Interest rate Derivatives


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1 Interest rate Derivatives There is a wide variety of interest rate options available. The most widely offered are interest rate caps and floors. Increasingly we also see swaptions offered. This note will deal with the valuation of these interest rate options. However, we have included a quick review of spot and forward rates and a brief discussion of the valuation of options with delayed payments. Both of these things play a major role in the valuation of interest rate derivatives. Spot rates and forward interest rates It is fairly common to think of a single interest rate. However, there are many interest rates and as such, it is necessary to be specific. First, in everything that follows in this note we will be referring to the riskfree rate. In addition, for interest rates it is necessary to specify the time period covered by that rate. Spot rates are interest rates that are used to discount cash received at one point in the future back to today. For example, a sixmonth spot rate, i 6 would be the appropriate rate to discount any riskfree payoff back to today. Hence, the term spot rate is the rate that begins today and ends at some future point in time. We use the term forward rates to refer to interest rates that apply for specific time periods in the future. For example, there are interest rates that apply specifically for the time period beginning six months from now and ending at nine months from now. This would be the forward rate 1 for months 6 through 9, f. 6 9 The relationship between the 9month spot, 6month spot rate and the forward rate for months 6 through 9 is as follows: e i 9 i f 9 e See Exhibit 1 for an example of forward and spot rates. We will use these rates as the market for interest rates through out the rest of this note. e 3 1 This forward rate could also be referred to as the 3month rate 6 months forward. This is a mouth full but it is the most common verbiage used concerning forward rates. For the purpose of this note, we will use the above descriptors to refer to forward rates for specific periods. This note was prepared by Professor Robert M. Conroy. Copyright 2003 by the University of Virginia Darden School Foundation, Charlottesville, VA. All rights reserved. To order copies, send an to No part of this publication may be reproduced, stored in a retrieval system, used in a spreadsheet, or transmitted in any form or by any means electronic, mechanical, photocopying, recording, or otherwise without the permission of the Darden School Foundation.
2 2 Options with delayed payment Before we actually deal with interest rate options, we need to address the issue of options with delayed payment. Consider a simple example. Suppose we have a European Call option on a share of stock (no dividends) with a maturity of 9months and an exercise price of $50. The current share price is $52, and the volatility estimate is.30. The payoff at the maturity of this option depends on the stock price at the end of month 9, S 9, and the payoff would be Payoff ( S X 0) Max( 50,0) Max S 9, 9 and we would value the option using the BlackScholes model with the following inputs: Call UAV Maturity $6.57 X R f σ $52 9months $ %.30 where the riskfree rate is the 9month spot rate from Exhibit 1. Now, we change the timing of the payoff on the option. Instead of the payoff being determined and paid at the end of month 9, let us set the terms of the option such that the payoff is determined at the end of month 9 but the actual payment is made at the end of month. If we assume that interest rates are constant, then the value of the payoff received at the end of month determined at the end of month 9 is the present value discounted for the 3 months of delay or ( S X ) 3 3 9f 9f Payoff Max 9 e,0 e. We can pull the discounting factor outside of the Max function and the payoff becomes Payoff e 3 9f Max [( S X ),0] 9. Since the discounting factor is outside of the Max function, we can value that payoff and then discount that option value for the delay. Hence the value of the option is delayed payment is
3 3 Call e 3 9 f UAV Maturity X R f σ $52 9months $50 e 1.24% f $6.57 e $6.57 $6.541 This is a nice result. It makes it easy to value options with delayed payments. The key is to value the option assuming that payoff is realized and paid at the maturity of the option. Then we account for the delay by discounting that option value for the delay using the forward rate applicable for the delay time period. Options on Bonds In principal we should not be able to value European call options on Bonds. The Black Scholes model specifically assumes that total volatility over some time period is an increasing function of time 2. Bonds really have a very different price process. Interest rate changes drive changes in Bond prices. While interest rates might get more volatile as the further out in time we go, bond price sensitivity decreases the further out in time that we go. Sensitivity decreases because Duration 3 decreases, as the bond gets closer to a maturity. In fact, if we ignore default risk, bond price risk goes to zero as the bond reaches maturity. The price at that point will be equal to the face value. One solution is to think differently about bond price volatility. If we are willing to assume that at any point in time that bond prices are log normally distributed, then we can assume that there is some total volatility of the bond price at the option maturity time T, σ T. Given this we can estimate a BlackScholes model volatility, σ bs, as σt σ bs T Let us look at a specific example. Suppose we had a 5year, 4% coupon bond with semiannual coupon payments. The current cash bond price 4 is per 100. Consider a call option on this bond with an exercise price 5 of $100 and 21 months to maturity. We estimate that the volatility of the bond price 21 months forward is σ T.08. Note that this is an estimate of the price volatility of a bond that has a maturity of not 5 years but 2 This means that the longer the time period the more variable the possible outcomes. 3 Modified duration is a measure of bond price sensitivity. The basic relationship is % Pr ice Duration ( yield ). As duration decreases, the %?price decreases for any given change in yield. 4 The cash price does include accrued interest. The quoted price excludes accrued interest. However, in this case, where it is exactly 6 months to the next dividend the accrued interest is 0 and the cash and quoted prices are the same. 5 In this simple case, the exercise price is the quoted price, which does not include accrued interest. If the exercise price were the cash price, the exercise price would be $100 plus 3 months of accrued interest, $1 ($100x.04x3/$1.00).
