Currency Swaps

Derivatives 4 min readModule 6 of 7Save

What you'll learn

LOS 28.f
Describe how currency swaps are priced, and calculate and interpret their no arbitrage value.

Currency Swaps

Determining the Fixed Rate and Foreign Notional Principal

Consider two currencies, the U.S. dollar ($) and the British pound (£), where the exchange rate is currently $2 per £ or £0.5 per $. The interest rates in a currency swap are simply the swap rates calculated from each country’s yield curve in the relevant country’s currency. Don’t forget: With currency swaps, there are two yield curves and two swap fixed rates, one for each currency.

The principal amounts of the fixed rate obligations must be adjusted for the current exchange rate. We need $2.00 to equal £1.00, so these are the notional amounts, exchanged at the inception of the swap and returned at the termination of the swap. For example, if the notional principal amount of the $ side of the swap is $25 million, the £ notional principal amount will be £12.5 million.

Worked example: Calculating the fixed rate and notional principal on a currency swap

In a previous example, we determined that the fixed rate on a 1 year quarterly $5,000,000 interest rate swap, given the following set of spot MRRs, was 1.1% quarterly, or 4.4% on an annual basis.

R_{90}^{$} = 0.030, R_{180}^{$} = 0.035, R_{270}^{$} = 0.040, R_{360}^{$} = 0.045

The comparable set of £ rates are:

R_{90}^{£} = 0.04, R_{180}^{£} = 0.05, R_{270}^{£} = 0.06, R_{360}^{£} = 0.07

The current exchange rate is £0.50 per $.

Determine the fixed rate on a 1 year £ interest rate swap. Then determine the notional £ principal amount and the quarterly cash flows on a Pay $ fixed, receive £ fixed currency swap.

Solution

First calculate the £ discount factors. Don’t forget to convert from annualized rates to per period rates.

Z_{90} Z_{180} Z_{270} Z_{360} = \frac{1}{1 + 0.04 ( 90/360 )} = 0.99010 = \frac{1}{1 + 0.05 ( 180/360 )} = 0.97561 = \frac{1}{1 + 0.06 ( 270/360 )} = 0.95694 = \frac{1}{1 + 0.07 ( 360/360 )} = 0.93458

The quarterly fixed rate on the £ swap is:

\frac{1 - 0.93458}{0.99010 + 0.97561 + 0.95694 + 0.93458} = 0.017 = 1.7%

The fixed rate on the £ swap in annual terms is:

1.7% \times \frac{360}{90} = 6.8%

The notional £ principal amount of the swap would be:

$5, 000, 000 \times £0.50 per dollar = £2, 500, 000

At the initiation of the swap, we would exchange £2,500,000 for $5,000,000. We would pay 1.1% quarterly on the $5,000,000 notional principal ($55,000) and receive 1.7% on £2,500,000 quarterly (£42,500). At the end of one year, we would exchange the original principal amounts.

The value to any party in a currency swap is the present value of the cash flows they expect to receive minus the present value of the cash flows they are obligated to pay.

Worked example: Calculating the value of a currency swap after initiation

Use the data on the $ and £ interest rate swaps in the previous examples to answer this question. After 300 days, the 60 day $ interest rate is 5.4%, the 60 day £ interest rate is 6.6%, and the exchange rate is £0.52 per $. Calculate the value of a $5,000,000 swap to the counterparty that receives $ fixed and pays £ fixed.

Solution

Recall that the $ fixed rate was calculated as 4.4% (or 1.1% per quarter) and the £ fixed rate was calculated as 6.8% (or 1.7% per quarter). After 300 days, the only cash flows remaining are the last interest payments and the principal payments in 60 days. We want to find the present value of those cash flows, so we need the 60 day discount factors based on the current 60 day rates:

Z_{60}^{$} Z_{60}^{£} = \frac{1}{1 + 0.054 ( 60/360 )} = 0.99108 = \frac{1}{1 + 0.066 ( 60/360 )} = 0.98912

First calculate the value after 300 days of the $ fixed payments. It is equivalent to the value of a $5,000,000 bond that matures in 60 days and makes a coupon payment in 60 days of $55,000 ($5,000,000 × 0.011). The value in $ of the $ fixed side is the present value of $5,055,000 discounted using the 60 day $ discount factor of 0.99108:

value of dollar fixed side in dollars = 0.99108 \times $5, 055, 000 = $5, 009, 909

Next, calculate the value after 300 days of the £ fixed payments. The value in £ of the £ fixed side is the present value of £2,542,500 discounted using the 60 day £ discount factor of 0.98912:

value of sterling fixed side in sterling = 0.98912 \times £2, 542, 500 = £2, 514, 838

This last step is to convert that amount into $ at the current spot exchange rate of £0.52 per $, to find the value of the £ fixed side in $:

value of sterling fixed side in dollars = \frac{£2 , 514 , 838}{0.52} = $4, 836, 227

Finally, the value of the receive $ fixed, pay £ fixed side of the swap is equal to the value of the $ fixed side minus the £ fixed side, or $5,009,909 − $4,836,227 = $173,682.

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