Chapter 12: Swaps

D. M. Chance An Introduction to Derivatives and Risk Management, 6th ed.

Ch. 12: 1

Chapter 12: Swaps

I once had to explain to my father that the bank didn’t really I once had to explain to my father that the bank didn’t really make its money taking deposits and lending out money to make its money taking deposits and lending out money to poor folk so they could buy houses. I explained that the poor folk so they could buy houses. I explained that the bank actually traded for a living.bank actually traded for a living.

Stan JonasStan JonasDerivatives StrategyDerivatives Strategy, April, 1998, p. 19, April, 1998, p. 19


Ch. 12: 2

Important Concepts in Chapter 12

The concept of a swapThe concept of a swap Different types of swaps, based on underlying currency, Different types of swaps, based on underlying currency,

interest rate, or equityinterest rate, or equity Pricing and valuation of swapsPricing and valuation of swaps Strategies using swapsStrategies using swaps


Ch. 12: 3

Definition of a swapDefinition of a swap Four types of swapsFour types of swaps

CurrencyCurrency Interest rateInterest rate EquityEquity Commodity (not covered in this book)Commodity (not covered in this book)

Characteristics of swapsCharacteristics of swaps No cash up frontNo cash up front Notional principalNotional principal Settlement date, settlement periodSettlement date, settlement period Credit riskCredit risk Dealer marketDealer market

See See Figure 12.1, p. 426Figure 12.1, p. 426 for growth in world-wide notional principal for growth in world-wide notional principal


Ch. 12: 4

Interest Rate Swaps

The Structure of a Typical Interest Rate SwapThe Structure of a Typical Interest Rate Swap Example: On December 15 XYZ enters into $50 Example: On December 15 XYZ enters into $50

million NP swap with ABSwaps. Payments will be million NP swap with ABSwaps. Payments will be on 15on 15thth of March, June, September, December for of March, June, September, December for one year, based on LIBOR. XYZ will pay 7.5% one year, based on LIBOR. XYZ will pay 7.5% fixed and ABSwaps will pay LIBOR. Interest based fixed and ABSwaps will pay LIBOR. Interest based on exact day count and 360 days (30 per month). In on exact day count and 360 days (30 per month). In general the cash flow to the fixed payer will begeneral the cash flow to the fixed payer will be

⎟⎠⎞⎜⎝

⎛365or 360

Daysrate) Fixed - (LIBORprincipal) (Notional


Ch. 12: 5

Interest Rate Swaps (continued)

The Structure of a Typical Interest Rate Swap The Structure of a Typical Interest Rate Swap (continued)(continued)

The payments in this swap areThe payments in this swap are

Payments are netted.Payments are netted. See See Figure 12.2, p. 428Figure 12.2, p. 428 for payment pattern for payment pattern See See Table 12.1, p. 429Table 12.1, p. 429 for sample of payments after- for sample of payments after-

the-fact.the-fact.

⎟⎠⎞⎜⎝

⎛360Days.075) - 00)(LIBOR($50,000,0


Ch. 12: 6


The Pricing and Valuation of Interest Rate SwapsThe Pricing and Valuation of Interest Rate Swaps How is the fixed rate determined?How is the fixed rate determined? A digression on floating-rate securities. The price A digression on floating-rate securities. The price

of a LIBOR zero coupon bond for maturity of tof a LIBOR zero coupon bond for maturity of tii days days isis

• Starting at the maturity date and working back, Starting at the maturity date and working back, we see that the price is par on each coupon date. we see that the price is par on each coupon date. See See Figure 12.3, p. 430Figure 12.3, p. 430..

/360))(t(tL11)(tB

ii0i0 +

=


Ch. 12: 7


The Pricing and Valuation of Interest Rate Swaps The Pricing and Valuation of Interest Rate Swaps (continued)(continued)

By adding the notional principals at the end, we can By adding the notional principals at the end, we can separate the cash flow streams of an interest rate separate the cash flow streams of an interest rate swap into those of a fixed-rate bond and a floating-swap into those of a fixed-rate bond and a floating-rate bond.rate bond.