4 years. As such, the volatility for the maturity of the option (21 months or 1.75 years) that we would use in the BlackScholes model is σt.08 σ bs.0605 T 1.75 The last necessary piece is the Underlying Asset Value or UAV. In order to deal with this we appeal to our dividend analogy once again. If I hold the bond, I get the coupon payments. The option holder does not get the coupon payments. Hence, the UAV for the option is the bond price less the present value of the coupon payments or using the rates in Exhibit 1, UAV $ $2 e UAV $ $2 e. i6.5 i1 i $2 e $2e $2e $2e $ Call UAV Maturity $3.41 X rf ( i21 ) σ bs $ months $ %.0605 If the exercise price had been in terms of the cash price ($101$100 + accrued interest 6 ), the option value would have been Call UAV Maturity $2.936 X rf ( i21) σ bs $ months $ %.0605 Bond Price Volatility and Yield volatility While it seems easy to say that we need the bond price volatility, σ T, it is much more difficult to actually estimate it. Since each bond is unique 7 and the maturity of a particular bond decreases over time, we cannot look at a series of bond prices and calculate a volatility of the bond price with a specific maturity. One way around this problem is to use Modified Duration 8, which is bond specific and the volatility of yields, which are not necessarily bond specific. 6 See footnote 4. 7 Each bond is unique in that the coupon rates are usually bond specific as is the maturity. 8 See Darden technical note, Duration and Convexity (UVA F38) for a fuller discussion of Modified Duration.
5 5 The basic relationship between bond prices and Modified Duration is % Pr ice ModDur Yield If we modify this slightly we can get the following: Yield % Pr ice ModDur Yield Yield % Pr ice ModDur Yield % Yield If we calculate the standard deviation of the %? of the above 9, we get the bond price volatility as a function of yield volatility. (% Pr ice) ModDur Yield σ (% Yield ) σ. Yield volatilities can be calculated directly from yields. Exhibit 2 lists the yields on US Treasury bonds with maturities of 3 years, 5 years, 7 years and 10 years. Volatility for each of these yields is also shown. Not surprisingly, shorterterm yields are more volatile than longterm yields. Note that the way we calculated the yield volatility is the same as we would have calculated the volatility on stock. As such, we can use this as a BlackScholes volatility. Let us use the same bond listed above to estimate the price volatility based on the forward and spot rates in Exhibit 1 and the yield volatilities in Exhibit 2. For both the Modified Duration and the Yield, we need to calculate these for a bond 1.75 years from today with a maturity of 3.25 years. We need the duration and the yield for the underlying bond as of the maturity of the option. Exhibit 3 shows the calculation of the bond price, yield to maturity and modified duration for the bond as of today given the forward and spot rates in Exhibit 1. Exhibit 4 shows the forward bond price, forward yield to maturity and forward modified duration for the same bond forward 1.75 years to coincide with the maturity of the option. This is the relevant data for calculating the BlackScholes bond price volatility. Using the data from Exhibit 4 and the yield volatility of the 3year maturity 10, we get a volatility of σ (% Pr ice) ModDur Yield σ (% Yield ) This is the volatility estimate that we would use in our BlackScholes model and given this volatility, σ bs , we would price a European call option on a 4%, 5 year maturity bond with a current value of $ , an exercise price of $100 (quoted price), a maturity of 21 months (1.75 years), and interest rates as shown in Exhibit 1 as, 9 Remember the definition of volatility is that it is the standard deviation of the percentage change in the underlying asset. 10 This is the time closest to the remaining time on the bond at the maturity of the option.
