Swaps and Other Derivatives

By Richard R. Flavell

"Richard Flavell has a strong theoretical perspective on swaps with considerable practical experience in the actual trading of these instruments. This rare combination makes this welcome updated second edition a useful reference work for market practitioners."
—Satyajit Das, author of Swaps and Financial Derivatives Library and Traders and Guns & Money: Knowns and Unknowns in the Dazzling World of Derivatives

Fully revised and updated from the first edition, Swaps and Other Derivatives, Second Edition, provides a practical explanation of the pricing and evaluation of swaps and interest rate derivatives.

Based on the author’s extensive experience in derivatives and risk management, working as a financial engineer, consultant and trainer for a wide range of institutions across the world this book discusses in detail how many of the wide range of swaps and other derivatives, such as yield curve, index amortisers, inflation-linked, cross-market, volatility, diff and quanto diffs, are priced and hedged. It also describes the modelling of interest rate curves, and the derivation of implied discount factors from both interest rate swap curves, and cross-currency adjusted curves.

There are detailed sections on the risk management of swap and option portfolios using both traditional approaches and also Value-at-Risk. Techniques are provided for the construction of dynamic and robust hedges, using ideas drawn from mathematical programming.

This second edition has expanded sections on the credit derivatives market – its mechanics, how credit default swaps may be priced and hedged, and how default probabilities may be derived from a market strip. It also prices complex swaps with embedded options, such as range accruals, Bermudan swaptions and target accrual redemption notes, by constructing detailed numerical models such as interest rate trees and LIBOR-based simulation. There is also increased discussion around the modelling of volatility smiles and surfaces.

The book is accompanied by a CD-ROM where all the models are replicated, enabling readers to implement the models in practice with the minimum of effort.

• Language: English
• Publisher: Wiley
• Release date: Mar 30, 2012
• ISBN: 9780470661802

Book preview

Chapter 1

Swaps and Other Derivatives

This is the second edition. Much has changed since the first was written in 2000. For the first seven years of the new century, the derivative market continued to grow at an exponential pace. From 2008, its growth reversed, albeit not by much, as the global economic recession bit. In terms of notional amount, it reduced by just over 13% in the second half of 2008, to just under USD600 trillion. Between the publication of the first edition and the writing of the second, there have been some major developments. Two in particular stand out: the growth in the credit transfer market, and the massive issuance of complex securities enabling investors to earn potentially higher returns by taking on more risks.¹ Hence the requirement for a second edition, which addresses both of these topics in considerable detail.

1.1 INTRODUCTION

In the 1970s there was an active Parallel Loan market. This arose during a period of exchange controls in Europe. Imagine that there is a UK company that needs to provide its US subsidiary with $100 million. The subsidiary is not of sufficiently good credit standing to borrow the money from a US bank without paying a considerable margin. The parent however cannot borrow the dollars itself and then pass them on to its subsidiary, or provide a parent guarantee, without being subject to the exchange control regulations which may make the transaction impossible or merely extremely expensive.

The Parallel Loan market requires a friendly US company prepared to provide the dollars, and at the same time requiring sterling in the UK, perhaps for its own subsidiary. Two loans with identical maturities are created in the two countries as shown. Usually the two principals would be at the prevailing spot FX rate, and the interest levels at the market rates. Obviously credit is a major concern, which would be alleviated by a set-off clause. This clause allowed each party to off-set unpaid receipts against payments due. As the spot and interest rates moved, one party would find that their loan would be cheap, i.e. below the current market levels, whilst the other would find their loan expensive. If the parties marked the loans to market—in other words, valued the loans relative to the current market levels—then the former would have a positive value and the latter a negative one. A topping-up clause, similar in today’s market to a regular mark-to-market and settlement, would often be used to call for adjustments in the principals if the rates moved by more than a trigger amount.

As exchange controls were abolished, the Parallel Loan became replaced with the back-to-back Loan market whereby the two parent organisations would enter into the loans directly with each other. This simplified the transactions, and reduced the operational risks. Because these loans were deemed to be separate transactions, albeit with an off-setting clause, they appeared on both sides of the balance sheet, with a potential adverse effect on the debt/equity ratios.

