Chance/BrooksAn Introduction to Derivatives and Risk Management, 7th ed.Ch. 12: 1 Chapter 12: Swaps Markets are an evolving ecology. New risks arise all the time. Andrew Lo CFA Magazine, March-April, 2004, p. 31

Chance/BrooksAn Introduction to Derivatives and Risk Management, 7th ed.Ch. 12: 2 Important Concepts n The concept of a swap n Different types of swaps, based on underlying currency, interest rate, or equity n Pricing and valuation of swaps n Strategies using swaps

Chance/BrooksAn Introduction to Derivatives and Risk Management, 7th ed.Ch. 12: 3 Nature of Swaps n A swap is an agreement to exchange cash flows at specified future times according to certain specified rules.

Chance/BrooksAn Introduction to Derivatives and Risk Management, 7th ed.Ch. 12: 4 n Four types of swaps u Currency u Interest rate u Equity u Commodity n Characteristics of swaps u No cash up front u Notional principal u Settlement date, settlement period u Credit risk u Dealer market n See Figure 12.1, p. 407 for growth in world-wide notional principal Figure 12.1, p. 407Figure 12.1, p. 407

Chance/BrooksAn Introduction to Derivatives and Risk Management, 7th ed.Ch. 12: 5 Interest Rate Swaps n In an interest rate swap, two parties agree to exchange or swap a series of interest payments. n In a “plain vanilla” interest rate swap, one party agrees to make a series of fixed interest payments and the other agrees to make a series of variable or floating interest payments.

Chance/Brooks 6 Example of a “Plain Vanilla” Swap An agreement by XYZ Corp 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.

Chance/Brooks Millions of Dollars LIBORFLOATINGFIXEDNet DateRateCash Flow Mar.5, % Sept. 5, %+2.10–2.50–0.40 Mar.5, %+2.40–2.50–0.10 Sept. 5, %+2.65– Mar.5, %+2.75– Sept. 5, %+2.80– Mar.5, %+2.95– Cash Flows to XYZ Corp

Chance/Brooks 8 Uses of 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

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10 Transforming a Liability ABCXYZ LIBOR 5% LIBOR+0.1% 5.2%

Chance/Brooks 11 When a Financial Institution is Involved F.I. LIBOR LIBOR+0.1% 4.985% 5.015% 5.2% ABCXYZ

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Chance/Brooks 15 Transforming an Asset ABC XYZ LIBOR 5% LIBOR-0.2% 4.7%

Chance/Brooks 16 When a Financial Institution is Involved ABC F.I.XYZ LIBOR 4.7% 5.015%4.985% LIBOR-0.2%

Chance/Brooks 17 Quotes By a Swap Dealer MaturityBid (%)Offer (%)Swap Rate (%) 2 years years years years years years

Chance/Brooks 18 The Comparative Advantage Argument PQR Corp wants to borrow floating RST Corp wants to borrow fixed FixedFloating PQR Corp4.00%6-month LIBOR % RST Corp5.20%6-month LIBOR %

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Chance/Brooks 20 The Swap PQR RST LIBOR LIBOR+1% 3.95% 4%

Chance/Brooks 21 When a Financial Institution is Involved PQR F.I. RST 4% LIBOR LIBOR+1% 3.93% 3.97%

Chance/Brooks 22 Criticism of the Comparative Advantage Argument The 4.0% and 5.2% rates available to PQR Corp and RST Corp in fixed rate markets are 5-year rates. The LIBOR+0.3% and LIBOR+1% rates available in the floating rate market are six- month rates. RST Corp’s fixed rate depends on the spread above LIBOR it borrows at in the future.

Chance/Brooks 23 The Nature of Swap Rates Six-month LIBOR is a short-term AA borrowing rate. The 5-year swap rate has a risk corresponding to the situation where 10 six-month loans are made to AA borrowers at LIBOR. This is because the lender can enter into a swap where income from the LIBOR loans is exchanged for the 5-year swap rate.