6 6 UAV $ $2 e UAV $ $2 e. i6.5 i1 i $2 e $2e $2e $2e $ Call UAV Maturity $3.33 X r f ( i21) σ bs $ months $ %.059 Interest Rate Caps and Floors Interest rate caps and floors are very popular vehicles used to modify interest rate risk. Often a firm will issue a floating rate note and at some future date decide to limit its exposure. Interest rate caps and floors are options designed to deal with this. Let s look at a specific example. Interest Rate Caps Today s date is December 31 st, The maturity date of the cap is in 6 months, June 30 th, If the 3month LIBOR rate on June 30 th is greater than 1.50%, the cap pays the holder 3 months worth of interest on $1,000,000 based on the interest rate differential between the 3month LIBOR rate on June 30 th, 2004 and 1.50%. If the 3month LIBOR 6/30 rate on June 30 th is less than 1.50%, the cap expires worthless. An additional feature of this cap is that while the payoff is determined on June 30 th the actual payment is made 3 months later on September 30 th, See table below for summary of payoff structure. Payoff Condition at maturity 6/30/04 Payoff determined 6/30/04 but paid 9/30/04 If 3month LIBOR 6/30 > 1.50% (3mon. LIBOR 6/ %) x $1,000,000 x.25 If 3month LIBOR 6/30 < 1.50% $0 In more traditional option terms the payoff would be Payoff Max 3 6f9 [( LIBOR X ) $1,000,000.25,0] e 6/ 30 Here we have incorporated the threemonth delay in the payment. We can value this using the BlackScholes model. From our discussion on call options with delayed payments, we can value the cap as a straight option and then discount the value by the time of the payment delay. For the straight option part, the key inputs are the UAV and the volatility. In this case, the underlying asset is a 3month LIBOR LIBOR $ 1,000, We can find the value payment on June 30 th, 2004 equal to [( ) ] T
7 7 of this payment today, by hedging the LIBOR payment with the sellside of a Eurodollar futures contract that matures on June 30 th, 2004 and lock in the futures rate. Once the rate is locked in, we can value it by discounting it at the riskfree rate 11. From Exhibit 5, the rate we can lock in is the Eurodollar futures rate for the contract with a maturity of June 2004, F 6/ %. This is the futures rate for the time period June to September and is essentially the same as the forward LIBOR rate for this time period. The value of this today is the present value at the 6month spot rate, 1.% from Exhibit 1. Note that there is a slight difference between the Eurodollar futures rates shown in Exhibit 5 and the US Treasury forward rates in Exhibit 1. Eurodollar futures are slightly higher. UAV F 6/ 30 e UAV 1.52% e i.5 UAV $250,000 UAV $3, $1,000, $1,000, Volatility is a bit more problematic. Exhibit 6 shows Eurodollar futures rates for different contract maturities over time. For example, the column labeled /03 is the Eurodollar futures contract with a maturity date of December, The first row labeled 60 months is the rate on that contract 60 months (5 years) prior to the maturity date on the contract, December The row labeled months is the rate that was quoted on the December 2003 contract twelve months prior, December Hence, as we go down each column row we have the rate that was quoted for a particular contract maturity date 60 months prior, 57 months prior, etc. If we calculate the volatility 13 of the rates across each row we have the volatility of the 3month rate x number of months prior. Figure 1 shows the volatility estimates of the 3month LIBOR rates x number of months prior. Here we see the usual result. Volatilities are humped. Near term futures rates have much more volatility than futures rates much further out in the future. Since the derivation of the BlackScholes model assumed that the volatility of the underlying asset was constant, this presents a problem for valuing caps using this model. For our example, the underlying 3month LIBOR rate starts 6 months in the future and this time period in the future decreases as we get closer to the maturity. From Figure 1 11 This is essentially the same approach we used in Valuation to value each of the hedged payments when valuing a swap. As you will see, the only major difference here is that we will use the US Treasury spot rates to discount the hedged payment. The underlying asset of a Eurodollar futures contract is an interbank US$ deposit. They are essentially riskfree but since these deposits do not have a government guarantee, they trade at a slightly higher rate. I have chosen to discount the Futures rate at the US riskfree spot rate. Sometimes you will see the LIBOR spot rate used. Generally, this does not result in a different cap value. 13 Volatility is defined as the standard deviation of the percentage changes or the natural logarithm of the price relatives. Note that since these are quarterly observations we would annualize the quarterly standard deviation by multiplying by the square root of 4.