The economic driving force behind back-to-back loans is an extremely important concept called comparative advantage. Suppose the UK company is little known in the US; it would be expensive to raise USD directly. Therefore borrowing sterling and doing a back-to-back loan with a US company (who may of course be in exactly the reverse position) is likely to be cheaper. In theory, comparative advantage cannot exist in efficient markets; in reality, markets are not efficient but are racked by varieties of distortions. Consider the simple corporate tax system: if a company is profitable, it has to pay tax; if a company is unprofitable, it doesn’t. The system is asymmetric; unprofitable companies do not receive negative tax (except possibly in the form of off-sets against future profits). Any asymmetry is a distortion, and it is frequently feasible to derive mechanisms to exploit it—such as the leasing industry.

Cross-currency swaps were rapidly developed from back-to-back loans in the late 1970s. In appearance they are very similar, and from an outside observer only able to see the cashflows, identical. But subtly different in that all cashflows are described as contingent sales or purchases, i.e. each sale is contingent upon the counter-sale. These transactions, being forward conditional commitments, are off-balance sheet. We have the beginning of the OTC swap market!

The structure of a generic (or vanilla) cross-currency swap is therefore:

initial exchange of principal amounts;

periodic exchanges of interest payments;²

re-exchange of the principal amounts at maturity.

Notice that, if the first exchange is done at the current spot exchange rate, then it possesses no economic value and can be omitted.

Interest rate, or single-currency swaps, followed soon afterwards. Obviously exchange of principals in the same currency makes no economic sense, and hence an interest swap only consists of the single stage:

periodic exchanges of interest payments;

where interest is calculated on different reference rates. The most common form is with one side using a variable (or floating) rate which is determined at regular intervals, and the other a fixed reference rate throughout the lifetime of the swap.

1.2 APPLICATIONS OF SWAPS

As suggested by its origins, the earliest applications of the swap market were to assist in the raising of cheap funds through the comparative advantage concept. The EIB-TVA transaction in 1996 was a classic example of this, and is described in the box below. The overall benefit to the two parties was about $3 million over a 10-year period, and therefore they were both willing to enter into the swap.

Comparative Advantage:

European Investment Bank-Tennessee Valley Authority swap

Date: September 1996

Both counterparties had the same objective: to raise cheap funds. The EIB, being an European lender, wanted deutschmarks. The TVA, all of whose revenues and costs were in USD, wanted to borrow dollars. Their funding costs (expressed as a spread over the appropriate government bond market) are shown in the matrix below:

Whilst both organisations were AAA, the EIB was deemed to be the slightly better credit.

If both organisations borrowed directly in their required currency, the total funding cost would be (approximately—because strictly the spreads in different currencies are not additive) 37 bp over the two bond curves.

However, the relative spread is much closer in DEM than it is in USD. This was for two reasons:

the TVA had always borrowed USD, and hence was starting to pay the price of excess supply;

it had never borrowed DEM, hence there was a considerable demand from European investors at a lower rate.

The total cost if the TVA borrowed DEM and the EIB borrowed USD would be only 34 bp, saving 3 bp pa.

The end result:

EIB issued a 10-year $1 billion bond;

TVA issued a 10-year DM1.5 billion bond; and

they swapped the proceeds to raise cheaper funding, saving roughly $3 million over the 10 years.

This was a real exercise in Comparative Advantage; neither party wanted the currency of their bond issues, but it was cheaper to issue and then swap.

It was quickly realised that swaps, especially being off-balance sheet instruments, could also be effective in the management of both currency and interest rate medium-term risk. The commonest example is of a company who is currently paying floating interest, and who is concerned about interest rates rising in the future; by entering into an interest rate swap to pay a fixed rate and to receive a floating rate, uncertainty has been removed.

To ensure that the risk management is effective, the floating interest receipts under the swap must exactly match the interest payments under the debt. Therefore the swap must mirror any structural complexities in the debt, such as principal repayment schedules, or options to repay early, and so on. Usually a swap entered into between a bank and a customer is tailored specifically for that situation. This book will provide details of many of the techniques used to structure such swaps.

