Chapter 12: Swaps Let us not forget there were plenty of financial disasters before quants showed up on Wall Street, and the subsequent disasters (including the current one) had plenty of help from the non-quants. Aaron Brown Risk Professional, April 2010, p. 18 Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Important Concepts in Chapter 12 n n n n The concept of a swap Different types of swaps, based on underlying currency, interest rate, or equity Pricing and valuation of swaps Strategies using swaps Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. n n u u u u n u u u u u n n Definition of a swap Four types of swaps Currency Interest rate Equity Commodity (not covered in this book) Characteristics of swaps No cash up front Notional amount Settlement date, settlement period Credit risk Dealer market See Figure 12.1 for growth in world-wide notional amount See Figure 12.2 for growth in world-wide gross market value Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. (Return to text slide) Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. (Return to text slide) Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Interest Rate Swaps u F The Structure of a Typical Interest Rate Swap Example: On December 15 XYZ enters into $50 million notional amount swap with ABSwaps. Payments will be on 15th of March, June, September, December for one year, based on LIBOR. XYZ will pay 7.5% fixed and ABSwaps will pay LIBOR. Interest based on exact day count and 360 days (30 per month). In general the cash flow to the fixed payer will be Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Interest Rate Swaps (continued) u F F F F The Structure of a Typical Interest Rate Swap (continued) The payments in this swap are Payments are netted. See Figure gure 12.3 for payment pattern See Table ble 12.1 for sample of payments after-the-fact. after-the- Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. (Return to text slide) Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. (Return to text slide) Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Interest Rate Swaps (continued) u F F The Pricing and Valuation of Interest Rate Swaps How is the fixed rate determined? A digression on floating-rate securities. The price of a LIBOR zero coupon bond for maturity of ti days is • Starting at the maturity date and working back, we see that the F price is par onn each coupon date. See Figure 12.4. Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. (Return to text slide) Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Interest Rate Swaps (continued) u F F F The Pricing and Valuation of Interest Rate Swaps (continued) By adding the notional amounts at the end, we can separate the cash flow streams of an interest rate swap into those of a fixed-rate bond and a floating-rate bond. See Figure 12.5. The value of a fixed-rate bond (q = days/360): Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. (Return to text slide) Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Interest Rate Swaps (continued) u F The Pricing and Valuation of Interest Rate Swaps (continued) The value of a floating-rate bond F At time t, between 0 and 1, F Thee value of the swap (pay fixed, receive floating) is, is therefore, Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Interest Rate Swaps (continued) u F F F F The Pricing and Valuation of Interest Rate Swaps (continued) To price the swap at the start, set this value to zero and solve for R See Table 12.2 2.2 for an example. Note how dealers ealers quote q ote as a spread over Treasury rate. To value a swap during its life, simply find the difference between the present values of the two streams of payments. See Table 12.3. Market value reflects the economic value, is necessary for accounting, and gives an indication of the credit risk. Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. (Return to text slide) Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. (Return to text slide) Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Interest Rate Swaps (continued) u F • • • • The Pricing and Valuation of Interest Rate Swaps (continued) A basis swap is equivalent to the difference between two plain vanilla swaps based on different rates: A swap to pay T-bill, receive fixed, plus A swap to pay fixed, receive LIBOR, equals A swap to pay T-bill, receive LIBOR, plus pay the difference between the LIBOR and T-bill fixed rates See Tables 12.4 and Table 12.5 for examples of pricing and valuation of a basis swap. Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. (Return to text slide) Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. (Return to text slide) Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Interest Rate Swaps (continued) u F F • • • Interest Rate Swap Strategies See Figure 12.6 for example of converting floating-rate loan into fixed-rate loan Other types of swaps Index amortizing swaps Diff swaps Constant maturity swaps Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. (Return to text slide) Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Currency Swaps u F u Example: Reston Technology enters into currency swap with GSI. Reston will pay euros at 4.35% based on NP of €10 million semiannually for two years. GSI will pay dollars at 6.1% based on NP of $9.804 million semiannually for two years. Notional amounts will be exchanged. See Figure 12.7. Note the relationship between interest rate and currency swaps in Figure 12.8. Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. (Return to text slide) Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. (Return to text slide) Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Currency Swaps (continued) u F F F F F F Pricing and Valuation of Currency Swaps Let dollar notional amount be NP$. Then euro notional amount is NP€ = 1/S0 for every dollar notional amount. Here euro notional amount will be €10 million. With S0 = $0.9804, NP$ = $9,804,000. For fixed payments, we use the fixed rate on plain vanilla swaps in that currency, R$ or R€. No pricing is required for the floating side of a currency swap. See Table 12.6. During the life of the swap, we value it by finding the difference in the present values of the two streams of payments, adjusting for the notional amounts, and converting to a common currency. Assume new exchange rate is $0.9790 three months later. See Table 12.7 for calculations of values of streams of payments per unit notional amount. Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. (Return to text slide) Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. (Return to text slide) Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Currency Swaps (continued) u F F F F Pricing and Valuation of Currency Swaps (continued) Dollars fixed for NA of $9.804 million = $9,804,000(1.01132335) = $9,915,014 Dollars floating for NA of $9.804 million = $9,804,000(1.013115) = $9,932,579 Euros fixed for NA of €10 million = €10,000,000(1.00883078)) = €10,088,308 Euros floating for NA of €10 million = €10,000,000(1.0091157) = €10,091,157 Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Currency Swaps (continued) u F • F • F • F • Pricing and Valuation of Currency Swaps (continued) Value of swap to pay € fixed, receive $ fixed $9,915,014 - €10,088,308($0.9790/€) = $38,560 Value of swap to pay € fixed, receive $ floating $9,932,579 - €10,088,308($0.9790/€) = $56,125 Value of swap to pay € floating, receive $ fixed $9,915,014 - €10,091,157($0.9790/€) = $35,771 15 Value of swap to pay € floating, receive $ floating $9,932,579 - €10,091,157($0.9790/€) = $53,336 Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Currency Swaps (continued) u F F Currency Swap Strategies A typical case is a firm borrowing in one currency and wanting to borrow in another. See Figure 12.9 for Reston-GSI example. Reston could get a better rate due to its familiarity to GSI and also due to credit risk. Also a currency swap be used to convert a stream of foreign cash flows. This type of swap wo would probably have no exchange of notional amounts. Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. (Return to text slide) Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Equity Swaps u F F F Characteristics One party pays the return on an equity, the other pays fixed, floating, or the return on another equity Rate of return is paid, so payment can be negative Payment is not determined until end of period Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Equity Swaps (continued) u F F The Structure of a Typical Equity Swap Cash flow to party paying stock and receiving fixed ple: IVM enters into a swap with FNS to pay S&P 500 T Example: Total 2 Returnn and receive a fixed rate of 3.45%. The index starts at 2710.55. d f r one year. Net payment will ill be b Payments every 90 days for Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Equity Swaps (continued) u F • F F The Structure of a Typical Equity Swap (continued) The fixed payment will be $25,000,000(.0345)(90/360) = $215,625 See Table 12.8 for example of payments. The first equity payment is So the first net payment is IVM pays $285,657. Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. (Return to text slide) Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Equity Swaps (continued) u F F The Structure of a Typical Equity Swap (continued) If IVM had received floating, the payoff formula would be If thee swap were structured so that IVM pays the return on on one stock w indexx and receives the return on another, the payoff formula would be Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Equity Swaps (continued) u F • • • – – – Pricing and Valuation of Equity Swaps For a swap to pay fixed and receive equity, we replicate as follows: Invest $1 in stock Issue $1 face value loan with interest at rate R. Pay interest on each swap settlement date and repay amount at swap termination date. Interest based on q = days/360. Example: Assume payments on days 180 and 360. ay On day 180, stock worth S180/S0. Sell stock and withdraw S180/S0 - 1 Owe interest of Rq Overall cash flow is S180/S0 – 1 – Rq, which is equivalent to the first swap payment. $1 is left over. Reinvest in the stock. Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Equity Swaps (continued) u F F F F Pricing and Valuation of Equity Swaps (continued) On day 360, stock is worth S360/S180. Liquidate stock. Pay back loan of $1 and interest of Rq. Overall cash flow is S360/S180 – 1 – Rq, which is equivalent to the second swap payment. The value of the position is the value of the swap. In general for n ta is payments, the value at the start Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Equity Swaps (continued) u F F Pricing and Valuation of Equity Swaps (continued) Setting the value to zero and solving for R gives which is the same me as the fixed rate on an a interest rate swap. See Table 12.9 for pricing g the IVM swap. wa Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. (Return to text slide) Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Equity Swaps (continued) u F F • • • Pricing and Valuation of Equity Swaps (continued) To value the swap at time t during its life, consider the party paying fixed and receiving equity. To replicate the first payment, at time t Purchase 1/S0 shares at a cost of (1/S0)St. Borrow $1 at rate R maturing at next payment date. At the next payment ddate (assume day 90), shares are worth (1/S0)S90. Sell the stock, generating (1/S0)S90 – 1 (equivalent to the equity payment on the swap), plus $1 left over, which is reinvested in the stock. Pay the loan interest, Rq (which is equivalent to the fixed payment on the swap). Do this for each payment on the swap. Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Equity Swaps (continued) u F F F • • F Pricing and Valuation of Equity Swaps (continued) The cost to do this strategy at time t is This is the value of the swap. See Table 12.10 for an example of the IVM swap. r To value the equity swap receiving floating and paying equity, note the equivalence to A swap to pay equity and receive fixed, plus A swap to pay fixed and receive floating. So we can use what we already know. Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. (Return to text slide) Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Equity Swaps (continued) u F • F • F • F • F • Pricing and Valuation of Equity Swaps (continued) Using the new discount factors, the value of the fixed payments (plus hypothetical notional amount) is 0.0345(90/360)(0.9971 + 0.9877 + 0.9778 + 0.9677) + 1(0.9677) = 1.00159884 The value of the floating payments (plus hypothetical notional amount) is 9 (1 + 0.03(90/360))(0.9971) = 1.00457825 The plain vanilla swap value is, thus, 1.00457825 – 1.00159884 = 0.00297941 For a $25 million notional amount, $25,000,000(0.00297941) = $74,485 So the value of the equity swap is (using -$227,964, the value of the equity swap to pay fixed) -$227,964 + $74,485 = -$153,479 Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Equity Swaps (continued) u F F Pricing and Valuation of Equity Swaps (continued) For swaps to pay one equity and receive another, replicate by selling short one stock and buy the other. Each period withdraw the cash return, reinvesting $1. Cover short position by buying it back, and then sell short $1. So each period start with $1 long one stock and $1 short the other. w pay the S&P and receive NASDAQ, For the IVM swap, suppose we which starts at 2710.55 and goes to 2739.60. The value of the swap is Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Equity Swaps (continued) u F • Pricing and Valuation of Equity Swaps (continued) For $25 million notional amount, the value is $25,000,000(0.03312974) = $828,244 Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Equity Swaps (continued) u F F F • • • Equity Swap Strategies Used to synthetically buy or sell stock See Figure 12.10 for example. Some risks default tracking error cash flow shortages Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. (Return to text slide) Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Some Final Words About Swaps u u F F F Similarities to forwards and futures Offsetting swaps Go back to dealer Offset with another counterparty Forward contract or option on the swap Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Summary Chance/Brooks An Introduction to Derivatives and Risk Management, 9th ed. Ch. 12: 0 © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 
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