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# Derivatives lecture

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### Derivatives lecture

2. 2. Copyright © 2003 Pearson Education, Inc. Slide 8-2 Chapter 8Chapter 8 SwapsSwaps
3. 3. Copyright © 2003 Pearson Education, Inc. Slide 8-3 Introduction to Swaps • A swap is a contract calling for an exchange of payments, on one or more dates, determined by the difference in two prices. • A swap provides a means to hedge a stream of risky payments. • A single-payment swap is the same thing as a cash-settled forward contract.
4. 4. Copyright © 2003 Pearson Education, Inc. Slide 8-4 An example of a commodity swap • An industrial producer, IP Inc., needs to buy 100,000 barrels of oil 1 year from today and 2 years from today. • The forward prices for deliver in 1 year and 2 years are \$20 and \$21/barrel. • The 1- and 2-year zero-coupon bond yields are 6% and 6.5%.
5. 5. Copyright © 2003 Pearson Education, Inc. Slide 8-5 An example of a commodity swap • IP can guarantee the cost of buying oil for the next 2 years by entering into long forward contracts for 100,000 barrels in each of the next 2 years. The PV of this cost per barrel is • Thus, IP could pay an oil supplier \$37.383, and the supplier would commit to delivering one barrel in each of the next two years. • A prepaid swap is a single payment today for multiple deliveries of oil in the future. \$ . \$ . \$ . 20 106 21 1065 37 3832 + =
6. 6. Copyright © 2003 Pearson Education, Inc. Slide 8-6 An example of a commodity swap • With a prepaid swap, the buyer might worry about the resulting credit risk. Therefore, a better solution is to defer payments until the oil is delivered, while still fixing the total price. • Any payment stream with a PV of \$37.383 is acceptable. Typically, a swap will call for equal payments in each year. – For example, the payment per year per barrel, x, will have to be \$20.483 to satisfy the following equation: • We then say that the 2-year swap price is \$20.483. x x 106 1065 37 3832 . . \$ .+ = a/A+b/B = (aB +Ab)/AB (1.065^2+1.06)/(1.06*1.065^2) = 1.825 37.383/1.825 = 20.483 = x
7. 7. Copyright © 2003 Pearson Education, Inc. Slide 8-7 Physical versus financial settlement • Physical settlement of the swap:
8. 8. Copyright © 2003 Pearson Education, Inc. Slide 8-8 Physical versus financial settlement • Financial settlement of the swap: – The oil buyer, IP, pays the swap counterparty the difference between \$20.483 and the spot price, and the oil buyer then buys oil at the spot price. – If the difference between \$20.483 and the spot price is negative, then the swap counterparty pays the buyer.
9. 9. Copyright © 2003 Pearson Education, Inc. Slide 8-9 Physical versus financial settlement • Whatever the market price of oil, the net cost to the buyer is the swap price, \$20.483: Spot price – Swap price – Spot price = – Swap price Swap Payment Spot Purchase of Oil • Note that 100,000 is the notional amount of the swap, meaning that 100,000 barrels is used to determine the magnitude of the payments when the swap is settled financially.
10. 10. Copyright © 2003 Pearson Education, Inc. Slide 8-10 Physical versus financial settlement • The results for the buyer are the same whether the swap is settled physically or financially. In both cases, the net cost to the oil buyer is \$20.483. Note: if spot is less than swap price, Oil buyer pays counterparty.
11. 11. Copyright © 2003 Pearson Education, Inc. Slide 8-11 • Swaps are nothing more than forward contracts coupled with borrowing and lending money. – Consider the swap price of \$20.483/barrel. Relative to the forward curve price of \$20 in 1 year and \$21 in 2 years, we are overpaying by \$0.483 in the first year, and we are underpaying by \$0.517 in the second year. – Thus, by entering into the swap, we are lending the counterparty money for 1 year. The interest rate on this loan is 0.517 / 0.483 – 1 = 7%. – Given 1- and 2-year zero-coupon bond yields of 6% and 6.5%, 7% is the 1-year implied forward yield from year 1 to year 2. If the deal is priced fairly, the interest rate on this loan should be the implied forward interest rate.