A well-known and very early example of the use of swaps is the one conducted between the World Bank and IBM in August 1981—described in the box below. This swap had the reputation of kick-starting the swap market because it was performed by two extremely prestigious organisations, and received a lot of publicity which attracted many other end-users to come into the market. It was the first long-term swap done by the World Bank, who is now one of the biggest users of the swap market.

World Bank-IBM Swap

Date: August 1981

This is a simplified version of the famous swap. The two counterparties have very different objectives.

IBM had embarked upon a world-wide funding programme some years earlier, raising money inter alia in deutschmarks and Swiss francs. The money was remitted back to the US for general funding. This had created a FX exposure, because IBM had to convert USDs into DEMs and CHFs regularly to make the coupon payments. Over the years the USD had significantly strengthened, creating a gain for IBM. It now wished to lock in the gain and remove any future exposure.

The World Bank had a policy of raising money in hard currency; namely DEM, CHF and yen. It was a prolific borrower, and by 1981 was finding that its cost of funds in these currencies was rising simply through an excess supply of WB paper. Its objective, as always, was to raise cheap funds.

Salomon Brothers suggested the following transactions:

(a) The WB could still raise USD at relatively cheap rates, therefore it should issue two euro-dollar bonds:

one matched the principal and maturity of IBM’s DEM liabilities equivalent to $210 million;

the other matched IBM’s Swiss franc liabilities equivalent to $80 million.

Each bond had a short first period to enable the timing of all the future cashflows to match.

(b) There was a 2-week settlement period, so WB entered into a FX forward contract to:

sell the total bond proceeds of $290 million;

buy the equivalent in DEM and CHF;

(c) IBM and WB entered into a two-stage swap whereby:

so that IBM converted its DEM and CHD liabilities into USD, and the WB effectively raised hard currencies at a cheap rate. Both achieved their objectives!

1.3 AN OVERVIEW OF THE SWAP MARKET

From these earliest beginnings, the swap market has grown exponentially. As the graph shows, the volume of interest rate swap business now totally dominates cross-currency swaps,³ suggesting that risk management using swaps is commonplace.

The graph is shown in terms of notional principal outstanding, i.e. the principals of all swaps transacted but not yet matured; for the cross-currency swap described above, this would be recorded as [$100m + £60m * S]/2 where S is the current spot rate. The market has shown a remarkable and consistent growth in activity.

It is arguable whether this is a very appropriate way of describing the current size of the market, although it certainly attracts headlines. Many professionals would use gross market value or total replacement cost of all contracts as a more realistic measure. This measure had been in broad decline as banks improve their risk management, and are unwilling to take on greater risks due to the imposition of capital charges. However, as can be seen from the figures below, the gross value increased in the second half of 2008, especially in interest rate and credit derivatives, due to the dramatic movements in these markets.

A brief overview of the OTC derivative market is shown in the table below. Probably the most important statistic is that, despite all the publicity given to more exotic transactions, the overwhelming workhorse of this market is the relatively short-term interest rate swap.

The derivative markets continue to grow at an astounding rate—why? There are two main sources of growth—breadth and depth:

financial markets around the world have increasingly deregulated over the past 30 years, witness activities in Greece and Portugal, the Far East and Eastern Europe. As they do, cash and bond markets first develop followed rapidly by swap and option markets;

the original swaps were done in relatively large principal amounts with high-credit counterparties. Banks have however been increasingly pushing derivatives down into the lower credit depths in the search of return. It is feasible to get quite small transactions, and some institutions even specialise in aggregating retail demand into a wholesale transaction.

A brief overview of the current state of the derivative market (in December 2008) (extracted from the semiannual BIS surveys)

The total OTC derivative market was estimated to be just under $600 trillion, measured in terms of outstanding principal amount, broken up as shown below (in US billions):

The table shows the fairly dramatic slowdown and then drop during 2008, especially with equity, commodity and credit-related derivatives, but also the increase in gross value.

The above table shows that the majority of FX derivatives, predominantly forwards, are under 1 year in maturity, interest rate derivatives are typically much longer, averaging between 5 and 10 years. The Eurozone, UK and US routinely now trade swaps out to 50 years. In terms of currencies, the major ones have little changed over the past 10 years. The main development is the increased rise in euro products, and the relative decline in USD.

