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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
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 : 2
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
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 : 3
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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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 : 4
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
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 : 5
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
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 : 6
Interest Rate Swaps (continued)
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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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 : 7
Interest Rate Swaps (continued)
The Pricing and Valuation of Interest Rate Swaps
(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.
See Figure 12.4, p. 431.
The value of a fixed-rate bond (q = days/360):
=
+=n
1i
n0i0FXRB )(tB)(tRqBV
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 : 8
Interest Rate Swaps (continued)
The Pricing and Valuation of Interest Rate Swaps
(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 =
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 : 9
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 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
Interest Rate Swaps (continued)
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.
D. M. Chance An Introduction to Derivatives and Risk Management, 6th ed. Ch. 12: 11
Interest Rate Swaps (continued)
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
D. M. Chance An Introduction to Derivatives and Risk Management, 6th ed. Ch. 12: 12
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 $9.804 million
semiannually for two years. Notional principals will beexchanged.
See Figure 12.6, p. 440.
Note the relationship between interest rate and currency
swaps in Figure 12.7, p. 441.
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D. M. Chance An Introduction to Derivatives and Risk Management, 6th ed. Ch. 12: 13
Currency Swaps (continued) Pricing and Valuation of Currency Swaps
Let dollar notional principal be NP$. 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.
D. M. Chance An Introduction to Derivatives and Risk Management, 6th ed. Ch. 12: 14
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
D. M. Chance An Introduction to Derivatives and Risk Management, 6th ed. Ch. 12: 15
Currency Swaps (continued)
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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D. M. Chance An Introduction to Derivatives and Risk Management, 6th ed. Ch. 12: 16
Currency Swaps (continued)
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.
D. M. Chance An Introduction to Derivatives and Risk Management, 6th ed. Ch. 12: 17
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
D. M. Chance An Introduction to Derivatives and Risk Management, 6th ed. Ch. 12: 18
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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D. M. Chance An Introduction to Derivatives and Risk Management, 6th ed. Ch. 12: 19
Equity Swaps (continued)
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 =
D. M. Chance An Introduction to Derivatives and Risk Management, 6th ed. Ch. 12: 20
Equity Swaps (continued)
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
D. M. Chance An Introduction to Derivatives and Risk Management, 6th ed. Ch. 12: 21
Equity Swaps (continued)
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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D. M. Chance An Introduction to Derivatives and Risk Management, 6th ed. Ch. 12: 22
Equity Swaps (continued)
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
D. M. Chance An Introduction to Derivatives and Risk Management, 6th ed. Ch. 12: 23
Equity Swaps (continued)
Pricing and Valuation of Equity Swaps (continued)
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
D. M. Chance An Introduction to Derivatives and Risk Management, 6th ed. Ch. 12: 24
Equity Swaps (continued)
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 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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D. M. Chance An Introduction to Derivatives and Risk Management, 6th ed. Ch. 12: 25
Equity Swaps (continued)
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, 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
D. M. Chance An Introduction to Derivatives and Risk Management, 6th ed. Ch. 12: 26
Equity Swaps (continued)
Pricing and Valuation of Equity Swaps (continued)
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
D. M. Chance An Introduction to Derivatives and Risk Management, 6th ed. Ch. 12: 27
Equity Swaps (continued)
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 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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D. M. Chance An Introduction to Derivatives and Risk Management, 6th ed. Ch. 12: 28
Equity Swaps (continued)
Pricing and Valuation of Equity Swaps (continued)
For $25 million notional principal, the value is
$25,000,000(0.03312974) = $828,244
D. M. Chance An Introduction to Derivatives and Risk Management, 6th ed. Ch. 12: 29
Equity Swaps (continued)
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
D. M. Chance An Introduction to Derivatives and Risk Management, 6th ed. Ch. 12: 30
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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D. M. Chance An Introduction to Derivatives and Risk Management, 6th ed. Ch. 12: 31
Summary
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