Chapter 14

1. Chapter 14 Swap Pricing

2. Outline • Swap pricing • Solving for the swap price • Valuing an off-market swap • Hedging the swap • Pricing a currency swap

3. Swap Pricing • Swaps as a pair of bonds • Swaps as a series of forward contracts • Swaps as a pair of option contracts

4. Swaps as A Pair of Bonds • If you buy a bond, you receive interest • If you issue a bond you pay interest • In a plain vanilla swap, you do both • You pay a fixed rate • You receive a floating rate • Or vice versa

5. Swaps as A Pair of Bonds (cont’d) • A bond with a fixed rate of 7% will sell at a premium if this is above the current market rate • A bond with a fixed rate of 7% will sell at a discount if this is below the current market rate • Swap values are impacted in a similar manner to bond values

6. Swaps as A Pair of Bonds (cont’d) • If a firm is involved in a swap and pays a fixed rate of 7% at a time when it would otherwise have to pay a higher rate, the swap is saving the firm money • swap has value • unwinding the swap captures the value • If because of the swap you are obliged to pay more than the current rate, the swap is beneficial to the other party • cost to the firm to unwind the swap

7. Swaps as A Series of Forward Contracts • A forward contract is an agreement to exchange assets at a particular date in the future, without marking-to-market • An interest rate swap has known payment dates evenly spaced throughout the tenor of the swap

8. Swaps as A Series of Forward Contracts (cont’d) • A six month swap with a single payment date six months hence is no different than an ordinary six-month forward contract • At that date, the party owing the greater amount remits a difference check

9. Swaps as A Pair of Option Contracts • Assume a firm buys a cap and writes a floor, both with a 5% striking price • At the next payment date, the firm will • Receive a check if the benchmark rate is above 5% • Remit a check if the benchmark rate is below 5%

10. Swaps as A Pair of Option Contracts (cont’d) • The cash flows of the two options are identical to the cash flows associated with a 5% fixed rate swap • If the floating rate is above the fixed rate, the party paying the fixed rate receives a check • If the floating rate is below the fixed rate, the party paying the floating rate receives a check

11. Swaps as A Pair of Option Contracts (cont’d) • Cap-floor-swap parity Write floor Long swap Buy cap + = 5% 5% 5%

12. Solving for the Swap Price • The role of the forward curve for LIBOR • Implied forward rates • Initial condition pricing • Quoting the swap price • Counterparty risk implications

13. Swap Pricing • The swap price is determined by fundamental arbitrage arguments • All swap dealers are in close agreement on what this rate should be

14. The Role of the Forward Curve for LIBOR • LIBOR depends on when you want to begin a loan and how long it will last • Similar to forward rates: • A 3 x 6 Forward Rate Agreement (FRA) begins in three months and lasts three months (denoted by ) • A 6 x 12 FRA begins in six months and lasts six months (denoted by )

15. The Role of the Forward Curve for LIBOR (cont’d) • Assume the following LIBOR interest rates:

16. The Role of the Forward Curve for LIBOR (cont’d) LIBOR yield curve % 5.62 0 x 12 5.57 0 x 9 5.50 0 x 6 5.42 spot 0 3 6 9 Months

17. Implied Forward Rates • We can use these LIBOR rates to solve for the implied forward rates • The rate expected to prevail in three months, 3f6 • The rate expected to prevail in six months, 6f9 • The rate expected to prevail in nine months, 9f12 • The technique to obtain the implied forward rates is called bootstrapping

18. Implied Forward Rates (cont’d) • An investor can • Invest in six-month LIBOR and earn 5.50% • Invest in spot, three-month LIBOR at 5.42% and re-invest for another three months at maturity • If the market expects both choices to provide the same return, then we can solve for the implied forward rate on the 3 x 6 FRA

19. Implied Forward Rates (cont’d) • The following relationship is true if both alternatives are expected to provide the same return:

20. Implied Forward Rates (cont’d) • Using the available data:

21. Implied Forward Rates (cont’d) • Applying bootstrapping to obtain the other implied forward rates: • 6f9 = 5.71% • 9f12 = 5.75%