1.4 THE EVOLUTION OF THE SWAP MARKET

The discussion below refers to the evolution of the early swap market in the major currencies during the 1980s. It is however applicable to many other generic markets as they have developed.

There are typically three phases of development of a swap market:

1. In the earliest days of a market, it is very much an arranged market whereby two swap end-users would negotiate directly with each other, and an advisory bank may well extract an upfront fee for locating and assisting them. This was obviously a slow market, with documentation frequently tailored for each transaction. The main banks involved are investment or merchant banks, long on people but low on capital and technology as of course they were taking no risk. Typical counterparties would be highly rated, and therefore happy to deal directly with each other.

The first swap markets in the major currencies were even slower, as there was considerable doubt about the efficacy of swaps. End-users were dubious about moving the activities off-balance sheet, and there was apprehension that the accounting rules would be changed to force them back on-balance sheet. The World Bank-IBM swap (described above) played a major role in persuading people that the swap market was acceptable.

2. In the second phase, originally early to mid-1980s, commercial banks started to take an increasing role providing traditional credit guarantees.

The counterparties now would both negotiate directly with the bank, who would structure back-to-back swaps but take the credit risk, usually for an on-going spread not an upfront fee. The normal lending departments of the bank would be responsible for negotiating the transaction and the credit spread. The documentation is now more standardised and provided by the bank. This role is often described as acting as an intermediary, taking credit but not market risk.

The role of intermediary may also be encouraged by external legislation. In the UK for example, if a swap is entered into by two non-bank counterparties, the cashflows are subject to withholding tax. This is not true if one counterparty is a bank.

3. The concept of a market-making bank originally developed by the mid to late 1980s, whereby a bank would provide swap quotations upon request. This would mean that they would be dealing with a range of counterparties simultaneously, and entering into a variety of non-matching swaps. With increased market risk, such banks required considerably more capital, pricing and risk management systems, and very standardised documentation. The swap market became dominated by the large commercial banks who saw it as a volume, commoditised business.

These banks would be typically off-setting the market risk by hedging in another market, usually the equivalent government bond market as this is the most liquid. Therefore banks with an underlying activity in this market are likely to be at a competitive advantage. Local domestic banks usually have close links with the local government bond market, and hence they are frequently dominant in the domestic swap market. Probably the only market where this is not the case is the USD market, where the markets are so large that a number of foreign banks can also be highly active and competitive.

It might be worth making the point here that banks frequently and misleadingly talk about trading swaps, as if a swap were equivalent to a spot FX transaction which is settled and forgotten about within two days. A swap is actually a transaction which has created a long-term credit exposure for the bank. The exposure is likely to remain on the bank’s books long after the swap trader has been paid a bonus and has left the bank. From this perspective, swaps fit much more comfortably within the traditional lending departments with all the concomitant credit-controlling processes and not within a treasury which is typically far more lax about credit.

This link with the bond market has meant that a bank may well adopt different roles in different markets. For example, a Scandinavian bank such as Nordea Bank would be a market-maker in the Scandinavian and possibly some of the Northern European currencies. On the other hand, it would act as an intermediary in other currencies. For example, if a customer wanted to do a South African rand swap, it would enter into it taking on the credit risk, but immediately laying off the market risk with a rand market-making bank.

In this context, the 1996 EIB-TVA swap was interesting. The deal was brokered by Lehmann Brothers, but who played no role in the swap. At one point the swap had been out for tender from a bank but (rumour has it) the bid was a 1 bp spread. Why, asked the two counterparties, do we need to deal with a bank at all, especially given that we are both AAA which is better than virtually all banks? So they dealt directly! As the relative credit standing of banks declines, the market may well see more transactions of this nature—back full circle.

One cannot really talk about a global swap market. There are obviously some global currencies, notably USD, yen and the euro, which are traded 24 hours a day, and when it would be feasible to get swaps. But most swap markets are tied into their domestic markets, and hence available only during trading hours.