See See Figure 12.4, p. 431Figure 12.4, p. 431.. The value of a fixed-rate bond (q = days/360):The value of a fixed-rate bond (q = days/360):

∑=

+=n

1in0i0FXRB )(tB)(tRqBV


Ch. 12: 8



The value of a floating-rate bondThe value of a floating-rate bond

At time t, between 0 and 1,At time t, between 0 and 1,

The value of the swap (pay fixed, receive floating) The value of the swap (pay fixed, receive floating) is, therefore,is, therefore,

date)payment aor 0 (at time 1VFLRB =

1) and 0 datespayment (between t)/360)(t(tL1)q(tL1V

11t10

FLRB −++

=

FXRBFLRB VVVS −=


Ch. 12: 9


The Pricing and Valuation of Interest Rate Swaps (continued)The Pricing and Valuation of Interest Rate Swaps (continued) To price the swap at the start, set this value to zero and solve for RTo price the swap at the start, set this value to zero and solve for R

See See Table 12.2, p. 433Table 12.2, p. 433 for an example. for an example. Note how dealers quote as a spread over Treasury rate.Note how dealers quote as a spread over Treasury rate. To value a swap during its life, simply find the difference between To value a swap during its life, simply find the difference between

the present values of the two streams of payments. See the present values of the two streams of payments. See Table 12.3, p. 434Table 12.3, p. 434. Market value reflects the economic value, is . Market value reflects the economic value, is necessary for accounting, and gives an indication of the credit necessary for accounting, and gives an indication of the credit risk.risk.

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛−

⎟⎟⎠⎞

⎜⎜⎝⎛

=

∑=

n

1ii0

n0

)(tB

)(tB11Rq


Ch. 12: 10



A basis swap is equivalent to the difference between A basis swap is equivalent to the difference between two plain vanilla swaps based on different rates:two plain vanilla swaps based on different rates:

• A swap to pay T-bill, receive fixed, plusA swap to pay T-bill, receive fixed, plus• A swap to pay fixed, receive LIBOR, equalsA swap to pay fixed, receive LIBOR, equals• A swap to pay T-bill, receive LIBOR, plus pay A swap to pay T-bill, receive LIBOR, plus pay

the difference between the LIBOR and T-bill the difference between the LIBOR and T-bill fixed ratesfixed rates

• See See Tables 12.4Tables 12.4 and and 12.5, p. 43612.5, p. 436 for examples of for examples of pricing and valuation of a basis swap.pricing and valuation of a basis swap.


Ch. 12: 11


Interest Rate Swap StrategiesInterest Rate Swap Strategies See See Figure 12.5, p. 437Figure 12.5, p. 437 for example of converting for example of converting

floating-rate loan into fixed-rate loanfloating-rate loan into fixed-rate loan Other types of swapsOther types of swaps

• Index amortizing swapsIndex amortizing swaps• Diff swapsDiff swaps• Constant maturity swapsConstant maturity swaps


Ch. 12: 12

Currency Swaps

Example: Reston Technology enters into currency Example: Reston Technology enters into currency swap with GSI. Reston will pay euros at 4.35% based swap with GSI. Reston will pay euros at 4.35% based on NP of on NP of €€10 million semiannually for two years. GSI 10 million semiannually for two years. GSI will pay dollars at 6.1% based on NP of $9.804 million will pay dollars at 6.1% based on NP of $9.804 million semiannually for two years. Notional principals will be semiannually for two years. Notional principals will be exchanged.exchanged.

See See Figure 12.6, p. 440Figure 12.6, p. 440.. Note the relationship between interest rate and currency Note the relationship between interest rate and currency

swaps in swaps in Figure 12.7, p. 441Figure 12.7, p. 441..


Ch. 12: 13

Currency Swaps (continued) Pricing and Valuation of Currency SwapsPricing and Valuation of Currency Swaps

Let dollar notional principal be NPLet dollar notional principal be NP$$. Then euro notional principal . Then euro notional principal is NPis NP€€ = 1/S = 1/S00 for every dollar notional principal. Here euro for every dollar notional principal. Here euro notional principal will be notional principal will be €10 million. With€10 million. With S S00 = $0.9804, NP = $0.9804, NP$$ = = $9,804,000.$9,804,000.

For fixed payments, we use the fixed rate on plain vanilla swaps For fixed payments, we use the fixed rate on plain vanilla swaps in that currency, Rin that currency, R$$ or R or R€€..

No pricing is required for the floating side of a currency swap.No pricing is required for the floating side of a currency swap. See See Table 12.6, p. 443Table 12.6, p. 443.. During the life of the swap, we value it by finding the difference 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 in the present values of the two streams of payments, adjusting for the notional principals, and converting to a common currency. the notional principals, and converting to a common currency. Assume new exchange rate is $0.9790 three months later.Assume new exchange rate is $0.9790 three months later.

