FIN 684 Professor Robert B.H. Hauswald Fixed-Income Analysis Kogod School of Business, AU Solutions 5 1. Forward Rate Curve. (a) Discount factors and discount yield curve: in fact, P t = 100 1 = 100 = 100d (1+y t) t (1+y t) t t for d t = 1 so that the discount factors are 0.9, 0.78, etc.; alternatively, one could have (1+y t) t use continuous discounting P t = 100e ytt with d t = e ytt. From this the discount yield curve is easily extracted as (in %) y 1 y y 3 y 4 y 5 11.11 13. 16.65 18.33 17.61 (b) he 1 year forward curve is given by f 1,t+1, t 1; from the fundamental forward rate equation (1 + f 1,t+1 ) t = (1+y t+1) t+1 (1+y 1 ) = d 1 d t+1 so that f 1,t+1 = (1+yt+1 ) t+1 1 1 t (1+y 1 ) 1 = d1 t d t+1 1, t 1 and f 1, f 1,3 f 1,4 f 1,5 15.37 19.5 0.83 16.409 (c) he forward rate curve is a predictor for the future yield curve under the pure expectations hypothesis that postulates that current forward rates are unbiased predictors for future spot rates.. Forward Contracts. (a) Loan commitment: f 3,4 = d 3 d 4 1 = 3.59%. 1 (b) Forward deposit: f 3,5 = d3 d 5 1 = 5.499%. (c) FRA: buying an FRA means receiving LIBOR and paying fixed. Assuming that the preceding yield curve represents LIBOR discount yields one needs to calculate l 4,48 f,4 = 1 d d 4 1 = 3.66%. In year s time this is worth (l 4,48 0.065) 70 that the fair price becomes F RA 0 = d (l 4,48 0.065) 70 (1 + l 4,48 ) 100 = (1+l 4,48 ) 100 = 1 (l 4,48 0.065) 70 (1 + y ) (1 + l 4,48 ) 100 (3.66 6.5) 70 (1.366) so = d 4 (l 4,48 0.065) 70 70 100 = 0.51 (0.366 6.5) 100 = 17.503 by the fundamental forward rate equation.
(d) FRN: let us now assume that the given yield curve represents -strips. Now, use f t 1,t = 1, t 1 for the reset rates and add the spread: d t 1 d t f 0,1 + s f 1, + s f,3 + s f 3,4 + s f 4,5 + s 11.46 15.73 4.16 3.88 7.8 5 One then has F RN 0 = 5 f t 1,t +s t=0 + 100 = 5 (1+y t) t (1+y t) 5 t=0 d t (f t 1,t + 0.35) + d 5 100 = 101.158. Notice that, once, again we implicitly appealed to the PEH to justify the pricing of the floater. (e) Buying a swap means paying fixed (and receiving floating). Under the PEH the reset rates are calculated as implied forward rates; in fact, since a swap is equivalent to a short and long position in a bond and a floater we can use the mechanics of the preceding two calculations to price the swap by the old-fashioned up-front fee method (for the swap rate method used nowadays see Assignement 3): c t (1 + y t ) t = he reset rates + spread are f t 1,t + s (1 + y t ) t + γ γ = d t c (f t 1,t + s). f 0,1 + s f 1, + s f,3 + s f 3,4 + s f 4,5 + s 11.61 15.89 4.30 3.0 so that γ = 9.47, i.e, the buyer will have to pay $9.47 per $100 par. 3. Bond Futures Contract. he easiest way to answer this question is to go to the CBO s website. 4. Up-Front Swap Fees. Up-front fees (all-in fees) are the old-fashioned way to price FX or interest rate swaps. (a) Semi-annual fixed 6.50% 5 year for floating PRIME + 15: the fair value ( up-front fee ) of the swap is calculated as V (0) = V c V F where V c is the present value of the fixed and V F the present value of the floating leg; here, V (0) = 7.7477497 3.1179318 = 4.84315704 per $100 face value so that the party paying floating receives $4.8431. (b) Annual floating LIBOR+80 bpts for fixed 6.5%: proceeding as before one finds that V (0) = V c V F =.6511 per $100 face value so that the party paying fixed receives $.6511. Notice that this exercise demonstrates a general payment rule: if the yield curve is sloping upward (and the spread over the index not too large) fixed pays floating; is the yield curve sloping downward, floating pays fixed! (c) In order to more sharply price swaps one one need to know future reset rates. However, at inception swaps are completely determined by the forward curve via an arbitrage argument which could be used to hedge (undo) the transaction. c f 5. Par Swaps and Swap Rates. It is quite straightforward to solve c : t 1,t = 0 (1+r t) t for the swap rate that equates the return of the floating and the fixed leg where f t 1,t is the floating rate gross of the spread over the relevant index: 1 1 1 f t 1,t c = (1 + r t ) t (1 + r t ) t = d t d t f t 1,t (1)
