Valuation of a Forward Rate Agreement (pp )

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Janis Waters
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1 Valuation of a Forward Rate Agreement (pp ) A forward-rate agreement (FRA) is a forward contract where it is agreed that a certain interest rate R K will apply to a certain principal to a specified future time period. Example: You agree to borrow $100 from September 1, 2002 to December 31, 2002 at 6% interest. The value of such a contract depends on the current forward rate R F. If the current September 90-day forward rate is 6%, then the contract just mentioned has value zero. If that forward rate is less than 6%, then the value of the FRA to the party receiving the 6% is positive. The value of the FRA to the party who is to receive R K is. L(R K R F )(T 2 T 1 )e -R 2 T 2. This is the present value of L(R K R F )(T 2 T 1 ) at time T 2. Why? You can get an FRA with interest rate R F for free. If you combine the paying side of this R F -FRA with the receiving side of the R K -FRA, your only cash flow will be L(R K R F )(T 2 T 1 ) at time T 2. So the value of the combination, which is also the value of the R K -FRA, is the present value of this time T 2 payment. Hull: An FRA can be valued by assuming that the forward interest rate is certain to be realized. 1

2 Valuation of a Swap A plain-vanilla swap can be thought of in two different ways, which lead to two different ways of valuing them: You are swapping a bond with fixed-rate payments for a bond with floating-rate payments. The fixed-rate bond can be valued as in the preceding chapter, discounting each coupon payment using the zero-coupon rate for its date. The floating-rate bond has its face value as its present value, immediately after the payment of a coupon. You are making a bundle of forward rate agreements, together with some agreements that have value zero to both parties. Say the interest payments are at 6-month intervals. Then on July 1, 2003, you receive a payment at the fixed rate, say R K, and you pay the floating rate in effect on January 1, The agreement to accept the fixed rate R K is a forward rate agreement, which can be valued as on the preceding page. The agreement to pay the floating rate has value zero. 2

3 Either way, it is customary to discount using LIBOR. The use of LIBOR rates hides an implicit assumption: both parties have the creditworthiness of large banks. If there is no risk of default, then the prices have the solidity of arbitrage arguments. But in practice, there may be substantial risk of default, and therefore the prices are less reliable than prices in an organized futures market. The risk is different for different parties. If Company A swaps with Company B, and Company A has a better credit rating, then Company A has greater risk. If a large bank plays the role of financial intermediary, it is assuming most of the risk. This has implications for the proper accounting of swaps. 3

4 Example 5.1 Bank has agreed to pay 6-month LIBOR and receive 8% (semiannual) on $100 million. Remaining life: 1.25 years. LIBOR (continuous) 10% for 3 months 10.5% for 9 months 11% for 15 months. The LIBOR 6-month rate at the last payment date (3 months ago) was 10.2% (semiannual) What is the value of the swap? Fixed payments are $4m. Floating payment due in 3 months is $5.1m. Value of fixed bond is B fix = 4e -0.1x e x e -0.11x0.25 = $98.24 m Value of floating bond is B fl = ( )e -0.1x0.25 = $ m Value of swap is = $4.27 million 4

5 Problem 5.19 Under the terms of an interest rate swap, a financial institution has agreed to pay 10% per annum and to receive three-month LIBOR in return on a notional principle of $100 million with payments being exchanged every three months. The swap has a remaining life of 14 months. The average of the bid-ask fixed rate currently being swapped for three-month LIBOR is 12% per annum. All rates are compounded quarterly. The three-month LIBOR one month ago was 11.8% per annum (quarterly compounding). What is the value of the swap? Hint: Convert the 12% to continuous compounding (this gives 11.82%) and use it as the LIBOR of all maturities to discount the future payments for both the fixed and floating rate bonds. But of course the 3-month LIBOR one month ago defines the payment on the floating rate bond one month from now. 5

YIELD CURVE GENERATION

1 YIELD CURVE GENERATION Dr Philip Symes Agenda 2 I. INTRODUCTION II. YIELD CURVES III. TYPES OF YIELD CURVES IV. USES OF YIELD CURVES V. YIELD TO MATURITY VI. BOND PRICING & VALUATION Introduction 3 A