12. 12. Copyright © 2003 Pearson Education, Inc. Slide 8-12 The swap counterparty • The swap counterparty is a dealer, who is, in effect, a broker between buyer and seller. • The fixed price paid by the buyer, usually, exceeds the fixed price received by the seller. This price difference is a bid-ask spread, and is the dealer’s fee. • The dealer bears the credit risk of both parties, but is not exposed to price risk.
13. 13. Copyright © 2003 Pearson Education, Inc. Slide 8-13 The swap counterparty • The situation where the dealer matches the buyer and seller is called a back-to-back transaction or “matched book” transaction.
14. 14. Copyright © 2003 Pearson Education, Inc. Slide 8-14 The swap counterparty • Alternatively, the dealer can serve as counterparty and hedge the transaction by entering into long forward or futures contracts. – Note that the net cash flow for the hedged dealer is a loan, where the dealer receives cash in year 1 and repays it in year 2. – Thus, the dealer also has interest rate exposure (which can be hedged by using Eurodollar contracts or forward rate agreements).
15. 15. Copyright © 2003 Pearson Education, Inc. Slide 8-15 The market value of a swap • The market value of a swap is zero at interception. • Once the swap is struck, its market value will generally no longer be zero because: – the forward prices for oil and interest rates will change over time; – even if prices do not change, the market value of swaps will change over time due to the implicit borrowing and lending. • A buyer wishing to exit the swap could enter into an offsetting swap with the original counterparty or whomever offers the best price. • The market value of the swap is the difference in the PV of payments between the original and new swap rates.
16. 16. Copyright © 2003 Pearson Education, Inc. Slide 8-16 Interest Rate Swaps • The notional principle of the swap is the amount on which the interest payments are based. • The life of the swap is the swap term or swap tenor. • If swap payments are made at the end of the period (when interest is due), the swap is said to be settled in arrears.
17. 17. Copyright © 2003 Pearson Education, Inc. Slide 8-17 Introduction (cont’d) Plain Vanilla Swap Example A large firm pays a fixed interest rate to its bondholders, while a smaller firm pays a floating interest rate to its bondholders. The two firms could engage in a swap transaction which results in the larger firm paying floating interest rates to the smaller firm, and the smaller firm paying fixed interest rates to the larger firm.
18. 18. Copyright © 2003 Pearson Education, Inc. Slide 8-18 Introduction (cont’d) Plain Vanilla Swap Example (cont’d) Big Firm Smaller Firm Bondholder s Bondholder s LIBOR – 50 bp 8.05% 8.05% LIBOR +100 bp
19. 19. Copyright © 2003 Pearson Education, Inc. Slide 8-19 Interest Rate SWAP Let’s assume: Borrowing Possibilities FIRM FIXED RATE FLOATING RATE AAA Current 5-YR T-bond +25 bp LIBOR BBB Current 5-YR T-bond +85 bp LIBOR + 30 bp Quality Spread 60 bp 30 bp
20. 20. Copyright © 2003 Pearson Education, Inc. Slide 8-20 Example Interest Rate SWAP Plain Vanilla Swap Example (cont’d) AAA BBB Bondholders Bondholders LIBOR Treas + 40 bp = 4.90% fixed Teas. +25 bp = 4.75% fixed LIBOR +30 bp LIBOR - 15 bp (compare to LIBOR) Treasury + 70 bp (compare to Treas. + 85 bp) Net rate is 15 bp less than without SWAP
21. 21. Copyright © 2003 Pearson Education, Inc. Slide 8-21 Introduction (cont’d) Plain Vanilla Swap Example (cont’d) A facilitator might act as an agent in the transaction and charge a 15 bp fee for the service.