1 Since P (0, t) = = d (1+r t) t t is the price of a zero-coupon bond or the discount factor one can see how swaps are a portfolio of zero bonds and interest rate forwards. From the above formula one has that the fixed leg s present value is V c (0) = Nc P (0, t) () for notional principal N. his expression is then equated to the present value of the floating leg calculated either from our usual forward rate argument or from the following observation on floating rate notes. A floater is nothing but a series of one period deposits that is rolled over until maturity with the interest being paid out instead of reinvested. Since its interest rate is set in advance at all the reset dates the fictitious one period deposit is valued at par at the beginning of each period: setting the return (coupon rate) on a deposit (bond) means that the instrument has to trade at par just after its inception for this period. ake a bond where Y M = c, the coupon rate: it has to sell at par, otherwise there are arbitrage opportunities. his simple observation yields a surprising but powerful method for pricing floaters and a swap s floating leg where the reset rates do not even show up. Although the explicit absence of reset rates in the calculation is initially quite mistifying it simply reflects both the mathematics of floaters and implied forward rates, and the arbitrage logic underlying our discussions. Since a floater s coupon is always (re)set at par by definition the preceding discussion implies that it has to be priced at par in its last period. Given that it is priced at par in its last period it has to be valued at par the period before, too, and so on where in our previous method the following periods reset rates are estimated as implied forward rates. herefore, the floater is priced at par conditional on knowing the reset rates! As it turns out the lack of foresight about rest rates is surprisingly not really a constraint: we can cut short our usual swap table procedure and directly to price the floating rate part by the following trick based on the par valuation observation. o see this consider reset date t = 1: the reset rate is known as r so that the floater s value at maturity is 100 (1 + r ); discounting this expression back to t = 1 at the appropriate discount rate r yields F RN 1 = 100. Now, repeat the argument for t = where one gets 100 (1 + r 1 ) which, similarly, needs to be discounted at the known rate r 1 to give F RN = 100, etc. down to F RN 0 = 100. Recall that a plain-vanilla swap is an appropriate combination of a short and long position in a floater and a coupon bond. So, view the swap s floating leg in terms of a floater that is priced at par since the variable coupon rate is determined at the beginning of each reset period. As a result, its value V F (0) is the notional principal N: V F (0) = N! However, contrary to a floater the interest rate swap s principal does not change hands. Hence, we need to subtract from V F (0) the notional principal s present value to get the value of the floating leg as the net present value of the notional floater. Let be the swaps maturity and L (0, ) the appropriate floating (e.g., LIBOR) discount factor derived from the floater s underlying discount yield curve: V F (0) = N L (0, ) N = N (1 L (0, )). (3) Put differently, a bond trades at par N for a set interest rate as is the case for a floater; however, since there is no principal payment in a swap one needs to deduct the present value of the principal in the form of L (0, ) N from par to get the floating leg s present value. Setting V c (0) = V F (0) then yields from the preceding two equations the swap rate 3
1 as c = P (0, t) (1 L (0, )) while the swap s value (up-front fee) is given as γ0 = V c (0) V F (0) = N P (0, t) (1 L (0, )). (a) From the above formula one has that the fixed leg s present value is V c (0) = N c 10 P (0, t) = N c 8.6965 for notional principal N. o get the present value of the floating leg you can either use our usual forward rate argument or use the preceding observations on floating rate notes and bonds selling at par. he value of the floating leg is now computed as the net present value of the notional floater where L (0, 5) is the appropriate LIBOR discount factor: V F (0) = N L (0, 5) N = N (1 0.730) = N (0.698). Setting V c (0) = V F (0) and cancelling N (or setting N = 1) yields the swap rate as c 8.6965 = 0.698 c = 6.048%. (b) Using the above reasoning and setting N = 0m yields the present value of the fixed rate payments as V c (1) = 0.065 N 10 s=1 P (1, s) = N ( ) 0.065 8.6965 = N (0.86) while the present value of the floating leg is, once again, V F (0) = N (0.698) so that V c (1) V F (1) = 0, 000, 000 (0.86 0.698) = $56, 000. Notice what a painless way to calculate swap rates and swap values the fictitious floater argument opens up! (c) From the preceding we now find that it has to be V c (1) V F (1) = 19, 500, 000 (0.86 0.698) = $49, 600. 