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Derivatives & Risk Management

Derivatives & Risk Management. Lecture 4: a) Swaps b) Options: properties and non-parametric bounds. Nature of Swaps. A swap is an agreement to exchange cash flows at specified future times according to certain specified rules. Example: a “Plain Vanilla” Interest Rate Swap.

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Derivatives & Risk Management

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

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

  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. ---------Millions of Dollars--------- LIBOR FLOATING FIXED Net Date Rate Cash Flow Cash Flow Cash Flow Mar.1, 1998 4.2% Sept. 1, 1998 4.8% +2.10 –2.50 –0.40 Mar.1, 1999 5.3% +2.40 –2.50 –0.10 Sept. 1, 1999 5.5% +2.65 –2.50 +0.15 Mar.1, 2000 5.6% +2.75 –2.50 +0.25 Sept. 1, 2000 5.9% +2.80 –2.50 +0.30 Mar.1, 2001 6.4% +2.95 –2.50 +0.45 Cash Flows to Company B

  5. 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 Typical Uses for an Interest Rate Swap

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

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

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

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

  10. Fixed Floating Company A 10.00% 6-month LIBOR + 0.30% Company B 11.20% 6-month LIBOR + 1.00% The Comparative Advantage Argument • Company A wants to borrow floating • Company B wants to borrow fixed

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

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

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

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

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

  20. Example III

  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, RK is exchanged for interest at the market rate

  22. Forward Rate Agreementcontinued (Page 100) • Capital R is the rate measured with compounding rate reflecting maturity, i.e. if the T2 – T1 is three months the rate is compounded quarterly etc. • The agreed cash flows are: • T1: - L • T2: L [1+ Rk (T2-T1)]

  23. FRA • Note if Rf = Rk the FRA is worth 0. Why? • To value the FRA, we can compare now two payments at time T2: • One that pays Rk and one that pays Rf • Note: we are not assuming anything more than no arbitrage

  24. Valuing future cash flows Hypothetical cash flow Cash settlement

  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. Swaps as FRA’s • Suppose an interest rate swap promises fixed rate payments C and receives floating payments P1flat 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. Swaps as FRA’s II

  28. Swaps as FRA’s III • The floating rate payment is computed based on the prevailing spot rate at T1 • Consider the second exchange of payments (the first is known)

  29. Swaps as FRA’s IV

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

  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. Swaps as FRA’s VI • We know that at T2, the floating rate payment will be worth • And thus T1, it must be worth

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

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

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

  39. 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 Typical Uses of a Currency Swap

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

  41. Valuationof 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. 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. 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

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