Swap brokers still play an important role in this market. Their traditional role has been to identify the cheapest suitable counterparty for a client, usually on the initial basis of anonymity. This activity creates liquidity and a uniformity of pricing, to the overall benefit of market participants. However, as the markets in the most liquid currencies continue to grow, the efficiency provided by a broker is less valued and their fees have been increasingly reduced to a fraction of a basis point. They are being forced to develop more electronic skills to survive.

1.5 CONCLUSION

The story of the swaps market has been one of remarkable growth from its beginnings only some 30 years ago. This growth has demonstrated that there is a real demand for the benefits swaps can bring, namely access to cheap funds and risk management, globally. Furthermore, the growth shows little sign of abating as swap markets continue to expand both geographically as countries deregulate and downwards into the economy. As we enter into 2009 and beyond, have derivatives suddenly become irrelevant?⁴ In my view, certainly not. The measurement and management of risk, whether it be interest rate, foreign exchange rate, credit and so on, is, and will remain, critical for all organisations. To suddenly deny the main mechanism for managing these risks is simply irrational. What is, of course, important is to ensure that users of derivatives understand and can assess derivatives, or at least employ people that do. I very much hope that this book will play some small role in the continued use of derivatives, and assisting the orderly development of the market, by ensuring that people are well-trained in their understanding of the pricing, structuring and risk management of swaps and related derivatives.

¹ Whether investors actually understood the risks they were taking on is an unanswered question, and very much outside the remit of this book.

² Remember: legally these cashflows are not interest but contingent sales, but for clarity of exposition they will be called interest as they are calculated in exactly the same way.

³ The source of these data is the Bank for International Settlement (BIS) which conducts a semi-annual survey of some 48 central banks and monetary authorities. It also does a more extensive triennial survey.

⁴ Or, as Warren Buffett famously described them, toxic waste.

Chapter 2

Short-term Interest Rate Swaps

OBJECTIVE

The main objective of this chapter is to provide an introduction to the construction and pricing of short-term IRS using futures contracts. However, because a simple swap may be regarded as an exchange of two streams of cashflows which occur at different points in the future, extensive use is made of the concept of discounting. The chapter therefore begins with a brief discussion on the time value of money, and demonstrates how implied discount factors may be derived from the cash market. Because rates are only available at discrete maturities, interpolation is a necessary technique; and there are a number of different approaches which end up with different results. The chapter then discusses how to estimate forward rates, and how to price FRAs first off the cash market and then off the futures market. This leads naturally to the pricing and hedging of short-term IRS off a futures strip. Examination of the hedging reveals a convexity effect which is discussed in more detail in Chapter 7 and its Appendix. Finally, an alternative approach to pricing swaps without discounting is briefly discussed.

2.1 DISCOUNTING, THE TIME VALUE OF MONEY AND OTHER MATTERS

Today’s date is Monday 4 February 2008; you have just been offered a choice of transactions:

Deal 1: to lend $10 million and to receive 3.25% for 3 months;

Deal 2: to lend $10 million and to receive 2.95% for 12 months.

Which do you find more attractive?

The current London rates at which you could normally deposit money are 3.145% pa and 2.89625% pa for 3 and 12 months, respectively;¹ we will assume that the creditworthiness of the counterparty is beyond question. Comparing the transactions with these market rates, the 3-month deal is 10.5 bp above the market, whilst the 12-month deal is only 5 bp. Intuitively you favour the first transaction, but wish to do some more analysis to be certain.

These market rates suggest that the following transactions are currently available:²

Note the following:

a. Whilst the rates are being quoted on 4 February, they are with effect from 6 February. In other words, there is a 2-day settlement period between the agreement of the transaction and its start. On 6 February, the counterparty’s bank account would be credited with $10 million. This is the normal convention in the USD market, although it is feasible to organise a same day transaction. Conventions vary between markets; for example, the GBP convention is normally same day.

b. Interest rates are invariably quoted on a per annum basis, even if they are going to be applied over a different period. It is therefore necessary to have a convention that translates the calendar time from, say, 6 May 2008 back to 6 February 2008 into years. The USD money market, in common with most money markets, uses an Actual/360 daycount convention, i.e. calculates the actual number of days:

equation

and then divides by 360 to convert into 0.2500 years. The other common convention is Actual/365, which is used in the sterling market and many of the old Commonwealth countries. The cashflow at the end of 3 months is given by:

equation

c. Payments can only be made on business days, and therefore a convention has to be applied to determine the appropriate date if the apparent cashflow date is a non-business day. The most popular is the modified following day convention, i.e. the operating date moves to the next business day unless this involves going across a month-end, in which case the operating date moves to the last business day in the month.