Chance/BrooksAn Introduction to Derivatives and Risk Management, 7th ed.Ch. 12: 24 Interest Rate Swaps u The Structure of a Typical Interest Rate Swap F Example: On December 15 XYZ enters into $50 million NP swap with ABSwaps. Payments will be on 15 th 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/BrooksAn Introduction to Derivatives and Risk Management, 7th ed.Ch. 12: 25 Interest Rate Swaps u The Structure of a Typical Interest Rate Swap (continued) F The payments in this swap are F Payments are netted. F See Figure 12.2, p. 409 for payment pattern Figure 12.2, p. 409Figure 12.2, p. 409 F See Table 12.1, p. 410 for sample of payments after- the-fact. Table 12.1, p. 410Table 12.1, p. 410

Chance/Brooks 26 Zero Rates A zero rate (or spot rate), for maturity T is the rate of interest earned on an investment that provides a payoff only at time T

Chance/Brooks 27 Example

Chance/Brooks 28 Forward Rates The forward rate is the future zero rate implied by today’s term structure of interest rates

Chance/Brooks 29 Formula for Forward Rates Suppose that the zero rates for time periods T 1 and T 2 are R 1 and R 2 with both rates continuously compounded. The forward rate for the period between times T 1 and T 2 is

Chance/Brooks 30 Calculation of Forward Rates Zero Rate forForward Rate an n -year Investmentfor n th Year Year ( n )(% per annum)

Chance/Brooks 31 Forward Rate Agreement Forward Rate Agreement (FRA) is an agreement where interest at a predetermined rate, R K is exchanged for interest at the market rate. An FRA can be valued by assuming that the forward interest rate is certain to be realized.

Chance/Brooks 32 FRA Valuation Value of FRA where a fixed rate R K will be received on a principal L between times T 1 and T 2 is Value of FRA where a fixed rate is paid is R F is the forward rate for the period and R 2 is the zero rate for maturity T 2

Chance/Brooks 33 Example: FRA Valuation Suppose that the three-month LIBOR rate is 5% and the six-month LIBOR rate is 5.5% with continuous compounding. Consider an FRA where you will receive a rate 7% measured with quarterly compounding, on a principal of $1 million between the end of month 3 and the end of month 6. The forward rate is 6% percent with continuous compounding or with quarterly compounding. The value of the FRA is $1,000,000 x (.07– ) x 0.25 x e x 0.5 = $2,322

Chance/Brooks 34 Valuation of an Outstanding Interest Rate Swap An interest rate swap is worth zero, or close to zero, when it is initiated. After it has been in existence for some time, its value may become positive or negative. Interest rate swaps can be valued as the difference between the value of a fixed-rate bond (B fix ) and the value of a floating-rate bond (B fl ). Alternatively, they can be valued as a portfolio of FRAs.

Chance/Brooks 35 Valuation in Terms of Bonds The fixed rate bond is valued in the usual way as present value of future cash flows. The floating rate bond is valued by noting that it is worth par immediately after the next payment date. Then the value of the swap ( V SWAP ) is V SWAP = B fix - B fl

Chance/Brooks 36 Example Suppose that PDQ Corp pays six-month LIBOR and receives 8% per annum (with semiannual compounding) on a swap with notional principal of $100 million and the remaining payment are in 3, 9, and 15 months. The swap has a remaining life of 15 months. The LIBOR rates with continuous compounding for 3-month, 9- month, and 15-month maturities are 10%, 10.5%, and 11%. The 6-month LIBOR rate at the last payment date was 10.2% (with semiannual compounding). What is the value of the swap?

Chance/Brooks 37 Swap Valuation B fix = 4e -0.1x3/12 + 4e x9/ e -0.11x15/12 = $98.24 million B fl = 5.1e -0.1x3/ e -0.1x3/12 = $ million The value of the swap is V SWAP = $98.24 million - $ million = - $4.27 million If PDQ Corp had been paying fixed and receiving floating, the value of the swap would be + $4.27 million.

Chance/Brooks 38 Valuation in Terms of FRAs Each exchange of payments in an interest rate swap is an FRA. The FRAs can be valued on the assumption that today’s forward rates are realized. The procedure is as follows: Calculate forward rates for each of the LIBOR rates that will determine swap cash flows. Calculate swap cash flows assuming that the LIBOR rates will equal the forward rates. Set the swap value equal to the present value of these cash flows.

Chance/Brooks 39 Swap Valuation as FRAs Consider again the situation in the previous example. The cash flows that will be exchanged in 3 months have already been determined. A rate of 8% will be exchanged for 10.2%. The value of the exchange to PDQ Corp is 0.5 x 100 x ( ) e -0.1 x 3/12 = To calculate the value of the exchange in 9 months, we first calculate the forward rate corresponding to the period between 3 and 9 months: [.105 x.75 –.10 x.25]/.5 =.1075 or 10.75%.

Chance/Brooks 40 Swap Valuation as FRAs Using the equation R m = m(e R c /m – 1) where R c is the rate of interest with continuous compounding and R m is the equivalent rate with compounding m times per annum, we convert 10.75% with continuous compounding into 2 x (e.1075/2 – 1) = or % with semiannual compounding.

