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D . M. C ha nc e A n I nt roduct ion to D er ivat ives and R is k Managem ent, 6th ed . C h. 12 : 1

Chapter 12: Swaps

I once had to explain to my father that the bank didnt reallymake its money taking deposits and lending out money to

poor folk so they could buy houses. I explained that the

bank actually traded for a living.

Stan Jonas

Derivatives Strategy, April, 1998, p. 19


Important Concepts in Chapter 12

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


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 principal

Settlement date, settlement period

Credit risk

Dealer market

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

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Interest Rate Swaps

The Structure of a Typical Interest Rate Swap

Example: On December 15 XYZ enters into $50million NP 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

365or360

Daysrate)Fixed-(LIBORprincipal)(Notional


Interest Rate Swaps (continued)

The Structure of a Typical Interest Rate Swap

(continued)

The payments in this swap are

Payments are netted.

See Figure 12.2, p. 428 for payment pattern

See Table 12.1, p. 429 for sample of payments after-

the-fact.

360

Days.075)-00)(LIBOR($50,000,0



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 t i daysis

Starting at the maturity date and working back,

we see that the price is par on each coupon date.

See Figure 12.3, p. 430.

/360))(t(tL1

1)(tB

ii0i0

+=

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(continued)By adding the notional principals 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.


The value of a fixed-rate bond (q = days/360):

=

+=n

1i

n0i0FXRB )(tB)(tRqBV




(continued)

The value of a floating-rate bond

At time t, between 0 and 1,

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

is, therefore,

date)paymentaor0(at time1VFLRB =

1)and0datespayment(betweent)/360)(t(tL1

)q(tL1V

11t

10FLRB

+

+=

FXRBFLRB VVVS =



The Pricing and Valuation of Interest Rate Swaps (continued)

To price the swap at the start, set this value to zero and solvefor R

See Table 12.2, p. 433 for an example.

Note how dealers quote as a spread over Treasury rate.

To value a swap during its life, simply find the differencebetween the present values of the two streams of payments.See Table 12.3, p. 434. Market value reflects the economicvalue, is necessary for accounting, and gives an indication ofthe credit risk.

=

=

n

1i

i0

n0

)(tB

)(tB11R

q

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D. M. Chance An Introduction to Derivatives and Risk Management, 6th ed. Ch. 12: 10


The Pricing and Valuation of Interest Rate Swaps(continued)

A basis swap is equivalent to the difference betweentwo 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 paythe difference between the LIBOR and T-billfixed rates

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



Interest Rate Swap Strategies

See Figure 12.5, p. 437 for example of converting

floating-rate loan into fixed-rate loan

Other types of swaps

Index amortizing swaps

Diff swaps

Constant maturity swaps


Currency Swaps

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 principals will beexchanged.


Note the relationship between interest rate and currency

swaps in Figure 12.7, p. 441.

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Currency Swaps (continued) Pricing and Valuation of Currency Swaps

Let dollar notional principal be NP$. Then euro notionalprincipal is NP = 1/S0 for every dollar notional principal.

Here euro notional principal will be 10 million. With S0 =$0.9804, NP$ = $9,804,000.

For fixed payments, we use the fixed rate on plain vanillaswaps in that currency, R$ or R.

No pricing is required for the floating side of a currency swap.

See Table 12.6, p. 443.

During the life of the swap, we value it by finding thedifference in the present values of the two streams ofpayments, adjusting for the notional principals, and convertingto a common currency. Assume new exchange rate is $0.9790three months later.

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


Currency Swaps (continued)

Pricing and Valuation of Currency Swaps (continued)

Dollars fixed for NP of $9.804 million =

$9,804,000(1.01132335) = $9,915,014

Dollars floating for NP of $9.804 million =

$9,804,000(1.013115) = $9,932,579

Euros fixed for NP of 10 million =

10,000,000(1.00883078) = 10,088,308

Euros floating for NP of 10 million =

10,000,000(1.0091157) = 10,091,157



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

Value of swap to pay floating, receive $ floating

$9,932,579 - 10,091,157($0.9790/) = $53,336

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Currency Swap Strategies

A typical case is a firm borrowing in one currency

and wanting to borrow in another. See Figure 12.8,

p. 448 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 would

probably have no exchange of notional principals.


Equity Swaps

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


Equity Swaps (continued)

The Structure of a Typical Equity Swap

Cash flow to party paying stock and receiving fixed

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 2710.55. Payments

every 90 days for one year. Net payment will be

periodsettlementoverstockonReturn

365or360

Daysrate)(Fixed

principal)(Notional

periodsettlementoverindexstockonReturn

360

90.034500)($25,000,0

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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, p. 451 for example of payments.

The first equity payment is

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

282,501$12710.55

2764.900$25,000,00 =



The Structure of a Typical Equity Swap (continued)

If IVM had received floating, the payoff formula

would be

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

periodsettlementoverstockonReturn

360

Days(LIBOR)

principal)(Notional

( )indexstockotheronReturn-indexstockoneonReturnprincipal)(Notional



Pricing and Valuation of Equity Swaps

For a swap to pay fixed and receive equity, we replicate asfollows:

Invest $1 in stock

Issue $1 face value loan with interest at rate R. Pay

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

Example: Assume payments on days 180 and 360.

On day 180, stock worth S180/S0. Sell stock andwithdraw S180/S0 - 1

Owe interest of Rq

Overall cash flow is S180/S0 1 Rq, which isequivalent to the first swap payment. $1 is left over.Reinvest in the stock.

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Pricing and Valuation of Equity Swaps (continued)

On day 360, stock is worth S360/S180.

Liquidate stock. Pay back loan of $1 and interest ofRq.

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 payments, the value at the start is

=

n

1i

i0n0 )(tBRq)(tB1




Setting the value to zero and solving for R gives

which is the same as the fixed rate on an interest rate

swap. See Table 12.9, p. 453 for pricing the IVM

swap.

=

=

n

1ii0

n0

)(tB

)(tB1

q

1R




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 date (assume day 90), shares areworth (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 loaninterest, Rq (which is equivalent to the fixed payment onthe swap).

Do this for each payment on the swap.

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The cost to do this strategy at time t is

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

example of the IVM swap.

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.

=

n

1i

itnt

0

t )(tBRq)(tBSS




Using the new discount factors, the value of the fixedpayments (plus hypothetical notional principal) is

.0345(90/360)(0.9971 + 0.9877 + 0.9778 + 0.9677) +1(0.9677) = 1.00159884

The value of the floating payments (plus hypothetical notionalprincipal) is

(1 + .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 principal,

$25,000,000(-0.00297941) = -$74,485

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

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




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 itback, and then sell short $1. So each period start

with $1 long one stock and $1 short the other.

For the IVM swap, suppose we pay the S&P and

receive NASDAQ, which starts at 2710.55 and goes

to 2739.60. The value of the swap is

03312974.055.2710

60.2739

24.1835

71.1915=

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For $25 million notional principal, the value is

$25,000,000(0.03312974) = $828,244



Equity Swap Strategies

Used to synthetically buy or sell stock

See Figure 12.9, p. 456 for example.

Some risks

default

tracking error

cash flow shortages


Some Final Words About Swaps

Similarities to forwards and futures

Offsetting swaps

Go back to dealer

Offset with another counterparty

Forward contract or option on the swap

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Summary


(Return to text slide)



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