1. 1 Derivatives & Risk Management Lecture 4: a) Swaps b) Options: properties and non- parametric bounds

2. 2 Nature of Swaps A swap is an agreement to exchange cash flows at specified future times according to certain specified rules

3. 3 Example: a “Plain Vanilla” Interest Rate Swap An agreement by “Company B” to receive 6-month LIBOR & pay a fixed rate of 5% per annum every 6 months for 3 years on a notional principal of $100 million

4. 4 ---------Millions of Dollars--------- LIBORFLOATINGFIXEDNet DateRateCash Flow Mar.1, 19984.2% Sept. 1, 19984.8%+2.10–2.50–0.40 Mar.1, 19995.3%+2.40–2.50–0.10 Sept. 1, 19995.5%+2.65–2.50+0.15 Mar.1, 20005.6%+2.75–2.50+0.25 Sept. 1, 20005.9%+2.80–2.50+0.30 Mar.1, 20016.4%+2.95–2.50 +0.45 Cash Flows to Company B

5. 5 Typical Uses for an Interest Rate Swap Converting a liability from –fixed rate to floating rate –floating rate to fixed rate Converting an investment from –fixed rate to floating rate –floating rate to fixed rate

6. 6 A and B Transform a Liability A B LIBOR 5% LIBOR+0.8% 5.2%

7. 7 Financial Institution is Involved A F.I. B LIBOR LIBOR+0.8% 4.985% 5.015% 5.2%

8. 8 A and B Transform an Asset A B LIBOR 5% LIBOR-0.25% 4.7%

9. 9 Financial Institution is Involved A F.I. B LIBOR 4.7% 5.015%4.985% LIBOR-0.25%

10. 10 The Comparative Advantage Argument Company A wants to borrow floating Company B wants to borrow fixed FixedFloating Company A10.00%6-month LIBOR + 0.30% Company B11.20%6-month LIBOR + 1.00%

11. 11 The Swap A B LIBOR LIBOR+1% 9.95% 10%

12. 12 The Swap when a Financial Institution is Involved A F.I. B 10% LIBOR LIBOR+1% 9.93% 9.97%

13. 13 Criticism of the Comparative Advantage Argument The 10.0% and 11.2% rates available to A and B in fixed rate markets are 5-year The LIBOR+0.3% and LIBOR+1% rates available in the floating rate market are six-month rates B’s fixed rate depends on the spread above LIBOR it borrows at in the future i.e. it is fixed only as long as its creditworthiness stays the same

14. 14 Alternatives Information asymmetry Flexible and liquid instruments Tax and regulatory arbitrage

15. 15 Valuation of an Interest Rate Swap Interest rate swaps can be valued as the difference between the value of a fixed-rate bond & the value of a floating-rate bond

16. 16 Valuation in Terms of Bonds The fixed rate bond is valued in the usual way The floating rate bond is valued by noting that it is worth par immediately after the next payment date

17. 17 Valuation as bonds K* is the floating rate know from at the beginning of the period

18. 18 Example A financial institution pays 6 month LIBOR and receives 8% (semi-annually) on $100 million notional principal. The FI has sold a floater and bought a fixed rate bond remaining life 1.25 years market rates for 3, 9, 15 months to go are 10%, 10.5% and 11%

19. 19 Example II The 6 month LIBOR when the swap was set up 3 months ago was 10.2%.

20. 20 Example III

21. 21 Forward Rate Agreement A forward rate agreement (FRA) is an agreement that a certain rate will apply to a certain principal during a certain future time period An FRA is equivalent to an agreement where interest at a predetermined rate, R K is exchanged for interest at the market rate

22. 22 Forward Rate Agreement continued (Page 100) Capital R is the rate measured with compounding rate reflecting maturity, i.e. if the T 2 – T 1 is three months the rate is compounded quarterly etc. The agreed cash flows are: T 1 : - L T 2 : L [1+ R k (T 2 -T 1 )]

23. 23 FRA Note if R f = R k the FRA is worth 0. Why? To value the FRA, we can compare now two payments at time T 2 : One that pays R k and one that pays R f Note: we are not assuming anything more than no arbitrage

24. 24 Valuing future cash flows Hypothetical cash flow Cash settlement

25. 25 Alternative Valuation of Interest Rate Swap: portfolio of FRA Swaps can be valued as a portfolio of forward rate agreements (FRAs) Each exchange of payments in an interest rate swap can be analyzed as an FRA The relevant interest rates are the fixed for one leg, and the forward associated with the period to be valued for the other leg

26. 26 Swaps as FRA’s Suppose an interest rate swap promises fixed rate payments C and receives floating payments P 1 fl at regular intervals We have seen that this could be valued as a portfolio of bonds What about valuating it as a package of FRA’s?

27. 27 Swaps as FRA’s II

28. 28 Swaps as FRA’s III The floating rate payment is computed based on the prevailing spot rate at T 1 Consider the second exchange of payments (the first is known)

29. 29 Swaps as FRA’s IV

30. 30 Swaps as FRA’s V So we want to compute the PV of Which can be written as

31. 31 Swaps V The value of the fixed part of this payment is obvious The value of the floating part less so because it involves a random interest rate

32. 32 Swaps as FRA’s VI We know that at T 2, the floating rate payment will be worth And thus T 1, it must be worth

33. 33 Swaps as FRA’s VII And thus at time t it must be worth Recall that by no arbitrage So that

34. 34 Swaps as FRA’s VIII Hence Changing compounding convention

35. 35 Swaps as FRA So to value a fixed for floating exchange, compute the present value of the exchange between the forward rate and the fixed rate

36. 36 An Example of a Currency Swap An agreement to pay 11% on a sterling principal of £10,000,000 & receive 8% on a US$ principal of $15,000,000 every year for 5 years

37. 37 Exchange of Principal In an interest rate swap the principal is not exchanged In a currency swap the principal is exchanged at the beginning &the end of the swap

38. 38 The Cash Flows Years DollarsPounds $ ------millions------ 0 –15.00 +10.00 1 +1.20 –1.10 2 +1.20 –1.10 3 +1.20 –1.10 4 +1.20 –1.10 5+16.20 -11.10 £

39. 39 Typical Uses of a Currency Swap Conversion from a liability in one currency to a liability in another currency Conversion from an investment in one currency to an investment in another currency

40. 40 Comparative Advantage Arguments for Currency Swaps Company A wants to borrow AUD Company B wants to borrow USD USDAUD Company A 5.0%12.6% Company B 7.0%13.0%

41. 41 Valuation of Currency Swaps Like interest rate swaps, currency swaps can be valued either as the difference between 2 bonds or as a portfolio of forward contracts

42. 42 Swaps & Forwards (continued) The value of the swap is the sum of the values of the forward contracts underlying the swap Swaps are normally “at the money” initially –This means that it costs NOTHING to enter into a swap –It does NOT mean that each forward contract underlying a swap is “at the money” initially

43. 43 Credit Risk A swap is worth zero to a company initially At a future time its value is liable to be either positive or negative The company has credit risk exposure only when its value is positive