 and 7 year rates 13.463% Act/365 n = 2 m = 365/98 Rn = 13.673% in Equation 5.10 15.17% Act/365 interpolated between 3 and 6 month rates 23-Aug-90 £361,472.33 £683,650.00 (£407,304.11)

Calculate the residuals (existing swap minus the comparison swap):

First fixed residual Subsequent fixed residuals of which there are: First floating residual

£186,486.57 (£131,150.00) 11 (£342,730.14)

Value the residuals by discounting first back to 23 August 1990 (using Equation 5.11) and then back to 17 May 1990 (using Equation 5.2):

80

CHAPTER 5.


PV of stream of fixed residuals First fixed residual First floating residual Total of above Discounted to 17-May-90

(£991,510.40)

£186,486.57 (£342,730.14) (£1,147,753.97) (£1,102,835.05)

N= 11 I = 6.8365% PMT = –131,150

R = 15.17% t = 98 in Equation 5.2

So the swap is worth a negative amount of nearly £1,103,000 to the fixed rate receiver. As a check that this result is approximately correct, note that the existing fixed rate is 11.05% and that the comparison swap rate is 13.673%. As a percentage of the principal amount the swap value is approximately given by the value of an annuity with: N = 12 PMT = (11.05 - 13.673)/2 = –1.3115 I = 13.673/2 = 6.8365% Annuity value = –10.51 On a principal of £10,000,000 this implies a negative value for the swap of £1,051,000. Because this approximate value is close to the result of £1.102,835.05 we can be reasonably confident that no large errors were made in the calculations.

Example 5.13:

The swap described below was originally a six year dollar interest rate swap with annual fixed payments. It includes a margin on the floating leg, which is based on three month LIBOR. Using the yield curve of Table 5.1 and a value date of 17 May 1990, value the deal from the perspective of the fixed rate receiver.

5.5.

81

WORKED EXAMPLES: VALUATION Amount Start date Maturity date Fixed rate Existing three month LIBOR rate Floating margin

$50,000,000 25-Jul-88 25-Jul-94 10.03% annual Act/360 8.50% Act/360 25 basis points

Solution 5.13:

Further details of the existing swap are:

Last fixed payment date Next fixed payment date Last floating payment date Next floating payment date Next fixed payment Subsequent fixed payments Next floating payment (including margin) Subsequent margin payments

A comparison swap is:

25-Jul-89 25-Jul-90 25-Apr-90 25-Jul-90 $5,084,652.78 $5,084,652.78 ($1,105,902.78) ($31,684.03)

82

CHAPTER 5.


Amount Start date Maturity date Term Fixed rate


First LIBOR rate Libor margin Next fixed payment date Next fixed payment Subsequent fixed payments Next floating payment date Next floating payment

$50,000,000 17-May-90 25-Jul-94 4 years 69 days 9.355% Act/360

interpolated between 4 and 5 year rates 9.015% Act/360 n = 1 m = 365/69 in Equation 5.10 and by Equations 5.5, 5.6 8.375% Act/360 interpolated between 2 and 3 month rates Zero 25-Jul-90 $863,937.50 $4,742,465.28 25-Jul-90 ($802,604.17)

Calculate the residuals (existing swap minus the comparison swap): First fixed residual Subsequent fixed residuals of which there are: First floating residual Subsequent floating residuals of which there are:

$4,220,715.28 $342,187.50 4 ($303,298.61) ($31,684.03) 16

Value the residuals by discounting first back to 25 July 1990 (us– ing Equation 5.11) and then back to 17 May 1990 (using Equation 5.2):

83

5.6. BASIS SWAPS PV of stream of fixed residuals PV of stream of floating residuals First fixed residual First floating residual Total of above Discounted to 17-May-90

$1,096,893.68 ($420,442.76)

$4,220,715.28 ($303,298.61) $4,593,867.59 $4,555,713.49

N=4 I = 9.48493% PMT = 342,187.50 N = 16 I = 2.291012% PMT = –31,684.03

R = 8.375% t = 69 in Equation 5.1

Hence the swap has a positive value to the fixed rate receiver of some $4,555,000. Again a simple check shows that this result is reasonable.

5.6

Basis swaps

Recall that currency swaps include exchanges of principal on the maturity date, and usually on the start date. A currency swap in which both legs are at floating rates is called a (cross currency) basis swap. Imagine that a bank is always able to borrow dollars at LIB OR and invest sterling at LIBOR minus 12.5 basis points. The bank trades a basis swap on which it receives six month dollar LIBOR on $10,000,000 against paying six month sterling LIBOR minus 12.5 basis points on £5,000,000. On the start date there is an initial exchange of principals under which the bank pays $10,000,000 and receives £5,000,000. In order to pay these dollars the bank borrows them for six months at LIBOR. It also invests the £5,000,000 at (sterling) LIBOR minus 12.5 basis points. Now on the next coupon date all the payments of interest cancel out. For instance, the bank receives dollar LIBOR on the swap and pays the same amount on its dollar borrowings. The bank can then reinvest the £5,000,000 and reborrow the $10,000,000 on subsequent coupon dates. On each date all the cashflows cancel. On the maturity date, the final dollar principal under the swap allows the

84


dollar borrowings to be repaid and the maturing sterling deposit allows the sterling principal to be paid under the swap. By using deposits, the bank has manufactured a basis swap to offset the original trade – albeit at the cost of putting loans and deposits on its balance sheet. In practice it is usually more efficient to use spot and forward foreign exchange trades in combination with deposits and loans. So the market price for basis swaps is bound to lie within certain limits. These are determined by the levels at which the above arbitrages work. At the time of writing, the sterling dollar basis swap market is quoted as flat to plus eight basis points. In other words, banks are prepared to pay sterling LIBOR against dollar LIBOR, or to receive sterling LIBOR plus eight basis points against paying dollar LIBOR. In the next section currency swaps are priced by considering them to be the combination of an interest rate swap and a cross currency basis swap.

5.7

Pricing currency swaps

Once the pricing of interest rate swaps has been mastered, pricing fixedfloating currency swaps is easy. A currency swap can be considered as an interest rate swap plus a basis swap, so that pricing a currency swap is a matter of pricing each component. There is just one pitfall known as “basis point conversion”. It is explained in the next example. Example 5.14: On 17 May 1990, the ECU-dollar basis swap market is quoted as plus five to minus six basis points. A bank is asked to paysix month ECU LIBOR plus 50.5 basis points against receiving six month dollar LIBOR plus some margin. The swap starts on 17 June 1990 and matures on 17 June 1993. Initial and final exchanges take place at a rate of ECU1 = $1.4 which is the prevailing one month forward exchange rate (the convention for forward start currency swaps). Assuming the ECU rates of Example 5.10 and the dollar rates of Table 5.1. what margin should the bank quote?

5.7.

PRICING CURRENCY SWAPS

85

Solution 5.14: By assumption, the bank is prepared to pay ECU LIBOR minus six basis points against receiving dollar LIBOR. Instead it must pay ECU LIBOR plus 50.5 basis points i.e. 56.5 basis points more. We have to calculate what margin over dollar LIBOR is adequate compensation for paying these 56.5 ECU basis points. This is the basis point conversion problem mentioned earlier. From Example 5.10. an appropriate discount rate is 10.591% (the semi-annual equivalent of 10.872%). A quick calculation from Table 5.1 gives a semi-annual Actual/365 discount rate of 9.188%. To value the 56.5 ECU basis points set: N=6 I = 10.591/2 = 5.2955% PMT = 56.2/2 = 28.1 Annuity value = 141.3 basis points Since the initial exchange under the swap is at the one month forward exchange rate, then the 142 ECU basis points are worth 142 dollar basis points of the dollar principal. To transform this into an annuity set: N=6 I = 9.188/2 = 4.594% PV = 142.0 PMT = 27.6 This is the size of each semi-annual payment so the dollar margin is 2 27.6 = 55.2 basis points Example 5.15: Suppose the ECU bond issuer of Example 5.9 had wanted to swap the bond proceeds into floating rate dollars. What margin over dollar LIBOR could have been achieved? Solution 5.15: The currency swap required is simply the combination of the basis swap of the previous example and the interest rate swap of Example 5.9. Hence the solution is again 55.2 basis points.

86

CHAPTER 5.


In practice swap traders often quote a more aggressive8 price for a basis swap when it forms part of a fixed-floating currency swap than when being asked for a price for a basis swap in isolation.

Question: 5.11. A swap trader is prepared to pay six month dollar LIBOR against receiving six month sterling LIBOR plus eight basis points on a 10 year basis swap. The 10 year dollar swap rate is 9.55% annual Actual/360 while the 10 year sterling swap rate is 13.10% semiannual Actual/365. What margin over sterling LIBOR should the trader quote against dollar LIBOR plus 80 basis points?

5.8

Valuing currency swaps

The comparison method can easily be adapted to value currency swaps. One complication is introduced: that the comparison swap and the existing swap are likely to use different rates for their final exchanges of principal 9. In fact the change in value of a currency swap is often due more to currency movements than to movements in interest rates. For fixed-floating currency swaps the best approach is to match the principal on the fixed leg. This implies that the floating leg principals on the existing and comparison swaps are unequal. Hence the floating leg residual cashflows are not all predetermined. Nevertheless they are easy to value as shown in the next example. Example 5.16: When traded on 21 February 1989 the swap below was a seven year fixed sterling against six month floating dollar currency swap. Assume that basis swaps are quoted as sterling LIBOR flat against dollar LIBOR flat and that £l = $1.9000 is the current exchange 8

That is. more advantageous to the client. To be fairly priced at prevailing swap rates, currency swaps must have initial and final exchanges at the same rate. When the initial exchange is missing it is replaced by a spot foreign exchange transaction. Hence the comparison swap, which has no initial exchange, must have a final exchange at today’s spot exchange rate. 9

5.8.

87

VALUING CURRENCY SWAPS

rate. With a value date of 17 May 1990 and the yield curves of Tables 5.1 and 5.3, calculate its worth to the fixed rate receiver.

Sterling amount Dollar amount Start date Maturity date Fixed rate Existing dollar LIBOR rate

£10,000,000 $17,100,000 23-Feb-89 23-Feb-96 11.05% semi-annual Act/365 8.625% Act/360

Solution 5.16:

Full details of the existing swap are:

Last payment date Next payment date Next sterling payment Subsequent sterling payments Next dollar payment Subsequent dollar payments

Final dollar principal exchange

23-Feb-90 23-Aug-90 £547,958.90 £552,500.00 ($741,534.38) (six month dollar LIBOR on $17,100,000) ($17,100,000)

Matching the sterling principal amount, a comparison swap is:

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CHAPTER 5.


Sterling amount Dollar amount Start date Maturity date Term Fixed rate


First dollar LIBOR rate Next payment date Next sterling payment Subsequent sterling

£10,000,000 $19,000,000 17-May-90 23-Feb-96 5 years 282 days 13.673% Act/365

interpolated between five and seven year rates 13.463% Act/365 n = 2 m = 365/98 Rn = 13.673% in Equation 5.10 8.383% Act/360 interpolated between three and six month rates 23-Aug-90 £361,472.33 £683,650.00

payments Next dollar payment Subsequent dollar payments Final dollar principal exchange

($433,587.39) (six month dollar LIBOR on $ 19,000,000) ($19,000,000)

The residuals (existing swap minus the comparison swap) are: First sterling residual Subsequent sterling residuals of which there are: First dollar residual Subsequent dollar residuals

Difference between final dollar exchange amounts

£186,486.57 (£131,150.00) 11 ($307,946.99) six month dollar

LIBOR on $1,900,000 $1,900,000

89

5.9. REMARKS

The unknown dollar residuals, together with the $1,900,000 mismatch in final exchange amounts can be replicated by a cashflow of $1,900,000 on 23 August 1990, since this amount could be repeatedly invested at LIBOR to produce the same cashflows as the unknown dollar residuals. Hence the residuals can be valued as follows:

PV of stream of sterling residuals

(£991,510.40)

First sterling residual Total of above Discounted to 17-May-90

£186,486.57 (£805,023.83) (£773,518.12)

First dollar residual Value as of 23-Aug-90 of subsequent dollar residuals Total of above Discounted to 17-May-90

($307,946.99) $1,900,000

Total value (using £1 = $1.9)

$1,592,053.01 ($1,556,532.34)

N = 11 I = 6.8365% PMT = –131,150

R = 15.17% t = 98 in Equation 5.2

R = 8.383% t = 98 in Equation 5.1

£45,709.43

The reader may have noticed that the swap of Example 5.16 was identical to that of Example 5.12 except for being against dollar LIBOR. Whereas the first of these two was worth minus £1,102,000, the second is worth plus £45,700. The difference is (mainly) due to the movement in the dollar-sterling rate from £ = $1.71 to £ = $1.9.

5.9

Remarks

Explaining the comparison method involves both describing a very simple underlying idea and displaying the detailed calculations required to carry through the valuation of a realistic swap. Once the spirit of the

90

CHAPTER 5.


method has been grasped most swaps can be priced. The next chapter describes zero coupon pricing, which is not only more flexible than the methods of this chapter, but also more logical and therefore more satisfying.

Chapter 6

Zero coupon pricing 6.1

In defence of zero coupon pricing

From now on. we shall be studying the application of zero coupon pricing to the valuation of swaps and. more generally, arbitrary collections of known, unknown and contingent (i.e. option related) cashflows. In view of this, the reader would expect a compelling reason to exist for choosing this method above all others. This section explains such a reason. Consider a shopkeeper who has 20 apples and 10 oranges for sale. He is prepared to sell each apple for 10 pence and each orange for 20 pence. Clearly, the value of his apples and oranges is 20 10 + 10 20 = £4.00. This trivial calculation illustrates the fundamental point that this valuation of apples and oranges is additive; the value of two apples is twice the value of one apple. Suppose that a nearby supermarket gives away one free apple for every two purchased, but each apple can still be purchased independently for 10 pence. If the supermarket has a stock of 60 apples, how much are they worth? Clearly there is a dilemma – if the apples were only ever purchased singly or in pairs, then the value would be 60 × 10 = £6.00. However, if they were all purchased in threes, the value would be 20 × 20 = £4.00. There also exist a whole gamut of possible combinations of purchases giving a value between £4 and £6 91

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CHAPTER 6. ZERO COUPON PRICING

– there is no single “true” value1. This second example demonstrates the idea of non-additive valuation – the value of three apples is not necessarily equal to three times the value of one apple. It is clear that the notion of an additive system depends on the rules used for defining the value of the collection of objects in the system. It is a convenient fact that, in almost all the financial situations that, arise in real life, additivity of value is observed. Thus, the investor with a portfolio of bonds can safely state that the value of the portfolio equals the sum of the value of the individual bonds. Situations can occasionally occur where additivity is not observed. Imagine a company with issued share capital. It issues free warrants which allow someone who has held one share for at least one year to purchase another one at a discount to the market price. However, people who have held a share for less than a year can only purchase an extra share at current market prices. Thus, the value of the warrant in isolation (i.e. to an investor who holds nothing else) is zero, whereas in a portfolio together with a share held for a year, it has a positive value. This type of situation has arisen for long term shareholders in UK companies that have been privatised. The reason for our digression is that there exists a theorem, proved rigorously in Appendix D, that states:

Theorem: Any additive cashflow valuation system can be shown to be a zero coupon cashflow valuation system, and every zero coupon cashflow valuation system is additive. Thus, the basic assumption that the value of swap A plus the value of swap B equals the value of swap A + B, leads inevitably to the zero coupon method. Indeed, so reasonable is this assumption, it may surprise some readers to know that non-additive swap valuation systems do exist. We now construct the discount function that defines the zero coupon pricing system for arbitrary cashflows, and explore its variations and subtleties. 1

Of course, the accounting answer is clear – the correct value is the prudent value i.e. £4.

6.2.

6.2

CONSTRUCTING THE DISCOUNT FUNCTION

93

Constructing the discount function

We shall be relying heavily on the use of the word “par” in much of the following. Consider how a swap market is defined. Taking as a concrete example the dollar market, rates are quoted as an annual Actual/360 fixed leg versus six month LIBOR. The standard maturities quoted are one, two, three, four, five, seven and 10 years. A “par five year swap” is defined as one starting at spot, maturing five years from spot, with an annual Actual/360 coupon at the current market rate, and a floating leg based on six month LIBOR paid semi-annually. In fact, the market rate used as the coupon to define the par swap is really the object that determines the “par-ness” of the swap. An off-market swap2 with the standard structure would not be a par swap. Thus we call the prevailing market rates the “par rates”. Clearly, a par swap exists, by definition, for one, two, three, four, five, seven and 10 year maturities. What about the “missing” maturities i.e. six, eight and nine years? The set of par swaps can be completed by defining a procedure for generating par rates in these maturities and using them as the relevant coupons. Next comes a crucial demand: A par swap must value to par. This means that the value of a par swap executed at the par rate should be zero – a reasonable requirement. In addition to par swap rates, money market rates are used to define the yield curve up to the one year point. We also call these par rates and insist, that the underlying instruments value to par.

6.2.1

Valuing LIBOR cashflows

We have conveniently glossed over one major obstacle to constructing the discount function. A swap is made up of a fixed leg plus a floating leg, and a par swap must value to par; how does one value LIBOR cashflows, given that they are unknown at the execution of the deal? To understand the solution to this dilemma, we must get ahead of ourselves slightly and assume that the discount factor for any date in 2

A swap with a coupon different from current market levels.

94


the future is known. Formally, we denote the value of the discount factor at a date Dt as Ft. Consider a single LIBOR cashflow due to have the rate set at a date D1 and paid on D2, after D1. Assume LIBOR is calculated on an Actual/360 basis. To make matters simple, let us assume that D2 is six months after D1, so that the relevant index is six month LIBOR. The problem is. we do not know what six month LIBOR will actually be on the date D1. The best solution, at any moment in time, is to use the market expectation of future rates to imply what LIBOR will be. This information is encapsulated in the forward rate This is the rate, derived from par rates, which the market believes will apply from D1 to D2 in the future. To calculate it, imagine that one has a cashflow C occurring at time D2. The forward rate is, by definition, the rate one uses to discount a cashflow from D2 to D1. Since we know the discount factor F1 at D1, we can discount the cashflow all the way back to the value date. From Equation 5.1. the present value of C is then: (6.1)

The denominator in Equation 6.1 looks a bit odd. The reason for the factor (D 2 — D1)/360 is that the forward rate is the rate applying over the period D1 to D2 – in this case, six months. The interest paid on a $1 deposit invested at the forward rate from D1 to D2 /100 (D2 – D1)/360 (as is expressed would therefore be $1 as a percentage). PV(C) can be re-expressed as:

(6.2) since the cashflow C occurs at D2. Equating Equations 6.1 and 6.2. and solving for gives:

(6.3) The part of Equation 6.3 dealing with day count basis depends on the market convention. One usually defines the accrual factor as

6.2.


95

the number of days between D1 and D2, divided by the number of days in the year, calculated according to the appropriate convention, divided by 100. Thus, for dollar swaps:

(6.4) where the number of days between D1 and D2 is calculated by actual number of days elapsed. We shall usually write as if the meaning of the subscripts is clear. Thus. Equation 6.3 becomes:

(6.5) Armed with a set of discount factors for any date required, and thus the implied forward rates for any period from Equation 6.5. it is possible to evaluate LIBOR cashflows. Under the assumption that market participants are able to both invest and borrow at LIBOR3. the market expectation for the value of LIBOR, calculated on $1,000,000 from D1 to D2 is Thus, the payment L implied from this is:

(6.6) However, since this payment is received at time D2, the present value of this cashflow is the right hand side of Equation 6.6 discounted by

F2:

(6.7) This value can be represented as a cashflow of $1,000,000 occurring at time D1 and a cashflow of –$1,000,000 at time D2 (see Figure 6.1). This makes sense because, if one actually did receive the principal at 3

This assumption is not true. However, the economic effect is insignificant due to the fact that within a generic swap portfolio, an approximately equal principal of swaps are paying LIBOR as are receiving LIBOR.

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Figure 6.1: LIBOR payment as principal flows D1, it could, in theory, be invested at LIBOR until D2, when it would mature as principal plus interest. Paying back the principal would leave the interest alone, which is the market estimate of the LIBOR payment. The valuation of LIBOR payments purely in terms of principals is very convenient. Taking a two year swap with LIBOR paid semiannually. the LIBOR payments can be represented as shown in Figure 6.2. So. except at the beginning and end of a swap, the principals cancel out 4 – leaving just two cashflows. Thus, the value of the LIBOR leg for this swap is just P (F s – F 2 Y ) . where P is the principal. Fs is the discount function at the start date of the swap, and F2Y is that at the two year point. Of course, the same result can be obtained by calculating all the forward rates and explicitly valuing the implied payments – in other words, calculating + ..., as the reader may verify.

6.2.2

Stripping the curve

Having described a framework for the evaluation of LIBOR cashflows. we can now demonstrate the idea of “stripping the curve” – generating 4

This is not. true of swaps where the principal changes through the life of the deal.

6.2.


97

Figure 6.2: LIBOR payments on a two year swap a zero coupon discount function from a yield curve. The necessary inputs for this process are: a. a set of par money market rates: b. a set of par swap rates. We presume that the structure of the associated instruments (i.e. their accrual basis, coupon frequency etc.) is denned. Later sections showhow to include other instruments in the stripping process. The conditions that the discount function must satisfy are: 1. par instruments must value to par: 2. the valuation process must be additive: 3. the resulting discount function must be “smooth”. The last condition is introduced for completeness – it is only relevant for denning interpolation procedures, a topic covered later in the chapter. To illustrate the construction of the discount function, we shall start by examining a one year par dollar swap with principal P. Let us assume that the one year par swap rate is R1Y = 8.5%. Interest is calculated on an Actual/360 basis, and the swap is against six month

CHAPTER 6.

98

ZERO COUPON PRICING

Figure 6.3: A one year par swap LIBOR. The cashflows on the swap are represented in Figure 6.3. Note that the LIBOR cashflows on the swap are incorporated using principal cashflows at the beginning and end of the swap. The value of the one year swap. S1Y, is given by:

(6.8) From condition (1) above:

(6.9) Since the instrument is par, the start date of the swap, D0, is spot, and so F0 = 1. Using R1Y = 8.5%, and assuming that D1Y = D0 = 365, 0.92066. gives F1Y Now consider a two year par swap, S2Y, with rate R2Y = 9%. Proceeding as before:

(6.10)

6.2.