8 8 we also see that the volatility of the underlying 3month LIBOR rate also decreases. In order to deal with this, most practitioners effectively cheat. They use average of the volatilities. Again for our example the volatility for the 3month LIBOR rate 6 months in the future is about.35 but this decrease to about.31 at maturity. Looking at Figure 1 we might use an average volatility of.331 to account for the decline in volatility over the life of the cap. This seems a bit ad hoc but we do similar things every time we estimate volatility for an option. Now that we have a UAV and volatility, we can value this cap. The UAV is $3,778.75, the exercise price 14 is $3,750, time to maturity is 6 months, the riskfree rate is the 6 month spot rate from Exhibit 1, 1.%, the volatility is.331 and again from Exhibit 1, the forward rate for months 6 through 9 to account for the delayed payment is 1.48%. Call e 3 6 f9 UAV Maturity X r f σ $3, months $3,750 e 1.% f $375. e $375. $ We could have valued the cap using the rates and then multiplied the result by the dollar amounts involved. With rates, the UAV would be UAV F 6/ 30 e i % e % Call e 3 6f9 UAV Maturity X rf σ months.015 e 1.% f e Call $1,000, $ This is the same as the dollar amount approach. Most often you see the cap valued and quoted with rates and then valued based on the dollar amount involved. Floors If interest rate caps are calls then floors are puts. Floors pay the holder the difference between the LIBOR rate and the exercise price when LIBOR is less than the exercise price. These are puts and we value them using putcall parity. 14 X 1.50% $1,000,
9 9 UAV + Put Call + X e Put Call + X e r T f r T UAV For example, a floor with the same parameters would have a value of f Put Call + X e T Put Floor $ Caps and Caplets r f UAV $ $3,750 e.01.5 $3, In the example above, we dealt with a single option that is termed a caplet. Usually a cap is a series of caplets with the same exercise price but different maturity dates. Suppose that we use the example above but now the cap is a series of 4 caplets that mature on the following dates: Mar. 31 st, 2004 Jun. 30 th, 2004 Sept. 30 th, 2004 Dec. 31 st, 2004 The UAV and other inputs for each caplet would be Maturity Mar. 31 st, 2004 Jun. 30 th, 2004 Sept. 30 th, 2004 Dec. 31 st, 2004 Maturity (T yrs.) Futures rate (F) (Exhibit 5) 1.26% 1.52% 1.74% 1.98% Spot rate ( i T ) (Exhibit 1) 1.00% 1.% 1.24% 1.36% i t T UAV F e % % % % Exercise Price 1.50% 1.50% 1.50% 1.50% Volatility???? R f (spot rate) 1.00% 1.% 1.24% 1.36% Call Value???? Payment delay Delay period 3/046/04 6/049/04 9/04/04 /043/05 Forward rate delay period (f) 1.24% 1.48% 1.72% 1.96% (Exhibit 1) Caplet Value???? Most of the inputs are readily available and fairly well defined. However, as is usually the case, the volatility estimate is the most problematical. The non constant volatility shown in figure 1 presents a real challenge in valuing caps. There are two approaches that people take. The first is to use an average or spot volatility which matches the
10 10 maturity of each caplet. The other is to use an average of the average and use the same flat volatility for each of the caplets. Figure 2 shows the individual volatilities, the spot volatilities (average) and the flat volatilities for data in exhibit 6. If we use the spot volatilities from Figure 2 to value the caplets we would get: Maturity Mar. 31 st, 2004 Jun. 30 th, 2004 Sept. 30 th, 2004 Dec. 31 st, 2004 Maturity (T yrs.) Futures rate (F) (Exhibit 5) 1.26% 1.52% 1.74% 1.98% Spot rate ( i T ) (Exhibit 1) 1.00% 1.% 1.24% 1.36% i t T UAV F e % % % % Exercise Price 1.50% 1.50% 1.50% 1.50% Volatility R f (spot rate) 1.00% 1.% 1.24% 1.36% Call Value Payment delay Delay period 3/046/04 6/049/04 9/04/04 /043/05 Forward rate delay period (f) 1.24% 1.48% 1.72% 1.96% (Exhibit 1) Caplet Value Caplet $ Value (x $250,000) $41.66 $ $ $1, Cap Value (sum of caplets) $2, The value 15 of the cap (4 caplets) is $2, Another alternative would have been to use a flat volatility. In this case, we would use the same volatility for all of the caplets. For example, if we used a flat volatility of.32176, the value of the cap would have been $2, Firms vary in their approach. Some use spot volatilities and others use flat volatilities.