6. Plain-Vanilla Swap. Once again, we appeal to the notional-floater-valued-at-par argument. (a) Swap rate: V c (0) = cn.4 = (1 0.7) N = V F (0) so that c = 1.50%. (b) he calculation of a swap s duration is quite straightforward since a swap is a portfolio of a bond and a floater position. But a portfolio s duration is the weighted average of its components duration. So, the only complication is to figure out a floater s duration. However, this also follows from the par-valuation argument: at reset dates, the floater just behaves like a discount security so that its duration equals the reset period. From this we have that from A s perspective A = F c = 1.675 = 1.675 while from B s perspective the duration is B = A = ( F c ) = 1.675. (c) As before, the swap s predicted value at t = 1 is V c (1) V F (1) = (1 0.9) (0.95 + 0.9) 0.15 N = 0.1315N. 7. Swap Hedging. Just use the observation that a swap is a combination of long and short positions in floaters and bonds. (a) Having sold a 7 year 6M-LIBOR swap means (from the investment bank s point of view) that they receive fixed and pay floating. his is equivalent to a long position in a bond a short one in a floater. o hedge the resulting exposure means to reverse the positions so that one would have to sell/ issue a 7 year -note and buy a 7 year floater indexed to 6M-LIBOR. 4
(b) Once again, one needs to reverse the resulting positions. Having bought a swap means paying fixed and receiving floating for the swap dealer which is equivalent to having issued a bond (short) and invested in a floater (long). So, a dealer might cover the position by buying an appropriate amount of 5 year -note futures contracts (not necessarily the swap s face value: remember basis risk and hedge ratio for futures?). he only complication here arises on the LIBOR side where one would need to replicate the short position in the floater by a collection of short positions in Eurodollar futures contracts with maturities of = 1,..., 5. However, the underlying contract is only a 3 months deposit so that one needs to string several of them together to get the yearly LIBOR deposit contracts. Unfortunately, this does not fully eliminate interest rate risk. one might incur timing mismatches and the market is quite illiquidity for delivery dates beyond 3 years. 8. Foreign Currency Swap. Here, we proceed in complete analogy with equations () and (3) in order to calculate the value of the swap s legs which, in turn, yield the swap rate or its value. he only difficulty is that the swap fee or rate computation proceeds from the British point of view in the formula given: either you turn everything around and look at it from an American point of view using the USD/GBP exchange rate S USD/GBP (0) = 1.6095 (this is, incidentally, how Sterling is quoted) or you use the fact that S GBP/USD (0) = S USD/GBP (0) 1, a well known no-arbitrage condition in FX calculus. (a) he USD swap rate is the rate on a fictitious USD loan that will equalize the present value of the two swap legs including exchange of principal. he Sterling leg present value in USD is S $/ (0) V GBP (0) = S $/ (0) N c P (0, t) + P (0, ) while the USD leg present value comes to V USD (0) = N c$ $ P $ (0, t) + P $ (0, ) so that from V USD (0) = S $/ (0) V GBP (0) the USD swap rate c $ follows as 1 { c $ = P $ (0, t) S $/ (0) N N $ (b) From the preceding one has c c V GBP (0) = N P (0, t) + P (0, ) c V USD (0) = N $ $ P $ (0, t) + P $ (0, ) } P (0, t) + P (0, ) P $ (0, ). = 50 0.075 5 c$ = 80.5 7.1503 + 0.8093 6.8697 + 0.7504 = 49.9713 so that V USD (0) = S $/ (0) V GBP (0) yields c $ from 80.5 c $ 7.1503 + 0.8093 = 1.6095 49.9713 as c $ = 5.31%. 9. Inverse Floaters and Swaps. Remembering that the value of a swap is calculated from 10 the difference of the fixed and floating legs one has net cashflows of N (c r t 1,t ) for swap rate c each period. Similarly, an inverse floater pays N (k r t 1,t ) each t over its lifetime. For c = 3.9465% = k the only difference between the two instrument s (net) cashflows is the principal payment of the inverse floater at = 3. Hence the replicating portfolio for the inverse floater is comprised of the swap and a 3Y zero: hence, the inverse floater s value is 5
P (0, 3) = 89.00 since the (par) swap value is set such that the swap has 0 value at inception. o summarize: the two instruments are valued at t = 0 as V IF (0) = N V S (0) = N d t (k r t 1,t ) + d 100 d t (c r t 1,t ) = 0; for c = k one has V S (0) = N d t (k r t 1,t ) = V IF (0) d 100 so that V IF (0) = V S (0) + d 100 = 0 + 89.00 = 89.00. 6