99

Once again, principal flows at the start and end of the swap have been used to represent LIBOR cashflows. The reader should note that the stripping, or “bootstrapping”, element enters via the fact that to calcu– late F1Y requires knowledge of F1Y. Similarly, calculating F3Y requires knowledge of F1Y and F2Y. Insisting that PV(S2Y) = 0, implies:

(6.11) The value of F1Y is known from Equation 6.9, and, if D2Y – D1Y = 365, 0.8394. then F2Y Similarly, for the discount function at the three year point: (6.12) This procedure generalises to give the discount factor for an n year point:

(6.13) The index i in the above equation runs from the first coupon on the swap (the one year point in this case) to the last but one coupon. So, in the case of a one year swap, the summation is zero. Notice that this equation only assumes that the floating leg starts when the first coupon begins accruing, and ends when the last coupon is paid. No assumption is made as to the frequency of the LIBOR payments. Thus we can generate a set of Fi’s, giving the value of the discount function at the maturities of each of the par swaps. As pointed out earlier, if R5Y = 9.5% and F7Y = 10.5%. we could generate a par six year swap rate by defining F6Y = (R5Y + R7Y ) = 10%. Using this procedure, a complete set of Fis for i = 1Y ... 10Y can be generated. So far. nothing has been said about the method used to define the yield curve up to one year. Equation 6.12 already contains the answer, since the money market instruments used have only a single coupon

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ZERO COUPON PRICING

at maturity. Thus, for such an instrument with rate Rm and maturity Dm:

(6.14) The set of discount factors formed by the Fm and FnY define the value of the discount function at a discrete number of points in time. To find the value of the discount function at dates between these points requires specifying a sensible interpolation procedure. This is covered in detail later on – for the moment, just assume that such a procedure exists. Then, the value of an arbitrary set of cashflows {Ci} occurring on dates Di is: (6.15) It is clear from this definition that the zero coupon valuation system is additive. Furthermore, by construction, the procedure values par instruments at par. It therefore satisfies our criteria. The reader may wonder why the term “zero coupon” arises. The answer lies in the fact that the Fi are the fair prices for zero coupon bonds maturing at Di. This is because a zero coupon bond is a single cashflow instrument – the single cashflow is repayment of the principal at maturity. The value of this cashflow, and thus of the bond, is therefore Fi times the face value.

6.2.3

A worked example

This section sets out a detailed example of a discount function under the following scenario: Currency: Dollars Grid point

O/N 1W 1M 2M 3M 6M

–

Date 18-May-90 24-May-90 18-Jun-90 17-Jul-90 17-Aug-90 19-Nov-90

Value date: 17 May 1990 Par rate

Instrument

8.3125

8.25 8.3125 8.375 8.375

8.5

Money Market

6.2.

CONSTRUCTING THE DISCOUNT FUNCTION 1Y 2Y 3Y 4F 5Y

17-May-91 18-May-92 17-May-93 17-May-94 17-May-95

8.75 9.08 9.24 9.34 9.42

101

Swap

A few points about the above data are worth making: – for the dollar market, we shall adopt the term “grid point” to mean any of the following dates: the maturity dates of overnight, one week, one, two, three and six month deposits and the maturity dates of swaps for a whole number of years; – the abbreviations for grid points are mainly self-evident. nM and mY refer to the n month and m year point respectively. O/N stands for “overnight” and is market terminology for the next business day. 1W is the one week point: – “value date” means the date, for valuation purposes, taken to be “today” – thus, for most currencies, it is spot i.e. two business days after today, whereas for sterling it actually is today; – the dates in the second column are adjusted for weekends on a modified following basis; – as before, all the par rates are on an Actual/360 basis. The one year swap has a floating leg based on three month LIBOR. All the other swaps have floating legs based on six month LIBOR. Using Equation 6.14. it is easy to generate all the discount factors from O/N up to and including 6M. For example:

102

CHAPTER 6.

Grid point O/N 1W 1M 2M 3M 6M 1Y 2Y 3Y 4Y 5Y

ZERO COUPON PRICING

Discount factor 0.9998 0.9984 0.9927 0.9860 0.9790 0.9579 0.9185 0.8379 0.7637 0.6953 0.6323

Table 6.1: Discount factors for a dollar curve The one year swap looks essentially the same as a money market instrument, since it only has a single coupon. Thus:

The two year swap has two coupons, and so:

The reader could calculate the remaining discount factors and check them against the results in Table 6.1. Having generated the discount function, let us check explicitly that the three year swap values to par. From Figure 6.4. the cashflows are:

6.2.


103

Figure 6.4: Cashflows on a three year par swap Time D0 D1Y D2Y D3Y

Cashflow –10,000,000.00 936,833.33 941,966.67 10,934,266.67

where we have assumed P = $10,000,000, and so the value of the swap is:

up to rounding errors.

6.2.4

A more complicated example

Although the generic dollar discount function displays most of the usual features of zero coupon stripping, there are a few subtleties which arise in more complicated situations. These can be illustrated using the example of sterling interest rate swaps. We first define the sterling instruments:

104


Money market instruments: The most commonly used maturities are O/N, 1W, 1M, 2M, 3M and 6M. As usual, these are single coupon instruments, accruing on an Actual/365(Fixed) basis. The “Fixed” means that, for periods including part of a leap year, the actual number of days is divided by 365, not 366. For the remainder of this chapter, we shall drop this label, and write Act/365. Swaps: Sterling swaps are quoted for 1Y, 2Y, 3Y, 4Y, 5Y, 7Y and 10Y maturities. The one year swap is quoted on an annual Act/365 basis for the fixed leg, versus three month LIBOR. The remaining swaps are semi-annual Act/365 versus six month LIBOR. All dates are generated on a modified following business day convention. Recall that sterling trades have a value date of today, not spot. The complications with sterling arise out of the fact that the par swaps are semi-annual coupon instruments. When creating the dollar curve, the iterative process for generating the five year point required knowledge of the discount factors at all the previous coupon dates i.e. the one, two, three and four year points. By analogy, for sterling, we will need to know the values of the discount function at each semi-annual point. For example, a two year swap will have coupons occurring at six months, one year and 18 months, in addition to two years. Therefore the missing par rates must be “filled in” , by generating an 18 month rate, a year rate, and so on. Leaving the 18 month rate for a moment, the most sensible way to generate a year rate is by straightforward linear interpolation: (6.16) The

year rate is likewise: (6.17)

since the two observed (i.e. usually quoted) par rates it lies between are those for seven and ten years.

6.2,


105

Figure 6.5: A one year semi-annual par swap The 18 month point requires a bit more thought. It lies between an annual Act/365 rate (the one year point) and a semi-annual Act/365 rate (the two year point). To interpolate, the rates must be expressed on the same basis. We choose to create the rate R1Ysa at which a semi-annual one year par swap would value to par (other choices are possible). The par swap cashflows are shown in Figure 6.5. The value of this swap is:

(6.18) This rate is a function of known discount factors. Having determined R1Ysa, the 18 month rate is defined5 as:

(6.19) Our example is based on the scenario outlined in Table 6.2. Determining 5

rate.

Another approach would be to treat the 18 month rate as an observed market

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CHAPTER 6.

ZERO COUPON PRICING

Value date: 17 May 1990 Grid point Date 18-May-90 O/N 1W 24-May-90 1M 18-Jun-90 2M 17-Jul-90 3M 17-Aug-90 6M 19-Nov-90 1Y 17-May-91 2Y 18-May-92 3Y 17-May-93 4Y 17-May-94 5Y 17-May-95 19-May-97 7Y 10Y 17-May-2000

Par rate 14.25 14.75 15 15.125 15.15625 15.375 15.55 14.25 13.95 13.85 13.75 13.55 13.10

Table 6.2: A sterling yield curve

6.2.


107

the discount factors from the single coupon instruments is easy. Equa– tion 6.14 applies, except with: (6.20) for sterling. Thus, for example:

The one year swap is not a money market instrument, but since it has only one coupon the same formula holds:

The 18 month par rate (for which the grid point date is 18 November 1991) follows from Equation 6.18:

and thus: (6.21) To find the discount function at this point, we need to rewrite Equation 6.13 for semi-annual coupon swaps:

(6.22)

108

CHAFTER 6. ZERO COUPON PRICING

where i = 1.5Y, 2Y, 2 . 5 Y , . . . , 10Y and j also runs in six month steps. So: (6.23) where the summation in the numerator is:

It is now a straightforward, although tedious, exercise to generate the discount factors at the remaining semi-annual points. The results up to five years are displayed in Table 6.3.

6.3

Interpolation

Before proceeding with the integration of futures and swap discount functions, we need to explain the interpolation process used. This is required in order to determine the value the discount function takes at a non-grid point date. For example, take the curve defined by Table 6.2. Suppose we require the discount factor implied by the curve at Da, 20 June 1990. A glance at the table shows that this date lies between the grid points D1M and D2M 16 June 1990 and 17 July 1990. The simplest approach is to draw a straight line between the discount functions at these dates as shown in Figure 6.6. The actual number of days from D1M to D2M is 31, and from D1M to Da is 4. Referring to Table 6.3, F1M is 0.9870 and F2M is 0.9753. Clearly: (6.24) Generally, if Db lies between D1 and D2 with discount factors F1 and F 2 , then: (6.25)

6.3.

109

INTERPOLATION

Grid point O/N 1W 1M 2M 3M 6M 1Y 1.5Y 2Y 2.5Y 3Y 3.5Y 4Y 4.5Y 5Y

Discount factor 0.9996 0.9972 0.9870 0.9753 0.9632 0.9273 0.8654 0.8088 0.7598 0.7120 0.6688 0.6261 0.5871 0.5503 0.5167

Table 6.3: Discount function for the curve in Table 6.2

Figure 6.6: A linearly interpolated function

110


This simple procedure is known as linear interpolation. For “back of the envelope” calculations, this method is adequate. However, more sophisticated approaches are possible. The key to these is to note that a generic discount curve has the look of an exponential function e–r (try plotting a graph of e–x for x taking values from 0 to 1). A mathematical proof of this assertion is given in Appendix A. Consequently, the discount factor between grid points is described better by an exponential curve than by a straight line. In view of this, we define a generic discount factor: (6.26) where:

(6.27) and is a number between 0 and 1. depending on the position of Da relative to the surrounding grid points D1 and D2:

(6.28) (6.29) and so:

(6.30)

(6.31) (6.32) is the exponentially interpolated discount function. Applying this formula to our example. (6.33)

6.4.

INCORPORATING FUTURES

111

Figure 6.7: Exponential interpolation which is slightly smaller than the value obtained from Equation 6.24 by the linear method. This is to be expected, since the exponential curve lies underneath the linear curve away from grid points (see Figure 6.7). Henceforth, “interpolation” is taken to mean exponential interpolation unless stated otherwise.

6.4

Incorporating Futures

So far. we have been concerned with generating a discount function using two sets of instruments: money market (i.e. loan and deposit rates) and interest rate swaps. In many currencies, other liquid instruments exist which can also be included in the stripping process. This section examines the most important example of these, futures. Consider the sterling market. In addition to money market and swap rates, there are short sterling and long gilt futures available. There are, at the time of writing, twelve short sterling contracts, covering a period of two years. There are four long gilt contracts in play at any one time, although usually only the nearest two trade actively.

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ZERO COUPON PRICING

The fact that some futures contracts are highly liquid means that the information they provide about interest rates is more useful than that provided by, say, money market rates. The inclusion of futures in the stripping process thus better reflects current market conditions. Unfortunately, gilt futures (and bond futures in general) are of no real use to us in generating the zero coupon discount function. The main reason is that, while they may give information about long term gilt yields, the swap rate will be determined as some margin over the gilt yield. However, swap spreads in this part of the yield curve are usually difficult to determine – especially those over notional gilts. Since the period of the yield curve over ten years is rarely traded, long term swap rates are usually quoted either as an outright rate, or as a margin over a liquid gilt.

6.5

The futures strip

Having excluded bond futures from the generation of the discount function, this section moves on to the case of three month interest rate futures, taking the sterling market as a specific example. The procedure generalises easily to other cases by adopting the appropriate accrual and date conventions. Consider the following scenario: Value date: 17 May 1990 Date Grid point Price D1F 20-Jun-90 P1 = 84.80 D2F 19-Sep-90 P2 = 85.05 19-Dec-90 P3 = 85.55 D3F D4F 20-Mar-90 P4 = 86.05 D5F 19-Jun-91 P5 = 86.53 18-Sep-91 P6 = 86.95 D6F D7F 18-Dec-91 P7 = 87.20 18-Mar-92 P8 = 87.35 D8F 17-Jun-92 D9F In addition. Rs = 15.125% is the money market rate applying from the value date to 20 June 1990 i.e. the one month and three day rate. The reason for its presence will soon become clear.

6.5.

113

THE FUTURES STRIP

P1 tells us about the forward rate from 20 June to 19 September 1990. From Equation 6.5 we have: (6.34) where F1F and F2F are the discount factors at D1F and D2F. Note that D1F is the settlement date of the first futures contract. So. finding F2F requires that F1F be known. From Equation 6.14: (6.35) which gives: (6.36)

In this manner subsequent discount factors are generated using: (6.37) where P n–1 is the price of the (n — l)th future. It is a simple matter to derive the following table: Grid point D1F D2F D3F D4F D5F D6F D7F D8F D9F

Discount factor 0.9861 0.9501 0.9160 0.8841 0.8544 0.8266 0.8006 0.7758 0.7521

114

6.6


Integrating the curves

Having established a procedure for interpolating points on the discount. curve, we can integrate the futures and swaps curves. The single most important parameter in this procedure is the number of futures to be used in generating the futures discount function. If none are used, the discount function is generated as before. The decision regarding how many futures to use is really a trading issue. The following example incorporates the first four futures only6. Going back to the example of the previous sections, the first step is to order the grid points, including the first five futures dates. These are set out in Table 6.4. The first thing to explain is what the column headed “Used?” means. Take the 6M grid point, for example. This lies between the 2F and 3F grid points. Since we are using the first four contracts, both the P1 and P2 prices are used in generating the curve. This implies that the 6M point found by interpolating the futures discount function overrides that defined by the cash curve. In fact, any grid points that lie on or between D1F and D5F are determined by the futures curve only. Thus, the 2M, 3M, 6M and 1Y discount factors are all generated from the futures curve. These points are labelled “N” (for “no”) in the table. We call them “redundant” grid points. Those labelled “Y” (for “yes”) are termed “non-redundant”. Consider now the two year discount factor. Recall that it depends on the discount factors at the 6M, 1Y and I.5Y points (see Equation 6.13). However, the 6M and 1Y points are now redundant – so the 1Y discount factor now depends on futures generated discount factors. This is true for all the swap discount factors (i.e. from two years onwards) since they are all functions of F6M and F1Y. The 18 month rate depends on the one and two year rates. However, in the above scenario, the one year point is redundant – it is generated from the futures discount function. By inference, the 18 month rate is also a function of the futures discount function. Our first task is to find the six month discount factor. It lies between 6

It is implicit in our discussion that a continuous strip of futures is used. Thus, we do not cover the case in which the first, second and fourth contracts are used. but not the third.

6.6. INTEGRATING THE CURVES

Grid point O/N 1W 1M 1F 2M 3M 2F 6M 3F 4F 1Y 5F 1.5Y 2Y 2.5Y 3Y 3.5Y 4Y 4.5F 5Y

Date 18-May-90 24-May-90 18-Jun-90 20-Jun-90 17-Jul-90 17-Aug-90 19-Sep-90 19-Nov-90 19-Dec-90 20-Mar-91 17-May-91 19-Jun-91 18-Nov-91 18-May-92 17-Nov-92 17-May-93 17-Nov-93 17-May-94 17-Nov-94 17-May-95

Used? Y Y Y Y N N Y N Y Y N Y Y Y Y Y Y Y Y Y

115

Discount factor 0.9996 0.9972 0.9870 0.9861 0.9752 0.9630 0.9501 0.9270 0.9160 0.8841 0.8649 0.8544 0.8085 0.7598 0.7120 0.6688 0.6261 0.5872 0.5504 0.5168

Table 6.4: Combined swap and futures dates

116


D2F and D3F and, using Equation 6.32:

Similarly, we obtain:

To obtain the 1.5Y rate, we use Equation 6.18:

and then:

Then from Equation 6.22:

Once F6M, F1Y and F1.5Y are known, the remaining discount factors are all straightforward to generate. These are presented in the fourth column of Table 6.4. We have now achieved our goal of integrating the curves. The general procedure for the case of futures used ( > 0) is:

6.7.

117

OTHER CURVES

1. Generate the discount function for all cash dates occurring on or before the first futures date, D1F. 2. Extend the discount function as far as the settlement date of the last futures contract being used, 3. Determine which, if any. cash or swap dates occur between D1F and interpolate them from the futures generated and discount function. 4. Imply a full set of rates R such that etc. 5. Strip the curve using the usual iterative formula. The above procedure will vary slightly for different currencies. For example, for the dollar curve using annual par swaps, we only need to ascertain the annual discount factors. Thus, the 18 month issue does not arise in this case.

6.7

Other curves

The cases covered so far – money market, swap and futures curves – are adequate for most situations one is likely to encounter7. However, a few other possibilities do exist. One such is constructing a discount function from an FRA curve. Consider the following set of prices for sterling FRAs: Value Date: 17 May 1990 3M 3-6 6-9 9-12 12-18 18-24 7

15.15625 15.05 14.60 14.10 13.60 12.90

The dollar market, which usually quotes swap rates as a spread to Treasuries, can be treated as in section 6.3. The par rates are created by adding the spread to the Treasury yield (ensuring that they are expressed in the same basis), and one then proceeds as before.

118


The price of a 9-12’s FRA is therefore 14.10%. The information in the above table presents us with a set of dates and forward rates between them (except for the three month rate which is from “today” to the three month point). Equation 6.5 then generates discount factors, just as it did for futures. Straightforwardly, F3M = 0.9632. where the 3M grid point occurs on 17 August 1990. Since the 6M point is 19 November 1990:

One point needs to be borne in mind about FRAs: by convention, for a 6-9’s FRA say. if the six month point is not a “good day”, i.e. 17 November 1990 is not a good business day, then the maturity of the FRA is denned as being three months from the start date adjusted under the modified following convention. Thus, in our example. 17 November is a bad day so the FRA date rolls to 19 November. The nine month point is therefore 19 February 1991, if this is a good day (it is). So:

The 9-12’s FRA runs from 17 February to 17 May 1991. However, 17 February is a bad day – it rolls the 18th. Thus, the end date is 18 May – also a bad day – which rolls to 20 May. Thus:

The 12-18 and 18-24 FRAs are treated analogously, the only difference being that they are six month, instead of three month. FRAs. The

6.7.

OTHER CURVES Grid point 3M FRA6M FRA9M FRA12M FRA18M FRA24M

119 Date 17-Aug-90 19-Nov-90 19-Feb-91 18-May-91 18-Nov-91 18-May-92

Discount factor 0.9632 0.9273 0.8944 0.8640 0.8083 0.7594

Table 6.5: An FRA generated discount function reader could verify Table 6.5. We have used the notation “FRA nM ” to denote the fact that the grid point is not the naive n month point, due to the FRA roll convention. One could now proceed as before to integrate this with other curves.

Chapter 7 Valuing a swap The techniques of the last chapter provide a framework for assigning a value to the discount function at any point in the future. All that remains, as far as valuing a swap goes, is to determine the cashflows which are to be discounted and to multiply each cashflow by the relevant discount factor. This chapter provides a brief guide to valuing all the standard, and most of the non-standard, swap structures. Although only relatively straightforward cases will be analysed, it should be possible to value any swap using the methods of this and the previous chapter.

7.1

The bid-offer spread

Throughout this chapter, the language of discount functions is used in valuing swaps. Implicit in the construction of this function are the input par rates. Clearly, one would not generally be prepared to pay and receive fixed at the same rate in a given maturity. For example, for a par five year swap a trader might pay (bid) 13.7% and receive (offer) 13.8%. The relevant side of the bid-offer spread would then be needed when building the discount function. In most of the examples in this book, mid-market rates have been used to simplify matters. In real life, the discount function will be constructed using bids and offers in different maturities as appropriate. Knowing which side to use in a given maturity requires knowledge of the delta vector – this breaks

121

CHAPTER 7.

122

VALUING A SWAP

down the problem into the language of equivalent swap positions in par maturities. The delta vector and equivalent positions, the central topics of the next chapter, tell you that a structure is equivalent to paying on par swaps in certain maturities and/or receiving on par swaps in others. For example, suppose one were pricing a swap that gave the following non-zero equivalent positions: Date 6M 1Y 2Y

Equivalent position (10,000,000) 5,000,000 (15;000,000)

This says that the swap being priced is equivalent to paying on a par £10,000,000 six month deposit and a par £15,000,000 two year swap, and receiving on a par £5,000,000 one year swap. Thus, the trader’s bid side would be used for the input 6M and 2Y rates, and his offered side for the 1Y rate.

7.2

The fixed leg

It will suit us, for the purpose of analysing more complex structures, to break a generic swap up into its fixed and floating legs. The former is by far the simplest of the two to construct, since every cashflow is known with certainty. What complexity there is arises in the type of accrual factor being used. As a reminder of the material of chapter two, the three most commonly used accrual factors are: Actual/360: actual number of days in the coupon period, divided by 360; Actual/365 (Fixed): actual number of days in the coupon period, divided by 365; 30/360: number of days in the coupon period, assuming1 all months have 30 days, divided by 360 – also known as “360/360” or “Eurobond basis”. 1

See page 12 for an exact definition of 30/360 basis days.

7.3.

THE FLOATING LEG

123

As an example of the different accrual factors, consider the 91 dayperiod from 11 November 1991 to 10 February 1992. Thus Actual/360 is 91/360 and Actual/365(Fixed) is 91/365. If we treat all months as having 30 days, the 30/360 accrual factor is (19 + 30 + 30 + 10)/360 = 89/360. The fixed leg can, of course, consist of any (known) cashflow one likes, together with the dates on which they occur. Whatever its form. the value is just with being the fixed cashflows occurring on dates Di.

7.3

The floating leg

As explained in chapter six, there are essentially two equivalent methods of valuing the floating leg of a swap. One is to calculate the implied floating payments throughout the life of the swap, and to value them as explicit cashflows. The other is to represent LIBOR cashflows as flows of principal at the relevant dates. Since the .second approach is less time consuming and more flexible, we have adopted this method. Section 6.2 explained the method used to value the floating leg on a standard swap – for example, if one were receiving the floating leg, its value over the period Da to Db would be P(Fa – Fb), P being the notional principal of the swap. If the principal remains the same over the life of the swap, the value of the floating leg is then P(Fstart – Fend), “start” and “end” referring to the start and end dates of the swap. It is not difficult to extend this to the case when the swap principal changes over the life of the swap. If the principal applicable from Di to Di+1 is Pi, then the value of the floating leg is: P 1 (F 1 – F2) + P2(F2 – F3) + . . . + Pn–1(Fn–1 – Fn)

(7.1)

D1 and Dn being the start and end dates of the floating leg of the swap. This is more commonly written as: F1P1 + F2(P2 – P1) + . . . + Fn–1(Pn–1 – P n – 2 ) – FnPn–1

(7-2)

which more clearly shows the cashflows that need to be valued. The only situation where one needs explicitly to calculate floating payments, is when the current LIBOR period has had its rate fixed.

CHAPTER 7.

124

Date

l8-Jun-90 17-Dec-90 17-Jun-91 17-Dec-91 17-Jun-92

Fixed cashflows

Cashflows to replicate LIB OR before first setting (10,000,000)

VALUING A SWAP Cashflows to replicate LIBOR after first setting (10,417,083)

940,333 945,500

10,000,000

10,000,000

Table 7.1: Cashflows to replicate LIBOR payments

Such payments are calculated on an Actual/360 or Actual/365 basis, depending on the currency (see chapter two). Note that once a floating payment is set. the remainder of the (unknown) floating cashflows are valued by “rolling forward” the principal. Table 7.1 illustrates this procedure by showing the cashflows required to replicate the floating leg of a two year dollar swap both before and after the first LIBOR has been set at 8.25% (Actual/360). The LIBOR setting requires two changes to be made to the replicating cashflows: first, the initial LIBOR payment of $417,083 is explicitlyentered on the date on which it is to be paid i.e. 17 December 1990; second, the swap principal of $10,000,000 is rolled forward to the start date of the next LIBOR period, again 17 December 1990.

