Pricing and hedging swaps

1. PRICING AND HEDGING SWAPS by Paul Miron and Philip Swannell Published by Euromoney Books

2. 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 Kings Lynn 3. 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 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 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 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 1 2 3 5 6 9 9 10 15 18 19 21 23 23 25 27 28 31 33 33 41 42 44 48 i 4. 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 5.6 Basis swaps 5.7 Pricing currency swaps 5.8 Valuing currency swaps 5.9 Remarks 6 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 ii CONTENTS 50 51 53 53 55 56 58 60 62 67 78 83 84 86 89 91 91 93 93 96 100 103 108 111 112 114 117 121 121 122 123 124 124 125 5. 7.4.3 Amortising and rollercoaster swaps 7.4.4 Currency swaps 7.5 Pricing the swap 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 8.10.4 The delta vector 8.10.5 Equivalent positions 8.11 The gamma matrix 8.10.2 Calculating 8.10.3 Calculating 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 CONTENTS iii 128 129 131 133 134 135 138 139 139 144 145 147 148 149 150 151 153 154 154 155 156 157 159 160 163 163 164 167 167 169 170 171 174 176 177 6. 9.8 Forward FX arbitrage 9.9 Fixed-fixed currency swaps 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 11 Managing a portfolio 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 A Continuous compounding B Example sterling curves C The delta of a par swap D Zero coupon and additive systems D.1 Definitions D.2 Proof E Answers to questions of chapter 5 Glossary of symbols Bibliography iv CONTENTS 179 181 185 186 188 197 201 207 209 213 213 217 218 219 223 225 227 231 233 233 235 237 240 245 7. List of Figures 2.1 Diagrammatic representation of an interest rate swap . . 2.2 Diagrammatic representation of a currency swap 4.1 Cashflow schedule of a generic bond 4.2 Cashflow schedule of a bond bought between coupon dates. 4.3 Graphical representation of the Newton-Raphson equation. 5.1 A simple annuity 5.2 A general annuity 5.3 Three swaps/bonds 5.4 A two and a half year bond 6.1 LIBOR payment as principal flows 6.2 LIBOR payments on a two year swap 6.3 A one year par swap 6.4 Cashflows on a three year par swap 6.5 A one year semi-annual par swap 6.6 A linearly interpolated function 6.7 Exponential interpolation 8.1 A picture of 8.2 A picture of part of 8.3 A picture of part of 8.4 An aid to calculating 10.1 Evolution of bond prices in a toy model 11.1 LIBOR exposure over time 11.2 Cross-currency position versus time v 14 19 35 37 38 58 59 63 64 96 97 98 103 105 109 111 146 150 161 162 186 217 221 8. List of Tables 3.1 Swap volumes for 1989 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 . 6.1 Discount factors for a dollar curve 6.2 A sterling yield curve 6.3 Discount function for the curve in Table 6.2 6.4 Combined swap and futures dates 6.5 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 8.1 A sterling yield curve 8.2 Change in portfolio value for a change in individual rates 8.3 The delta vector 8.4 The delta of a 10,000,000 par five year swap 8.5 Deltas for 10,000,000 par swaps 8.6 Portfolio equivalent positions 8.7 The ordering of the grid points 9.1 PVBP for par sterling swaps vii 26 69 70 71 73 76 102 106 109 115 119 124 126 127 129 132 134 136 137 140 141 142 157 171 9. 9.2 Forward dollar/sterling points 10.1 A toy model for option hedging B.1 A sterling yield curve B.2 Discount function generated from swaps and deposits . . B.3 Sterling futures prices B.4 Discount function generated from futures B.5 Discount function generated from swaps, deposits and futures . viii LIST OF TABLES 181 211 228 228 229 229 230 10. 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 NatWest. 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. Johns 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 11. 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 Hender- son and Charmaine Ferris at Euromoney Publications for their encour- agement and help in pursuing this project. Adrian Fitch for a plentiful supply of nourishing lunches, and an un- printable suggestion for the title of this book. xi This book was typeset by the authors using the document preparation system. 12. To Lucy and Marie-Vronique 13. Chapter 1 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. Neverthe- less, 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 occu- pies 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 head- lines 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 Introduction 14. 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. 15. 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 in- house 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 16. 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. How- ever, in North American universities there is now a movement towards a more technical treatment of all their subject matter. 17. 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, its the traders 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 presup- pose 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. Dont 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. 18. 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 19. 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. 20. 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 trou- blesome 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 de- pend on the future course of interest rates. Typically, the floating rate payments will track rates for six month interbank deposits in London 9 21. 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 22. 