22. Implied Forward Rates (cont’d) LIBOR Implied forward rate curve % 5.77 9 x 12 5.71 6 x 9 5.58 3 x 6 5.42 spot 0 3 6 9 Months

23. Initial Condition Pricing • An at-the-market swap is one in which the swap price is set such that the present value of the floating rate side of the swap equals the present value of the fixed rate side (equilibrium) • The floating rate payments are uncertain • Use the implied forward rate curve (derived from the spot yield curve) as a proxy for the floating rate payments

24. Initial Condition Pricing (cont’d) At-the-Market Swap Example A one-year, quarterly payment swap exists based on actual days in the quarter and a 360-day year on both the fixed and floating sides. Days in the next 4 quarters are 91, 90, 92, and 92, respectively. The notional principal of the swap is $1. Convert the future values of the swap into present values by discounting at the appropriate zero coupon rate contained in the forward rate curve.

25. Initial Condition Pricing (cont’d) At-the-Market Swap Example (cont’d) First obtain the discount factors:

26. Initial Condition Pricing (cont’d) At-the-Market Swap Example (cont’d) First obtain the discount factors:

27. Initial Condition Pricing (cont’d) At-the-Market Swap Example (cont’d) Next, apply the discount factors to both the fixed and floating rate sides of the swap to solve for the swap fixed rate that will equate the two sides:

28. Initial Condition Pricing (cont’d) At-the-Market Swap Example (cont’d) Apply the discount factors to both the fixed and floating rate sides of the swap to solve for the swap fixed rate that will equate the two sides:

29. Initial Condition Pricing (cont’d) At-the-Market Swap Example (cont’d) Solving the two equations simultaneously for X gives X = 5.62%. This is the equilibrium swap fixed rate, or swap price. ...this is the starting point for a swap dealer

30. Quoting the Swap Price • Common practice to quote the swap price relative to the U.S. Treasury yield curve • interest rates are constantly changing so dealers will quote relative to or off the U.S. treasury or Govt. Of Canada yield curve • Maturity should match the tenor of the swap • There is both a bid and an ask associated with the swap price • The dealer adds a swap spread to the appropriate Treasury yield

31. Counterparty Risk Implications • From the perspective of the party paying the fixed rate • Higher when the floating rate is above the fixed rate • From the perspective of the party paying the floating rate • Higher when the fixed rate is above the floating rate

32. Valuing an Off-Market Swap • The swap value reflects the difference between the swap price (fixed rate) and the interest rate that would make the swap have zero value • As soon as market interest rates change after a swap is entered, the swap has value to one party or the other

33. Valuing an Off-Market Swap (cont’d) • An off-market swap is one in which the fixed rate is such that the fixed rate and floating rate sides of the swap do not have equal value - the present value of the two cash flow streams is different. • Thus, the swap has value to one of the counterparties

34. Valuing an Off-Market Swap (cont’d) • If the fixed rate in our at-the-market swap example was 5.75% instead of 5.62% • The value of the floating rate side would not change • The value of the fixed rate side would be higher than the floating rate side • The swap has value to the floating rate payer and fixed rate receiver ...in a stand alone situation or if interest rates moved from 5.62% to 5.75 %

35. Hedging the Swap- by the Dealer • If interest is predominantly in one direction (e.g., everyone wants to pay a fixed rate), then the dealer stands to suffer a considerable loss • E.g., the dealer is a counterparty to a one-year, $10 million swap with quarterly payments and pays floating • The dealer is hurt by rising interest rates

36. Hedging the Swap by the Dealer • The dealer can hedge this risk in the futures market • If the deal is Libor based - then use Euro-dollar futures • If the dealer faces the risk of rising rates, he could sell Eurodollar futures and benefit from the decline in value associated with rising interest rates

37. Pricing A Currency Swap • To value a currency swap: • Solve for the equilibrium fixed rate on a plain vanilla interest rate swap for each of the two countries • Determine the relevant spot rates over the tenor of the swap • Determine the relevant implied forward rates • Find the equilibrium swap price for an interest rate swap in both currencies - in effect there are two swap prices in a currency swap