# Fixed-Income Securities. Assignment

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2 (e) Deliverables and cheapest-to-deliver. (f) Conversion factors: formula and explanation. (g) Cash/futures price equivalents: cash-and-carry (C&C) arbitrage. (h) Duration hedging with futures. 4. Up-Front Swap Value. As the risk manager of a commercial bank you need to neutralize certain interest rate exposures. Up-front fees to or from the counterparty are determined in terms of fair values of swaps. (a) Suppose you wish to swap your semi-annual fixed 6.50% 5-year liability into a floating PRIME + 15 bpts liability. Given the following discount yield curve calculate the fair value of the swap. Are you getting paid or do you need to compensate the counterparty? Payment dates (years) Discount yields r t (%) (b) Suppose you wish to swap your annual floating LIBOR+80 bpts liability into a fixed 6.25% 5 year liability. Given the following discount yield curve calculate the fair value of the swap. Are you getting paid or do you need to compensate the counterparty? Payment dates (years) Discount yields r t (%) (c) What are two different ways to conceptualize swaps? 5. Par Swaps and Swap Rates. In class, we determined the value of a swap by calculating an up-front fee that would equate the present expected value of the floating cashflows to the present expected value of the fixed ones. However, there is an alternative definition of swaps that solves for the fixed rate in terms of the other variables rather than the fee from floating and fixed rates. Here, the fixed rate is set in such a way so as to equate the floating with the fixed leg in expected present value terms. This means that one has to solve for c = c t = fixed in c t (1 + r t ) t = f t 1,t (1 + r t ) t c : c f t 1,t (1 + r t ) t = 0. (1) In swap terminology, c is the swap rate: it is the fixed rate that has the same present value as the forward (reset) rates. Swaps where the fixed rate c is set so that the expected present value of floating and fixed payments are equalized are called par swaps. Using basic annuity mathematics you should derive an analytic solution to the swap rate in (1); otherwise, use an appropriate spreadsheet function. (a) On the basis of the following information, determine the swap rate on a five year floatingfor-fixed swap (i.e., receiving fixed, paying floating Eurodollar deposit); the floating payments are determined from the zero-libor curve: 2

3 Payment Dates T-Bill Prices Euro-USD Deposit (years) P (0, T ) L (0, T ) (b) A commercial bank has sold a 6 year T-bill-for-Eurodollar (LIBOR) swap precisely one year ago with a swap rate of 6.5%; as a consequence, the bank receives fixed and pays floating. Being in charge of risk-management at this bank you wonder what the value of the swap (notional principal: USD 20m) is given the information in the above table. (c) Your counterpart phoned you about the preceding 6.5%, 5 years remaining maturity swap and proposes to reduce the notional principal by USD 500,000. Before giving a final answer, you would like to know what the new value of the swap will be. 6. Plain-Vanilla Swap. Consider a 3 year plain-vanilla fixed-for-floating swap with annual resetting where the variable rate is equal to the annualized future one-year spot rate. Annual payments are netted and the notional amount is USD 10m. A receives floating, B receives fixed. (a) Swap rate: given the following strip prices, calculate swap rate A should pay B: Maturity T (years) Strip prices P (0, T ) (b) Swap duration: given the preceding strip information, what are the dollar durations ( \$ ) of the swap from A s and B s perspective, respectively? (c) Future swap value: calculate the fair value of the swap at t = 1 from A s perspective given that A s yield curve model predicts the following strip prices in one year s time: Maturity T (years) Strip prices P (1, T ) Swap Hedging. You are a swap dealer at a major investment bank. Recently, you have been less than successful in finding offsetting deals to unload the risks that you warehoused in buying and selling swaps (in terms of paying or receiving fixed, what does it mean to buy or sell a swap?). You are currently evaluating alternatives to deal with this problem. (a) Spot market: you sold a 7 year 6M-LIBOR swap. How would you hedge the resulting exposure in the Treasury and Eurodollar spot markets? Explain. (b) Futures market: you bought a 5 year 1Y-LIBOR swap. How would you hedge the resulting exposure in the Treasury and Eurodollar futures markets? Explain. 3

4 8. Foreign Currency Swap Value. A further variant of the generic swap is the FX swap. Here, two parties borrow in different currencies and then agree to swap both principal and future loan payments. There are four basic variants that arise from the combination of fixedfor-floating and the two currencies. Take a plain vanilla currency swap between parties A and B and currencies GBP and USD. A borrowed in Sterling but needs dollars, B borrowed in dollar but needs Sterling. Hence, the two parties agree to swap principals and liabilities at t = 0: 7.25% (GBP) GBP 50m USD 80.5m A B r USD % (USD) swap rate (USD) 7.25% (GBP) At maturity, the principals are returned to the respective creditors: USD 80.5m A GBP 50m B An FX swap s valuation is straightforward: just use the up-front fee method. Incidentally, FX swaps preceded and inspired interest rate swaps, which explains why the latter were initially valued by the up-front fee approach. The required data is obvious: the spot FX rate, the respective loans specifications and the two yield curves. Let S GBP/USD (0) be the current price of 1 USD in terms of GBP and V F X (0) the present value of the loan in F X using the appropriate F X yield curve. Then the value of the preceding FX swap is V GBP/USD (0) = V GBP (0) S GBP/USD (0) V USD (0) (2) where the GBP party receives or pays an appropriate fee. In reality, principals are adjusted so that V GBP/USD (0) = 0 at the inception of the swap or, once again, an FX swap rate is computed. Also notice that the maturities of the two loans are assumed to coincide but that the loans could carry either a fixed or a variable rate. (a) On the basis of the relation (2) and in complete analogy with the plain-vanilla interest rate swap, define the swap rate for USD payments. Derive an analytic expression for the USD swap rate and explain what this rate represents. (b) Suppose that swap payments are made semi-annually, the maturity of the GBP/USD swap is 4 years and that the current USD/GBP spot exchange rate is USD /GBP (careful: which spot rate do you need and how do you get it?). Determine the USD swap rate from the following information: Payment Dates T-Bill Prices Sterling Gilt-Strip Prices (years) P USD (0, T ) P GBP (0, T )

5 9. Inverse Floaters and Swaps. The relationship between floaters and swaps is quite obvious: a floater is one of the two underlying instruments of a swap. Hence, it should be quite intuitive that a similar relationship exists between inverse floaters and swaps. Recall that inverse floaters make variable coupon payments N (k r t 1,t ) at t for some fixed rate k. Suppose that P (0, 3) = and that the k of a 3Y LIBOR inverse floater with annual resetting is set at k = %. Also, suppose that a newly negotiated 3Y plain-vanilla 1Y LIBOR swap pays a fixed par swap rate of c = %. What is the value of the inverse floater at t = 0? You do not need to use a replication argument that synthetically creates the inverse floater from the swap and the 3Y zero similar to our usual replication equations: a simple cashflow diagram should be sufficient. 5

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