7.4

Special features

7.4.1

LIBOR margins

Since corporates usually have to raise funds at a margin over LIBOR, it is often necessary for them to enter into a swap whereby they pay a fixed rate in return for receiving the same margin over LIBOR. To build this feature into the construction of the cashflows is easy – since the margin is known with certainty, each of the margin cashflows can

7.4. SPECIAL FEATURES

125

be valued as if it were a fixed cashflow. As an example, consider a sterling swap with principal P, on which one pays a floating rate of six month LIBOR plus 25 basis points. The cashflows corresponding to this are a normal floating leg of LIBOR flat (in other words, just the normal principal flows), plus a “pseudofixed leg”, paying a fixed rate of 25 basis points semi-annually on an Actual/365 basis. Of course, there will probably be a normal fixed leg, constructed in the standard way, and added to the margin cashflows. If the fixed leg accrues on the same basis as the floating leg (as is the case for generic sterling swaps), the margin can simply be subtracted from the fixed rate. For our example, the value of each margin will be:

7.4.2

Back-set and compounded LIBOR

While the overwhelming majority of swaps have the standard “set in advance, paid in arrears” type of floating leg, one occasionally encounters more esoteric methods of payment. In this section, the two most common variations – back-set and compounded LIBORs – are examined. Back-set LIBOR payments are ones for which the rate is fixed at the end of the calculation period, and paid for value the same date. In other words, they are “set and paid in arrears”. For example, consider a six month period Da to Db, with six month LIBOR fixed, on date Db, at R%. The payment amount due at time Db is P a,b R, for a swap principal P. The rate R, being the rate prevailing at Db, would, normally have applied to the period from Db to Dc, six months after Db. Thus, in order to value a general back-set LIBOR payment for the period Da to Db in the future, one must evaluate the forward rate for Then using: one period further on:

(7.3) the value of this back-set LIBOR payment is:

(7.4)

CHAPTER 7.

126

Grid point 6M 1Y 1.5Y 2Y 2.5Y 3Y 3.5Y 4Y 4.5Y 5Y 5.5Y

Date 19-Nov-90 17-May-91 18-Nov-91 18-May-92 17-Nov-92 17-May-93 17-Nov-93 17-May-94 17-Nov-94 17-May-95 17-Nov-95

Discount factor 0.9273433 0.8654262 0.8088405 0.7597755 0.7119726 0.6687732 0.6260895 0.5871113 0.5503178 0.5167149 0.4849783

VALUING A SWAP

Forward rate 14.58882 13.80273 12.95114 13.39161 13.02607 13.52385 13.38803 13.26270 13.11414 12.98117

Table 7.2: Implied sterling forward rates

The part of the above equation involving discount factors is:

(7.5) This is not linear in the Fi’s2. By comparison, Equation 6.7 on page 95 was linear and therefore allowed standard LIBOR payments to be represented by two principal flows. Hence, for back-set LIBOR payments, one must resort to explicitly calculating the payments. Equation 7.4 shows that the last floating cashflow depends on the discount function at a date later than the maturity on the swap. The fair price for a swap with back-set LIBOR payments can be substantially different from that for a standard swap. The following example demonstrates that, with the sterling yield curve of chapter six. the fair price for a five year swap is reduced by approximately 30 basis points if LIBOR is set in arrears. Table 7.2 sets out the grid points, discount factors and implied forward rates for this yield curve. The implied forward rates are calculated 2

In other words, it is not of the form:

for

constants.

7.4

SPECIAL FEATURES Date

Fixed cashflows

19-Nov-90 17-May-91 18-Nov-91 18-May-92 17-Nov-92 17-May-93 17-Nov-93 17-May-94 17-Nov-94 17-May-95

686,080 660,260 682,392 671,326 675,014 667,637 678,703 667,637 678,703 667,637

127 Implied floating cashflows (743,430) (676,901) (656,428) (667,746) (653,088) (670,635) (674,903) (657,685) (661,096) (643,724)

Net cashflows

PV

(57,350) (16,641) 25,964 3,580 21,927 (2,998) 3,800 9,952 17,606 23,913

(53,183) (14,401) 21,001 2,720 15,611 (2,005) 2,379 5,843 9,689 12,356

Table 7.3: The value of a back-set LIBOR deal

using the usual formula:

(7.6) (7.7) So for example, the implied 6-12 forward rate will be:

Table 7.3 displays the fixed cashflows and implied floating cashflows for a £10,000,000 five year swap, valued at 17 May 1990. LIBOR is set in arrears and the fixed rate is 13.4643%. The implied floating cashflow occurring on the six month date, 19 November 1990 is calculated using the market’s expectation of what six month LIBOR will be on that date i.e. the implied 6-12 forward rate. So the first implied floating cashflow is equal to:

128

CHAPTER 7. VALUING A SWAP

The right hand column of Table 7.3 shows the PV of the implied net cashflows. The total of these is £10, which shows that 13.4643% is indeed the fair rate for the swap. As stated earlier, this is 30 basis points below the par five year rate of 13.75%. This is not surprising since an inverse yield curve embodies the market’s expectation of falling interest rates. Hence the market expects back-set LIBOR settings to be below standard LIBOR settings (which would be set six months earlier). Therefore the fair fixed rate, which is compensation for paying LIBOR. is lower for the swap against back-set LIBOR. Compounded LIBOR payments tend to arise only within the dollar swap market. An example would be three month LIBOR, compounded quarterly and paid semi-annually. To explain, imagine a semi-annual swap with two consecutive payment dates Da and Dc. The date three months after Da is Db. If three month LIBOR is set on Da at La and on Db, at Lb then, by definition, the payment due on the date Dc is:

where P is the principal of the swap. payment is:

Before Da the value of this

But this is exactly the same as the value of a six month LIBOR payment to be set at Da and paid on Dc. Hence, prior to Da, six month LIBOR is equivalent to three month LIBOR compounded quarterly and paid semi-annually. The difference lies in the value of the payment as calculated between Da and Db. The compounded LIBOR has a value which depends on Fb and Fc whereas the standard LIBOR (which would have already been set) has a value which depends only on Fc.

7.4.3

Amortising and rollercoaster swaps

Swaps in which the notional principal decreases throughout the life are called “amortising”. These have become increasingly popular as a means of hedging leasing transactions. Similarly, those on which the

7.4. SPECIAL FEATURES Date

Principal

17-May-90 19-Nov-90 17-May-91 18-Nov-91 18-May-92 17-Nov-92 17-May-93

10,000,000 8,000,000 6,000,000 7,000,000 8,000,000 9,000,000

129

Fixed cashflow

714,953 550,437 426,666 489,705 562,738 626,161

Cashflow to replicate LIBOR (10,000,000) 2,000,000 2,000,000 (1,000,000) (1,000,000) (1,000,000) 9,000,000

Total

(10,000,000) 2,714,953 2,550,437 (573,334) (510,295) (437,262) 9,626,161

Table 7.4: A rollercoaster swap

principal increases over the life are called “accreting”. These are both special cases of “rollercoaster” swaps. As their name implies, these have the principal increasing and decreasing over the life of the swap. As an example, consider the swap shown in Table 7.4 which has an initial principal of £10,000,000. The principal reduces by £2,000,000 for the next two periods, and then increases by £1,000,000 for the last three. The fixed rate is 14.03%. The important point to note is how the LIBOR cashflows are modelled by the cashflows shown in the fourth column. Each cashflow shown is the principal on the previous period minus the principal on the current period. This is an application of Equation 7.2. The right hand column of Table 7.4 shows the total cashflow to be valued. With the familiar sterling discount function, the total PV of these cashflows is £137, which shows that 14.03% is very near to the fair swap rate.

7.4.4

Currency swaps

The final category of swap structure that we shall examine is the currency swap. These come in essentially three types: Fixed-Fixed: Each set of known fixed cashflows is valued on the relevant yield curve to give present values in different currencies. One

130

CHAPTER 7. VALUING A SWAP of the values is then converted into the other currency using the current spot exchange rate. Note that arbitrage between fixedfixed currency swaps and the forward foreign exchange market may be possible and is discussed in section 9.8.

Floating-Floating: Floating-Floating currency swaps are also called basis swaps. Given the fact that LIBOR legs value to par. valuing such swaps is easy. For a swap of LIBOR flat in one currency against LIBOR flat in another, the value is zero (until the first settings are made). In reality, as for all swaps, there is a bid-offer spread for basis swaps. For most pairs of currencies, the bidoffer spread would straddle a price of LIBOR flat against LIBOR flat. For example, the sterling-dollar basis swap might be bid at sterling LIBOR minus two basis points against dollar LIBOR flat and offered at sterling LIBOR plus six basis points against dollar LIBOR flat. However, basis swaps sometimes trade away from “flat-flat” – an example being the yen-dollar market. Typically, this market might be quoted as yen minus three to yen minus ten3. The reason has traditionally been an imbalance in the capital markets, with a great many yen bond issues being swapped into dollar LIBOR – this means that banks will have paid yen against receiving dollar LIBOR. and will therefore be keen to pay dollar LIBOR against receiving yen. But even this market cannot trade too far away from LIBOR flat against LIBOR flat – for at some stage, arbitrage against loans and deposits becomes economic (see section 5.6). Fixed-Floating: Under the assumption that cross currency swaps are trading at LIBOR flat against LIBOR flat, fixed-floating currency swaps are easy to value. Take, for example, a fixed sterling against floating dollar swap. The sterling cashflows, including the principal exchanges, are valued using the sterling discount curve. The unknown dollar LIBOR payments are. as usual, replicated by imaginary payments of principal. In most cases, these imaginary 3

In other words, banks are prepared to pay yen LIBOR minus ten basis points against dollar LIBOR flat and to receive yen LIBOR minus three basis points against dollar LIBOR flat.

7.5. PRICING THE SWAP

131

payments cancel with the actual initial and final exchanges of dollar principal. Any payments that do not cancel (e.g. dollar margin payments) are valued using the dollar discount function. The sterling value of the entire swap is the value of the fixed leg plus the sterling equivalent of the value of the floating leg. When the associated basis swap market is not quoted at LIBOR fiat against LIBOR flat, then a fixed-floating currency swap can be valued by breaking it down into an interest rate swap plus a basis swap that does value to zero. For example, suppose that the yen-dollar basis swaps are trading at yen LIBOR minus seven basis points against dollar LIBOR flat. Then a five year swap of fixed yen at 7.20% against dollar LIBOR fiat is equivalent to a five year yen interest rate swap at 7.13% plus a basis swap of yen LIBOR minus seven basis points against dollar LIBOR flat. The interest rate swap can be valued using the yen discount curve and, by assumption, the basis swap has zero value.

7.5

Pricing the swap

In almost all trading situations, it is not the present value itself which is of most relevance to either the trader or customer - it is the fixed rate at which the trader will deal. Occasionally, the fixed rate might be given and the trader be asked to quote a margin on the floating leg. The fair rate is defined, for our purposes, as the rate R which generates swap cashflows with a present value of zero. Now most swaps have zero value when the fixed and floating legs have equal and opposite PV. Of course, the PV of the floating leg is independent of the fixed rate and the PV of the fixed leg is directly proportional to the fixed rate. Hence, if R% is the fair rate for an interest rate swap:

R

PV(Fixed leg with fixed rate = 1%) + PV(Floating leg) = 0 – PV(Floating leg) (7.8) R= PV(Fixed leg with fixed rate = 1%)

For standard (non-rollercoaster) interest rate swaps, this becomes:

(7.9)

CHAPTER 7.

132

Date

Fixed cashflow

PV

if R = 1% 18-May-92 17-Nov-92 17-May-93 17-Nov-93 17-Mav-94 17-Nov-94 17-May-95 Total PV

50.137 49,589 50,411 49,589 50,411 49,589

35,696 33,164 31,562 29.114 27,742 25,623 182,901

VALUING A SWAP

Cashflow to replicate LIBOR

PV

(10,000,000)

(7,597,755)

10,000,000

5,167,149 (2,430,606)

Table 7.5: A three year sterling swap two years forward where the index i runs from the first fixed coupon date to the maturity date. As an example, Table 7.5 displays the pricing, with the usual sterling yield curve, of a three year sterling swap starting two years after the value date of 17 May 1990. From Equation 7.8 the fair rate for this swap is:

R=

2,430,606 = 13.289% 182,901

Similarly, when the fixed rate is given, and the swap price is to be quoted in terms of a LIBOR margin, then the fair margin, M basis points, is given by: M=

PV(Swap with zero margin) PV(Margin cashflows when margin is 1 basis point)

Chapter 8

Interst

rate exposure

Imagine that as a swap trader you own a portfolio of swaps. You will be most concerned with two things: the value of the portfolio and how that value is likely to change. Earlier chapters explained in some detail how to calculate the value of a swap, and hence the value of a portfolio of swaps. This chapter is about the effect of movements in rates and the “delta vector” for a swap portfolio. The delta vector measures the manner in which the value of a swap (or interest rate option) portfolio depends upon interest rates. For example, is a given portfolio more sensitive to a one basis point movement in three year rates than to an equal movement in seven year rates? This is exactly the type of question to which swap traders and other sophisticated users of swaps require an answer. Next the delta vector is re-expressed as a set of “equivalent positions”. Using this idea, any swap portfolio is shown to have precisely the same exposure to interest rates as a small set of par swaps. The concept of equivalent position is an extremely powerful tool. It states that a portfolio of perhaps several thousand swaps, with off market coupons and various maturities, is equivalent to a set of a dozen or so par swaps. No matter what deformations occur in the yield curve’s shape, the actual portfolio and the collection of par swaps both display equal instantaneous changes in value1. Hence by calculating the 1

Perhaps this overstates the power of equivalent positions. See section 8.5 for a fairer explanation of how equivalent positions replicate the price behaviour of a swap portfolio.

133

134

CHAPTER 8.

INTEREST RATE EXPOSURE

Value date: 17 May 1990 Maturity O/N 1W 1M 2M 3M 6M 1Y 2Y 3Y 4Y 5Y 7Y 10Y

Par rate 14.25 14.75 15 15.125 15.15625 15.375 15.55 14.25 13.95 13.85 13.75 13.55 13.10

Table 8.1: A sterling yield curve equivalent positions of a portfolio, traders can immediately see how to insulate the value of their portfolio from the effect of any further movements in rates. Equally, they can see how best to take on a particular risk which suits their own view on the likely future course of interest rates. This topic is covered in chapter nine, which discusses swap trading strategies.

8.1

A simple example

The fact that a swap gives rise to interest rate risk is intuitive – the following example illustrates why. Sterling interest rates are as in Table 8.1. Convinced that five year sterling rates are set to rise, you have just paid the fixed rate on a £25,000,000 five year semi-annual swap at 13.75%. For the time being this is a par swap and hence has zero value. To your dismay, almost immediately after you deal, sterling swap rates start to fall. The rally continues, and by late in the day you decide to

8.2. AN EXPERIMENT

135

close out your position by receiving the fixed rate on a matching swap. The best bid in the market is then 13.65%. You do the deal. You now have two matching swaps. What are the net cashflows? What is the size of your loss? Since the two swaps have identical floating legs the net cashflows are purely the differences in the fixed legs i.e. 10 payments of:

Equation 5.11 for the value of an annuity, with a discount rate of 13.65% gives a loss of £88,509.2

8.2

An experiment

Imagine that you know exactly how to value your portfolio of swaps. You have even written some software to do the work for you. The software asks you to enter rates for overnight, one week, one, two, three and six month deposits and for swaps of maturities one, two. three, four. five, seven and 10 years. Knowing these, the software returns a value of your portfolio. Unfortunately, it tells you nothing of your sensitivity to interest rates. You expect that a movement of 10 basis points in the seven year rate relative to the 10 year rate will occur, but have no idea as to its effect on the value of the portfolio. How can you find out? Suppose the rates in Table 8.1 imply a portfolio value of £132,993. You increase in turn each of the input rates by one basis point and recalculate the value. Five of the yield curves and the resulting portfolio values are shown in Table 8.2. Comparing the values you obtain in the thirteen test runs with the portfolio’s current value of £132,993 gives Table 8.3. This is the delta vector for the swap portfolio3, that is to say it tells us exactly how much the portfolio will change in value for independent movements of one basis point in each of the input rates. 2

Using zero coupon pricing, and assuming that all rates have fallen by 10 basis points, your loss is £88,073. 3 Strictly. the delta vector is defined as the rate of change (i.e. derivative) of PV with respect to interest rates. We have calculated the actual change in PV for a small change in rates.

CHAPTER 8.

136

Maturity O/N 1W 1M 2M 3M 6M 1F 2Y 3Y 4Y 5Y 7Y 10Y Value

Trial 1 14.26 14.75 15 15.125 15.15625 15.375 15.55 14.25 13.95 13.85 13.75 13.55 13.10 £132.979

Trial 2 14.25 14.76 15 15.125 15.15625 15.375 15.55 15.25 13.95 13.85 13.75 13.55 13.10 £132,966


Trial 3 14.25 14.75 15.01 15.125 15.15625 15.375 15.55 15.25 13.95 13.85 13.75 13.55 13.10 £133,224

Trial 12 14.25 14.75 15 15.125 15.15625 15.375 15.55 14.25 13.95 13.85 13.75 13.56 13.10 £119,445

Trial 13 14.25 14.75 15 15.125 15.15625 15.375 15.55 14.25 13.95 13.85 13.75 13.55 13.11 £145,323

Table 8.2: Change in portfolio value for a change in individual rates

8.2.

137

AN EXPERIMENT

Rate changed O/N 1W 1M 2M 3M 6M 1Y 2Y 3Y 4Y 5Y 7Y 10Y

Change in value (14) (27) 231 77 (150) 193 501 1,300

(1,400) (3,006) (5.927) (13,548) 12,330

Table 8.3: The delta vector However it yields more information than that. For instance, it answers the question about the spread between the seven and 10 year rates which was posed at the beginning of this section. Suppose the yield curve distorts so that seven year rates increase by five basis points to 13.60% while the 10 year rate falls by five basis points to 13.05%. Assume for simplicity that other rates remain unchanged. Very closely, the change in value of the portfolio will be: (5 × –£13,548) + (–5 × £12,330) = –£129,390. So such a movement in rates would wipe out the portfolio's value. We have just made two implicit assumptions. First that the effect of a five basis point move in a particular rate will be five times the effect of a one basis point move, second that the combined effect of two simultaneous movements in different rates will be equal to the sum of the effect of each move in isolation4. It turns out that for swaps our assumption holds well for small (say less than 20 basis points) movements in rates. It holds, but progressively less well, for larger 4

Equivalently. that PV is a linear function of input rates.

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CHAPTER 8. INTEREST RATE EXPOSURE

movements.

8.3

The nature of the delta vector

There is an important conclusion to be drawn from our experiment: the delta vector for a given portfolio of swaps and a given yield curve is logically implied by the algorithm chosen to value that portfolio. The experiment assumed only a knowledge of swap valuation, and yet we derived a unique delta vector. In particular no mention was made of differential calculus or of any method of allocating cashflows to grid point dates. These techniques are valid only if they yield the same results as the method just outlined. This has implications in the field of swap valuation software. If the system purports to calculate a delta vector (as we have defined one) then it is simple to check that the results it produces are correct. For a given swap, change the input yield curve and test that the reported PV changes by the amount the system predicts. In fact this is a test of the internal consistency of the system. In the experiment, the three year delta was calculated to be – £1,400. This was the change in value of the swap portfolio as the three year par rate was changed leaving other par rates unaltered. Recall that the discount function was constructed as far as the two year point without reference to the three year swap rate. Hence a movement in the three year rate leaves this area of the discount function unaltered. The only part of the discount function that changes is that after the two year point. Therefore the three year delta is equal to: • The change in the value of cashflows occurring after the two year grid point as the three year par rate moves, other par rates being held constant. In general the three year delta is not equal to any of: • The change in value of cashflows occurring in three years time as the three year par rate changes. • The change in value of cashflows occurring in three years time as all input rates change.

8.4.

PAR SWAPS AND OTHER PAR RATES

139

• The change in value of the portfolio if the discount function is changed at the three year point.

8.4

Par swaps have no exposure to other par rates

Recall the simple example of section 8.1. You paid 13.75% on a par £25,000,000 five year swap. Swap rates, including the five year rate, subsequently fell and you lost £88,509. Suppose instead that the three, four and seven year rates were to fall but the five year rate remain unchanged. What now would be the value of the swap? The answer is that it would be unchanged at zero. This is because your swap is still a par swap, being at the still prevailing five year market rate. By definition, par swaps value to par. Notice that this is not to claim that the PV of the constituent cashflows remains constant – in fact they do not. Taken together the changes cancel out and the swap continues to have zero PV. So the delta vector for a five year par swap has only one non zero element5, the five year delta. For the time being, the delta of a five year swap is calculated by valuing a five year par deal and experimentally changing the five year rate. The result is illustrated in Table 8.4.

8.5

Equivalent positions

We define the five year equivalent position for a swap portfolio to be that amount of par five year swap which has a five year delta equal to the five year delta of . From Table 8.4, a £1,000,000 five year swap has a five year delta of –£351.5. From section 8.2. the five year delta for the portfolio is –£5,927. Hence: Five year equivalent position in millions 5

–351.5

=

–5,927

Assuming that the first LIBOR setting has not been made, otherwise the six month delta will also be non-zero – see chapter 11.

140

CHAPTER 8.



Delta 0 0 0 0 0 0 0 0 0 0 (3,515) 0 0

Table 8.4: The delta of a £10,000,000 par five year swap Five year equivalent position

1,000,000

–5,927 –351.5

(8.1)

16,860,000 Each par swap (or deposit) has one non-zero element to its delta vector. These are set out in Table 8.5. For all the maturities, equivalent positions can be calculated just as in Equation 8.1. The results so obtained are shown6 in Table 8.6. Here, a positive number denotes that the equivalent position is to receive the fixed rate. For instance, the portfolio is equivalent to receiving the fixed on a £30,757,008 par seven year swap and paying the fixed on a £22,809,112 par 10 year swap. The fixed rate receiver on a swap loses money as interest rates rise. Hence for each maturity the deltas and equivalent positions shown in Table 8.6 are of opposite signs. 6

In writing this chapter the authors constructed and analysed an imaginary portfolio of swaps and deposits. Equivalent positions are shown to a greater accuracy than the reader will obtain using the deltas of Table 8.3.

8.5.

EQUIVALENT POSITIONS

Maturity

O/N 1W 1M 2M 3M 6M 1Y 2Y 3Y 4Y 5Y 7Y 10Y

Non-zero element of delta vector of associated par swap (£10,000,000) (3) (19) (87) (163) (243) (473) (865) (1,686) (2,374) (2,981) (3,515) (4,405) (5,406)

Table 8.5: Deltas for £10,000,000 par swaps

141

142

CHAPTER 8.


Delta (14) (27) 231 77 (150) 193 501 1,300 (1,400) (3,006) (5,927) (13,548) 12,330


Equivalent position 49,900,315 14,257,233 (26,715,016) (4,696,519) 6,175,281 (4,085,589) (5,792,118) (7,713,238) 5,898,134 10,084,192 16,863,429 30,757,008 (22,809,112)

Table 8.6: Portfolio equivalent positions Having calculated the equivalent positions, a natural question arises: suppose the portfolio were to be hedged by trading the reverse of each of its equivalent positions. What would be the sensitivity to interest rates of the combined portfolio (i.e. the original portfolio plus the hedging trades)? For brevity, call the original portfolio O. the portfolio of (minus) the equivalent positions E and the combined portfolio C. Swap valuation is additive, so: PV(O) + PV(E) = PV(C)

(8.2)

But E is a portfolio of par swaps7 and so: PV(E) = 0

PV(O) = PV(C)

(8.3) (8.4)

Consider next the delta vector for C. Equation 8.2 holds for any yield curve, in particular for one basis point movements in par rates. So, in 7

The par deposits in E are assumed to have counterbalancing cash positions.

8.5.

EQUIVALENT POSITIONS

143

common with portfolio values, portfolio deltas are additive. Hence: Delta(O) + Delta(E) = Delta(C)

(8.5)

But E was constructed to have a delta vector exactly cancelling out that of O. Hence: Delta(C) = 0

(8.6)

Equations 8.4 and 8.6 together state that the value of O is “locked in”. That is, the hedging swaps maintain O’s value and cancel out its exposure to interest rates. What can be concluded from the fact that C has zero delta? Certainly that a one basis point movement in any one of the par rates will not affect its value. Unfortunately, this implies neither that C will be immune to the effect of larger movements in interest rates nor that C will continue to have zero delta as time passes. For larger movements in rates, “second order” effects come into play. As rates change Delta(E) and Delta(O) change, in general not in the same way. Thus Delta(C) becomes non-zero, which implies that PV(C) starts to change. In practice, these second order effects of value with respect to interest rates do not present a major problem in swap book running. The passage of time has two identifiable effects. First, LIBOR rollovers will occur on the swaps within C. If C has zero delta today but a £20 million swap is due to have a LIBOR set tomorrow. then the six month delta for C will certainly be non-zero tomorrow. We comment on how to deal with this in chapter 11, which discusses swap portfolio management. The following example illustrates the second effect of the passage of time. Again imagine the yield curve to be as in Table 8.1. Suppose you execute a £10 million six year swap at 13.65%. Its equivalent positions consists solely8 of five and seven year swaps of size £5,671,125 and £4,525,375 respectively. You execute these five and seven year swaps to give a portfolio with zero delta. Wait a year and re-examine your portfolio deltas and equivalent positions. By now you have three swaps, probably with off-market coupons, of maturities four, five, and six years. The four and five year swaps make no contribution to your 3

Can you see why the other equivalent positions are exactly zero?

144


seven year delta. By contrast the six year swap does. It contributes a seven year equivalent position of approximately £2,500,000. You certainly no longer have a zero delta. Section 8.8.7 describes how to construct larger replicating portfolios of swaps. These will reduce (but not eliminate) the type of problem just seen. For the time being, note that equivalent positions constitute a dynamic hedge for a swap portfolio – that is a hedge which must be adjusted as time passes and as rates move.

8.6

Analytic deltas

We have explained how to calculate deltas and equivalent positions by “toggling” rates and observing what happens to portfolio values. This is to regard the swap valuation process as a “black box”; change the inputs and see what happens to the output. The method has two advantages: it is easy to understand and involves no mathematics. However it also has two disadvantages: it yields little understanding as to why deltas are as they are and is computationally expensive - in our example, calculating the swap deltas involved valuing the portfolio a total of 14 times. A better approach is to calculate deltas and equivalent positions analytically. This involves a small change in our definition of the delta vector. The elements of the delta become the rate of change (or derivative) of the value of a swap as input rates change. As before, deltas are expressed as the change in portfolio value per basis point movement in input rate. Maintaining the sterling market as the example, most of the rest of this chapter is devoted to explaining how to calculate deltas and equivalent positions analytically. First, the simple case not involving short sterling contracts in the valuation process. Later, the complications that arise when they are included. In both cases, before the mathematics, there is a “preview to the argument”. These explain the ideas which underlie the maths. For those who wish to follow the mathematics through, it will set the route our argument takes. We encourage others to at least read the previews; they give a flavour of the analysis without subjecting the reader to the rigours of differential calculus.

8.7. THE CASE OF NO FUTURES: A PREVIEW

8.7

145

The case of no futures: a preview

Given that LIBOR payments can be modelled as two (predetermined) flows of principal, a portfolio of swaps can be represented as a set of predetermined cashflows. Since deltas are additive it will be enough to calculate the delta for a single cashflow. Recall that cashflow valuation was a four step process: 1. From the mark to market rates interpolate the “missing” rates such as the five and a half year swap rate. 2. From the complete set of rates calculate the value that the discount function takes on the grid point dates. This is sometimes called a “stripping” or “bootstrapping” process. 3. Interpolate between adjacent grid point dates to imply values for the discount function at all other dates. 4. Value a cashflow occurring on a date D by multiplying the cashflow by the value of the discount function at D. It is clear that the value of a cashflow is completely determined by the value of the discount function at the two surrounding grid point dates. So a change in rates has a “two pronged” effect on the value of a given cashflow. First, via the effect on the discount function at the prior grid point date: second, via the subsequent grid point date. So it will be enough to calculate the effect that a change in rates has on the value of the discount function at grid point dates. This is achieved by feeding the effect of a change through the first two steps of our four step evaluation procedure. For example, a change in the five year rate will produce changes in the implied four and a half, five, five and a half, six and six and a half year swap rates. Each of these in turn will have an effect on the discount function at (some, but not all of) the grid points. The maths is developed in terms of vectors of input rates and an output vector F. The notation of vectors (which are usually printed in bold type) is no more than a convenient shorthand for talking about a collection of objects which are considered as a whole. So the collection

146

CHAPTER 8.


Figure 8.1: A picture of of values that the discount function takes at all grid points is considered as a unit and labelled F. The set of rates observed in the market is called R´ (read as R prime). The elements of R´ are independent i.e. knowledge of some of their values implies no knowledge of any of the others. There is also a second set of rates, R. This is the complete set needed for the stripping process, namely the cash rates RO/N, . . . , R6M and all the swap rates R1Y, R1.5Y, . . . , R10Y. Again, when calculating the effect of R on F, we consider the elements of R to be independent. At a later stage it is recognised that the elements of R do not move independently but are determined by the elements of R´. The two stage process appears again. To calculate the effect on F of a change in R´ we first calculate the change in R and feed this through to the implied change in F. In differential notation: (8.7)

The core of the maths is the effect that R has on F. Since a general element of R affects a general element of F a matrix is required to describe the process. The matrix is called and is illustrated in Figure 8.1. Each represents some non-zero element of It is important to understand what the elements of mean. For example, the asterisked element is where the column labelled R1.5Y meets the

8.8.

USING NO FUTURES: THE MATHS

147

row labelled F2Y The element is written Encapsulated within this notation is the fact that F2Y is determined by a collection of objects jointly called R. One of these objects is called R1.5Y and if R1.5Y is changed while holding its companions within R constant then simply measures the speed of this change. F2Y will change. Notice that is zero, since the discount function at one year is determined solely by the one year rate. One last fact: for sensible yield curves, the on-diagonal elements of will be negative and the non-zero elements below the diagonal will be positive. What does this mean? Why is it so?

8.8

Using no futures: the maths

For ease of reference, notation is listed here. First there are four indexing sets. Most frequently, the indexing sets appear in order to state when a particular equation holds true. For example, Equation 8.8 defines the discount function at grid points less than or equal to one year. This means that the equation is To its right is the statement true whenever i is a member of the set C. But by the definition of C, means “for i a grid point less than or equal to one year”. The indexing sets are:

The remaining notation is mostly familiar: DO

The value date.

Di

The ith grid point date.

Fi

The value of the discount function at Di.

148


F

As discussed in section 8.7, a vector of variables denned as:

Ri

These form a “complete” set of rates so the subscript i can be any element of V. Not all of the Ri are usually quoted in the market. R O / N , . . . , R1Y are Actual/365 cash rates and R1.5Y, . . . , R10Y are semi-annual Actual/365 swap rates.

R

A vector consisting of all the Ri described above.

The elements of R are considered to be independent. The observed market rates. Hence the subscript i is restricted to be in A vector consisting of all the observed market rates.

These are the accrual factors defined (for our example market, sterling) as:

In order to prevent the accrual factors from cluttering up all the equations, wherever possible is abbreviated to read Further. is abbreviated to read

8.8.1

How F changes as R changes

The equations that define F from R are:

(8.8)

(8.9)

8.8.


149

Remember that the elements of R are independent. Hence Equations 8.8 and 8.9 can be differentiated to obtain all the non-zero elements of (8.10)

(8.11)

(8.12)

Notice how Equation 8.12 defines the elements of recursively. For example, we can calculate using Equation 8.11. Then using Equation 8.12 we can in turn calculate and so on. In fact this is equivalent to working down a column of the matrix which was illustrated in Figure 8.1.

8.8.2

How R changes as R´ changes

describes how the implied par rates change as the obThe matrix served par rates change. Now, as set out in chapter six. R is defined in terms of R´ . Except for the 18 month rate this is done very simply, for example:

Which implies that:

That is to say that a change of one basis point in the observed four year rate produces a change of half a basis point in the implied rates for four and a half and five years.

CHAPTER 8.

150


Figure 8.2: A picture of part of The 18 month rate. R1.5Y, depends upon R´6Y and R´1Y (which are used to construct R1Ysa) and also on R´2Y :

(8.13) where: (8.14) Differentiating gives: (8.15) (8.16) With the yield curve of Table 8.1, we can draw a small part of the This is shown in Figure 8.2. The full matrix has 25 rows matrix and 13 columns but only 38 non-zero entries. Try drawing it.

8.8.3

How F changes as R´ changes

So far we have calculated , which tells us how the implied rates change as the observed rates change. From section 8.8.1, we also know

8.8.


151

, which tells us how the discount function at grid points changes as the implied rates change. Multiplying the two expressions gives:

(8.17) There is nothing mysterious about Equation 8.17, it is just a compact way to list a set of common sense statements. For example: “If the observed four year rate changes by one basis point then, by implication, the three and a half and the four and a half year rates also change. The effect on the discount function is the sum of the effects of: 1. a half basis point change in the three and a half year rate 2. a one basis point change in the four year rate 3. a half basis point change in the four and a half year rate”

8.8.4

Between the grid points

We can now calculate the effect of movements in observed rates on the discount function at grid points. The delta vector for a single cashflow occurring on a grid point follows immediately. What of cashflows occurring on other dates? Equation 6.32 defines a method of interpolating the discount function between grid points: if there are adjacent grid points D1 and D2 and a date Da between them then Fa is given by: (8.18) where: (8.19) So Fa is determined by F1 and F2 . Differentiating the relationship of Equation 8.18 gives: (8.20) (8.21)

152


Equations 8.20 and 8.21 allow a very useful trick: suppose C is a cashflow occurring on the date Da. Our aim is to replace C by a pair of cashflows occurring on D1 and D2. Call them First(C) and Second(C) respectively. The replacement is required to preserve interest rate sensitivity (i.e. to preserve deltas). Now, any movement in interest rates can only affect the value of C via its effect on F1 and F2. So it is enough to ensure that First (C) and Second(C) together have the same sensitivity as C to small movements in F1 and F2. It follows that: (8.22) (8.23) To understand why Equations 8.22 and 8.23 are true note that PV(C) = CFa and hence that the sensitivity of the value of C to movements in F is:

But this is exactly the same as the sensitivity of the pair of cashflows First(C) and Second(C). since:

(8.24) (8.25)

Notice that the substitution of First (C) and Second(C) for C does not preserve PV. as is not difficult to verify, by showing that, in general:

8.8.


153

So. it is impossible9 to both preserve PV and ensure that the delta vector of the pair is equal to the delta vector of C. This is of no great concern, since PV is calculated before any allocation of cashflows takes place.

8.8.5

Portfolio deltas

Finally we are able to calculate the delta vector for a portfolio of swaps . Think of this as a five step process. 1. Represent all unknown LIBOR flows by payments of principal. 2. Replace all cashflows by their “equivalent cashflows” on surrounding grid points. 3. Sum the cashflows allocated to each grid point to produce a list (or vector) of equivalent cashflows. Call this vector In fact, just as in Equations 8.24 and 8.25 for a single cashflow. is the ' derivative of s value with respect to F:

4. Given R´, the observed market rates, calculate in section 8.8.3.

as explained

5. The delta vector is found by multiplying the matrix vector

by the

(8.26) The factor appears because the delta vector is defined as the change in value per basis point rather than per percent. 9

Except for linear interpolation, where

154

8.8.6



As explained in section 8.5. for each par swap we must calculate the single non-zero element of its associated delta vector. There is a simple expression for this (which is proved in Appendix C). An n-year par swap (n 2) with notional principal P has an n-year delta equal to:

(8.27) In fact this is an intuitive result. Imagine the yield curve was the familiar one shown in Table 8.1. We receive the fixed on a £10,000,000 five year swap at 13.75%. Subsequently the five year rate moves up by a basis point to 13.76%. After this move a five year swap at 13.76% must value to par. Our swap differs from the new par swap only to the extent of a semi-annual annuity of one basis point. But Equation 8.27 is simply the value of this annuity. The minus sign appears to keep a consistent sign convention. At last we can add a sixth and final step to our description of portfolio analysis: 6. Calculate the deltas for par swaps and re-express the portfolio deltas as portfolio equivalent positions.

8.8.7

Expanded equivalent positions

Section 8.5 discussed the way in which a portfolio of equivalent positions fails to serve as a hedge for a complex portfolio of swaps. The problems were threefold: first the effect of large interest rate movements on the delta vector (a “gamma” effect). Second, the effect of LIBOR settings. Lastly, the effect of the passage of time, which was illustrated with the example of a six year swap. Hedging efficiency can be improved (but with increased hedging costs) by expanding the family of equivalent positions. There is an obvious way to do this: For a portfolio consider:

8.9.

INCORPORATING FUTURES: A DISCUSSION

155

This is the delta vector that would be obtained were all the elements of R allowed to be independent. For example the four and a half year rate would no longer be constrained to be the average of the four and five year rates. This expanded delta vector has 25 instead of 13 elements. In a natural way, the new delta vector translates into a new set of equivalent positions. The expanded set would indeed be a more accurate hedge for a portfolio P. In particular it would maintain its efficiency more accurately as time passes.

8.9

Incorporating futures: a discussion

Section 6.4 described a procedure for incorporating any number of futures contracts within the construction of the discount function. Each futures price gave us information about the forward rate between its settlement date and the settlement date of the subsequent contract. The discount function was constructed to incorporate these forward rates. This necessarily involved overriding some information from the swap curve. So when using futures contracts, only par swaps of maturity beyond the settlement date of the ( + l)th future (i.e. beyond value to par. In calculating the delta vector, precisely the same methodology serves as in the case with no futures. There is a set of observed inputs futures prices and which is again called R´. R´ includes the first those observable swap and deposit rates which are not “overwritten” by the futures contracts. As before, from R´ we construct a complete set of rates R. R includes all the elements of R´ and implied swap and deposit rates for all the cash and swap grid points. Again the elements of R are considered to be independent so it is straightforward to calculate how the discount function depends on R. . how changes The delicate part of the analysis is to calculate in the observed rates and futures prices feed through to changes in the is known the problem is all but solved. implied swap rates. Once since: (8.28)

156


is, in effect, a choice of a different valuation method Each choice of by one adds an taking a different set of inputs. Obviously, increasing extra futures price to R, and it may also remove a swap rate. We could explain how to calculate the delta vector no matter what value takes. However the exposition has to cover a host of possibilities concerning which swap rates are implied and which are observed. Clarity suffers as generality increases. Instead we take a different route. Again using the sterling market, we shall refer to the first six futures prices and a value date of 17 May 1990. Hopefully this approach will demonstrate most of the subtleties involved without burdening the reader with too many caveats.

8.10

Futures: the maths

First the indexing sets are redefined:

lists the input rates and so forms an index for R´. Notice The set that because the first futures settlement date lies between the one and two month grid points, 2M and 3M are abandoned. So R´ has sixteen elements. is a still larger set designed to list the grid points for which a forms an index for value of the discount function is calculated i.e. F. The 2M and 3M points are overwritten by futures and need not be included in By contrast, although 6M, 1Y and 1.5Y are also overwritten by futures, F6M, F1Y and F1.5Y are needed in the construction of subsequent values of the discount function. Hence 6M. 1Y and 1.5Y are included in In addition to our earlier notation we use: The number of futures used. In what follows

Pi

= 6.

The futures prices. The subscript i runs from 1 to

8.10.

FUTURES: THE MATHS

157

Value date: 17 May 1990 Grid point O/N 1W lM 1F 2F 6M 3F 4F 1Y 5F 6F 1.5Y 7F 2Y 2.5Y 3Y 10Y

Date 18-May-90 24-May-90 18-Jun-90 20-Jun-90 19-Sep-90 19-Nov-90 19-Dec-90 20-Mar-91 17-May-91 19-Jun-91 18-Sep-91 18-Nov-91 18-Dec-91 18-May-92 17-Nov-92 17-May-93 17-May-100

Table 8.7: The ordering of the grid points R0F Another name for Rs which, in chapter six. denoted the market rate for a deposit maturing on the first futures settlement date. R´

Some other elements of R´ are new: names for the futures prices

8.10.1

are just new

Building the discount function

Remember that we are assuming a value date of 17 May 1990 and using six futures prices. The grid point dates needed are therefore ordered as in Table 8.7. Building the discount function is the five step process of

158


section 6.6. Step 1 Use the cash rates to construct the discount function up to and including the first futures date. So for i = O/N, 1W, 1M : (8.29)

and: (8.30) Step 2 Extend the discount function to the next six futures dates. So for i = 2,. . . ,7: (8.31) Explicitly in terms of the futures prices: (8.32) Step 3 Interpolate, using Equation 8.18: F6M between F2M and F3M F1M between F4M and F5M F1.5M between F6M and F7M Step 4 Imply the full set of rates R: (8.33) (8.34) (8.35) (8.36)

and so on.

8.10. FUTURES: THE MATHS

159

Notice that it was not necessary to construct the one year semiannual rate R1Ysa. This is because the 18 month rate R1.5Y is implied purely from futures prices via step three above and Equation 8.35.

Step 5 Use the familiar stripping process to calculate the discount function at D2Y, . . . , D10Y. For i = 2Y, . . . , 10Y:

(8.37)

8.10.2

Calculating

Bearing in mind that the elements of R are, as ever, independent we can differentiate Equations 8.29 to 8.37. This produces the slightly daunting block of equations given below. Don’t be intimidated by them. The first four appeared in section 8.8.1. We discussed how they allow us to work down the columns of the matrix calculating the value of each element in turn. The last two. Equations 8.42 and 8.43 describe how changes in futures prices change the discount function on subsequent

160


(but not prior) futures settlement dates.

(8.38) (8.39)

(8.40) (8.41) (8.42) (8.43)

8.10.3

Calculating

as a large matrix. Now it is Section 8.8.2 explained how to regard a still larger matrix, having 30 rows and 16 columns. It turns out that of the 480 elements of the matrix, only 53 are not zero and a mere 15 are not constants. The 15 describe the way in which changes in the first six futures prices (and Rs) cause changes in the implied swap rates R6M, R1M and R1.5Y. Figure 8.3 shows a sketch of the interesting part of . Here the symbols represent the 15 non-constant elements. Rather than give general equations describing how to calculate all of them we take a particular example, namely how changes in the first futures price. P´1, lead to changes in the implied 18 month rate R1.5Y. This is the element marked with an asterisk in Figure 8.3. To see how the process works, imagine a change in the first futures price. This feeds directly through to changes in the discount function at D2F, . . . , D7F. In turn these lead to changes in F6M, F1Y and F1.5Y. Each of the changes in F6M, F1Y and F1.5Y is responsible for a change in

8.10.

FUTURES: THE MATHS

161

Figure 8.3: A picture of part of R1.5Y. Figure 8.4 represents the process as a flow diagram. Each arrow in the diagram means that a change in the contents of the first box produces a change in the contents of the second box. So for instance:

means that a change in F2F produces a change in F6M, in this case because of the interpolation procedure. This idea can be expressed another way: that each arrow represents a derivative. The diagram above Fortunately, all arrows on the flow diagram represent represents derivatives that we know how to calculate. To calculate notice that there are six “routes” from the top box to the bottom box. Each route travels down three arrows. We need to multiply the derivatives

162

CHAPTER 8.


Figure 8.4: An aid to calculating

8.10. FUTURES: THE MATHS

163

represented by each arrow in a route and add up the result for each of the six routes. This would give a rather large equation were it written out completely. The first term (represented by the leftmost route) is as below:

(8.44) The remaining 14 non-constant elements of (the s) can be calculated in much the same manner described above. Therefore the whole of is known to us.

8.10.4

The delta vector

Let us take stock. For the case involving futures we now know how the full set of rates changes as the observed rates and futures prices change (i.e. we know Also we know how changes in the full set of rates cause changes in the discount function at grid points. Just as in section 8.8.5, the case with no futures, to calculate the delta of allocate the cashflows of a portfolio to grid points to form the Then: vector

(8.45)

8.10.5


Equivalent positions are easy to define. Remember that to do so involves dividing each element of the delta vector by the delta10 of the associated swap, deposit or future. For the equivalent swap and deposit positions the situation is exactly as in section 8.8.6. Of course, the delta for a futures contract is just its tick value, which for short sterling contracts is £12.50. So if for a portfolio the delta with respect to the first futures price is £1,250 then the equivalent position in the first futures contract is 1.250/12.5 = 100 contracts. Thus both a long position of 100 contracts and display the same sensitivity to the first futures price. 10

Strictly, the non-zero element of the delta vector of the associated swap, deposit or future.

164

8.11


The gamma matrix

The delta vector for a portfolio of swaps depends on the input rates R´. In general, a change in each input rate will produce a change in each element of the delta vector. The gamma matrix describes this process. An example element of this matrix would describe the rate at which the three year delta changes as the two year rate is changed. As was the case for the delta vector itself, there are two ways to calculate the gamma matrix. The straightforward way is to change each of the input rates in turn and observe the resultant change in the delta vector. A second approach is to calculate the gamma matrix analytically. The analytic approach is not a trivial task and there is space here only for a sketch of how the process is carried through. For simplicity, we examine the case not involving futures contracts. Differentiating them with respect Equations 8.10 to 8.12 define This describes how the dependency of F on to R gives a matrix R itself depends on R (i.e. the second order dependency of F on R). The value of a single cashflow. C, depends on F. In fact, it depends on only two elements of F: the values of the discount factor at the two surrounding grid points. Equation 8.18 defines this, and was differentiated to give Equations 8.20 and 8.21. It is easy to differentiate a second time to give four terms:

and

the last two of which are equal. These four terms are the only non-zero elements of a matrix which describes the second order dependency of the discount function between grid points on the discount function at gridpoints. When evaluating the delta vector, we first calculated an expanded delta vector before recognising that the elements of R are not all independent, but are determined by the observed market rates. The situation is analogous for the gamma matrix – an expanded gamma matrix must first be calculated. Each element of the matrix is

8.11.

