Investments, tenth edition

  Swap Pricing 

 How can the fair swap rate be determined? For example, how would we know that an 

exchange of LIBOR is a fair trade for a fixed rate of 6%? Or, what is the fair swap 

rate between dollars and pounds for a foreign exchange swap? To answer these ques-

tions we can exploit the analogy between a swap agreement and forward or futures 

contract. 

 Consider a swap agreement to exchange dollars for pounds for one period only. Next 

year, for example, one might exchange $1 million for £.5 million. This is no more than a 

simple forward contract in foreign exchange. The dollar-paying party is contracting to buy 

British pounds in 1 year for a number of dollars agreed to today. The forward exchange 

rate for 1-year delivery is  F  

1

   5  $2.00/pound. We know from the interest rate parity rela-

tionship that this forward price should be related to the spot exchange rate,  E  

0

 , by the for-

mula  F  

1

   5   E  

0

 (1  1   r  

US

 )/(1  1   r  

UK

 ). Because a one-period swap is in fact a forward contract, 

the fair swap rate is also given by the parity relationship. 

 Now consider an agreement to trade foreign exchange for two periods. This agreement 

could be structured as a portfolio of two separate forward contracts. If so, the forward price 

for the exchange of currencies in 1 year would be  F  

1

   5   E  

0

 (1  1   r  

US

 )/(1  1   r  

UK

 ), while the 

forward price for the exchange in the second year would be  F  

2

   5   E  

0

 [(1  1   r  

US

 )/(1  1   r  

 UK 

 )] 

2

 . 

As an example, suppose that  E  

0

   5  $2.03/pound,  r  

US

   5  5%, and  r  

UK

   5  7%. Then, using the 

parity relationship, prices for forward delivery would be  F  

1

   5  $2.03/£  3  (1.05/1.07)  5  

$1.992/£ and  F  

2

   5  $2.03/£  3  (1.05/1.07) 

2

   5  $1.955/£.  Figure 23.8, panel A  illustrates this 

sequence of cash exchanges assuming that the swap calls for delivery of one pound in each 

year. Although the dollars to be paid in each of the 2 years are known today, they differ 

from year to year.  

 In contrast, a swap agreement to exchange currency for 2 years would call for a fixed 

exchange rate to be used for the duration of the swap. This means that the same number of 

dollars would be paid per pound in each year, as illustrated in  Figure 23.8, panel B . Because 

the forward prices for delivery in each of the next 2 years are $1.992/£ and $1.955/£, the 

fixed exchange rate that makes the two-period swap a fair deal must be between these two 

values. Therefore, the dollar payer underpays for the pound in the first year (compared to 

the forward exchange rate) and overpays in the second year. Thus, the swap can be viewed 

as a portfolio of forward transactions, but instead of each transaction being priced indepen-

dently, one forward price is applied to all of the transactions. 

 Given this insight, it is easy to determine the fair swap price. If we were to purchase 

one pound per year for 2 years using two independent forward agreements, we would 

pay  F  

1

  dollars in 1 year and  F  

2

  dollars in 2 years. If instead we enter a swap, we pay a 

constant rate of  F * dollars per pound. Because both strategies must be equally costly, we 

conclude that   

F

1

1

1 y

1

1

F

2

(1

1 y

2

)

2

5

F*

1

1 y

1

1

F*

(1

1 y

2

)

2

 

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