Currency Swaps
Value a currency swap during its life as two bonds, one in each currency, with the foreign bond converted at today's spot rate. Find each currency's par fixed rate and the spot rate at which the swap is worth zero.
Learning focus: receiving one currency and paying the other is like holding a bond in the first and having issued one in the second, so the swap gains when the currency you receive strengthens or its interest rates fall.
List only the payments still to come, in order. Time is years from today; τ is the fraction of a year the payment covers (1 annual, 0.5 semiannual). Both currencies pay on the same dates. On a reset date the first payment is one full period away (time = τ); between resets it falls within the current period (time less than τ), and a floating leg pays the MRR already set on that date.
A new swap sets the EUR notional at USD notional ÷ spot, so the initial exchange is fair. With each fixed rate at its own currency's par rate, both legs are worth their notionals and the swap starts at zero value. The two par rates differ because the two currencies' interest rates differ, not because one side overpays.
| Date | Time | USD DF | EUR DF | USD rate | EUR rate | USD flow | EUR flow | PV in USD |
|---|---|---|---|---|---|---|---|---|
| Initial exchange (at the start) | start | −110,000,000.00 | 100,000,000.00 | past | ||||
| Payment 1 | 1.00 | 0.9662 | 0.9804 | 3.4000% | 2.1000% | 3,740,000.00 | −2,100,000.00 | 1,276,804.60 |
| Payment 2 + re-exchange | 2.00 | 0.9311 | 0.9597 | 3.4000% | 2.1000% | 113,740,000.00 | −102,100,000.00 | −5,310,080.95 |
| Swap value to you (sum of PVs) | −4,033,276.35 | |||||||
Positive flows are received, negative ones paid. The initial exchange happened when the swap began, so it has no value today. The rate columns show the rate each payment uses. A floating payment not yet set is shown at the forward rate from that currency's discount factors, (DFᵢ₋₁ ÷ DFᵢ − 1) ÷ τᵢ, and its actual size is set at its reset; between resets the first floating payment uses the MRR already set. Each PV converts the EUR flow at today's spot of 1.1350.
The value is a straight line in the spot rate whose slope is −B(EUR): you pay EUR, so a stronger EUR lowers your value. It crosses zero at B(USD) ÷ B(EUR).