The concept of discounting will be used extensively throughout this book. The time value of money suggests that the value of money depends upon its time of receipt; for example, $1 million received today would be usually valued more highly than $1 million to be received in 1 year’s time because it could be invested today to generate interest or profits in the future. If Ct represents a certain cashflow to be received at time t > 0, then a discount factor DFt relates this cashflow to its value today (or present value) C0 by

equation

Note that this does not presuppose any source or derivation of the discount factor.

The present value of each of these two market-based transactions may be easily calculated as:

equation

and

equation

where DF3 and DF12 are the 3 and 12-month discount factors, respectively. The market rates are obviously freely negotiated, and we will assume that, at the moment of entering into the transactions, the transactions represent no clear profit to either party. In other words, at inception the transactions would be deemed to be fair to both parties, and hence have a zero net value. Hence, we can solve for the two discount factors, i.e. DF3 = 0.992199 and DF12 = 0.971397, respectively. A general formula for discount factors from the money markets is:

(2.1)

equation

where dt is the length of time (in years) and rt is the rate (expressed as a % pa).

Turning back to the two original transactions, these will generate the following cashflows:

The present values are determined using the discount factors derived from the market rates. Thus we can see, perhaps against our intuition, that the second transaction would be the more profitable of the two. This is of course because the deal is longer: 10.5 bp over 3 months is roughly half of 5 bp over a year.

This is of course ignoring market realities such as bid-offer or bid-ask spreads (or doubles as they are frequently called). In practice, most analysis uses mid-rates, i.e. the arithmetic average between bid and offer, simply to enable the statement of fairness to be made, and subsequently adjusted for various spreads. These issues will be discussed in more detail later; for the current discussion they will be ignored.

The current money market data readily available is:

Discount factors at each of the maturities can be easily calculated as above, i.e.:

You are now offered the opportunity to purchase today a riskless $100m due to be paid on 6 November 2008. What value would you place on this transaction? To answer this question, the discount factor on 6 November is required—but how to calculate it? The obvious approach is interpolation, but this raises two questions:

What is interpolated: cash rates or discount factors?

How is the interpolation calculated: linear, polynomial, exponential, etc.? with associated questions do the answers change the valuation? and are there any ‘right’ answers?. The simple answers to the latter questions are yes and no, but some are better than others! The results from some popular methods are shown below:

where:

linear is simply straight-line interpolation;

cubic implies fitting a cubic polynomial of the form a + b. t + c. t² + d. t³ through the four neighbouring points and solving for {a, b, c, d};

log-linear is the straight line interpolation of the natural logarithm of the discount factors (this last one is often suggested as a discount curve is similar to a negative exponential curve).

The deal value fluctuates by some $50,000 or roughly 5 bp, which, whilst perhaps not significant, is certainly worthwhile. It is more common practice to interpolate rates rather than discount factors at the short end of the curve. This is probably because it would be perfectly feasible to get a quote for a rate out to 6 November for depositing, and of course the two transactions should be arbitrage-free.

Cash rates are of course spot rates, i.e. they all start out of today. The cash curve may be used to estimate forward rates, i.e. rates starting at some point in the future. For example, if we knew that we would receive $100 million on 6 May 2008 for, say, 3 months, we could lock in the investment rate today by calculating the 3/6 rate.³ Forward rates are usually estimated using an arbitrage argument as follows:

1. We could borrow $100 million for 3 months at 3.1450%, the repayment cashflow⁴ would be 100m * (1 + 3.1450% * 0.250) = 100,786,250.00.

2. The $100 million could then be lent out for 6 months at 3.0975%, this would generate a cashflow of 100m * (1 + 3.0975% * 0.506) = 101,565,958.33.