22. 22. Copyright © 2003 Pearson Education, Inc. Slide 8-22 Introduction (cont’d) Plain Vanilla Swap Example (cont’d) Big Firm Smaller Firm Bondholder s Bondholder s 8.05% LIBOR +100 bp Facilitator LIBOR -50 bp 8.05% 8.20% LIBOR -50 bp
23. 23. Copyright © 2003 Pearson Education, Inc. Slide 8-23 An example of an interest rate swap • XYZ Corp. has \$200M of floating-rate debt at LIBOR, i.e., every year it pays that year’s current LIBOR. • XYZ would prefer to have fixed-rate debt with 3 years to maturity. • XYZ could enter a swap, in which they receive a floating rate and pay the fixed rate, which is 6.9548%.
24. 24. Copyright © 2003 Pearson Education, Inc. Slide 8-24 An example of an interest rate swap On net, XYZ pays 6.9548%: XYZ net payment = – LIBOR + LIBOR – 6.9548% = –6.9548% Floating Payment Swap Payment
25. 25. Copyright © 2003 Pearson Education, Inc. Slide 8-25 Computing the swap rate – Suppose there are n swap settlements, occurring on dates ti, i = 1,… , n. – The implied forward interest rate from date ti-1 to date ti, known at date 0, is r0(ti-1, ti). – The price of a zero-coupon bond maturing on date ti is P(0, ti). – The fixed swap rate is R. • The market-maker is a counterparty to the swap in order to earn fees, not to take on interest rate risk. Therefore, the market-maker will hedge the floating rate payments by using, for example, forward rate agreements.
26. 26. Copyright © 2003 Pearson Education, Inc. Slide 8-26 Computing the swap rate • The requirement that the hedged swap have zero net PV is (8.1) • Equation (8.1) can be rewritten as (8.2) where Σn i=1 P(0, ti) r(ti-1, ti) is the PV of interest payments implied by the strip of forward rates, and Σn i=1 P(0, ti) is the PV of a \$1 annuity when interest rates vary over time. R P t r t t P t i i ii n ii n = = = ∑ ∑ ( , ) ( , ) ( , ) 0 0 11 1 - P t R r t ti i ii n ( , ) [ ( , )]0 00 11 − ==∑ -
27. 27. Copyright © 2003 Pearson Education, Inc. Slide 8-27 Computing the swap rate • We can rewrite equation (8.2) to make it easier to interpret: Thus, the fixed swap rate is as a weighted average of the implied forward rates, where zero-coupon bond prices are used to determine the weights. R P t P t r t ti jj ni n i i=        = = − ∑ ∑ ( , ) ( , ) ( , ) 0 01 1 1
28. 28. Copyright © 2003 Pearson Education, Inc. Slide 8-28 Computing the swap rate • Alternative way to express the swap rate is (8.3) This equation is equivalent to the formula for the coupon on a par coupon bond. Thus, the swap rate is the coupon rate on a par coupon bond. R P t P t n ii n = − =∑ 1 0 0 0 01 ( , ) ( , )
29. 29. Copyright © 2003 Pearson Education, Inc. Slide 8-29 The swap curve • A set of swap rates at different maturities is called the swap curve. • The swap curve should be consistent with the interest rate curve implied by the Eurodollar futures contract, which is used to hedge swaps. • Recall that the Eurodollar futures contract provides a set of 3-month forward LIBOR rates. In turn, zero-coupon bond prices can be constructed from implied forward rates. Therefore, we can use this information to compute swap rates.
30. 30. Copyright © 2003 Pearson Education, Inc. Slide 8-30 The swap curve For example, the December swap rate can be computed using equation (8.3): (1 – 0.9485)/ (0.9830 + 0.9658 + 0.9485) = 1.778%. Multiplying 1.778% by 4 to annualize the rate gives the December swap rate of 7.109%. Swap rate curve Given [(91/90)*(1-.9318)]/4 +(1/.01724) +(1/.01782) +(1/0.1827) =.9485 (1-.9314)/ (+.9830 +.9658 +.9485 +.9314) = .017917 or annual rate of 7.166%
31. 31. Copyright © 2003 Pearson Education, Inc. Slide 8-31 The swap curve • The swap spread is the difference between swap rates and Treasury-bond yields for comparable maturities.