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 holi- day 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; Preced- ing 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 : 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 For example, if D2 were 17 May 1991 then D2 - (17,5,1990). 2 Fixed here refers to the fact that 365 is used regardless of leap years. Fixed Amount Notional Principal Fixed Rate Fixed Rate Day Count Fraction 23. 12 CHAPTER 2. DEFINING THE SWAP rate is paid on this basis: Hence: Fixed Rate Day Count Fraction 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 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, money market basis means Actual/365(Fixed). Here. 24. 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 issuers 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 25. 14 CHAPTER 2. DEFINING THE SWAP Fixedcashflows Start date Maturity 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: 26. 2.3. NON STANDARD INTEREST RATE SWAPS 15 Principal Fixed rate Fixedrate day- count fraction Trade date Effective date Termination date Fixed rate payer payment dates $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. Payment date 05-Feb-91 05-Feb-92 05-Feb-93 07-Feb-94 06-Feb-95 05-Feb-96 Fixed payment 2,494,166.67 2,501,000.00 2,507,833.33 2,487,333.33 2,487,333.33 No. of days 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. The first cashflow is given by: This and subsequent fixed cashflows are: 25,000,000 9.84 365 100 360 2,494,166.67 27. 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 non- standard structures are fairly common, these are considered next. An example non-standard swap: Imagine a property devel- oper 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 uneco- nomic. 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. 28. 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 Fed- eral 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: 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 Floating Amount Notional Principal Floating Rate LIBOR margin Floating Rate Day Count Fraction 29. 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. 30. 2.4. CURRENCY SWAPS 19 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 exchange amount for clarity. Sterlingprincipal + final couponFixedsterlingcashflows Dollar principal Sterling principal Floatingdollarcashflows Dollar principal + final coupon 31. 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 banks position is then: 7 Currency swaps with no final exchange do exist, however their price will generally be very different from the price for a standard deal. Suppose now that sterling strengthens against the dollar, moving to $1.705. The banks position is now worth: 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. 10,000,000 + $17,000,000 17,000,000 = 10,000,000 1.70 0 17,000,000 10,000,000+ 1.705 = 29,325.51 32. 2.4. CURRENCY SWAPS 21 2.4.2 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. 33. 3.1 The development of the market The origins of the swap market lie back in the 1970s - a time when many countries imposed restrictions on the cross-borderflowof 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-to- back 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 transac- 23 Chapter 3 to the swap market Background 34. 24 CHAPTER 3. BACKGROUND TO THE SWAP MARKET tion, with the added benefit of flexibility and (relative) liquidity. How- ever, the main reason for the sudden explosion of swap activity in the early 1980s 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 mid- 1980s, 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 zero- coupon, dual-currency, warrant and index-linked bonds. In addition, swap markets started in many of the smaller currencies, the most no- table of recent years being ECU (the European Currency Unit). By the late 1980s, the swap market had reached such a developed stage, that in some currencies swaps were regarded almost as commodi- ties. The secondary swap market became of major importance, with 35. 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 1990s. Many large corporates, supranationals and sovereign entities have become increasingly selective about the creditworthiness of their swap counterparties, pre- ferring 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. 36. 26 CHAPTER 3. BACKGROUND TO THE SWAP MARKET Interest rate swap volumes Currency Dollars Yen Sterling Deutschemarks Australian dollars French francs Canadian dollars Swiss francs ECU Dutch guilders Number of contracts 36,627 5,259 6,361 6,608 9,029 4,131 1,936 1,987 1,054 446 Notional principal local currency (millions) 993,746 17,420,529 60,611 157,529 84,851 265,460 34,563 46,218 17,075 12.556 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 Currency swap volumes Currency Dollars Yen Swiss francs Australian dollars Deutschemarks ECU Sterling Canadian dollars Dutch guilders French francs Number of contracts 12,492 6,364 2,186 3,733 1,629 1,222 952 914 315 281 Notional principal local currency (millions) 354,166 28,041,825 106,567 82,215 103,106 36,923 20,047 41,361 21,897 55,447 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 Table 3.1: Swap volumes for 1989 Source: ISDA 37. 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- 38. 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 months 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 com- paratively 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. 39. 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 societys 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 swaps pricing. 40. Chapter 4 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 effec- 1 A general discussion of hedging swap positions is given in chapter 11. 31 InstrumentsHedging 41. 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 at- tempt 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 investors 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. 