THE GAMMA MATRIX

165

found by multiplying out terras that are already known:

(8.46)

Equation 8.46 is quite general – it describes how, starting with: (1) the first and second order dependency of PV on F; (2) the first and second order dependency of F on R the second order dependency of PV on R can be obtained. To calculate the gamma matrix is a similar process. The first and second order dependency of PV on R is now known, as is It remains to calculate which has very few non-zero terms. Once that is known, follows by a relationship analogous to that of Equation 8.46. Even using no futures, there are 13 elements of R. Hence, the gamma matrix has, in this case. 169 elements. This is too large to be of any practical use to the trader, even though the matrix is symmetric and only the elements near the diagonal are significant. The answer is to group the input rates in some way – perhaps as short (overnight to six months), medium (one to four years) and long (five to ten years). Adding terms in the gamma matrix then shows how movements in the short rates effect the long delta (i.e. the sum of the five, seven and ten year deltas). Of course, this could equally be done using the toggling method. In other words, one could experimentally change all of the short rates and observe the resultant change in the long delta. For all the work involved in calculating the gamma matrix, it turns out that, for a portfolio of swaps, it can safely be ignored. Other factors are far more important in determining how deltas, and hence hedge ratios, change. In contrast, for a portfolio of interest rate options, the delta vector can depend crucially on interest rates. Calculating the gamma matrix for a cap is exactly like calculating the gamma matrix for a swap, except with different expressions for We shall return to this in chapter 10.

Chapter 9 Hedging and trading swaps 9.1

Why and what to hedge

This chapter is about hedging interest rate risk from the swap bookrunner’s point of view. The previous chapter has given a recipe for evaluating the “first order risk”, or the delta vector, for any given set of cashflows, represented by the sensitivity of the portfolio in pounds sterling (say) per basis point to a shift in any input rate (or price, in the case of futures). Earlier in the book, the various instruments available to the prospective hedger were explained, and it was shown how to calculate their sensitivity, or PVBP (price value of a basis point). With this information, hedging becomes a straightforward process. To understand what hedging means, consider a swap trader who concludes a sterling swap with a client; the trader receives fixed in five years. Broadly speaking, he now has four choices: 1. Do nothing in the belief that the five year rate will fall. 2. Cover the position immediately in the market by paying fixed in five years to another book-runner. Clearly, in order for this to be economical, he needs to make a (positive) margin, or bidoffer spread, between the deals. In liquid markets, and with progressively more sophisticated and aware counterparties, this has become almost impossible to achieve in the majority of cases. 3. Although the book-runner’s delta exposure is to five year rates. 167

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CHAPTER 9. HEDGING AND TRADING SWAPS he could achieve a net zero delta across the whole yield curve by paying fixed in a different maturity. In this situation, the size of the new deal will be different from that of the original – paying into a longer maturity will require a smaller nominal, while a shorter maturity will require a larger nominal. Calculating exactly how much is required (in other words, the hedge ratio of swaps of different maturities) is considered later in the chapter. This course of action, while achieving a total zero delta, will give the book-runner a “yield curve position”. Thus, if he receives in five years and pays in two years, he would need the spread R5Y – R2Y (the five year minus the two year swap rate) to decrease (in the case of a negatively sloped – “inverse” – yield curve to become more negative). Alternatively, had he paid in ten years, he would need the spread R5Y – R10Y to decrease.

4. Hedge the swap using instruments other than swaps. The only such instruments which will reduce the delta and not introduce non-interest rate exposures are FRAs, bonds and interest rate or bond futures (the possibility of actual loan or deposit transactions, which most swap groups are not at liberty to take, is excluded - for the example of a five year swap, they are irrelevant anyway). Of the above possibilities, only the second option perfectly hedges the original deal, thus “locking in” any profit or loss made. Option one (doing nothing) is tantamount to an outright bet on the direction of the market, and no amount of theoretical analysis will help if this direction is chosen. The decision to take a yield curve position (option three) usually depends on two factors. First, the trader will have to believe that the shape of the yield curve will move in a favourable way. or not move in a disadvantageous way. However, another factor is the relative volatility of the spread (for example R5Y – R2Y) versus that of the five year rate itself. For example, suppose the five year rate was extremely volatile. Maintaining the five year swap position in the hope that rates will move favourably is clearly dangerous. However, suppose the spread R5Y – R2Y is much less volatile – in other words, movements in R5Y approximately track movements in R5Y. In this situation, one is less at risk by paying fixed on a two year swap than by doing nothing.

9.2.

HEDGING WITH BONDS

169

In reality, such a situation is more likely to occur for swaps with close by maturities than with widely spaced maturities.

9.2

Hedging with bonds

By far the most common method of reducing absolute interest rate exposure is by hedging with government bonds, which have the advantage of liquidity. In the US treasury market, it is possible to trade tranches of several hundred million dollars worth of bonds with on-therun maturities1. Furthermore, the bid-offer spread is usually only one basis point in yield. The treasury market is something of an exception – the UK gilt market is comparatively illiquid, with bid-offer spreads typically at least two pence, even for relatively small amounts. This market also suffers from a restrictive shorting system, whereby only designated gilt market makers are allowed to sell gilts they do not own. In contrast, the repurchase system for treasuries is advanced and liquid, allowing any swap market maker to take short positions. When there is a liquid bond market, the bond-swap spread (usually referred to as the spread) is less volatile than the corresponding absolute swap rate, making the bond a natural hedging instrument. Chapter three explained how to obtain the modified duration of a bond. Recall that this quantity is the proportional change in the dirty price of the bond per unit change in the yield:

where yH is the yield expressed in H compounding periods per year, and Pd is the dirty price. In order to hedge one bond with another, the required hedge ratio is proportional to the ratio of the PVBPs of the two bonds. This implicitly assumes that the yields of the two bonds will move in parallel. We now wish to determine the hedge ratio between a bond and a swap. The exposure for a generic swap is expressed through the delta vector – the change in PV for a basis point movement in the relevant input rate. The corresponding amount for a basis point 1

This and other terms relating to bonds and futures are explained in chapter four.

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CHAPTER 9. HEDGING AND TRADING SWAPS

movement in the bond yield is not the modified duration, but simply d P d / d y H . If the swap delta is expressed in units of currency per basis point, and the bond PVBP in units of currency per million bonds, then the required hedge amount is:

(9.1) represents an element of the delta vector. So as long as yH where and Ri move in parallel, any rise in value of the swap will be exactly offset by a fall in value of the bond, or vice versa. For example, suppose that a trader receives the fixed rate on a five year sterling swap, with a resulting negative five year delta of (£4,000). If the five year gilt had a PVBP of £400 per million stock, then he would need to sell exactly ten million of the gilts as a hedge. So if rates subsequently rise, the swap will lose money while the short gilt position will make money. The five year delta exposure has been transformed into an equal exposure to the spread. Now he will lose £4,000 for every basis point increase in the spread. Presumably, he believes that this exposure is less dangerous than an exposure to the underlying five year swap rate. In the same way. it is possible to hedge a five year swap with, say, a two year bond. However, the spread between five year swap rates and two year gilt yields is likely to be more volatile than the five year swap spread. Once a position has been taken, the only way to reduce the spread exposure is by doing another spread trade the other way round, since there is. at present, no hedge for spreads.

9.3

Calculating the swap PVBP

Section 8.8.6 explained why the PVBP of a par swap is equal to the present value of a basis point of the swap coupon (a result proved in Appendix C). In the case of sterling, the result holds true for the halfyearly par swaps, such as a four and a half year swap. The sterling yield curve of chapter six gives the swap PVBPs listed in Table 9.1. It is now clear how to hedge a par swap of one maturity with another of a different maturity. Suppose you had received the fixed rate on a

9.4.

TRADING AGAINST THE FUTURES STRIP Maturity 1Y 1.5Y 2Y 2.5Y 3Y 3.5Y 4Y 4.5Y 5Y 5.5Y 6Y 6.5Y 7Y 7.5Y 8Y 8.5Y 9Y 9.5Y 10Y

171

PVBP 87 131 169 204 237 269 298 326 351 376 399 420 440 459 477 495 511 526 541

Table 9.1: PVBP for par sterling swaps (Pounds per million notional)

£10,000,000 par five year swap with associated delta of (£3.510). and wished to pay in two years against it. To give a zero delta across all maturities, requires a trade of size £10,000,000 3,510/169 £20,800,000. These numbers will not be exactly correct for slightly off-market swaps, but are a good approximation to the size of the hedge.

9.4

Trading against the futures strip

One of the most liquid instruments in the hedger’s armoury is the deposit futures contract. Not only is there a huge daily turnover in the

172


most liquid contracts, but the Eurodollar and short sterling contracts go out as far as four and three years respectively. For this reason, many swap traders prefer to price and hedge short dated swaps via the discount function generated by futures prices. In practice, it can sometimes prove difficult to execute large orders in contracts having more than a year before delivery, so futures hedging is generally restricted to swaps up to three years in dollars, and two years in sterling. At present, no other currency offers a sufficient number of contracts for futures hedging. Chapter six set out how to calculate the discount function from a futures strip and. from this, how to imply swap rates. Chapter eight developed the mathematics for decomposing a collection of cashflows into a set of exposures (the delta vector) to cash rates, swap rates, and futures prices. In general, the precise number of contracts required to hedge are not particularly straightforward to derive. As an example, consider again the sterling discount function defined in Table 6.4. This was obtained by integrating the first four futures contracts with the cash and swap discount function of Table 6.3. In the integrated curve, the one year swap rate is determined by the futures strip2: (9.2)

This is the annual Actual/365 swap rate. In fact, the market will often not trade at quite the same level as is implied by the strip - indeed, the spread between the two is exactly what a trader is interested in. A glance at Table 6.2 shows that, in this case, the spread is 15.614 — 15.55 6.4 basis points. Now consider the cashflows of a £10,000,000 par one year swap with the fixed rate. 15.614%. being that implied from the futures strip. Modelling the LIBOR leg as two payments of principal, the swap has two cashflows: –£10,000,000 on 17 May 1990, and £11,561,400 on 17 May 1991. The cashflow occurring “today” can be ignored – it values 2

Tb obtain the same result, the reader will need to use a more accurate version of F1Y, namely 0.86494763.

9.4.

TRADING AGAINST THE FUTURES STRIP

173

to par. One now needs to follow the prescription of section 8.10 as applied to this one cashflow. Since 17 May 1991 is not a valid grid point in this scenario, the cashflow is allocated to the two surrounding good dates – both of which are futures dates. These are 20 March 1991 and 19 June 1991, the settlement dates of the third and fourth contracts. The standard exponential allocation explained in section 8.8.4 gives a cashflow of size £4,876,640 on 20 March 1991, and of size £6,841,288 on 19 June 1991. These are the values of the derivatives dPV/dF 3F and d P V / d F 4 F respectively. The deltas are obtained by multiplying these numbers into Equation 8.42, the expression for dF/dR. For example, the delta for the first futures contract is given by:

In actual fact, the first bracketed expression in the above equation should be: (9.3)

but all the other terms are zero, due to the way the allocation of the cashflows worked out. The same prescription can be followed for the other dates to give the following delta vector for the one year swap3: Contract Rs 1F 2F 3F 4F 3

PVBP –93 244 244 244 141

Hedge –20 –20 –20 –11

In the table, the first element is negative, but subsequent ones are positive. In both cases, rising interest rates cause the swap’s value to fall. This is because rising rates are equivalent to falling futures prices.

174


With the above information, the futures equivalent position is obtained by dividing the PVBP of the swap by the PVBP of the futures contract, which is £12.50 for short sterling. The hedge is minus the equivalent position, thus obtaining the third column in the table. Note that no hedge amount is shown on the line corresponding to Rs – the deposit rate to the first futures expiry date. This is because the only exact hedge against this exposure is to take out a loan to the same date. This is impractical and therefore one may decide to add this exposure to that of the 1F point, and hedge it all with the first contract. Once the futures hedge has been put on against the swap, the delta exposure to the one year swap rate becomes an exposure to the spread between the swap rate and the implied rate from the strip. This is analogous to the case with bond hedging, where the delta exposure became a spread exposure.

9.5

The swap-FRA arbitrage

As explained in section 4.3, FRAs are essentially equivalent to single period swaps. The only difference is that the settlement on FRAs is paid discounted in advance (on the fixing date), whereas swap settlements are paid undiscounted in arrears (at the end of the calculation period). However, from a valuation perspective, this does not introduce any differences. Clearly, one should be able to reproduce the cashflows on any given swap by choosing an appropriate strip of FRAs. This section explains how to do so. FRAs are regarded as money market instruments, and. as such, are usually only traded with maturities of up to two years. FRAs longer than this impinge directly onto swap market territory, and so tend to remain untraded. However, for swap traders, FRAs provide a very useful hedging mechanism for short swaps. Their main advantage is that, unlike bonds and futures, they do not introduce any spread risk. In other words, a suitable choice of FRA strip can eliminate the delta on a swap without introducing any other risks. Thus, such a hedge “locks in” the value of the swap. Whether this value is positive or negative depends on the spread between the swap rate and the rate implied by FRA prices.

175

9.5. THE SWAP-FRA ARBITRAGE

Consider again a one year sterling swap on which one pays fixed at 15.55% annual Actual/365 against three month LIBOR on £10,000,000 principal. Furthermore, suppose that the LIBOR has already been set for the first period at the prevailing three month rate of 15.15625%4. Having paid on the swap, one would obviously need to receive fixed on the FRA hedge – in other words, sell a strip of FRAs. Imagine one can execute the FRA hedges at the prices listed in section 6.7. The swap is an annual fixed coupon versus a three monthly floating leg – thus, the natural hedge is a strip of three month FRAs – 3-6, 6-9 and 9-12, with prices 15.05%, 14.60% and 14.10%. The cashflows resulting from a sale of £10,000,000 of each of these, and the underlying swap are shown below. Date 17-May-90 17-Aug-90 19-Nov-90 18-Feb-91 17-May-91

Swap cashflows

3-6 FRA cashflows

10,382,020

(10,000,000) 10,387,589

6-9 FRA cashflows

(10,000,000) 10,364,000

(11,555,000) Date 17-May-90 17-Aug-90 19-Nov-90 18-Feb-91 17-May-91

9-12 FRA cashflows

(10,000,000) 10,339,945

Net cashflows 382,020 387,589 364,000 (1,215,055)

Having obtained the net cashflows, the question now arises of how to value them. This involves multiplying each of the cashflows by a discount factor – but which one? There is a choice of either the discount function generated in the normal way by cash and swap rates – as in Table 6.3 – or the curve generated by stripping the FRAs, as in Table 4

This assumption is for the sake of simplicity – ignoring it introduces some further simple calculations, but no new techniques.

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CHAPTER 9.

HEDGING AND TRADING SWAPS

6.5. Naturally, these two curves give slightly different answers. The swap curve gives a PV of £1,612. and the FRA curve a PV of about £3,127. In reality, this problem is usually solved by the practical issue of how one values one’s book. Most often it will be valued off swap rates rather than FRA prices, which is the route followed here. The outcome of the exercise is a small profit – it turns out to be about two basis points. As time marches on, the hedge may need slight readjustment– however, these will probably be small, and so one can regard the PV as having been locked in. If all the cashflows corresponding to the FRAs and the swap were put into a standard cashflow valuation system, one would find that the exposures do not exactly cancel out. This is because we are expecting to receive some small cashflows every three months, but only pay a large cashflow in a years time. To compensate for this factor, one could hedge with slightly different amounts of FRAs. It is possible to make all except the first net cashflow zero by trading £10,380,292 of 3-6’s, £10,782,620 of 6-9’s. and £11,175,108 of 9-12’s. In this case, the only non-zero cashflow would be £1,728 on 17 August 1990. These are not particularly important refinements, and indeed most swap traders prefer to ignore them and hedge with the equivalent notional of the swap.

9.6

Bond futures hedging

As explained in chapter three, long bond futures fail to provide a useful hedging function for most swaps. The fact that the most liquid bond futures have an underlying maturity usually greater than fifteen years, coupled with the fact that most swaps have a maturity of less than ten years means that there is usually a large basis risk between the two. In fact, basis risk is a more general concept measuring the degree by which two numbers – usually yield or prices – fail to track each other. For example, hedging a dollar swap with a sterling swap would probably exhibit huge basis risk. Similarly, hedging a swap with a bond exhibits a smaller basis risk – namely, the spread exposure. Even so. occasions do arise in which bond futures provide the best available hedge – twenty year swaps, for example, or sterling swap books

9.7.

REINVESTMENT RISK

177

which cannot sell gilts short. The hedge required is then simply determined by dividing the PVBP of the swap by the product of the PVBP of the cheapest to deliver gilt and its price factor (see section 4.2). Note that, for a generic ten year sterling swap, this implies that one would need to buy or sell about 15 contracts per million notional on the swap. As the maturity of the swap gets longer, this number will get even bigger. Thus, executing a futures hedge against a reasonably large long term swap can prove problematic.

9.7

Reinvestment risk

Consider a generic sterling interest rate swap. At six monthly intervals, payments are made between the counterparties according to the movement between the fixed rate and six month LIBOR. Suppose, for example, one entered into a swap, receiving 14% for two years – furthermore, assume that rates fall for the period of the swap. In this case, one will always be receiving net payments on each payment date. The issue addressed in this section is what to do physically with the cash receipts or payments over the lifetime of the swap. It might be assumed that the problem is hardly worth worrying about. After all. unless rates fall a huge amount, these payments will not be a significant percentage of the notional amount of the swap. Moreover, since a general portfolio will contain a mixture of deals where one is paying and receiving in or out of the money swaps, such cashflows will tend to cancel each other out to a greater degree. The above critique is a fair representation of the situation that occurs for most interest rate swap portfolios. However, there are a few structures which upset this argument. An extreme case is a currency swap, where physical payments of principal are made at the start and end of the swap. This is a slightly different situation, considered in chapter 11. A less extreme, but still important, problem arises in the case of zero coupon or reinvestment swaps. A zero coupon swap is precisely what it sounds like – a swap where the fixed rate is set to 0%. If we are receiving the fixed on a zero coupon swap and paying LIBOR every six months, some additional payment is needed to make the deal economic. What is normally added is a one-off

178


payment – if it were at the start of the swap, the deal would essentially be a reinvestment contract, with a return to the investor of six month LIBOR (possibly plus or minus some margin). The investor would usually be able to achieve a similar sort of return by a normal cash deposit, reinvested every six months. However, the deposit requires him to part with the entire notional principal for the full term of the deal. A more common type of structure has a swap book paying a zero coupon against receiving LIBOR. and paying out either an up-front or balloon (back-end) payment. The first of these two cases (where an up-front payment is made) is, in effect, a loan – the only difference being that there is no repayment of principal at maturity. As such, the risk to the “lender” is considerablyreduced, since the repayments are a relatively even stream. The second case (a balloon payment at maturity) is a reinvestment contract. One has to invest each LIBOR payment on behalf of the investor and roll them up six monthly in order to produce the lump sum at maturity. When expressed in the above manner, the problem is clear. These types of swaps leave significant cashflows which need to be invested or funded (depending on whether we are receiving or paying LIBOR). When these deals are priced, one usually represents the LIBOR payments as simple flows of principal. This method implicitly assumes that all such payments can be rolled up (i.e. reinvested) at the then prevailing six month forward rate. Thus, if six month LIBOR in one year’s time is 14%. it is assumed that any payments so far received will be able to be reinvested in one years time, for six months, at 14%. In reality, it can prove difficult to invest funds at LIBOR – even swap books investing with their own in-house treasury will often have problems in this regard. In view of this, it is essential that some leeway is included in the pricing of the deal to take account of this fact. Depending on the currency, an additional l/16th to 1/8th of a percent should be allowed for. This will enter via the par rates used to generate the discount function. An n year zero coupon swap, where we pay fixed, will generally give a positive delta in n years, and a negative delta at previous grid points. In other words, one uses the bid side rate at the n year point, and the offered side rates in others. The margin for reinvestment risk is then included by decreasing the n year rate and increasing the others by the relevant margin.

9.8. FORWARD FX ARBITRAGE

179

Using the sterling yield curve of chapter six to price a £10,000,000 five year zero coupon deal against six month LIBOR, the balloon payment turns out to be £9,353,031 on 17 May 1995 (the reader is invited to check these results – they simply involve valuing a cashflow in five years time). If one now adds in a margin of 12.5 basis points, so that the five year rate is 13.625%. and the four year rate, for example, is 13.975%, the balloon becomes £9,168,705. a difference (our “reinvestment risk premium”) of some £184,000. The mirror image structure to this – receiving zero coupon and paying LIBOR – is priced similarly. Bids and offers will be interchanged, but the idea is the same. It should be pointed out that the problem with these structures is not predominantly technical – rather, they can involve a swap book taking deposits or making loans. This type of business is, strictly speaking, “on-balance sheet” – the physical loan/deposit has a direct impact on the institution’s balance sheet. This is in contrast with normal swap activity – “off-balance sheet” – where principal flows do not happen. For this reason, many swap groups may be precluded from loan/deposit business.

9.8

Forward FX arbitrage

Our main concern so far has been with issues relating to hedging and trading interest rate swaps. We now turn to the possibilities offered by multi-currency swap books. The most significant of these is the forward foreign exchange arbitrage. The foreign exchange market splits into two distinct parts: the spot and the forward markets. The spot market deals in currencies for value spot (two business days after the trade date) – the forward market trades in currencies for delivery some time in the future. In the major traded currencies (for example, dollars against deutschemarks, yen and sterling) liquid forward markets exist out to about five years, although it is sometimes possible to get quotes out to ten years. The forward market trades in two different ways: as an “outright” trade, or as an “FX swap”. An outright forward trade is a simple contract to exchange two currencies at a rate determined today at a given date in the future. Since the most basic transaction type is the outright trade, it is this

180


that we shall concentrate on. The connection between the outright FX trade and the swap market is the zero coupon discount function. This function enables us to put a present value on a payment or receipt at any future time. This is precisely how a forward contract is valued. As an example, suppose one wishes to look at a five year dollar/sterling outright, buying sterling for dollars. The five year discount factors for both the dollar and sterling yield curves are needed, obtained from the example of chapter six – Tables 6.1 and 6.3. Furthermore, the spot exchange rate is $1.70=£l. To represent the cashflows resulting from the deal, one simply needs to value +£1 in five years on the sterling yield curve, and – $1 in five years on the dollar curve. These are +£0.5167 and –$0.6323 respectively. Using the spot FX rate, the pound received in five years is worth $0.5167 1.70 = $0.8784 today - in turn, this is worth $0.8784/0.6323 = $1.3892 in five years. Thus, the forward outright for five year dollar/sterling is £l=$1.3892. Outrights are usually quoted as a discount, in one-hundredths of a cent, to the spot rate. This is called the “forward point”. In this example, it is 10,000 (1.70 – 1.3892) = 3,108. To generalise, given two discount factors Fa and Fb, for currencies a and b at the same point in time, and with a spot exchange rate of (in units of currency a), the forward point (expressed as a percentage) is

(9-4) expressed in units of currency a. In the real world, the above formula is complicated a little by bidoffer spreads. In the example above, one should properly use the offered side of the five year sterling curve, and the bid side of the dollar curve5. Similarly, if we had been interested in selling sterling for dollars forward. the bid side in sterling and the offered side in dollars would have been used. This tends to create quite a wide bid-offer spread in the forward points, since it involves the ratio of two discount factors, and so the probability that an arbitrage will occur is small. 5

What is actually required, is the discount factor representing where we would be prepared to pay or receive on a zero coupon swap to the relevant date.

9.9.

FIXED-FIXED CURRENCY SWAPS

Spot 1Y 2Y 3Y 4Y 5Y

181

Rate: 1.70 990-965 1590-1540 2200-2050 2690-2490 3200-2950

Table 9.2: Forward dollar/sterling points

9.9

Fixed-fixed currency swaps

One of the benefits of our discussion of forward FX arbitrage is an alternative way of looking at fixed-fixed cross-currency swaps. These have fixed rate structures on both legs – there being no floating leg – usually accompanied by initial and final exchanges of principal6. For example, suppose you are asked to quote on a swap whereby you pay five year fixed sterling at 14.22%. (the annual equivalent of 13.75% semiannual) on £10,000,000. Against this you receive fixed rate dollars on a notional principal of $17,000,000, i.e. the sterling amount converted at a spot exchange rate of 1.70. What is the fixed dollar rate? We shall return again to the discount functions of Tables 6.1 and 6.3 and assume that the forward dollar/sterling points quoted in the market are as given in Table 9.2. There are now three different ways to approach the pricing of this deal: 1. Insert a floating dollar leg in each deal. In other words, the sterling swap book would pay fixed sterling against receiving floatingdollars, while the dollar book would receive the fixed on a normal interest rate swap. 2. Value the sterling cashflows as an up-front amount, converted into dollars, and use this to fund the dollar payments. 6

Fixed-fixed swaps where there is no final exchange of principal are known as “coupon swaps”.

182


3. Buy each of the sterling coupons forward for dollars, and smooth out the dollar cashflows. Each of these is dealt with in turn. Option 1: Inserting a dollar leg is the easiest of the three options to evaluate. We are prepared to receive fixed dollars on a straight interest rate swap at 9.42% (see section 6.2.3 for the dollar yield curve). Furthermore, we are prepared to pay 14.22% annually in sterling against sterling LIB OR flat – see Table 6.2. Thus, the only extra ingredient is the cost of the basis swap; where would the sterling book pay sterling LIBOR against receiving dollar LIBOR? As emphasized many times, the LIBOR legs value to par – there is no intrinsic value attached to the basis swap; the only element of pricing that enters the discussion is how keen one is to do it. This will usually reflect the markets bias towards the structure7 and one’s own position. Let us presume, for simplicity, that we were happy to do the basis swap for no margin. In this case, the dollar fixed rate that we would be prepared to receive is simply 9.42%. Had we been less keen to do the basis swap, ten basis points, say. might have been charged on the dollar side. In this case, one would need to convert ten semi-annual basis points of margin to annual basis points on the fixed side, and add this to 9.42%. Option 2: This is actually the same as option 1. The physical cashflows that take place in each currency take the place of the imaginary principal payments used to replicate LIBOR flows. For example, in sterling, the fixed coupons are at 14.22%. The final exchange of principal involves selling sterling for dollars, obtaining a cashflow of size – £10,000,000 at maturity. Similarly, there will be an initial exchange the other way round. The resulting cashflows value to par, since we have ended up with what looks like a five year swap at the annual equivalent of the par rate. Thus, the dollar side must value to par. again giving a result of 9.42%. Option 3: What we now intend to do is buy sterling forward for dollars, using the forward points in Table 9.2. in order to cover the sterling coupons to be paid. This will tell us how many dollars need 7

See section 7.4.4 for a brief discussion of why basis swaps might trade at a price other than LIBOR flat against LIBOR flat.

9.9.

183

FIXED-FIXED CURRENCY SWAPS

to be sold in the future. One can value these payments, and then find which dollar coupon gives the same result. The right hand side of the forward points must be used to give the larger outright exchange rates: Date 17-May-90 17-May-91 18-May-92 17-May-93 17-May-94 17-May-95

Sterling cashflows 10,000,000 (1,422,000) (1,429,792) (1,418,104) (1,422,000) (11,422,000)

Forward Implied dollar cashflows outrights (17,000,000) 1.7000 2,280,177 1.6035 2,210,458 1.5460 2,120,066 1.4950 2,063,322 1.4510 16,047,910 1.4050 PV = $147,301

Discount factors 1.0000 0.9185 0.8379 0.7637 0.6953 0.6323

It is a simple trial and error exercise to find that the required dollar coupon is approximately 9.64%: Date 17-May-90 17-May-91 18-May-92 17-May-93 17-May-94 17-May-95

Dollar swap Discount factors cashflows 1.0000 (17,000,000) 0.9185 1,661,785 0.8375 1,670,891 0.7637 1,657,232 0.6953 1,661,785 0.6323 18,661,785 PV = $146,441

In this case, the arbitrage doesn’t work, and the first option turns out to be the optimal route.

Chapter 10 Interest rate options Thus far. we have been concerned with two types of cashflow: “known” cashflows, such as those on the fixed leg of a swap, and “unknown”, such as those on the floating leg. There is one further category to introduce: the “contingent” cashflow. A contingent cashflow is one whose size and/or occurrence depends upon another event. In this chapter, this event will be related to various future LIBOR or swap rates being greater or less than a given rate. Instruments with these characteristics are called “options”. More generally, an option is a contract giving, for a fixed period of time, the right, but not the obligation, to buy or sell an instrument at a fixed price. Options to buy are called “calls”, and those to sell. “puts”. The fixed price at which one can buy or sell (“exercise”) the object is called the “strike” or “exercise” price, and the fixed date, the “expiry date”. Options are divided further into two categories: an “American” option is one that can be exercised at any time between the purchase and the expiry date. A “European” option can only be exercised on the expiry date. Almost all interest rate options traded are European, and we shall only be examining this type. American options introduce a considerable degree of complexity which is beyond the aims and scope of this book. As noted in the introduction, this chapter requires a higher level of mathematical ability on the part of the reader. In particular, some knowledge of probability and statistics is assumed, as well as the ability to carry through some detailed algebra.

185

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CHAPTER 10. INTEREST RATE OPTIONS

Figure 10.1: Evolution of bond prices in a toy model

10.1

A toy model

This section introduces the valuation of options by constructing an exceedingly simple model of the movements in the price of a security. Those readers who are already familiar with the basics of option pricing could skip this section. Suppose you are offered a contract giving you the right, but not the obligation, to buy a bond at some date in the future (say four weeks) at a price of 94.00. The bond price at the moment is 93.00. What is the value of this “option” ? Let us divide the next four weeks into one week time intervals, and assume that, in each interval, the bond price can either move up by a point or down by a point. Further, suppose that the probability of an up movement (denoted by u) is p, so that the probability of a down movement d is 1 – p. One can then construct a tree of possible price movements over the four week period, as shown in Figure 10.1. The

10.1. A TOY MODEL

187

probability of four up movements is simply p4. However, the probability of three ups and one down is 4p3 (1 — p), since there are four possible paths leading to this result: uuud, uudu, uduu and duuu. It should be clear that the probability of j upward movements in four periods is given by the coefficients of the binomial expansion:

(10.1) are the binomial coefficients [7]. where the There are only two results of interest, namely when the price is 95 or 97, since then the option is actually worth something – if the price is 95, its value is one point, and if the price is 97, three points. Thus, the value of the option is just: (10.2) which, for p = 0.5 (an equal chance of up or down movements) is 43.75 basis points. In fact, this value should really be discounted back to today, since the payoff occurs in a month’s time. Supposing that the relevant discount rate is 10%, the value of the option is approximately: (10.3) This is the theoretical price of the option according to a simple model. There are a number of points worth making in relation to more realistic models: 1. The formula. Equation 10.1. for the probability of j upward movements is called the probability distribution of the model. The type of distribution chosen is one of the key features in model building. Clearly a binomial distribution is a gross oversimplification of the true nature of bond markets. One might wish to improve on this by also allowing the bond price to remain unchanged at each point of the tree. This would correspond to a trinomial distribution. 2. Equation 10.1 could be generalised to the case of I time intervals: (10.4)

188

CHAPTER 10. INTEREST RATE OPTIONS There is a well known result, due to Laplace and DeMoivre [8], the binomial distribution converges to that says that as a normal distribution: (10.5) for Z a normal variable. This particular distribution occurs in the next section.

3. The expected value and variance of a binomial distribution X is:

In option theory, the standard deviation. is usually called the volatility of the random variable X.

10.2

The standard model

This section presents a brief explanation of the standard model used to price interest rate options. This has its origins in a paper by Professors Black and Scholes. published in 1973 [9], which presented a model for pricing stock options. This was subsequently developed for pricing options on commodities such as futures and forward contracts [10]. Many books provide a rigorous derivation of the Black-Scholes equation [11]. and the interested reader is referred to these if he wishes to fill in some of the gaps we have left. All the options considered here are assumed to depend on some random variable. Indeed, if this were not the case, all cashflows would be reducible to either known or LIBOR type cashflows. There is a particular type of random variable in which we shall be interested, called a “Wiener variable”, which is usually denoted by z. This is defined over an infinitesimal time period t as satisfying: 1.

where is a random sample from a normal distribution (in other words, a random variable) with mean zero and standard deviation 1 (thus the variable is “normalised”): and

10.2. THE STANDARD MODEL

189

2. the set of variables are independent for different property makes the variable “Markov”). The fact that

(this

belongs to a normal distribution means that: mean of

standard deviation of variance of One can move to a “generalised” Wiener process by adding a drift term. t, and a constant multiplier, 6, to the random term to create: a, b constant

(10.6)

which has the properties:

The obvious further generalisation is to make the constants a and b functions of the random variable, x, and time – one therefore defines an “Itô process” as: (10.7) Investors expect to earn a return on an instrument which is independent of its current price. S. Thus, the assumption that S has a constant drift rate is incorrect; the proportional price must have constant drift t. Assume further that the variance of the return over a period t is independent of the price – this is consistent with the previous assumption. Then. t is the variance of the actual change in price over t (so that is the variance of the proportional change). Then. the Itô process becomes: (10.8) (10.9) (10.10)

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INTEREST RATE OPTIONS

where (m, n) is the normal distribution with mean m and standard deviation n. There is a powerful result due to Itô: Itô’s Lemma If a variable x undergoes an ltô process: (10.11) with dz a Wiener process, then a function

(x, t) follows the process: (10.12)

where dz is the same Wiener process as in Equation 10.11, and we have moved to differential notation. So. with the model described by Equation 10.8: (10.13) Denning: (10.14) so that: (10.15) gives: (10.16) so that A has drift rate and standard deviation and over a period of time t is normally distributed with T, the change in mean and standard deviation Then, if 5, is the price at time i: (10.17)

Thus. ST has a lognormal distribution. To derive the Black-Scholes equation for stocks requires various assumptions:

10.2. THE STANDARD MODEL

191

• the stock price follows an Itô process; • there are no dividends, taxes or transaction costs; • there are no arbitrage possibilities; • the risk free rate, r, is constant and the same for all maturities; • short sales are allowed: • trading is continuous. Consider an option, whose price is on an underlying stock S; function of S and t. The stock undergoes the stochastic process:

is a

(10.18) and since

is a function of S and t, it undergoes the process: (10.19)

where we have just replaced A by in Equation 10.13. From Itô’s Lemma, the random variable z is the same in Equations 10.18 and 10.19. Considering the linear combination: (10.20) This represents a long position of option. From this:

shares and a short position in one (10.21)

and substituting from Equations 10.18 and 10.19: (10.22) Since the random variable has been eliminated. free rate r:

must earn the risk (10.23)

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CHAPTER 10.


and. equating Equations 10.22 and 10.23 gives the Black-Scholes equation: (10.24) The standard deviation, a. is usually referred to as the “volatility” of the underlying security. Boundary conditions must be imposed on this equation in order to solve it. Consider a call option – at the exercise date, it entitles the holder to buy the underlying instrument at the strike price K. Clearly, this will only be beneficial if the price at the exercise date of the underlying. ST, is greater than the strike (transaction costs are ignored). If this is not the case, the option will expire worthless. Thus the “payoff” at exercise is: (10.25)

max[0, ST - K] Similarly, for a put option the payoff will be:

(10.26)

max[0, K - ST]

The Black-Scholes equation, together with the relevant boundary condition, can be solved to give the value of a call option [9]: (10.27) (10.28) (10.29)

(10.30) (x) is the cumulative normal distribution function mentioned earlier: (10.31) Although the integral can be performed numerically, it is usual to use one of the standard polynomial approximations [7] such as: where

10.2.

THE STANDARD MODEL

b1 b2 63 64 b5 and:

=

193

0.3193815

= –0.356563 = 1.7814779 = –1.821255 = 1.3302744

is the normal density. The two types of options in which we shall have special interest are called “caps” and “floors”. A cap is a strip of (European) call options on a given LIBOR index. Thus, a one year cap on three month sterling LIBOR, L3, is a set of four options (individually called “caplets”) – if the strike is K, the buyer receives money if, at the maturity of any of the four options, three month LIBOR exceeds K. The amount payable is calculated as: (10.32) where P is the notional principal of the cap, D1 is the expiry date of the option, and D2 is three months after D1. The convention is to pay any amount due at D2. A floor is simply a strip of puts on a LIBOR index, and so a payout occurs if the relevant index is less than K. From the above definition of a cap. it should be clear that a single caplet is equivalent to an option on a single period swap. One may therefore be tempted to think of it as an option on an FRA, but this is not strictly correct, since the payoff on an FRA is not the same as that on a swap (remember that an FRA pays discounted in advance, a swap undiscounted in arrears). So, a buyer of a caplet is essentially purchasing an option to pay fixed on a single period swap, and vice versa for a “floorlet”. One may try to extend this analogy to a cap comprised of more than one caplet. In other words, is a two year cap. say. equivalent to an option on a two year swap? The answer is a categorical no. If one exercises an option into a two year swap, one has contracted to exchange the difference between the fixed rate and

194

CHAPTER 10.


LIBOR on every period of the swap. However, on the cap, each caplet acts as an option in its own right. Thus, the price of a cap is the sum of the prices of the constituent caplets. An option on a swap, on the other hand, cannot be decomposed in this way. We now come to the main point of this discussion: namely, it costs nothing to enter into a swap at market rates (or into an FRA for that matter). However, the formula derived in Equation 10.27 assumed that the underlying instrument (a stock) did cost a non-zero amount to purchase – indeed, this fact enters explicitly as S, the price of the security, via Equation 10.20. The modifications introduced by this feature were first studied by Black in relation to options on commodities and futures [10]. These are also “zero-cost” instruments. Black followed the same argument as in the case of stocks, but the derivative term in Equation 10.20 is missing, so that , the value of the portfolio, is just – , the value of the option. One can then follow through the same process to derive the differential equation that must be satisfied by an option on a zero-cost security Z: (10.33) where r is the riskless rate. This equation was solved by Black to give the price of call options on zero-cost securities. For a caplet, his solution becomes: (10.34) where the coefficients in the normal function are: (10.35)

(10.36) t is the time in years to the exercise date of the option. Here. which replaces S in Equation 10.27, is a forward price (in the case of options on futures, it is the price of the future) or rate (in the case of a caplet). We have also introduced the discount function Fb at the end of the underlying instrument. So, for a caplet on three month LIBOR which expires in two months. Fb will be the discount factor at the five month point. Da is the date at the start of the underlying index (two months in the example). The reason for this term is that the payment on the

10.2.

THE STANDARD MODEL

195

caplet is made at Db, and is calculated as the accrual factor times the payoff function max[0, Li — K], with Li the underlying index, times the principal. The forward rate can be written for a caplet as: (10.37) The volatility that appears in Equation 10.35 is that of the forward Black’s model assumes that this will remain constant over the rate life of the option. While this is not a serious problem for options which have a long exercise period in comparison to the underlying instrument, it is inaccurate for, say, a three month option on three month LIBOR. In practice, this effect is either ignored, or some “manual adjustment” is made to the price. It turns out that errors in estimating the volatility overshadow this problem. We shall now try to crystallise these formulae with a specific example. Assume that the yield curve is that defined in Table 6.2. Thus, the value date is 17 May 1990. Consider a three month option on three month LIBOR. with a strike of 15%. The exercise date of the option is 17 August 1990, and the end of the underlying index is 19 November 1990 (17 November is a Saturday). From Table 6.3, Fa = 0.9632036, Fb = 0.9273432, and = 0.0025753. t is 92/365 = 0.252055 (the extra precision in the discount factors is required for calculating an accurate forward rate). We shall assume that the volatility is 17%. Then: (10.38)

and so: (10.39)

You will need to work out all the numbers to as high a degree of accuracy as possible to get the same answer. From this: (10.40)

196


The polynomial approximation for the normal distribution gives:

and so the value of the caplet is:

This is expressed as a fraction of the nominal. Thus, a £10,000,000 caplet would cost £10,000,000 0.0012389% = £12,389. To determine the value of a floorlet. consider a portfolio consisting of a long position in a floorlet with strike K, and a short position in a caplet with the same strike, underlying security and dates. Assume that the volatilities of the two options are the same. If the underlying index is Li, the payoff from this combination is:

This is precisely the same payoff as occurs on a one period swap, with the fixed rate equal to the strike K, and the floating index equal to the index Li. The swap begins at the start date of the option, Da, and ends at the end date. Db. The cashflows on the swap are minus the principal (assumed to be 1) at Da, and principal plus interest of K at Db. By the methods of chapter six: Value of swap

(10.41)

(10.42) So, the value of the floorlet is given by: (10.43)

10.3. CAP AND FLOOR EXPOSURES

197

where is shown in the option argument as a reminder that the value of a floorlet with volatility a must be calculated by using the same value of sigma in the caplet formula. The reason this must be done is that both the caplet and floorlet are assumed to be options on the same underlying single period swap. Using different volatilities would introduce an inconsistency. The above relation is an example of put-call parity. This type of formula occurs in all option theory – the idea is that one can always form a combination of call and put options (and possibly some default free bonds or cash) to give a payoff identical to that of the underlying security. Before concluding this section, we mention one more instrument which has become popular: namely, the interest rate “collar”. A buyer of a 15% – 12% collar has purchased a cap struck at 15% and, simultaneously, sold a floor with a strike of 12%. Thus, the buyer has hedged himself against movements of the relevant index outside the collar range. While the index stays within this range, no payments take place under the collar. He has also offset some (or all) of the cost of the cap by writing the floor. Indeed, the strike on the floor is often chosen so as to exactly cancel the cost of the cap, creating a “zero-cost collar”.

10.3

Cap and floor exposures

Having established a method for valuing caps and floors, the next task is to express their exposures in a manner which enables them to be easily hedged. There are two distinct types of exposure that arise: interest rate, at both the first derivative (delta) level and higher order (gamma), and volatility exposure. We shall call the first order volatility exposure the kappa (some people call it “vega”). Ideally, interest rate exposure should be expressed in a manner comparable with that for swaps – in other words, as the swap equivalent positions (or. possibly, futures or deposit equivalent positions). Since swaps have no volatility exposure, we are at liberty to interpret vega in whatever manner best suits us. It is worth reiterating a point made earlier: namely, that once the value of the option is determined by a differentiable function, all rele-

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CHAPTER 10.


vant exposures are determined by examining the derivatives of the PV with respect to the input parameters (the par rates and the volatility in this case). We first evaluate the delta vector of a caplet. Differentiating Equation 10.34 with respect to the vector of grid point discount factors, F, gives: (10.44) Deriving Equation 10.44 involves differentiating the normal distribution function ( y ) . There is a standard result [7]: (10.45) where (y) is the normal density defined earlier. Another useful equality arises when looking at:

It is a simple matter to derive the delta for fioorlets: (10.46) When Da and Db are grid points, the terms and in the above formulae are essentially “delta functions” – in other words:

otherwise

10.3.

CAP AND FLOOR EXPOSURES

199

so that they are vectors with a single non-zero entry. However, if the date Di falls between grid points, the discount factor Fi is a function of the two surrounding grid point discount factors1. The form of this function depends on the type of interpolation procedure used. Note that we have only taken the derivative with respect to discount factors, not par rates. This last step can be achieved by using the chain rule: (10.47) where R is the complete set of rates. The matrix was evaluated in chapter eight as part of the calculation of swap deltas. One simply into this matrix to give the sensitivities needs to multiply the vector to changes in par rates. Going back to the simple example discussed earlier in the section – a three month option into three month LIBOR capped at 15% – there are two non-zero terms in the delta vector. Because Fa and Fb are both grid points, Equation 10.44 simplifies to give for the three month point: (10.48) and for the six month point: (10.49)

Using: (10.50) gives: (10.51) for a one basis point shift in a short position in a £10,000,000 caplet. 1

This assumes either a linear or exponential interpolation procedure.

200


There is a convenient way of evaluating the exposures on a caplet which explicitly shows the relationship of the option to the underlying instrument (a one period swap). One can easily rewrite the delta as: (10.52) Exactly the same delta is generated by the cashflows on a one period swap with notional principal (h 1 ) (times the principal of the option), and fixed rate with the start and end dates of the swap the same as those of the caplet. This representation of the delta clearly shows the equivalence between an option and a position in the underlying instrument. Having obtained the formulae for the deltas, it is a simple, although very tedious, matter to derive the second order derivative – the gamma matrix – for caplets: (10.53) Unfortunately, although obtaining the second order derivative with respect to discount factors is relatively straightforward, obtaining the second order sensitivity to rates involves using the chain rule and the matrix: (10.54) This matrix is much more tedious to evaluate, as was discussed in chapter eight. However, for our simple example, it is relatively straightforward to find:

so that:

10.4. SWAPTIONS

201

for a one basis point squared shift on a £10.000,000 caplet. The gamma matrix is a measure of how great a degree of rehedging will be required as rates move – being very long options, whether they be caps or floors, will give a large gamma position. The remaining derivatives of interest are those with respect to the volatility and time to exercise. These are simple to derive: (10.55) (10.56) The derivative with respect to (“kappa”) is for a one percent shift in volatility, and that with respect to t (“theta” or time decay) is for a one year movement in time to expiry. It is usual to divide theta by the number of days in the year (or possibly business-days) to obtain a more useful measure.