Chance/Brooks 41 Swap Valuation as FRAs The value of the FRA corresponding to the exchange in 9 months is therefore 0.5 x 100 x ( ) e x 9/12 = To calculate the value of the exchange in 15 months, we first calculate the forward rate corresponding to the period between 9 and 15 months. This is [.11 x 1.25 –.105 x.75]/.5 =.1175 or 11.75%. This value becomes % with semiannual compounding.

Chance/Brooks 42 Swap Valuation as FRAs The value of the FRA corresponding to the exchange in 15 months is therefore 0.5 x 100 x ( ) e x 15/12 = The total value of the swap is – 1.41 – 1.79 = -$4.27 million

Chance/BrooksAn Introduction to Derivatives and Risk Management, 7th ed.Ch. 12: 43 Interest Rate Swaps u Interest Rate Swap Strategies F See Figure 12.5, p. 418 for example of converting floating-rate loan into fixed-rate loan Figure 12.5, p. 418Figure 12.5, p. 418 F Other types of swaps Index amortizing swapsIndex amortizing swaps Diff swapsDiff swaps Constant maturity swapsConstant maturity swaps

Chance/BrooksAn Introduction to Derivatives and Risk Management, 7th ed.Ch. 12: 44 Currency Swaps u In a currency swap, the parties make either fixed or variable payments to each other in different currencies. 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. F See Figure 12.6, p Figure 12.6, p. 421Figure 12.6, p. 421 u Note the relationship between interest rate and currency swaps in Figure 12.7, p Figure 12.7, p. 422Figure 12.7, p. 422

Chance/Brooks 45 Exchange of Principal In an interest rate swap the principal is not exchanged. In a currency swap the principal is usually exchanged at the beginning and the end of the swap’s life.

Chance/Brooks 46 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. This is a fixed-for-fixed currency swap.

Chance/Brooks 47 The Cash Flows Year DollarsPounds $ millions – – – – – −11.10 £

Chance/Brooks 48 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.

49 Comparative Advantage Arguments for Currency Swaps Shell wants to borrow UK £ BP wants to borrow US $ US $ UK £ Shell 5.0%12.6% BP 7.0%13.0%

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Chance/Brooks 51 Financial Institution is Involved F.I. £ 11.9% £ 13.0% $ 5.0% $ 6.3% $ 5.0% ShellBP

Chance/Brooks 52 BP Bears FX Risk F.I. £ 11.9% £ 13.0% $ 5.0% $ 5.2% $ 5.0% ShellBP

Chance/Brooks 53 Shell Bears FX Risk F.I. £ 13.0% $ 6.1% $ 6.3% $ 5.0% ShellBP

Chance/Brooks 54 Valuation of Currency Swaps Like interest rate swaps, currency swaps can be valued either as the difference between two bonds or as a portfolio of forward contracts. If we define V SWAP as the value in US dollars of a swap where dollars are received and a foreign currency is paid, then V SWAP = B D – S 0 B F where B F is the value, measured in foreign currency, of the foreign-denominated bond underlying the swap, B D is the value of the US dollar bond underlying the swap, and S 0 is the spot exchange rate (expressed as number of units of domestic currency per unit of foreign currency).

Chance/Brooks 55 Example Suppose that the term structure of interest rate is flat in both Japan and US at 4% and 9% per annum, respectively (both with continuous compounding). ABM Corp has entered into a currency swap to receive 5% per annum in yen and pay 8% per annum in dollars once a year. The principals in two currencies are $10 million and 1,200 million yen. The swap will last for another three years, and the current exchange is 110 yen = $1. What is the value of the swap in dollars?

Chance/Brooks 56 Swap Valuation B D = 0.8e -0.09x e -0.09x e -0.09x3 = million dollars B F = 60e -0.04x1 + 60e -0.04x2 + 1,260e -0.04x3 = 1, million yen The value of the swap in dollars is 1,230.55/110 – = $1.543 million If ABM Corp had been paying yen and receiving dollars, the value of the swap would have been -$1.543 million.

Chance/Brooks 57 Currency Swap as FRAs Consider the situation in the previous example. The current spot rate is 110 yen per dollar, or dollar per yen. Using the equation F 0 = S 0 e (r - r f ) T we calculate the one-year, two-year, and three- year forward rates as e ( ) x 1 = e ( ) x 2 = e ( ) x 3 =

Chance/Brooks 58 Currency Swap as FRAs The exchange of interest involves receiving 60 million yen and paying $0.8 million. The risk-free interest rate in dollars is 9% per annum. The value of the forward contracts corresponding to these exchanges are as follows: (60 x – 0.8)e x 1 = (60 x – 0.8)e x 2 = (60 x – 0.8)e x 3 =

