FIN 472
Fixed-Income Securities
Professor Robert B.H. Hauswald Kogod School of Business, AU
Assignment
Please be reminded that you are expected to use contemporary computer software to solve the following problems.
1. Forward Rate Curve. You are given the following prices for T-strips Pt where t refers to
the remaining maturity in years
P1 P2 P3 P4 P5
90 78 63 51 40
(a) Extract discount factors and a discount yield curve from this data. (b) Calculate and plot the one year forward rate curve, i.e., f1,t+1, t ≥ 1.
(c) Under what assumption is the forward rate curve a predictor for the future spot rate curve?
2. Forward Contracts. Using the previous problem’s data, i.e., the T-strip prices Pt, t =
1, . . . , 5 consider the following questions:
(a) Loan commitment. You are a loan officer in a local bank. What rate would you quote a customer who wishes to lock in the interest rate for a loan commencing in three years and ending four years from to day?
(b) Forward deposit. Your company concluded an agreement to deliver an assembly line whose final payment of USD 100m (upon completion) will be received at the end of year 3. You would like to invest this payment for two years. What kind of interest rate can you lock-in by engaging in a forward transaction?
(c) FRA. In order to offset a future variable interest rate liability you would like to buy a 24/48 6.50% FRA. How much would you offer for this security? Explain.
(d) FRN. You are offered a 5 year floater paying Treasury + 35; how much are you willing to pay for this security? Explain the mechanics and hypotheses underlying your calculation. (e) Fixed-for-floating swap. You wish to buy a five year 7.00% for Treasury + 50 bpts interest rate swap (careful: buying a swap means that you wish to pay what and receive what?). What price would you consider fair?
3. Bond Futures Contract. Use resources on the internet (where are the instruments traded? 2 American, 1 European exchange!) to answer the following questions. For each answer indicate the URL (internet address starting with http://) where you found the relevant in-formation.
(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) 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Discount yields rt (%) 4.50 5.00 5.25 5.75 6.25 6.50 6.75 7.00 7.25 7.75
(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) 1.0 2.0 3.0 4.0 5.0 Discount yields rt (%) 8.00 7.50 6.50 6.25 6.00
(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 = ct= fixed
in T X t=1 ct (1 + rt)t = T X t=1 ft−1,t (1 + rt)t ⇒ c : T X t=1 c − ft−1,t (1 + rt)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.
Payment Dates T-Bill Prices Euro-USD Deposit (years) P (0, T ) L (0, T ) 0.5 97.72 0.9748 1.0 95.39 0.9492 1.5 93.02 0.9227 2.0 90.63 0.8960 2.5 88.21 0.8687 3.0 85.78 0.8413 3.5 83.35 0.8136 4.0 80.93 0.7857 4.5 78.51 0.7577 5.0 76.11 0.7302
(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) 1.0 2.0 3.0 4.0
Strip prices P (0, T ) 90.00 80.00 70.00 60.00
(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) 1.0 2.0 3.0 4.0 Strip prices P (1, T ) 95.00 90.00 85.00 80.00
7. 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.
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 fixed-for-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:
GBP 50m −−−−−−−→
Counterparty USD 80.5m←−−−−−−−− Counterparty 7.25% (GBP)
←−−−−−−−−− A B r−−−−−−−−−−→U SD% (USD)
swap rate (USD) −−−−−−−−−−−−→
7.25% (GBP) ←−−−−−−−−−
At maturity, the principals are returned to the respective creditors: Counterparty USD 80.5m−−−−−−−−→ Counterparty
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 SGBP /U SD(0) be the current
price of 1 USD in terms of GBP and VF 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
VGBP /U SD(0) = VGBP(0) − SGBP /U SD(0) VU SD(0) (2)
where the GBP party receives or pays an appropriate fee. In reality, principals are adjusted so that VGBP /U SD(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 1.6095/GBP (careful: which spot rate do you need and how do you get it?). Determine the USD swap rate from the following information:
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 − rt−1,t) at t for some fixed rate k.