11 11 Swaption A swap is an exchange of payments. Swaps exist in many forms. You can see a range of swaps from plainvanilla interest rates swaps, where there is a simple exchange of fixed rate payments for floating rate payments based on a nominal amount, to more complicated swaps, which offer exchange of returns on commodities in exchange for returns on exotic stock indexes. In either case, these contracts have value. Let us examine the valuation of a plainvanilla interest rate swap 16. Consider a 3year swap where you receive a fixed rate of 2.328% and pay 3month LIBOR. Today is December 31, 2003 and the first payment is due March 31 st, This means that the first fixed payment will be F $1,000, / 04 $5, and the floating Rate LIBOR payment made on March 31 st, 2004 will be based on the current 3month spot LIBOR rate of 1.02% and would be.0102 L 3 / 04 $1,000,000 $2, The June LIBOR payment would be based on the 3month LIBOR rate, LIBOR 3/04, as of March 31 st, 2004 and the payment would be L 6/ 04 LIBOR 3/ 04 $1,000, The value of the fixed payment is just the present value of the payments discounted 17 at the LIBOR spot rates. The value of the LIBOR payments is determined by assuming that the LIBOR payments are hedged using Eurodollar futures contracts and locking in the Eurodollar futures rate. These hedged payments are discounted using the LIBOR spot rates. Exhibit 7 shows the Eurodollar futures rates from Exhibit 5 and the corresponding LIBOR spot rates derived 18 from those rates. We use the Eurodollar futures to hedge and 16 For a fuller explanation, see Valuation of PlainVanilla Interest Rate Swaps (UVA F11). 17 In valuing swaps, the convention is to value the swaps using the LIBOR spot rates derived from LIBOR Eurodollar futures contracts. 18 The relationship between LIBOR spot and the Futures rates is as follows: t t 1 1 spott spot t 1 + t ft and solving for the Libor spot rates in Exhibit 7 yields,
12  this results in the hedged payments shown in Exhibit 8. These payments are discounted at the LIBOR spot rate and the value of the LIBOR payments is then $67,740. Also included in Exhibit 8 are the fixed payments, and discounting these at the LIBOR spot rates yields a value of $67,740. The values of the LIBOR payments and the Fixed payments are equal. This is usually the result when a swap is first initiated or priced. A swaption gives the holder the right to enter into a swap at a set fixed rate at some future date. For example, consider a swaption that gave the holder the right one year from today to enter into a 3year swap to pay fixed of 2.328% and receive LIBOR on a notional amount of $1,000,000. It is essentially the same as the swap above except that the start date of the swap is December 31 st, This swaption will have value at maturity if at maturity, December 31 st, 2004, the 3year swap rate is higher than 2.328%. Suppose that the 3year swap rate at maturity is 3.00%. The holder of the swaption would realize the value of the swaption by exercising the swaption and paying 2.328% and receiving LIBOR. At the same time, the holder would enter into a new swap agreement to pay LIBOR and receive the fixed rate of 3.00%. This would result in a fixed payment of 0.672% (3.00% %0.672%), every 3 months for the next three years. The key to valuing a swaption is to look at the payoff and realize that this is a series of call options that pay off the difference between the 3year swap rate on December 31 st, 