Chance/Brooks 59 Currency Swap as FRAs The final exchange of principal involves receiving 1,200 million yen and paying $10 million. The value of the forward contract corresponding to the exchange is (1,200 x – 10)e x 3 = The total value of the swap is – – – = $1.543 million

Chance/BrooksAn Introduction to Derivatives and Risk Management, 7th ed.Ch. 12: 60 Currency Swaps u Currency Swap Strategies F A typical case is a firm borrowing in one currency and wanting to borrow in another. See Figure 12.8, p. 429 for Reston-GSI example. Reston could get a better rate due to its familiarity to GSI and also due to credit risk. Figure 12.8, p. 429Figure 12.8, p. 429 F Also a currency swap be used to convert a stream of foreign cash flows. This type of swap would probably have no exchange of notional principals.

Chance/BrooksAn Introduction to Derivatives and Risk Management, 7th ed.Ch. 12: 61

Chance/BrooksAn Introduction to Derivatives and Risk Management, 7th ed.Ch. 12: 62 Equity Swaps u In an equity swap, at least one of the two parties makes payments determined by the price of a stock, the value of a stock portfolio, or the level of a stock index. u The other party’s payment can be determined by another stock, portfolio or index, or by an interest rate, or it can be fixed. u Characteristics F One party pays the return on an equity, the other pays fixed, floating, or the return on another equity F Rate of return is paid, so payment can be negative F Payment is not determined until end of period

Chance/BrooksAn Introduction to Derivatives and Risk Management, 7th ed.Ch. 12: 63 Equity Swaps u The Structure of a Typical Equity Swap F Cash flow to party paying stock and receiving fixed F Example: IVM enters into a swap with FNS to pay S&P 500 Total Return and receive a fixed rate of 3.45%. The index starts at Payments every 90 days for one year. Net payment will be

Chance/BrooksAn Introduction to Derivatives and Risk Management, 7th ed.Ch. 12: 64 Equity Swaps u The Structure of a Typical Equity Swap (continued) F The fixed payment will be $25,000,000(.0345)(90/360) = $215,625$25,000,000(.0345)(90/360) = $215,625 F See Table 12.8, p. 431 for example of payments. The first equity payment is Table 12.8, p. 431Table 12.8, p. 431 F So the first net payment is IVM pays $285,657.

Chance/BrooksAn Introduction to Derivatives and Risk Management, 7th ed.Ch. 12: 65 Equity Swaps u The Structure of a Typical Equity Swap (continued) F If IVM had received floating, the payoff formula would be F If the swap were structured so that IVM pays the return on one stock index and receives the return on another, the payoff formula would be

Chance/BrooksAn Introduction to Derivatives and Risk Management, 7th ed.Ch. 12: 66 Equity Swaps u Equity Swap Strategies F Used to synthetically buy or sell stock F See Figure 12.9, p. 437 for example. Figure 12.9, p. 437Figure 12.9, p. 437 F Some risks defaultdefault tracking errortracking error cash flow shortagescash flow shortages

Chance/BrooksAn Introduction to Derivatives and Risk Management, 7th ed.Ch. 12: 67 Synthetic Trading of a Portfolio n Currently the investor owns a portfolio of S&P 500 stocks worth $1M. n Sell the portfolio for $1M and reinvest in Treasury for 3 months. n After 3 months: S&P 500 return : -1.77%; Treasury: 1.23%. n Buy back S&P 500 stocks portfolio. n Alternatively, the investor can enter into an equity swap to receive Treasury rate and pay S&P 500 return for 3 months u without selling and buying back the portfolio u thus saving round trip transaction cost.

Chance/BrooksAn Introduction to Derivatives and Risk Management, 7th ed.Ch. 12: 68 Some Final Words About Swaps u Similarities to forwards and futures u Offsetting swaps F Go back to dealer F Offset with another counterparty F Forward contract or option on the swap

Chance/Brooks 69 Swaps & Forwards A swap can be regarded as a convenient way of packaging forward contracts. The “plain vanilla” interest rate swap consists of a series of FRAs. The “fixed for fixed” currency swap in consists of a cash transaction & a series of forward contracts.

Chance/Brooks 70 Swaps & Forwards 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.

Chance/Brooks 71 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.

Chance/Brooks 72 Other Types of Swaps Floating-for-floating interest rate swaps, amortizing swaps, step up swaps, forward swaps, constant maturity swaps, compounding swaps, LIBOR-in-arrears swaps, accrual swaps, diff swaps, cross currency interest rate swaps, equity swaps, extendable swaps, puttable swaps, swaptions, commodity swaps, volatility swaps……..

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