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Use a series of forward exchange rates to value a currency swap.

FRM® Part I · Financial Markets and Products · Swaps · FRM1.FM.20.k

The idea

A currency swap can also be valued as a portfolio of forward contracts, one for every exchange of cash flows, including the final principal exchange. Quote exchange rates as domestic units per foreign unit and use continuous compounding. The forward rate for time t is then Ft = S0 × e^((rD − rF) × t). Here S0 is the spot rate, and rD and rF are the domestic and foreign rates. For the party receiving domestic cash flows and paying foreign ones, each exchange is worth: Value = (domestic cash flow − Ft × foreign cash flow) × e^(−rD × t). Each foreign amount is converted at its forward rate, and the net amount is discounted at the domestic rate. Summing these values gives the swap value, which equals the two-bond value. Each foreign payment is converted at the forward rate for its own date, not at spot, because that rate can be locked in today. The typical errors are using the spot rate for every date, writing the exponent with the rates reversed, and discounting at the foreign rate after already converting at the forward.

Formulas

Forward exchange rate for a currency swap cash flow

Ft = S0 × e^((rD − rF) × t)

S0
the spot rate, domestic units per foreign unit
rD
the domestic continuously compounded rate
rF
the foreign continuously compounded rate
t
the time of the cash flow, in years

Quote in domestic per foreign. Writing the exponent as rF − rD is the usual slip.

Value of one exchange in a currency swap

value = (domestic cash flow − Ft × foreign cash flow) × e^(−rD × t)

domestic cash flow
the amount received in domestic currency
Ft
the forward rate for date t
foreign cash flow
the amount paid in foreign currency
rD
the domestic continuously compounded rate
t
the time of the exchange, in years

For the party receiving domestic. Discount at the domestic rate after converting at the forward, and include the final principal exchange.

A practice question

From the question bank, with the reason each wrong option is wrong.

The spot exchange rate is 1.10 USD per EUR, with USD and EUR continuously compounded rates of 4% and 3%. What is the 2-year forward rate (USD per EUR)?

  1. 1.0782Why not: Writes the exponent as (r_F - r_D) x T, with the rates reversed.
  2. 1.1916Why not: Uses only the USD rate in the exponent, ignoring the EUR rate.
  3. 1.1222 - correct
  4. 1.1449Why not: Uses the rate differential for one year rather than for two years.

F = S0 x exp((r_D - r_F) x T) = 1.10 x exp(0.01 x 2) = 1.1222.

A harder one

The spot rate is 1.10 USD per EUR, with continuously compounded USD and EUR rates of 4% and 3% (flat). What is the USD present value of a EUR 2.0 million payment due in 3 years, valued with the forward rate and discounted at the USD rate (USD million)?

  1. 2.01 - correct
  2. 2.27Why not: Converts at the forward rate but does not discount.
  3. 2.07Why not: Converts at the forward rate but discounts at the EUR rate instead of the USD rate.
  4. 1.95Why not: Converts at spot but discounts at the USD rate, ignoring the forward rate.

F3 = 1.10 x exp(0.01 x 3) = 1.1335. PV = 2.0 x F3 x exp(-0.04 x 3) = 2.01 USD million; the exponents combine to exp(-0.03 x 3), so this equals the spot rate times the EUR present value at the EUR rate.

The hardest tier

You pay USD 4.5% on USD 50 million and receive EUR 3.0% on EUR 44 million, annually for 2 years, with principals exchanged at maturity. Forward rates are 1.1450 and 1.1900 USD per EUR for years 1 and 2, and USD discount factors are 0.9615 and 0.9246. What is the swap's value to you (USD million), closest to?

  1. -0.84Why not: Reverses the sign, valuing the swap for the party that receives USD.
  2. -2.98Why not: Converts the EUR flows at the 1.10 spot rate instead of the forward rates.
  3. 0.94Why not: Does not discount the forward values at the USD discount factors.
  4. 0.84 - correct

Year 1: (1.1450 x 1.32 - 2.25) x 0.9615 = -0.710. Year 2: (1.1900 x 45.32 - 52.25) x 0.9246 = 1.554. Total = 0.84.

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