Currency Swaps

Currency Swap Valuation

Currency swap involves exchanging both, principal and interest payments (unlike interest rate swap which involves only exchanging the interest payments, i.e. principal is notional) in one currency for principal and interest payments in another currency. The principal amounts are usually chosen to be approximately equivalent using the spot exchange rate.

For the purpose of currency swap valuation, a currency swap can be decomposed into a position in two bonds:

Vswap = BD – S0BF

BD – value in domestic currency of the domestic bond underlying the swap
BF – value in foreign currency of the foreign-denominated bond underlying the swap
S0 – current spot exchange rate

Value of the bond depends on the type of the bond:

1. fixed-rate bonds value can be calculated as:

n
Bfix = ∑ ke-riti + Le-rntn
i = 1

L – principal in swap agreement
k – fixed payment made on each payment date
ti – time when ith payments are exchanged (1 ≤ i ≤ n)
ri – interest rate when ith payments are exchanged

2. floating-rate bonds value can be calculated as:

Bfl = (L + k*)e-r1t1

L – principal in swap agreement
k* – floating rate payment (already known, defined at the time of previous payment) that will be made on the next payment date
t1 – time until the next payment
r1 – interest rate

There are three types of currency swaps regarding the interest rate for two different currencies, i.e. fixed-fixed, floating-floating, and fixed-floating. The latter is also called cross-currency swap.

For illustration see the attached example of currency swap valuation (fixed-fixed type):

The Japanese interest rate is 4% p.a. and the U.S. interest rate is 9% p.a. (both with continuous compounding). A financial institution has entered into currency swap where it receives 5% p.a. in yen and pays 8% p.a. in dollars once a year. The principals of the two currencies are $10 million and 1,200 million yen. The swap will last for another three years and the current exchange rate is 110 yen = $1.

BD = 0.8e-0.09x1 + 0.8e-0.09x2 + 10.8e-0.09x3
= 9.644 million dollars
BF = 60e-0.04x1 + 60e-0.04x2 + 1,260e-0.04x3
= 1,230.55 million yen

The value of the swap is:

1,230.55 / 110 – 9.644 = $1.543,

in case the financial institution receives yen and pays dollars (and -$1.543 vice versa).

Sources:
Hull, John, C.; Options, futures and other derivatives; Prentice-Hall International, Upper Saddle River, NJ, 2000
Boenkost, W.; Schmidt, W., M.; Cross currency swap valuation, HfB – Business School of Finance and Management, November 2004

Topic was processed by Tibor Gerliczy.

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