See See Table 12.7, p. 444Table 12.7, p. 444 for calculations of values of streams of for calculations of values of streams of payments per unit notional principal.payments per unit notional principal.


Ch. 12: 14

Currency Swaps (continued) Pricing and Valuation of Currency Swaps (continued)Pricing and Valuation of Currency Swaps (continued)

Dollars fixed for NP of $9.804 million = Dollars fixed for NP of $9.804 million = $9,804,000(1.01132335) = $9,915,014$9,804,000(1.01132335) = $9,915,014

Dollars floating for NP of $9.804 million = Dollars floating for NP of $9.804 million = $9,804,000(1.013115) = $9,932,579$9,804,000(1.013115) = $9,932,579

Euros fixed for NPEuros fixed for NP of €10 million = of €10 million = €10,000,000(1.00883078) = €10,088,308€10,000,000(1.00883078) = €10,088,308

Euros floating for NP of €10 million = Euros floating for NP of €10 million = €10,000,000(1.0091157) = €10,091,157€10,000,000(1.0091157) = €10,091,157


Ch. 12: 15

Currency Swaps (continued) Pricing and Valuation of Currency Swaps (continued)Pricing and Valuation of Currency Swaps (continued)

Value of swap to pay Value of swap to pay €€ fixed, receive $ fixed fixed, receive $ fixed• $9,915,014 - $9,915,014 - €10,088,308($0.9790/€) = $38,560€10,088,308($0.9790/€) = $38,560

Value of swap to pay € fixed, receive $ floatingValue of swap to pay € fixed, receive $ floating• $9,932,579 - €10,088,308($0.9790/€) = $56,125$9,932,579 - €10,088,308($0.9790/€) = $56,125

Value of swap to pay € floating, receive $ fixedValue of swap to pay € floating, receive $ fixed• $9,915,014 - €10,091,157($0.9790/€) = $35,771$9,915,014 - €10,091,157($0.9790/€) = $35,771

Value of swap to pay € floating, receive $ floatingValue of swap to pay € floating, receive $ floating• $9,932,579 - €10,091,157($0.9790/€) = $53,336$9,932,579 - €10,091,157($0.9790/€) = $53,336


Ch. 12: 16

Currency Swaps (continued) Currency Swap StrategiesCurrency Swap Strategies

A typical case is a firm borrowing in one currency A typical case is a firm borrowing in one currency and wanting to borrow in another. See and wanting to borrow in another. See Figure 12.8, p. 448Figure 12.8, p. 448 for Reston-GSI example. for Reston-GSI example. Reston could get a better rate due to its familiarity Reston could get a better rate due to its familiarity to GSI and also due to credit risk.to GSI and also due to credit risk.

Also a currency swap be used to convert a stream of Also a currency swap be used to convert a stream of foreign cash flows. This type of swap would foreign cash flows. This type of swap would probably have no exchange of notional principals.probably have no exchange of notional principals.


Ch. 12: 17

Equity Swaps CharacteristicsCharacteristics

One party pays the return on an equity, the other One party pays the return on an equity, the other pays fixed, floating, or the return on another equitypays fixed, floating, or the return on another equity

Rate of return is paid, so payment can be negativeRate of return is paid, so payment can be negative Payment is not determined until end of periodPayment is not determined until end of period


Ch. 12: 18

Equity Swaps (continued) The Structure of a Typical Equity SwapThe Structure of a Typical Equity Swap

Cash flow to party paying stock and receiving fixedCash flow to party paying stock and receiving fixed

Example: IVM enters into a swap with FNS to pay Example: IVM enters into a swap with FNS to pay S&P 500 Total Return and receive a fixed rate of S&P 500 Total Return and receive a fixed rate of 3.45%. The index starts at 2710.55. Payments 3.45%. The index starts at 2710.55. Payments every 90 days for one year. Net payment will be every 90 days for one year. Net payment will be

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

⎟⎠⎞⎜⎝

⎛

period settlementover stock on Return 365or 360

Daysrate) (Fixedprincipal) (Notional

⎟⎟⎠⎞

⎜⎜⎝⎛

−⎟⎠⎞⎜⎝

⎛ period settlementover index stock on Return 36090.034500)($25,000,0


Ch. 12: 19

Equity Swaps (continued) The Structure of a Typical Equity Swap (continued)The Structure of a Typical Equity Swap (continued)

The fixed payment will beThe fixed payment will be• $25,000,000(.0345)(90/360) = $215,625$25,000,000(.0345)(90/360) = $215,625

See See Table 12.8, p. 451Table 12.8, p. 451 for example of payments. for example of payments. The first equity payment isThe first equity payment is

So the first net payment is IVM pays $285,657.So the first net payment is IVM pays $285,657.