42. 4.1. GOVERNMENT BONDS 33 4.1 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. 43. 34 CHAPTER 4. HEDGING INSTRUMENTS 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 Or putting it another way: $1 in n years Value of $1 in one year = $(1 + R) 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) $1/(1 + R)n today. $(1 + R)n in n years. 44. 4.1. GOVERNMENT BONDS 35 -90 +110 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 +10 +10 +10 +10 o 1 2 3 4 5 t 45. 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 gener- alised 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 Eu- robond 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. 46. 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) 47. 38 CHAPTER 4. HEDGING INSTRUMENTS Figure 4.3: Graphical representation of the Newton-Raphson equation. 48. 4.1. GOVERNMENT BONDS 39 Setting: (4.6) so that: (4.7) Then, the better approximation. Equation 4.5. becomes: Calculating y1 for the case c = 12%, Pd = 111 (Pc = 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: Where Vc is the result of putting y = c in Equation 4.6 i.e. Vc = 49. 40 CHAPTER 4. HEDGING INSTRUMENTS 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 ef- fects 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. 50. 4.1. GOVERNMENT BONDS 41 4.1.2 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. 51. 42 CHAPTER 4. HEDGING INSTRUMENTS 4.1.3 Modified duration and convexity Imagine a bond with yield yH and a security S whose price P(yH) 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 and hedge ratio 7 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. 52. 4.1. GOVERNMENT BONDS 43 (4.19) (4.20) where is the ratio of the change in yield for the securities. To assume that the yields of the two securities move in parallel is equivalent to setting the ratio Under this assumption, for another 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 convexity8 . 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 con- venient 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 ex- pansion 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. 53. 44 CHAPTER 4. HEDGING INSTRUMENTS This is a rather large equation as follows: 4.2 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. (4.23) (4.24) with as before. = 4.26474, with the previous values of Pd = 111, c = 12% and y = 11.97%. So, for a basis point change in yield. to 11.98%, the dirty price changes 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: Going back to the earlier example of Equation 4.3, gives 54. 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 es- sentially over-the-counter markets. In other words, each transaction has to be tailored to suit the clients 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. 55. 46 CHAPTER 4. HEDGING INSTRUMENTS 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 LIBOR. 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 500,000 Expiry months March. June, September and December. Delivery day First business day after last trading day. Last trading day Third Wednesday of expiry month. Quotation 100.00 minus rate of interest. Tick size/value 0.01/12.50 Delivery Cash settlement based on 100.00 minus three month sterling LIBOR on last trading day. Three month Eurodollar interest rate future Contract size $500,000 Expiry months March. June, September and December. Delivery day First business day after last trading day. Last trading day Third Wednesday of expiry month. Quotation 100.00 minus rate of interest. Tick size/value 0.01/S12.50 Delivery Cash settlement based on 100.00 minus three month dollar LIBOR on last trading day. 56. 4.2. FUTURES CONTRACTS 47 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: 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 Long gilt future Contract size Expiry months Deliveryday Last trading day Quotation Tick size/value Delivery 50,000 of a notional 9% gilt. March. June. September and December. Any business day in delivery month at sellers choice. Two business days prior to last business day in delivery month. Per 100.00 nominal. 15.625 Delivery can be of any gilt with redemption dates 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). 57. 48 CHAPTER 4. HEDGING INSTRUMENTS 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 markets 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 3-6s 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 58. where Dy is the day count basis relevant to the currency (Dy = 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 D2). 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 0-6s plus a 6-12s FRA (in fact, a 0-6s 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 3-6s 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. 4.3. FORWARD RATE AGREEMENTS 49 month cashflow. In the case of a 6-12s 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) 59. 50 CHAPTER 4. HEDGING INSTRUMENTS 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. 60. 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 Chapter 5 princing A simple approach to swap 61. 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 swaps 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 62. 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: 63. 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 con- verted to an Actual/365 rate by multiplying by 365/360 simply because a year has 365 days. 64. 5.1. BASIC CONCEPTS 55 5.1.2 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 2,494,166.67 2,501,000.00 2,507,833.33 2,487,333.33 2,487,333.33 No. of days 365 366 367 364 364 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 swap

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