2004 and the exercise fixed rate. All mature on December 31 st, 2004 but one has a delay of 3 months (pays off March 31 st, 2005), another has a delay of 6 months (pays off June 30 th, 2005), and so on. Hence, the key is to value the basic call option and incorporate the payoff delay in each call. We will use the BlackScholes model to value the basic call. Our approach will be to do everything in terms of rates. As usual, the inputs are Underlying Asset Value, Exercise Price, Time to Maturity, Riskfree rate, and volatility. The easy inputs are X2.328%, T1 year, and from Exhibit 1, r f 1.36%. The exercise price, time to maturity and riskfree rate are fairly straight forward. The UAV is a bit more difficult. What we need here is the one year forward swap rate. This is the rate quoted today for a 3year swap that begins one year from today. Exhibit 9 shows the calculation of the oneyear forward 3 year swap rate. Based on the current Futures rates and the implied LIBOR spot rates, the rate today that should be quoted is 3.284%. Using the same analogy as we did for the interest rate caps, the UAV for the swaption should be UAV 3.284% e i months % e % swaption. spot t spot t 1 1 t t f t 1 1 t 1 4
13 13 The other difficult input is volatility. Figure 3 shows the 1year, 3year and 5year interest rate swap rates for the last 5 years. There has been a great deal of volatility. This is reflected in the volatilities calculated from this data. The 1yr swap rate volatility is.277, the 3year swap rate volatility is.350 and the 5year is.298. Since the underlying asset is a 3year swap rate 19, we can use the.350. With this, we can value the basic call as follows: Call UAV Maturity X rf σ % months 3.00% %.35 The $ value of the basic call option would be $1, Now if the payment differentials were all paid at the maturity of the option, December 2004, the value of the swaption would be just times the basic call option value. However, while the payment differential is determined on December 2004, each of the twelve payments is made at some future date. Hence, the value of the swaption is actually the value of a series of these basic call options each with a different delayed payment. The underlying swap involves payments. The first begins March 2005, and each payment has a different delay. Exhibit 10 shows the series of swap payment dates and the delay relative to December Each payment date contributes an amount equal to the basic call value discounted for the delay. For example, the September 2005 payment date is the basic call value of $1, discounted back for 9 months to December The discount rate is based on the forward rates 20 from December 04 to March 05, March 05 to June 05 and June 05 to September 05. If we do this for each payment, the value of the swaption is $16, Summary In this note we have discussed the valuation of options on bonds, interest rates and swaps. In each case, there are certain adaptations that are required but the basic process remains the same. 19 The underlying asset is actually the oneyear forward 3year swap rate. From our discussion of interest rate caps, we know that the volatility of the spot rate, like the 3year swap rate, can be different than the volatility of the 3year swap rate one year forward. For our purposes here, we will ignore this distinction. 20 The forward rates are the riskfree continuously compounded rates from Exhibit 1.