282,501$12710.552764.900$25,000,00 =⎟⎠

⎞⎜⎝⎛ −


Ch. 12: 20

Equity Swaps (continued) The Structure of a Typical Equity Swap (continued)The Structure of a Typical Equity Swap (continued)

If IVM had received floating, the payoff formula If IVM had received floating, the payoff formula would bewould be

If the swap were structured so that IVM pays the If the swap were structured so that IVM pays the return on one stock index and receives the return on return on one stock index and receives the return on another, the payoff formula would beanother, the payoff formula would be

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

⎟⎠⎞⎜⎝

⎛

period settlementover stock on Return 360Days(LIBOR)principal) (Notional

( )indexstock other on Return -index stock oneon Return principal) (Notional


Ch. 12: 21

Equity Swaps (continued) Pricing and Valuation of Equity SwapsPricing and Valuation of Equity Swaps

For a swap to pay fixed and receive equity, we replicate as follows:For a swap to pay fixed and receive equity, we replicate as follows:• Invest $1 in stockInvest $1 in stock• Issue $1 face value loan with interest at rate R. Pay interest on Issue $1 face value loan with interest at rate R. Pay interest on

each swap settlement date and repay principal at swap each swap settlement date and repay principal at swap termination date. Interest based on q = days/360.termination date. Interest based on q = days/360.

• Example: Assume payments on days 180 and 360.Example: Assume payments on days 180 and 360.– On day 180, stock worth SOn day 180, stock worth S180180/S/S00. Sell stock and withdraw . Sell stock and withdraw

SS180180/S/S00 - 1 - 1– Owe interest of RqOwe interest of Rq– Overall cash flow is SOverall cash flow is S180180/S/S00 – 1 – Rq, which is equivalent to – 1 – Rq, which is equivalent to

the first swap payment. $1 is left over. Reinvest in the the first swap payment. $1 is left over. Reinvest in the stock.stock.


Ch. 12: 22

Equity Swaps (continued) Pricing and Valuation of Equity Swaps (continued)Pricing and Valuation of Equity Swaps (continued)

On day 360, stock is worth SOn day 360, stock is worth S360360/S/S180180.. Liquidate stock. Pay back loan of $1 and interest of Liquidate stock. Pay back loan of $1 and interest of

Rq. Rq. Overall cash flow is SOverall cash flow is S360360/S/S180180 – 1 – Rq, which is – 1 – Rq, which is

equivalent to the second swap payment.equivalent to the second swap payment. The value of the position is the value of the swap. The value of the position is the value of the swap.

In general for n payments, the value at the start isIn general for n payments, the value at the start is∑=

−−n

1ii0n0 )(tBRq)(tB1


Ch. 12: 23


Setting the value to zero and solving for R givesSetting the value to zero and solving for R gives

which is the same as the fixed rate on an interest which is the same as the fixed rate on an interest rate swap. See rate swap. See Table 12.9, p. 453Table 12.9, p. 453 for pricing the for pricing the IVM swap.IVM swap.

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛−

⎟⎟⎠⎞

⎜⎜⎝⎛

=

∑=

n

1ii0

n0

)(tB)(tB1

q1R


Ch. 12: 24


To value the swap at time t during its life, consider the party To value the swap at time t during its life, consider the party paying fixed and receiving equity.paying fixed and receiving equity.

To replicate the first payment, at time tTo replicate the first payment, at time t• Purchase 1/SPurchase 1/S00 shares at a cost of (1/S shares at a cost of (1/S00)S)Stt. Borrow $1 at . Borrow $1 at

rate R maturing at next payment date.rate R maturing at next payment date.• At the next payment date (assume day 90), shares are At the next payment date (assume day 90), shares are

worth (1/Sworth (1/S00)S)S9090. Sell the stock, generating (1/S. Sell the stock, generating (1/S00)S)S9090 – 1 – 1 (equivalent to the equity payment on the swap), plus $1 (equivalent to the equity payment on the swap), plus $1 left over, which is reinvested in the stock. Pay the loan left over, which is reinvested in the stock. Pay the loan interest, Rq (which is equivalent to the fixed payment on interest, Rq (which is equivalent to the fixed payment on the swap).the swap).