At the end of the 3 months, the borrowing has to be repaid. Assume the repayment is to be financed by borrowing for another 3 months at the rate r3/6, thus generating a new liability:

equation

where d3/6 = 0.256 is the length of time at the end of 6 months. For the transactions to break even:

equation

The implied 3/6 rate is 3.0272%.

A general expression for a forward rate Ft/T, from t to T, is:

(2.2)

equation

However, to use this expression, the rates must be zero-coupon spot rates with maturities t and T. These are generally available when T is under 1 year, but are unlikely to be available for longer maturities. A more widely used expression for longer dated forward rates is:

(2.3) equation

using discount factors estimated off the discount curve (which of course is synonymous for cash rates).

Returning to the cash curve above, we want to estimate the 3-monthly forward rates, 3/ 6, 6/9 and 9/12. To do this, we need to estimate the 9-month discount factor DF9. The table below shows it being estimated in a variety of ways, and the resulting forward rates.

The impact of the different methods on the forward rates is quite dramatic, showing differences of up to 20 bp. Contrast this with the difference in the discount factors, which in the previous example only reached 5 bp.

To understand why, rewriting eq. (2.3) as:

equation

highlights the fact that a forward rate is related to the gradient of the discount curve and is therefore much more sensitive to small differences in the estimates. To demonstrate this more clearly, the table below calculates a 15-day forward rate curve using all the five different methods of interpolation:

The average difference between the highest and lowest curves is 11.4 bp.

In practice, whilst there is no right method, most people interpolate the cash rates using either linear if the cash curve is relatively flat, or polynomial if the curve is quite steep.

2.2 FORWARD RATE AGREEMENTS (FRAs) AND INTEREST RATE FUTURES

A FRA is an agreement between two counterparties whereby:

seller of FRA agrees to pay a floating rate interest and to receive a fixed interest rate;

buyer of FRA agrees to pay the fixed interest and to receive the floating interest

on an agreed notional principal amount, and over an agreed forward period.

For example, a company is a payer of 3-month floating interest on $100 million of debt. The company is concerned about interest rates rising, and on 4 February 2008 it buys a $100 million 3/6 FRA at a fixed rate of 3.0272% from a bank. The following operations occur:

This is shown from the point of view of the company, and will be positive if L > 3.0272% or negative if L < 3.0272%. Hence, the company is locked into the fixed rate even if rates do rise over the period from 4 February to 4 May.

In practice, the net amount is discounted back to 6 May 2008 using the recent Libor fixing, i.e.:

equation

and paid then. The usual reason given for this market convention is a reduction in the credit exposure between the two parties:

a. On 4 February, the current exposure is assumed to be zero, i.e. the FRA would have a zero valuation for both parties.

b. However, there is a potential future exposure over the period from 4 February to 4 May which would fluctuate as the estimate of the Libor fixing on 4 May varies. If the estimate rises, then the FRA has a negative value for the bank and hence the company has a credit exposure on the bank; conversely, if the estimate falls, then the FRA has a positive value for the bank, and it has a credit exposure on the company.

c. On 4 May, the official Libor fixing is known, which then fixes the net settlement amount and crystallises the residual credit exposure.

d. The two parties could wait until 6 August with one of them having this known residual exposure. By making the payment immediately on 6 May, this 3-month residual risk is removed.

As banks are required to place capital against all credit exposures, and capital has a cost, retaining the residual exposure could be expensive. Discounting the net settlement amount therefore appears to favour the bank, as it implies that for a given credit limit and amount of capital, the bank could effectively do twice the total business in 3/6 FRAs. The impact of discounting on reducing the total credit exposure obviously declines as the time to the fixing date lengthens.

The benefit to the company is less clear. Whilst the value of the net settlement remains constant whether discounted or not, most companies neither mark-to-market nor are overly concerned about credit exposures. The cashflows from the FRA and from the underlying debt are not on the same dates, therefore creating a mismatch which may cause accounting and tax problems. It is highly unlikely that the company could reproduce the undiscounted net settlement, as it would not be able to deposit or borrow at Libor flat for an odd cashflow, irrespective of its credit worthiness. It is perfectly feasible for banks to provide non-discounted FRAs⁵ at a price, but this is seldom