10.4

Swaptions

Swaptions (or swoptions, depending on one’s view of which product is more important) are options on interest rate, or sometimes currency, swaps. Although less developed than the cap and floor market, a large swaption market exists in dollars, with deutschemarks and sterling still developing. They originally arose as a component of structured bond issues, where the bond was puttable at some time prior to its maturity. Since the bond issuer often required an interest rate swap to achieve a sub-LIB OR funding requirement, he also needed the option to terminate all or part of the swap in the event that the bonds were put. Such a facility is offered by the swaption. This section presents the two easiest ways of pricing swaptions. Later on in this chapter, we shall discuss the failures of these models, together with some possible alternatives. The most obvious approach to pricing is to try to simulate the route taken in the Black model. For caps and floors, the relevant stochastic variable was the implied forward rate, Analogously, we could regard the underlying price for a forward start swap as the stochastic variable for a swaption. Thus, a six month option into a two year swap is an

202


option on a two year swap, six months forward. The calculation of this forward rate depends on the type of swaption under consideration: a “payers swaption” gives the buyer the right to pay the fixed rate at an agreed strike, while a “receivers swaption” gives the right to receive. Since the payers option gives the buyer the right to “purchase” a swap at a fixed price, it is a call option on the swap – similarly, the receivers option is a put on the swap. This makes sense when one considers that a cap is a call on a set of forward rates. It should be appreciated that there is a very important difference between, say. an 18 month cap on six month LIBOR (the first option starts in six months’ time), and a six month option into a one year swap. The difference lies in the fact that the cap is made up of two separate options on six month LIBOR; the swaption is a single option into a swap. Even though the cashflow dates on the swap coincide with those on the cap. each caplet in the cap is treated independently, whereas, once exercised, all the cashflows on the swap must occur. So, to summarise, a cap is a set of options on a LIBOR index, whereas a swaption is a single option on a complete swap. Returning to the question of pricing, we can write down the Black formula for a payers swaption by extending Equation 10.34 in the obvious way: (10.57) with h1 and h2 defined as in Equations 10.35 and 10.36. Note that the discount factor in Equation 10.57 is that for the expiry of the option, not the expiry of the underlying swap. This is because, were the buyer of the option to exercise, any resulting cash settlement would be for value Da. is now the price of the forward start swap, which, for a standard interest rate swap is just: (10.58) where i runs over the coupons on the swap, and the last coupon is at Dn. This formula is fatally flawed. To understand why, consider a one day payers option into a two year swap. Suppose the yield curve is flat at 10%, and the strike of the option is. say, 9%. Furthermore, suppose

10.4. SWAPTIONS

203

that the volatility of the underlying price is very small. Now. the price of the forward swap, calculated from the above formula, will be almost exactly 10% – a two year swap, one day forward will have almost the same price as a par two year swap. Furthermore, since the volatility is very small, the normal functions and will be almost exactly one. Fa in Equation 10.57 will also be almost one. Thus, the value of the option is approximately 1[10 1–9 1] = 1%. So if the option was on a £1,000,000 swap, the value would be about £100,000. The following day, the option is exercised – if rates have not changed (as is essentially assumed by taking a small volatility), then the value of the swap into which one has exercised is not just one percent of the notional. It is actuallv: Notional

(10.59)

where i runs over the four coupons (for a generic sterling swap) of the two year swap. This is clearly more than one percent of the notional > 1. and is quite possibly significantly larger. The reason if for this discrepancy lies in the fact that the Black model regards the underlying swap as simply a forward rate; it does not encapsulate the structure of the swap, such as the maturity, coupon frequency etc. Indeed, one can envisage options on complex cashflow structures where the answer would be vastly different from the actual value. Therefore we must reject this model as a description of options on multi-cashflow instruments. The problem can be most easily envisaged by looking at the payoff – as described by Equation 10.59. More generally, as the time to expiry of the swaption goes to zero, Equation 10.57 becomes max[0, – K], where is now the spot swap rate. The expression which captures the true value of the swap to be exercised into is: (10.60) But. for a forward start swap starting at the expiry date Dt: (10.61)

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CHAPTER 10.


so that the payoff can be written as: (10.62) where the floating and fixed legs are respectively: (10.63)

From this representation of the payoff, the equation for a call (payers) option becomes: Payers =

(10.64)

where: (10.65) The central assumption in deriving the above equation is that it is the value of the floating leg, Lt, which is the stochastic variable. Of course, this is determined by a collection of implied forward rates, and so the volatility of the entire leg will be a complicated function of the volatilities of the individual forward rates plus some correlation coefficients between them. This makes it somewhat difficult to estimate in this form – however, the form of the floating leg represented in Equation 10.63 shows that it can be evaluated from the volatilities of the discount factors at the beginning and end of the swap, in addition to their correlation coefficient. It is a simple exercise to show that, for an option into a one period swap, the formula Equation 10.64 reduces to that of a caplet (Equation 10.34), as it should. The usual put-call parity argument gives the value of a receivers (put) option as: Receivers = Payers –

(10.66)

10.4. SWAPTIONS where

205

is the implied forward start swap rate: (10.67)

The delta vector of the payers option is straightforward to evaluate:

for j = t for j = i + 1 for j = n

(10.68)

and that of the receivers is:

for j = t for j = i + 1 for j = n

(10.69)

Although this looks rather complicated to implement, one can obtain the correct sensitivities for a call, by analysing a swap with principal equal to the notional of the option times (d 1 ), and coupon equal The cashflows occur on the same dates as to that of the underlying swap. This swap, when analysed by the means explained in chapter eight, gives the correct sensitivities and present value of the option. The other sensitivities are obtained in the usual way: (10.70) (10.71) We shall calculate the price of a six months payers option into a par two year swap, with a strike of 13%. The expiry date of the option is 19 November 1990 (the 17th and 18th being bad days), and the maturity is 17 November 1992. Using the sterling curve of Table 6.3 and a volatility of 12% gives: Lt = 0.21537 Lf = 0.20412

CHAPTER 10.

206

= = Payers = Receivers =

INTEREST RATE OPTIONS 0.66894 0.74823 1.41% 0.29%

As an exercise, the reader could verify that the swap which replicates the delta of the option has the following cashflows (assuming a notional principal of £10,000,000): Date 19-Nov-90 17-May-91 18-Nov-91 18-May-92 17-Nov-92

Cashflow 7,482,317 (459,117) (474,507) (466,812) (7,951,694)

As usual, brackets indicate a negative cashflow. An alternative form for the payoff can be obtained from Equation 10.62:

Ft max

(10.72)

It should be noted that this payoff only applies to options into swaps that start at the expiry date of the option. A modified formula is needed for options into forward start swaps. Equation 10.72 resembles closely the payoff on a bond option. In fact, we shall denote the second term as B: (10.73) It is essentially the forward price at time Dt of a one pound bond with coupon K occurring on the dates Di and redeeming at Dn at par. So the payoff at expiry is: Ft max [0,1 – B]

(10.74)

It is now easy to work back to the bond equivalent version of Equation 10.57: (10.75)

10.5.

OTHER MODELS

207

where: (10.76) Since a portfolio of a long position in a payers option and a short position in a receivers option is equivalent to the underlying swap: Receivers = Payers – F t (1 – B)

(10.77)

Taking the example used previously and choosing the volatility to be three percent (this may seem very low. but the “bond price” volatility is lower than the “rate” volatility), a straightforward calculation gives the bond price as B 0.9890, and the normal variable h 0.50523. Finally: Payers =

0.9273 [0.700781 – 0.9890 140 basis points

0.693302]

The main problem with this approach is the volatility. Instead of the volatility of either a forward rate, as for caps, or a floating leg, as for the previous model, a volatility of the price of a synthetic bond is now needed. Clearly, this is not a quantity that is readily available. For this reason, this second formulation of the Black model is seldom used for swaptions.

10.5

Other models

Although the modified Black-Scholes model expounded in the previous section is essentially the market standard pricing tool, it is nonetheless theoretically imperfect. The most important such imperfection relates to the number of random variables used – the Black model allows for only one stochastic variable. Unfortunately, such a restriction is at odds with the real world – each swap rate, for example, is, in theory, an independent random variable. Of course, in practice one rarely sees the different rates move in opposite directions – there is a certain degree of correlation between them. However, the correlation between a three month and a ten year rate is small. The idealised option model would therefore allow for as many random variables as input rates - some of

208


these would perhaps be related through correlation coefficients. However, the prospect of solving a set of perhaps ten differential equations, let alone computing any useful numbers within a realistic time frame, is remote. In view of this, a number of academics have attempted to construct models with two or more stochastic variables, in the hope that more information about the yield curve can be accommodated. An example of a “two factor” model is that proposed by Brennan and Schwartz [12]. Their stochastic variables are the instantaneous riskless rate and the yield on an undated default free bond (a “consol”). They then assume that each of these variables undergoes a stochastic process similar to that in Equation 10.11, with the two related by some correlation coefficient. One subsequently obtains a partial differential equation which can be solved to give the price of a bond maturing at any time in the future – in other words, the discount curve is determined. One would then wish to make this derived curve compatible with the term structure which is observed. This has to be achieved by fine tuning of the parameters of the model to give a “best fit” of the discount curve – not necessarily an easy process. This idea of obtaining the discount function, as well as option prices, from a particular model is common to a number of approaches. However, as just commented, it can be cumbersome to fit the real yield curve. Ho and Lee [13] found one way around this problem – they managed to derive arbitrage free restrictions on movements in the term structure, taking the entire curve as an input of the model. Essentially, they derive a binomial tree, where each node on the tree is a complete discount function. In this way. the possible allowed curves can be obtained for any point in the future, and thus option values obtained in the manner of section 10.1. Unfortunately, there are two parameters in the model which need to be estimated from other option prices, which could provide problems for obscure or esoteric products. Furthermore, the model allows for unbounded and negative interest rates. The Ho-Lee model has been generalised by Heath, Jarrow and Morton [14] to allow for two or more stochastic variables in continuous time, and avoids negative interest rates. Their approach is to impose a stochastic process on all forward rates, rather than the discount function. The model is still able to incorporate details of the initial term

10.6.

HEDGING OPTIONS

209

structure. Furthermore, it can accommodate any specified volatility structure. However, like Ho and Lee, the model is path dependent, and is extremely time consuming. Given the choice of models available, the reader would be forgiven for feeling somewhat overwhelmed. The ultimate decision regarding the model to be used on the trading floor will depend critically on ease of use and flexibility. One must, for example, be able to obtain not only the price, but also the full delta vector and swap equivalent positions of the option. This condition alone restricts the number of models which can be implemented. However, with the advent of extremely powerful desk-top computers, it may not be long before multifactor models are being used for live trading.

10.6

Hedging options

When considering swaps alone, the process of hedging becomes a relatively straightforward one. One decomposes the swap into either swap equivalent and/or futures positions and then hedges the resulting delta by grid point. The gamma for swaps is usually very small, and can be ignored. For options, however, life is not as simple. In addition to the deltas, the gamma will usually not be as small, and there is an added dimension to hedging in the form of volatility exposure. Little has been said thus far about representing the volatility exposure. For caps/floors alone, one is exposed to the volatility of forward rates. One can reasonably argue that the three month rate three months forward has a different volatility to a three month rate two years forward. It is a simple enough procedure to incorporate this in the pricing (just use a different volatility for the longer caplet elements), but how does one reproduce this effect in a volatility vector (we shall refer to it as the kappa)? One approach is to analyse the kappa in the same manner as one treats the delta. This involves allocating kappa to the grid points surrounding the caplet dates. This will indeed give a kappa vector which has non-zero entries over the full period of the option. The procedure for weighting these elements would be analogous to that for weighting cashflows on grid points. When one introduces swaptions into the picture, matters become

210


more complicated. The trouble lies in the fact that the volatility in this case is that of a floating leg – not a forward rate. Thus, the volatilities of caps and swaptions are not the same. Of course, there is some extremely complicated relationship, since the forward rates and the floating leg can all be decomposed into discount factors. However, it is unlikely that this way of viewing the volatility will suit many traders. Probably, the only solution is to use a different volatility curve for swaptions. One then has to decide on its form in the same manner as for caps. For swaptions. one approach would be to decompose the kappa vector by maturity of the underlying swap. Thus, a six month option into a two year swap would give non-zero kappa elements at the two and three year points. Let us assume that the trader has decided on some form of representation for the exposure to cap and swaption volatility. How does he go about hedging them? Clearly, it is only possible to hedge kappa exposure by using other options. The trivial solution is to hedge caps and floors with other caps and floors, and swaptions with other swaptions. In fact, for long term options, this is probably the only realistic alternative. One might try using options on government bonds. However, one is once again frustrated by the fact that the volatilities are intrinsically different. In addition, there is a spread between the two which usually remains unhedgeable. In the shorter end, however, the trader can find some solace by using options on three month interest rate futures. These have the great advantage that they are exchange traded, and thus much more liquid than other options. The value of a call option on a futures contract is not difficult to write down: (10.78) where: (10.79) with P the price of the underlying future and K the strike. The volatility is clearly that of the underlying contract – i.e. the volatility of a forward rate. The above formula is the obvious extension of Equation

10.6.

HEDGING OPTIONS

Option portfolio 0 1Y delta –100 2Y delta 200 3Y delta 0 1Y kappa 20 2Y kappa 100 3Y kappa 50 Total gamma

211

Hedging instruments a –100 100 0 –20 20 0 0

b c –100 100 0 50 50 150 50 100 50 0 50 –100 –20 50

d –100 150 0 50 –50 0 20

e –100 50 50 100 –50 –50 20

f –100 20 120 0 0 0 20

Table 10.1: A toy model for option hedging 10.75. The delta of the call is simply Ψ(h), and the kappa is given by: (10.80) A put option on a future is easy to derive from the put-call parity relationship: Put = Call + K – P (10.81) Futures options are well suited for hedging the volatility exposure of short dated caps and floors. Their usefulness for swaptions is limited. The process of hedging all the exposures that arise in an option portfolio is rather tedious. The general method relies on a branch of mathematics called linear algebra. Rather than present a general argument, a toy example is given. Suppose a portfolio of options has non-zero delta and kappa exposures at the one, two and three year points, and has a total gamma summed over all gridpoints. Now assume that seven hedging instruments are available – each of these will introduce exposures in various maturities. These are given in Table 10.6. Thus, one unit of hedging instrument “c” generates a three year delta of 50, one year kappa of 100. and so on. The problem to be solved is: how many units of each hedging instrument are required to exactly cancel every one of the seven non-zero option portfolio exposures (three deltas, three kappas and one gamma)? The answer is obtained by a procedure called “matrix inversion”. That

g –100 130 0 0 0 0 –20

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CHAPTER 10.


is. the problem is rephrased in terms of vectors and matrices as:

p = Mh

h = M–1p

where p is the vector formed by the option portfolio exposures (the first column of figures in the table), M is the matrix formed by the seven rows and columns to the right of p, and h lists the amount of each hedging instrument required to solve the problem (so that its first entry is the number of units of instrument “a” required). In order to obtain the answer, the matrix M must be inverted [15]. Once this has been achieved, it is multiplied into p to get:

h=

–1.700 –0.693 0.427 –0.465 –0.188 2.433 1.040

So. – 1.700 units of “a”, –0.693 units of “b” etc. are required for the exact hedge. The reader will by now appreciate that this process is unwieldy – indeed, it may not be possible to invert a larger matrix at all. In general, the more hedging instruments are available, the better the chances of a solution being found.

Chapter 11 Managing a portfolio Our main concern in this book has been with the technical aspects of pricing and hedging interest rate swaps and options. There are. however, some practical issues related to running a swap and option portfolio which can have a significant effect on profitability. This chapter concerns these issues.

11.1

LIBOR exposures

Most of the hedging performed in the process of running a portfolio relates to the fixed leg of the swap, while tacitly ignoring the floating leg. For example, it is quite common for a five year swap against six month LIBOR to be hedged with a five year swap the other wayround, but against three month LIBOR. In such a situation, there is no exposure to five year rates, but there remains a mismatch on the floating leg. What is the effect of such LIBOR exposures? In a swap portfolio consisting of perhaps thousands of deals, there will, in general, be LIBOR sets occurring every working day. For a generic swap, this means that a deal will have its six month LIBOR set for the next period, to be paid in six months time. However, as we move in time through this period, six month rates will almost certainly change. For example, suppose LIBOR is set at 15% to be paid by us in six months time. If rates subsequently fall, the deal will show a PV loss. It is this risk that we wish to hedge, if possible. 213

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MANAGING A PORTFOLIO

The most obvious instrument to use as a hedge against LIBOR exposure is an interest rate futures contract. This is because the contract is set, at expiry, at the three month LIBOR settlement rate for that day. Unfortunately, there are relatively few liquid interest rate futures – Eurodollars and short sterling being the best. To explain the hedging process, consider a par five year sterling swap, on which we pay fixed at 13.75% against six month LIBOR. At 11:00am on the morning of the deal, six month LIBOR is set at 15.375%. Analysing the swap cashflows against the sterling yield curve of Table 6.2 gives a delta vector with two non-zero entries: Grid point 6M 5Y

Equivalent position

PVBP

10,000,000 (10,000,000)

(473) 3,515

This is easy to understand. The swap cashflows are equivalent to a five year par swap without the LIBOR set for the first period, plus a par six month loan. Since these are both par deals, the equivalent positions are equal in size to the swap principal. If six month LIBOR had not been set at 15.375%. the six month equivalent position would have been different from £10,000,000. The naive approach is now to isolate the exposure (PVBP) due to the LIBOR set. divide this by the tick value on the future, and sell that number of June short sterling futures (the nearest contract). In this case, this would entail selling approximately 473/12.5 38 June contracts. The hedge involves selling contracts, since the original swap loses value as LIBOR rates rise. To offset this loss, the hedge must gain value as LIBOR rates rise i.e. it must be a short position. An immediate problem presents itself. If June contracts are used as a hedge, what happens on the 20th June when the contract expires? Clearly, the hedge cannot be good, since the LIBOR period runs from 17 May to 19 November 1990. Instead, the hedge must involve contracts that cover this entire period. The correct answer is provided by the futures curve. If the exposure is re-expressed in terms of sensitivities to each futures contract, the exact hedge is obtained. Stripping off the futures curve of section 6.5,

11.1.

215

LIBOR EXPOSURES

the following non-zero exposures arise1: Grid point Rs 1F 2F 5Y

PVBP (92) (240) (159) 3.515

All the exposures are to rates – thus, the 1F and 2F exposures must have their signs changed to give exposure to futures prices. The reader may notice something slightly unusual here. The exposure created by the LIBOR set appears to have changed from (473) to (92) + (240) + (159) = (491). This can be explained by noting that we have re-expressed the exposure in terms of different instruments. In particular, futures are three monthly contracts, as opposed to the six month deposit instrument. Thus, the difference can be accounted for by looking at the relationship between three and six month equivalent rates: (11.1) where S and Q are semi-annual and quarterly rates. Differentiating this expression with respect to Q gives: (11.2) This is the conversion factor between the deltas: 473 x 1.0375 491. Having obtained the exposures to the futures contracts the exact hedge is just the PVBP divided by the tick value. As was explained in chapter nine, the exposure to Rs is normally hedged by adding it in to the 1F exposure. In this case, the hedge is to sell (92 + 240)/12.5 26 June contracts, and sell 159/12.5 13 September contracts. Unfortunately, one can rarely escape the problems of basis risk, and hedging LIBOR exposure is no exception. The technique outlined above hedges against movements in the six month rate implied from the futures strip. As long as this implied rate and the actual six month 1 Recall that Rs is the market rate for a deposit maturing on the settlement date of the first futures contract.

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CHAPTER 11. MANAGING A PORTFOLIO

rate move in parallel, the hedge is effective. This problem tends to be more severe for short-term hedges – for example, maintaining a hedge overnight (which might occur if you unwound the above deal the next day). In part, this is caused by the fact that cash rates usually move in sixteenths, whereas futures prices move in one-hundredths. Thus the futures often have to move more than six ticks before the cash rate moves. A more profound reason, at least in the sterling market, is the lack of cash-futures arbitrage activity, which is required if actual and implied cash rates are to track one another. This is because physical loans and deposits attract punitive capital requirements from the UK regulatory authorities, which tend to make the arbitrage uneconomic. Let us now move three months forward in time to 17 August 1990. Assume, for simplicity, that the input rates and futures prices remain the same as for 17 May 1990. Recalculating the delta vector with respect to futures prices gives: Grid point Rs 1F 2F 3F 4F 5F 6F 7F 8F 3Y 4Y 5Y

PVBP (89) (162) (2) 3 (2) 3 (2) 3 (1) (7) 828

2,556

The small exposure to the 2F to 8F contracts arise because the deal is no longer par. They are not related to the LIBOR exposure – this is encapsulated in the Rs and 1F exposures, which now total (251). less than at the instigation of the deal. This is due to the fact that a three month exposure is less than a six month exposure with equal principal. So, as the next LIBOR roll date approaches, the futures hedge must gradually be removed until, on the roll date, no hedge remains. The hedge does not decrease in a simple manner – it depends

11.2.

CROSS-CURRENCY AND CASH POSITIONS

217

Figure 11.1: LIBOR exposure over time on the movement of rates as well as time. However, the general trend is one of a “saw-tooth” function, as shown in Figure 11.1. The situation for a portfolio of swaps is essentially no different from that of a single swap. Of course. LIBOR sets occur more frequently, and the total LIBOR exposure will be more complicated in structure. However, the general procedure and properties that have been discussed still hold. One of the problems that can arise in a large portfolio is build-up of short term positions. For instance, if one received the fixed on a large volume of swaps against six month LIBOR, and paid the fixed on an equally large volume against three month LIBOR, the resulting 3-6’s position may be quite hard to hedge with futures. In this situation. it may be possible to alleviate the problem either through the FRA market, or by trading a single currency basis swap.

11.2

Cross-currency and cash positions

Perhaps the least appreciated part of swap book running is cash management. A swap trader’s cash positions build up as a result of swap principal flows on cross-currency swaps and any other cashflows. The cost involved in funding or investing this money can sometimes have a major impact on the profitability of a portfolio. This section discusses

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CHAPTER 11. MANAGING A PORTFOLIO

methods of analysing and managing the problem.

11.2.1

Cash payments on swaps

Within a swap portfolio, the commonest form of cashflow are fixed and floating interest payments. The size of these payments, in relation to the swap principal, depends largely on whether payments can be netted. For example, in a standard sterling swap, both fixed and floating payments occur on the same semi-annual date, so that any payment will generally be, at most, a few percent of the notional. However, a dollar swap paid annually versus six month LIBOR will have every other cashflow un-netted, and the net payments will be larger. What is ideally required is a way of predicting future payments, netted or not. on the entire portfolio. One is unable, of course, to say with certainty what LIBOR payments will be until they are set. However, the method used to price LIBOR legs on swaps is equivalent to implying future LIBOR rates and discounting the resultant cashflows back to today. If future LIBOR rates can be implied, so too can future cash balances. The accuracy of this method depends upon the time horizon being considered: as LIBOR setting dates move closer, one becomes able to predict the setting rate with greater accuracy. As an example, take a time horizon of one week (admittedly, too short to be of much practical value, but it serves to illustrate the technique). With the sterling scenario of chapter six, and the period 17 to 24 May 1990. imagine that the following settings and payments are due: 18 May Receive 14% on £10,000,000 from 20-Nov-89 to 18-May-90. and pay six month LIBOR set at 15%. 21 May Receive six month LIBOR at 15.125% on £5,000,000 from 21Nov-89 to 21-May-90. 23 May Pay 12% on £10,000,000 from 23-May-89 to 23-May-90; receive LIBOR at 15% from 23-Nov-89 to 23-May-90. The cashflows resulting from these deals are:

11.2.