• Do this for each payment on the swap.Do this for each payment on the swap.


Ch. 12: 25


The cost to do this strategy at time t isThe cost to do this strategy at time t is

This is the value of the swap. See This is the value of the swap. See Table 12.10, p. 454Table 12.10, p. 454 for an for an example of the IVM swap.example of the IVM swap.

To value the equity swap receiving floating and paying equity, To value the equity swap receiving floating and paying equity, note the equivalence tonote the equivalence to

• A swap to pay equity and receive fixed, plusA swap to pay equity and receive fixed, plus• A swap to pay fixed and receive floating.A swap to pay fixed and receive floating.

So we can use what we already know.So we can use what we already know.

∑=

−−⎟⎟⎠⎞

⎜⎜⎝⎛ n

1iitnt

0

t )(tBRq)(tBSS


Ch. 12: 26


Using the new discount factors, the value of the fixed payments (plus Using the new discount factors, the value of the fixed payments (plus hypothetical notional principal) ishypothetical notional principal) is

• .0345(90/360)(0.9971 + 0.9877 + 0.9778 + 0.9677) + 1(0.9677) = .0345(90/360)(0.9971 + 0.9877 + 0.9778 + 0.9677) + 1(0.9677) = 1.001598841.00159884

The value of the floating payments (plus hypothetical notional principal) isThe value of the floating payments (plus hypothetical notional principal) is• (1 + .03(90/360))(0.9971) = 1.00457825(1 + .03(90/360))(0.9971) = 1.00457825

The plain vanilla swap value is, thus,The plain vanilla swap value is, thus,• 1.00457825 – 1.00159884 = -0.002979411.00457825 – 1.00159884 = -0.00297941

For a $25 million notional principal, For a $25 million notional principal, • $25,000,000(-0.00297941) = -$74,485$25,000,000(-0.00297941) = -$74,485

So the value of the equity swap is (using -$227,964, the value of the equity So the value of the equity swap is (using -$227,964, the value of the equity swap to pay fixed)swap to pay fixed)

• -$227,964 - $74,485 = -$302,449-$227,964 - $74,485 = -$302,449


Ch. 12: 27


For swaps to pay one equity and receive another, For swaps to pay one equity and receive another, replicate by selling short one stock and buy the replicate by selling short one stock and buy the other. Each period withdraw the cash return, other. Each period withdraw the cash return, reinvesting $1. Cover short position by buying it reinvesting $1. Cover short position by buying it back, and then sell short $1. So each period start back, and then sell short $1. So each period start with $1 long one stock and $1 short the other.with $1 long one stock and $1 short the other.

For the IVM swap, suppose we pay the S&P and For the IVM swap, suppose we pay the S&P and receive NASDAQ, which starts at 2710.55 and goes receive NASDAQ, which starts at 2710.55 and goes to 2739.60. The value of the swap isto 2739.60. The value of the swap is

03312974.055.271060.2739

24.183571.1915

=⎟⎠⎞⎜⎝

⎛−⎟⎠⎞⎜⎝

⎛


Ch. 12: 28


For $25 million notional principal, the value is For $25 million notional principal, the value is • $25,000,000(0.03312974) = $828,244$25,000,000(0.03312974) = $828,244


Ch. 12: 29

Equity Swaps (continued) Equity Swap StrategiesEquity Swap Strategies

Used to synthetically buy or sell stockUsed to synthetically buy or sell stock See See Figure 12.9, p. 456Figure 12.9, p. 456 for example. for example. Some risksSome risks

• defaultdefault• tracking errortracking error• cash flow shortagescash flow shortages


Ch. 12: 30

Some Final Words About Swaps Similarities to forwards and futuresSimilarities to forwards and futures Offsetting swapsOffsetting swaps

Go back to dealerGo back to dealer Offset with another counterpartyOffset with another counterparty Forward contract or option on the swapForward contract or option on the swap


Ch. 12: 31

Summary


Ch. 12: 32

(Return to text slide)


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Chapter 12: Swaps

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