32. 32. Copyright © 2003 Pearson Education, Inc. Slide 8-32 The swap’s implicit loan balance • Implicit borrowing and lending in a swap can be illustrated using the following graph, where the 10-year swap rate is 7.4667%:
33. 33. Copyright © 2003 Pearson Education, Inc. Slide 8-33 The swap’s implicit loan balance • In the above graph, – Consider an investor who pays fixed and receives floating. This investor is paying a high rate in the early years of the swap, and hence is lending money. About halfway through the life of the swap, the Eurodollar forward rate exceeds the swap rate and the loan balance declines, falling to zero by the end of the swap. – Therefore, the credit risk in this swap is borne, at least initially, by the fixed-rate payer, who is lending to the fixed-rate recipient.
34. 34. Copyright © 2003 Pearson Education, Inc. Slide 8-34 Deferred swap • A deferred swap is a swap that begins at some date in the future, but its swap rate is agreed upon today. • The fixed rate on a deferred swap beginning in k periods is computed as (8.4) Equation (8.4) is equal to equation (8.2), when k = 1. R P t r t t P t i i ii k T ii k T = −= = ∑ ∑ 0 0 10 0 ( , ) ( , ) ( , )
35. 35. Copyright © 2003 Pearson Education, Inc. Slide 8-35 Why swap interest rates? • Interest rate swaps permit firms to separate credit risk and interest rate risk. – By swapping its interest rate exposure, a firm can pay the short- term interest rate it desires, while the long-term bondholders will continue to bear the credit risk.
36. 36. Copyright © 2003 Pearson Education, Inc. Slide 8-36 Amortizing and accreting swaps • An amortizing swap is a swap where the notional value is declining over time (e.g., floating rate mortgage). • An accreting swap is a swap where the notional value is growing over time. • The fixed swap rate is still a weighted average of implied forward rates, but now the weights also involve changing notional principle, Qt: (8.7)R Q P t r t t Q P t t i i ii n t ii n i i = −= = ∑ ∑ ( , ) ( , ) ( , ) 0 0 11 1
37. 37. Copyright © 2003 Pearson Education, Inc. Slide 8-37 Currency Swaps • A currency swap entails an exchange of payments in different currencies. • A currency swap is equivalent to borrowing in one currency and lending in another.
38. 38. Copyright © 2003 Pearson Education, Inc. Slide 8-38 An example of a currency swap • Suppose a dollar-based firm enters into a swap where it pays dollars and receives euros. • The position of the market-maker is summarized below: The PV of the market-maker’s net cash flows is (\$2.174 / 1.06) + (\$2.096 / 1.062 ) – (\$4.664 / 1.063 ) = 0
39. 39. Copyright © 2003 Pearson Education, Inc. Slide 8-39 Currency swap formulas – Consider a swap in which a dollar annuity, R, is exchanged for an annuity in another currency, R*. – There are n payments. – The time-0 forward price for a unit of foreign currency delivered at time ti is F0,ti . – The dollar-denominated zero-coupon bond price is P0,ti .
40. 40. Copyright © 2003 Pearson Education, Inc. Slide 8-40 Currency swap formulas • Given R*, what is R? (8.8) This equation is equivalent to equation (8.2), with the implied forward rate, r0(ti-1, ti), replaced by the foreign-currency-denominated annuity payment translated into dollars, R* F0,ti . R P R* F P ti n t ti n i i i = = = ∑ ∑ 01 0 01 , , ,
41. 41. Copyright © 2003 Pearson Education, Inc. Slide 8-41 Currency swap formulas – When coupon bonds are swapped, one has to account for the difference in maturity value as well as the coupon payment. – If the dollar bond has a par value of \$1, the foreign bond will have a par value of 1/x0, where x0 is the current exchange rate expressed as dollar per unit of the foreign currency. • The coupon rate on the dollar bond, R, in this case is (8.9)R P R* F x P F x P t t t ti n ti n i i n n i = + −= = ∑ ∑ 0 0 0 0 0 01 01 1, , , , , / ( / )
42. 42. Copyright © 2003 Pearson Education, Inc. Slide 8-42 Other currency swaps • A diff swap, short for differential swap, is a swap where payments are made based on the difference in floating interest rates in two different currencies, with the notional amount in a single currency. • Standard currency forward contracts cannot be used to hedge a diff swap. – We can’t easily hedge the exchange rate at which the value of the interest rate change is converted because we don’t know in advance how much currency will need to be converted.