14 14 Forward End of month Exhibit 1. Forward Rates and Spot Rates (US Treasury) (continuous compounded) Spot Rates From To Forward rate Time (months) time (years) as of the end of Spot rate Dec03 Mar % Mar % Mar04 Jun % Jun04 1.% Jun04 Sep % Sep % Sep04 Dec % 1.00 Dec % Dec04 Mar % Mar % Mar05 Jun % Jun % Jun05 Sep % Sep % Sep05 Dec % Dec % Dec05 Mar % Mar % Mar06 Jun % Jun % Jun06 Sep % Sep % Sep06 Dec % Dec % Dec06 Mar % Mar % Mar07 Jun07 4.% Jun % Jun07 Sep % Sep % Sep07 Dec % Dec % Dec07 Mar % Mar % Mar08 Jun % Jun % Jun08 Sep % Sep % Sep08 Dec % Dec % Dec08 Mar % Mar % Mar09 Jun % Jun % Jun09 Sep % Sep % Sep09 Dec % Dec % Dec09 Mar % Mar % Mar10 Jun % Jun % Jun10 Sep % Sep10 4.% Sep10 Dec % Dec %
15 15 Exhibit 2. Yield Volatilities US Treasury Constant Maturity 3year 5year 7year 10year Yield (%) %? Yield (%) %? Yield (%) %? Yield (%) %? Jan Feb (0.0929) 4.78 (0.0421) 5.01 (0.0291) 5.1 (0.0039) Mar (0.0240) 4.67 (0.0230) 4.88 (0.0259) 4.87 (0.0451) Apr (0.0246) 4.66 (0.0021) May Jun (0.0022) Jul (0.0045) 4.88 (0.01) 5.21 (0.0057) 5.37 (0.0037) Aug (0.0850) 4.62 (0.0533) 4.9 (0.0595) 5.11 (0.0484) Sep (0.0440) 4.46 (0.0346) 4.72 (0.0367) 4.85 (0.0509) Oct (0.1867) 3.9 (0.56) 4.33 (0.0826) 4.55 (0.0619) Nov (0.0849) 3.66 (0.0615) 4.01 (0.0739) 4.24 (0.0681) Dec Jan Feb (0.0023) 4.78 (0.04) 5.02 (0.0099) Mar (0.0080) Apr May (0.1189) 4.49 (0.0892) 4.86 (0.0813) 5.08 (0.0662) Jun (0.0159) 4.36 (0.0290) 4.75 (0.0226) 5.06 (0.0039) Jul (0.0995) 4.08 (0.0642) 4.54 (0.0442) 4.85 (0.0415) Aug (0.2239) 3.46 (0.1520) 4.04 (0.1101) 4.47 (0.0784) Sep (0.0385) 3.22 (0.0694) 3.78 (0.0644) 4.14 (0.0738) Oct (0.1560) 2.75 (0.1460) 3.34 (0.1164) 3.72 (0.1014) Nov Dec Jan (0.2072) 2.78 (0.1601) 3.36 (0.1362) 3.83 (0.0924) Feb Mar (0.90) 2.66 (0.79) 3.21 (0.1034) 3.68 (0.0823) Apr May Jun (0.1658) 2.37 (0.1596) 2.94 (0.76) 3.43 (0.1160) Jul Aug Sep (0.0025) Oct (0.2311) 2.84 (0.1792) 3.4 (0.1500) 3.96 (0.1101) Nov Dec Monthly Yield Volatities Annual Yield Volatities
16 16 Exhibit 3 Price, ytm and Modified Duration for 5 year 4% coupon bond time beg. time end Forward rate time (years) as of the end of Spot rate Bond PV* PV** Cash Flow Cash Flow CC ytm Duration Dec03 Mar % 0.25 Mar % Mar04 Jun % 0.50 Jun04 1.% Jun04 Sep % 0.75 Sep % Sep04 Dec % 1.00 Dec % Dec04 Mar % 1.25 Mar % Mar05 Jun % 1.50 Jun % Jun05 Sep % 1.75 Sep % Sep05 Dec % 2.00 Dec % Dec05 Mar % 2.25 Mar % Mar06 Jun % 2.50 Jun % Jun06 Sep % 2.75 Sep % Sep06 Dec % 3.00 Dec % Dec06 Mar % 3.25 Mar % Mar07 Jun07 4.% 3.50 Jun % Jun07 Sep % 3.75 Sep % Sep07 Dec % 4.00 Dec % Dec07 Mar % 4.25 Mar % Mar08 Jun % 4.50 Jun % Jun08 Sep % 4.75 Sep % Sep08 Dec % 5.00 Dec % *This present value (PV) is based on continuous compounding (CC). **This present value is based on a semiannual ytm. Bond Price YTM 3.228% Modified Duration (Yrs) 4.45
17 17 time beg. time end Exhibit 4 Forward Price, Forward ytm and Forward Modified Duration for 5 year 4% coupon bond Bond PV PV Forward rate time (years) as of the end of Spot rate Cash Flow Dec03 Mar % 0.25 Mar % Mar04 Jun % 0.50 Jun04 1.% 2 Jun04 Sep % 0.75 Sep % Sep04 Dec % 1.00 Dec % 2 Dec04 Mar % 1.25 Mar % Mar05 Jun % 1.50 Jun % 2 Jun05 Sep % 1.75 Sep % Time forward spot Cash Flow CC ytm Duration Sep05 Dec % 2.00 Dec % % Dec05 Mar % 2.25 Mar % % Mar06 Jun % 2.50 Jun % % Jun06 Sep % 2.75 Sep % % Sep06 Dec % 3.00 Dec % % Dec06 Mar % 3.25 Mar % % Mar07 Jun07 4.% 3.50 Jun % % Jun07 Sep % 3.75 Sep % % Sep07 Dec % 4.00 Dec % % Dec07 Mar % 4.25 Mar % % Mar08 Jun % 4.50 Jun % % Jun08 Sep % 4.75 Sep % % Sep08 Dec % 5.00 Dec % % Forward Bond Price Forward YTM 4.13% Forward Modified Duration (Yrs) 2.99
18 18 Exhibit 5. Eurodollar Futures Maturity Futures rate Mar % Jun % Sep % Dec % Mar % Jun % Sep % Dec % Mar % Jun % Sep % Dec % Mar % Jun % Sep % Dec % Mar08 5.% Jun % Sep % Dec %
19 19 Exhibit 6. 3month Eurodollar Futures Rates Months Maturity date of Eurodollar Futures contract (mon/year) Forward /03 09/03 06/03 03/03 /02 09/02 06/02 03/02 /01 09/01 06/01 03/01 /00 09/00 06/00 03/00 /99 09/99 06/99 03/ Spot
20 20 Forward End of month Exhibit 7. Eurodollar Futures Rates and LIBOR Spot Rates (Quarterly compounded) Spot Rates From To Forward rate Time (months) time (years) as of the end of Spot rate Mar % Mar04 Jun % Jun % Jun04 Sep % Sep % Sep04 Dec % 1.00 Dec % Dec04 Mar % Mar % Mar05 Jun % Jun % Jun05 Sep % Sep % Sep05 Dec % Dec % Dec05 Mar % Mar % Mar06 Jun % Jun % Jun06 Sep % Sep % Sep06 Dec % Dec % Dec06 Mar % Mar % Mar07 Jun % Jun % Jun07 Sep % Sep % Sep07 Dec % Dec % Dec07 Mar % Mar % Mar08 Jun08 5.% Jun % Jun08 Sep % Sep % Sep08 Dec % Dec %
21 21 Exhibit 8. Swap Valuation Fixed Rate 2.328% time beg. as of the end of PV Hedged LIBOR ('000's) Fixed Payments ('000's) PV Fixed Payment ('000's) time end Future rate time (Q) Spot rate Hedged LIBOR payments ('000's) Mar04 1 Mar % $ 2.55 $ 2.54 $ 5.82 $ 5.81 Mar04 Jun % 2 Jun % 3.15 $ 3.13 $ 5.82 $ 5.79 Jun04 Sep % 3 Sep % Sep04 Dec % 4 Dec % Dec04 Mar % 5 Mar % Mar05 Jun % 6 Jun % Jun05 Sep % 7 Sep % Sep05 Dec % 8 Dec % Dec05 Mar % 9 Mar % Mar06 Jun % 10 Jun % Jun06 Sep % 11 Sep % Sep06 Dec % Dec % Dec06 Mar % 13 Mar % Mar07 Jun % 14 Jun % Jun07 Sep % 15 Sep % Sep07 Dec % 16 Dec % Dec07 Mar % 17 Mar % Mar08 Jun08 5.% 18 Jun % Jun08 Sep % 19 Sep % Sep08 Dec % 20 Dec % Total Present Value as of Dec. 31, 2003 ('000's) $ $ 67.74
22 22 Exhibit 9. Forward Swap Rate Forward Swap rate 3.284% Hedged LIBOR payments ('000's) Forward Spot Rate discount to /2004 PV Hedged LIBOR ('000's) Fixed Payments ('000's) PV Fixed Payment ('000's) time beg. time end Future rate time (Q) as of the end of Spot rate Mar04 1 Mar % Mar04 Jun % 2 Jun % Jun04 Sep % 3 Sep % Sep04 Dec % 4 Dec % Dec04 Mar % 5 Mar % % Mar05 Jun % 6 Jun % % Jun05 Sep % 7 Sep % % Sep05 Dec % 8 Dec % % Dec05 Mar % 9 Mar % % Mar06 Jun % 10 Jun % % Jun06 Sep % 11 Sep % % Sep06 Dec % Dec % % Dec06 Mar % 13 Mar % % Mar07 Jun % 14 Jun % % Jun07 Sep % 15 Sep % % Sep07 Dec % 16 Dec % % Dec07 Mar % 17 Mar % Mar08 Jun08 5.% 18 Jun % Jun08 Sep % 19 Sep % Sep08 Dec % 20 Dec % Total Present Value as of Dec. 31, 2004 ('000's) $ $ 94.09
23 23 Exhibit 10 Valuation of Swaption Time period Forward Rates for time period Exhibit 1 Swap Payment Date Delay Discount Factor for Delay* Basic Option Value Option Value with Delay Dec 04  Mar % Mar05 Dec. 04Mar $1, $ 1, Mar 05  Jun % Jun05 Dec. 04Jun , , Jun 05  Sep % Sep05 Dec. 04Sep , , Sep 05  Dec % Dec05 Dec. 04Dec , , Dec 05  Mar % Mar06 Dec. 04Mar , , Mar 06  Jun % Jun06 Dec. 04Jun , , Jun 06  Sep % Sep06 Dec. 04Sep , , Sep 06  Dec % Dec06 Dec. 04Dec , , Dec 06  Mar % Mar07 Dec. 04Mar , , Mar 07  Jun 07 4.% Jun07 Dec. 04Jun , , Jun 07  Sep % Sep07 Dec. 04Sep , , Sep 07  Dec % Dec07 Dec. 04Dec , , Swaption Value $ 16, * DF * DF Dec04 Jun05 Dec04 Sep05 e e e e e
24 24 Figure1. Annualized Volatility 3month Eurodollar Futures Rates volatility Months prior
25 25 Figure 2. Cap Volatilities volatility 0.2 volatility Spot Flat months forward
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