CROSS-CURRENCY AND CASH POSITIONS Date

Pay

Receive

Running total

18-May-90 21-May-90 23-May-90

735,616

686,575 375,017 743,836

(49,041) 325,976. (456,164)

1,200,000

219

Thus, we will have to fund a cash deficit on the 18th and 23rd May. while investing a surplus on 21st May. In addition to these payments, there will also be LIBOR sets made on these days. The best guess, as at the 17 May, for these rates are the forward rates. These are:

Dates

Rate

18-May-90 to 19-Nov-90 21-May-90 to 21-Nov-90 23-May-90 to 23-Nov-90

15.375% 15.372% 15.365%

which give the following implied cashflows:

Date

Cashflow

19-Nov-90 21-Nov-90 23-Nov-90

(779,281) 387,459 774,564

These can then be netted off against fixed payments as appropriate, and a running balance calculated. The usefulness of such an analysis depends on the results obtained. If the implied cash balance is either positive or negative continuously for some period of time, then one could look to borrow or invest the maximum amount possible for this period, rather than continually borrowing the balance overnight. One problem is that this takes no account of any new deals transacted in the intervening period, and their impact on cash balances – one needs to weigh up the advantages of locking in longer term funding versus possible costs incurred in variations from expected balances.

11.2.2

Cross-currency cashflows

Although the day-to-day variations in a cash balance tend to be small, larger movements are possible. These are mostly due to the exchanges of principal on cross-currency swaps.

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MANAGING A PORTFOLIO

Consider, for example, a sterling/dollar cross-currency swap, on which we pay the fixed on £10,000,000 sterling for five years, and receive dollar LIBOR on $17,000,000. The initial principal flows involve us receiving the sterling principal and paying the dollar principal. These principal payments are reversed at maturity. If no further deals were done, one would need continually to reinvest the sterling and fund the dollars for the remaining life of the deal. How frequently one does this depends on our view of rate movements at the short end of the yield curve. The obvious way for us to avoid this reinvestment risk is to do another cross-currency swap the other way round. This would involve receiving sterling against paying dollars. If one were to offset a five year currency swap with a two year currency swap of equal size but the other way round, the cash balance problem would have been alleviated for two years, but would recur after that. The key to a well balanced cross-currency book is to try and offset such exposures to matching maturities. One way to monitor this is to plot a graph of the running balance of principal flows in each currency against time. For example, suppose a book runner trades the following swaps: 1. Paying fixed on £10 million for two years. 2. Receiving fixed on £50 million for three years. 3. Paying LIBOR on £100 million for five years. The running cash balance is: Period 0-2 years 2-3 years 3-5 years

Cash balance + £60 million +£50 million + £100 million

The corresponding graph is shown in Figure 11.2. A good strategy might be to receive sterling against another currency in £50 million for five years, thus offsetting most of the first three years balance.

11.2.

CROSS-CURRENCY AND CASH POSITIONS

Figure 11.2: Cross-currency position versus time

221

Chapter 12 Conclusions The past decade has seen considerable development in the derivatives market. The interest rate swap has firmly established itself as an indispensable tool of the capital markets, with almost every major financial institution establishing swap teams in the major centres. In addition, the trading of complex option-based instruments have become an everyday occurrence, with a whole new armoury of equity derivatives for the borrower or investor to choose from. From a technical perspective, there has been considerable standardisation in the methods used to price derivative products. The zero coupon technique has been adopted by almost all software developers for pricing swaps. The principles on which it is founded are generally accepted as sound, and the flexibility which it offers has rendered older methods redundant – it appears unlikely that this approach will be replaced in the foreseeable future. Although the principles of zero coupon pricing have been generally accepted, it is surprising how little agreement there is over the precise measurement and hedging of risk. As we have attempted to emphasize. interest rate risk is determined solely by the algorithm used to price an instrument. What ambiguity exists relates to the form in which this exposure is expressed, and this is usually determined by the individual’s preference for certain hedging instruments. This determinist approach is the most important theme in the book, and it is the hope of the authors that it leads practitioners to cast a more critical eye on their risk analysis software, and move away from ac223

224

CHAPTER 12. CONCLUSIONS

ceptance of “black-box” pricing. This is especially important when one considers the huge costs incurred in buying a complete risk-management system. Furthermore, such systems are rarely developed with a trader’s needs in mind – as a result, they can sometimes be awkward or slow to use. A competent programmer should, having read this book, be able to write his or her own swap and option pricing system – spreadsheets are especially useful in this respect. The move towards a more technologically advanced market, both in pricing theory and computing speed, has also had an impact on profitability. With most traders able to price highly complex deals accurately, and hedge them with liquid instruments, the big profit margins that such deals used to bring have gradually been eroded. In addition, bid-offer spreads in many currencies have tightened to such an extent that they hardly cover a bank’s cost of capital. Perhaps these developments signal a move towards trading swaps as commodities. In many respects this would be a pity; one of the attractions of swaps and options is that they require a little more thought than most instruments, as swap traders are only too ready to confirm. However, the days when derivatives were purely the realm of the superintelligent “rocket scientist” are long gone!

Appendix A Continuous compounding The discount curve only really becomes an exact exponential in the case of continuous compounding. To see this, suppose that one invests $1 at an annual rate of R%, compounded m times per year. In other words, interest is paid and reinvested at m equal periods during a year. So after one period, the value becomes 1 + R/m, after two (l + R/m) 2 , and after one year (1 + R/m)m. Hence, the discount factor is (1 + R/m) – m . Moving to the continuously compounded limit is the same as letting m go to infinity. In fact: (A.1)

as can be verified by using the binomial expansion: (A.2)

Thus:

(A.3) Now, as m gets very large: (A.4) and so [16]:

(A.5) 225

226

APPENDIX A.

CONTINUOUS COMPOUNDING

Appendix B Example sterling curves We give here the input rates, futures prices and the discount factors resulting from using cash only, futures only, and cash rates integrated with all eight futures. Where appropriate, figures are given to ten decimal places.

227

228

APPENDIX B.

EXAMPLE STERLING CURVES

Value date: 17th May 1990 Grid point O/N 1W 1M 2M 3M 6M 1Y 2Y 3Y 4Y 5Y 7Y 10Y

Date 18-May-90 24-May-90 16-Jun-90 17-Jul-90 17-Aug-90 19-Nov-90 17-May-91 18-May-92 17-May-93 17-May-94 17-May-95 19-May-97 17-May-2000

Par rate 14.25 14.75 15 15.125 15.15625 15.375 15.55 14.25 13.95 13.85 13.75 13.55 13.10

Table B.1: A sterling yield curve

Grid point O/N 1W 1M 2M 3M 6M 1Y 1.5Y 2Y 2.5Y 3Y 3.5Y 4Y 4.5Y 5Y

Discount factor 0.9996097414 0.9971792122 0.9870200108 0.9753457968 0.9632036417 0.9273432885 0.8654262224 0.8088405335 0.7597755335 0.7119726241 0.6687732130 0.6260895398 0.5871112577 0.5503178043 0.5167149164

Table B.2: Discount function generated from swaps and deposits

229

Value date: 17th May 1990 Grid point DRs = D1F D1F D2F D3F D4F D5F D6F D7F D8F D9F

Date 20-Jun-90 20-Jun-90 19-Sep-90 19-Dec-90 20-Mar-90 19-Jun-91 18-Sep-91 18-Dec-91 18-Mar-92 17-Jun-92

Price Rs = 15.125 P1 = 84.80 P2= 85.05 P3 = 85.55 P4 = 86.05 P5 = 86.53 P6 = 86.95 P7 = 87.20 P8 = 87.35

Table B.3: Sterling futures prices

Grid point D1F D2F D3F D4F D5F D6F D7F D8F D9F

Discount factor 0.9861067021 0.9501017503 0.9159614819 0.8841104931 0.8543950996 0.8266344501 0.8005868630 0.7758283729 0.7521081191

Table B.4: Discount function generated from futures

230

APPENDIX B.

Date Grid point O/N 18-May-90 1W 24-May-90 18-Jun-90 1M 20-Jun-90 1F 17-Jul-90 2M 17-Aug-90 3M 2F 19-Sep-90 19-Nov-90 6M 3F 19-Dec-90 4F 20-Mar-91 1Y 17-May-91 5F 19-Jun-91 6F 18-Sep-91 1.5Y 18-Nov-91 7F 18-Dec-91 8F 18-Mar-92 2Y 18-May-92 9F 17-Jun-92 2.5Y 17-Nov-92 3Y 17-May-93 3.5Y 17-Nov-93 4Y 17-May-94 4.5Y 17-Nov-94 5Y 17-May-95

EXAMPLE STERLING CURVES

Used? Y Y Y Y N N Y N Y Y N Y Y N Y Y N Y Y Y Y Y Y Y

Discount factor 0.9996097414 0.9971792122 0.9870200108 0.9861067021 0.9752467617 0.9629634413 0.9501017503 0.9270210375 0.9159614819 0.8841104931 0.8649476338 0.8543950996 0.8266344501 0.8089862495 0.8005868630 0.7758283729 0.7597791288 0.7521081191 0.7119911620 0.6688141790 0.6261276860 0.5871468240 0.5503509372 0.5167458220

Table B.5: Discount function generated from swaps, deposits and futures

Appendix C The delta of a par swap In this appendix, we prove the formula for the delta of a par swap, a result heuristically derived in chapter eight. We shall take the sterling swap market as our example, although the result is readily extendable to other currencies. Using the notation of section 8.8. assume that i S, and i 1.5Y (the result holds for shorter maturities, and is easy to prove). Suppose an i year par swap is executed at the par rate and that the value of this swap is V;. Since the swap has been executed. is now regarded as a constant whereas Ri is a variable. The value of the swap is given by:

(C.1)

Then, the sensitivity of the i year par swap to the i year par rate is:

(C.2)

is zero when i > j since From section 8.8.1. and using = Ri, Ri is not involved in the construction of Fj. So the second term in

231

232

APPENDIX C.

THE DELTA OF A PAR SWAP

Equation C.2 is zero. Then, from Equation 8.11:

(C.3) Since the swap is par. Vi — 1. so that:

(C.4) which gives:

(C.5)

but since:

(C.6) we have the desired result:

(C.7)

Appendix D Equivalence of zero coupon and additive systems Theorem: Any additive cashflow valuation system can be shown to be a zero coupon cashflow valuation system, and every zero coupon cashflow valuation system is additive. The first step in the proof of any theorem is to ensure that the theorem is stated unambiguously. This involves distilling the important features of real portfolios and valuation systems to create a precise definition of ideal portfolios and valuation systems. Using these definitions the theorem can be rigorously proved. Notice that the definition below defines a cashflow to be predetermined. So the theorem holds for the fixed legs of swaps. It also holds for floating legs if these can be modelled by payments of principal. This is discussed in chapter six.

D.1

Definitions

• A cashflow is simply a number. We assume that each cashflow is a multiple of some minimum possible size, called a cent. • A portfolio is a (finite) list of cashflows. Each cashflow has an associated date on which it occurs. We write:

233

234

APPENDIX D.

ZERO COUPON AND ADDITIVE SYSTEMS

where the Ci are the cashflows, the Di are the dates and indexing set.

is an

• A cashflow valuation system S is a process which takes three inputs: 1. a value date, V; 2. a yield curve, ϒ, i.e. a list of particular market interest rates and swap rates; 3. a portfolio . By assumption all the cashflows within occur on or after the value date, V. From the three inputs, the process yields a single value, called the and written PV [S, V, ϒ] ( ). Everything that present value of follows holds for any given value date and yield curve and so, for simplicity we write the present value of as PV S ( ).

and . They can be amalgamated to Imagine two portfolios form a larger portfolio R. Write R = For a given system S all three portfolios can be processed to give three values: PVs( ), PVs( ) and PVs(R). A cashflow valuation system is additive if for any portfolios and

• A cashflow valuation system is zero coupon if there exists an associated discount function, F, with the following properties: 1. For a given value date V and yield curve ϒ, F assigns a number to each date on or after V. The value of the function is written F [S, V, ϒ] (D i ). Where possible, this notation is abbreviated to read Fs(Di). 2. For any portfolio , the value produced by S for P is equal to the value obtained by discounting the cashflows of using the discount function FS. In other words:

For all

(D.1)

D.2. PROOF

235

Notice that nothing has been said about the internal workings of the system S, only that there exists a discount function Fs with the property described in Equation D.1. It could be that S operates by calculating a discount function and applying it in turn to the cashflows of . Equally, S might go through a far more convoluted process but nevertheless be a zero coupon system as defined here.

D.2 Proof Suppose that S is a zero coupon system. It is not difficult to show that S is also additive: let and be two portfolios where:

Since S is zero coupon, there is a discount function Fs such that:

but:

since S is zero coupon Hence S is indeed additive. Now suppose that S is an additive system. Our aim is to show that an associated discount function Fs exists. For any date Di, let Ui be a cashflow of size 1 occurring on Di. For any value date V, yield curve ϒ and date Di define the discount function at Di to be the value that S returns for the unit cashflow Ui. In other words:

F s (A) = PV S (U i )

(D.2)

236

APPENDIX D.

ZERO COUPON AND ADDITIVE SYSTEMS

We need to show that Fs satisfies Equation D.1 for any portfolio

by Lemma 1 below by Lemma 2 below by the definition of F So Fs does indeed satisfy Equation D.1 This completes the proof.

Lemma 1: For an additive system. S and portfolio

(D.3) Proof: The proof is very simple, by induction on the number of elements in . It is easy to show that if the lemma is true when has n elements it is also true when has n + 1 elements. So since the lemma is trivially true when n = 1 it is also true when n = 2. Hence also when n = 3 and so on.

Lemma 2: For an additive system, S and cashflow Ci: PVs(C i ) = C i PVs(U i )

(D.4)

where Ui is a unit cashflow occurring on the same date as Ci. Proof: The proof follows from lemma 1 above and the fact that all cashflows considered (including the unit cashflows) are multiples of one cent. In fact, the condition that cashflows must be multiples of one cent can be dropped if additive systems are also required to be continuous. In other words, as the size of a cashflow tends to zero, its value also tends to zero.

Appendix E

Answers to questions of chapter 5 5.1

1,000,000 1 + 15.375 x 186/36500

£927,343.29

5.2 Discount rate = 9.42 x 365/360 = 9.551% Present value = $633,755.25 5.3 10 x 365/360 = 10.14% 5.4 From Equation 5.7: R30/360

366 360

360 365

15

15.04% 5.5 10.25% 5.6 n = 2, m – 4 in Equation 5.10 implies: R4 = 9.390% 5.7 Actual/365 equivalent of 10.33% = 10.4735% Semi-annual equivalent = 10.2127% Actual/360 equivalent = 10.073% 5.8 First annuity: N=5

237

238

APPENDIX E. ANSWERS TO QUESTIONS OF CHAPTER 5 I = 10.25% PMT = 1,000 ⇒ Annuity value = $3,766.70 Second annuity: N = 10 I = 5% PMT = 489 ⇒ Annuity value = $3,775.93 So the second annuity is worth more.

5.9 Using the usual four step approach: (a) The interpolated 7.5 year rate is 13.475% semi-annual Act/365. (b) No decompounding is necessary for the first period, since it is six months. (c) Six month LIBOR is 15.375%. implying a net first cashflow on the comparison swap of: (13.475 – 15.375)/2 = –0.95

(d) Set: PV = –0.95 N = 14 I = 13.475/2 = 6.7375% ⇒ PMT = –0.1069 So the solution swap has cashflows of size 13.475/2–0.1069 = 6.6306, implying a rate of 6.6306 × 2 = 13.261%. So the forward start costs 48.9 basis point. 5.10 A comparison swap is constructed as follows:

239 Fixed rate:

9.24 + (9.34 – 9.24) = 9.265% Annual Act/360 9.394% Annual 30/360 9.183% Semi-annual 30/360 Decompounded first rate: 9.080% Quarterly 30/360 First fixed payment: 2.27 Subsequent fixed payments: 4.591% First floating rate: 8.375% Actual/360 First floating payment: 2.123 Net first payment: 0.147 There are six further residuals of size x, whose value discounted back to the first payment date must equal 0.147. N=6 PMT = 1 I = 9.183/2 = 4.591% giving an annuity value of 5.143. Thus: 0.147 x 0.0286 5.143 So, the fixed payments on the solution swap are of size: 9.183 + 0.0286 ≈ 9.21% 5.11 To value the 80 dollar basis points in sterling terms, use a discount rate of 9.46% (the semi-annual Actual/365 equivalent of 9.55%). Then: N = 20 I = 9.46/2 = 4.73% PMT = 80/2 × 365/360 = 40.55

giving an annuity value of 517.2 basis points. On the sterling side: N = 20 I = 13.10/2 = 6.55% PV = 517.2 ⇒ PMT = 47.1 So 80 dollar basis points are equivalent to 47.1 × 2 ≈ 94 sterling basis points. Since the trader will pay dollar LIBOR against sterling LIBOR plus eight basis points, he will pay dollar LIBOR plus 80 basis points against sterling LIBOR plus 102 basis points.

240

GLOSSARY OF SYMBOLS

Glossary of symbols Symbols that are used only once are defined where they appear in the text. Other symbols are defined below. The index denoting the one week grid point. Accrued interest on a bond – defined on page 36. Accrual factor between dates D1 and D2 – defined on page 95. Abbreviated form of accrual factor. Shorthand form of accrual factor for sterling market, as used in chapter eight (see page 148). Coupon on a bond. A general cashflow: also used in chapter 10 as the value of a cap. An indexing set – see section 8.8. A general spot foreign exchange rate, used in chapter nine. A fixed cashflow, indexed by i – see page 123. The standard binomial coefficients, used in section 10.1. A “down” movement in a binomial tree – see section 10.1. The delta vector for a portfolio P. 360(D1,

D2) The number of days from D1 to D2, calculated assuming that all months have 30 days – see page 12.

241

An element of the delta vector.

A

A general date of the form (day, month, year). A random variable with a normal distribution, as used in chapter 10.

E ( X ) The expectation of a distribution X – used in chapter 10.

F

The price of an FRA, as defined in section 4.3 – also used in chapter 10 as the value of a floor.

f

For a bond, the time, expressed as a fraction of the coupon period, from settlement to the first coupon date: defined on page 39.

F

The discount function on grid point dates, expressed as a vector, first discussed on page 145.

First(C) One of a pair of equivalent cashflows, defined in section 8.8.4.

Fi

The discount factor at the date Di. The convexity of a bond – defined in section 4.1.3.

H

The coupon frequency of a bond, used in chapter four. An indexing set defined in section 8.8, and redefined in section 8.10.

I

Interest rate used to discount an annuity on a financial calculator; see chapter five.

K

The strike on an option – used in chapter 10. A number between 0 and 1 used to describe the position of a date relative to the surrounding grid points; defined on page 110.

Lf

The fixed leg of a swap – used in chapter 10. The floating leg of a swap – used in chapter 10. An indexing set defined in section 8.10.

MD The modified duration of a bond – see section 4.1.3.

GLOSSARY OF SYMBOLS

242 µ

The mean of a distribution as used in chapter 10.

N

The number of coupons yet to be paid on a bond, used in chapter four. The number of futures contracts used when integrating futures into the construction of the discount function – see section 8.9.

N

The number of payments on an annuity whose value is calculated on a financial calculator; see chapter five.

nF

The index used to label futures grid points. DnF is the settlement date of the nth futures contract.

nM

The index used to label one, two, three and six month grid points.

nY

The index used to label annual grid points.

O/N The index used to label the overnight grid point.

P

The probability of an “up” movement in a binomial tree; see section 10.1.

P

The notional principal of a swap. A general portfolio of cashflows.

Φ

A shorthand form for the forward rate, used in chapter 10.

φ(x) The normal density, defined on page 193. ΦD1, D2 The forward rate between dates D1 and D2, defined on page 94.

PMT The size of each payment when valuing an annuity on a financial calculator: see chapter five. Ψ(µ, σ) The normal distribution with mean µ and standard deviation σ; see chapter 10.

PV

The present value of an annuity calculated on a financial calculator; see chapter five.

243 PV(C) The present value of a cashflow C. The derivative of the value of a portfolio page 153.

with respect to F; see

PC

The clean price (i.e. excluding accrued interest) of a bond ; see chapter four.

pd

The dirty price (i.e. including accrued interest) of a bond: see chapter four.

Pi

The price of the ith futures contract.

R

A general interest rate. Also used in section 4.1.1 to mean the redemption amount of a bond. The risk free interest rate used in option mathematics – see chapter 10.

R

The complete set of rates, expressed as a vector, needed for the stripping process; see page 145. The hedge ratio for two bonds; see page 42.

R´

The vector of input rates used for the stripping process; see page 145. An element of the vector R´ defined on page 148.

Ri

An element of the vector R, defined on page 145.

Rm

An interest rate with m compounding periods per year; see page 57.

Rs

The rate applicable for a deposit maturing at the settlement date of the first futures contract.

S

The price of a stock used in the Black-Scholes equation – see chapter 10; also used as a semi-annual rate.

S

An indexing set defined in section 8.8. and redefined in section 8.10. Also used in Appendix D as a general cashflow valuation system.

244

GLOSSARY OF SYMBOLS

Second(C) One of a pair of equivalent cashflows, defined in section 8.8.4. The standard deviation of a distribution – see chapter 10.

ST

The stock price at time T, used in option theory; see chapter 10. The price of an option; see chapter 10.

u

An “up” movement in a binomial tree – see section 10.1.

V

Defined on page 39, used for calculating bond yields. Also used in Appendix D as a value date. An indexing set defined on page 147.

Var(X) The variance of a distribution X – see chapter 10.

y

The-yield to maturity of a bond – see chapter four.

yH

The yield of a bond compounded H times per annum; see page 39.

z

A Wiener variable used in chapter 10.

Bibliography [1] J. Hull. Options, Futures and Other Derivative Securities, Prentice-Hall (1989). [2] ISDA, Interest Rate and Currency Exchange Definitions, ISDA (1987). [3] ISDA/Andersen Consulting, ISDA Market Survey Highlights; Year End 1989, ISDA/Andersen Consulting (1990). [4] R. Brealey & S. Myers, Principles of Corporate Finance, McGrawHill (1984). [5] R. Burden & J. Faires, Numerical Analysis, Prindle, Weber & Schmidt (1985). [6] F. Fisher III, Eurobonds, Euromoney (1988). [7] M. Abramowitz & I. Stegun, Handbook of Mathematical Functions, Dover (1965). [8] W. Feller, An Introduction to Probability Theory tions, Wiley (1968).

Its Applica-

[9] F. Black & M. Scholes, The Pricing of Options Corporate Liabilities, Journal of Political Economy, Vol.81, p637 (1973). [10] F. Black, The Pricing of Commodity Contracts, Journal of Finance, Vol.3, p167 (1976). [11] R. Jarrow & A. Rudd, Option Pricing, Dow Jones-Irwin (1983). 245

246

BIBLIOGRAPHY

[12] M. Brennan & E. Schwartz, Alternative Methods for Valuing Debt Options, Finance, Vol.4, p119 (1983). [13] T. Ho & S. Lee, Term Structure Movements and Pricing Interest Rate Contingent Claims, Journal of Finance, Vol.41, p1011 (1986). [14] D. Heath, R. Jarrow & A. Morton, Bond Pricing and the Term Structure of Interest Rates; a New Methodology. Cornell University preprint (1987). [15] C. Tranter & C. Lambe, Advanced Level Mathematics (Pure & Applied), Hodder & Stoughton (1980). [16] W. Ferrar, A Textbook of Convergence, Oxford University Press (1980).

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