43. 43. Copyright © 2003 Pearson Education, Inc. Slide 8-43 Commodity Swaps • The fixed payment on a commodity swap is (8.11) The commodity swap price is a weighted average of commodity forward prices. F P t F P t i ti n ii n i = = = ∑ ∑ ( , ) ( , ) ,0 0 01 1
44. 44. Copyright © 2003 Pearson Education, Inc. Slide 8-44 Swaps with variable quantity and prices • A buyer with seasonally-varying demand (e.g., someone buying gas for heating) might enter into a swap, in which quantities vary over time. • The swap price with seasonally-varying quantities is , (8.12) where Qti is the quantity of gas purchased at time ti . When Qt = 1, the formula is the same as equation (8.11), when the quantity is not varying. F Q P t F Q P t t i ti n ti n i i i i = = = ∑ ∑ ( , ) ( , ) ,0 0 01 1
45. 45. Copyright © 2003 Pearson Education, Inc. Slide 8-45 Swaps with variable quantity and prices • It is also possibly for prices to be time-varying. – For example, a gas buyer who needs gas for heating can enter into a swap, in which the summer price is fixed at a low value, and the winter price is then determined by the zero present value condition.
46. 46. Copyright © 2003 Pearson Education, Inc. Slide 8-46 Swaptions • A swaption is an option to enter into a swap with specified terms. This contract will have a premium. • A swaption is analogous to an ordinary option, with the PV of the swap obligations (the price of the prepaid swap) as the underlying asset. • Swaptions can be American or European.
47. 47. Copyright © 2003 Pearson Education, Inc. Slide 8-47 Swaptions • A payer swaption gives its holder the right, but not the obligation, to pay the fixed price and receive the floating price. – The holder of a receiver swaption would exercise when the fixed swap price is above the strike. • A receiver swaption gives its holder the right to pay the floating price and receive the fixed strike price. – The holder of a receiver swaption would exercise when the fixed swap price is below the strike.
48. 48. Copyright © 2003 Pearson Education, Inc. Slide 8-48 Total Return Swaps • A total return swap is a swap, in which one party pays the realized total return (dividends plus capital gains) on a reference asset, and the other party pays a floating return such as LIBOR. • The two parties exchange only the difference between these rates. • The party paying the return on the reference asset is the total return payer.
49. 49. Copyright © 2003 Pearson Education, Inc. Slide 8-49 Total Return Swaps • Some uses of total return swaps are: – avoiding withholding taxes on foreign stocks, – management of credit risk. • A default swap is a swap, in which the seller makes a payment to the buyer if the reference asset experiences a “credit event” (e.g., a failure to make a scheduled payment on a bond). – A default swap allows the buyer to eliminate bankruptcy risk, while retaining interest rate risk. – The buyer pays a premium, usually amortized over a series of payments.
50. 50. Copyright © 2003 Pearson Education, Inc. Slide 8-50 Summary • The swap formulas in different cases all take the same general form. Let f0(ti) denote the forward price for the floating payment in the swap. Then the fixed swap payment is (8.13)R P t f t P t i ii n ii n = = = ∑ ∑ ( , ) ( ) ( , ) 0 0 01 1
51. 51. Copyright © 2003 Pearson Education, Inc. Slide 8-51 Summary • The following table summarizes the substitutions to make in equation (8.13) to get various swap formulas: