Discounting Currency Swap Cash Flows at Current Yields
Summary
The document explains why a currency swap’s future payments are valued by discounting them at current market yields. Under annual compounding, a future cash flow is converted to present value by dividing by one plus the relevant yield raised to the time remaining, measured in years. Multiplying by that factor instead compounds a present amount forward and therefore answers a different question.
The worked expressions value sterling and dollar legs separately using their respective current yields. The example has an early coupon payment discounted over a quarter-year and a final coupon-plus-principal payment discounted over the swap’s remaining life. The response also clarifies that coupon amounts are calculated from the stated notional and coupon rate. This is a simplified illustration: it does not discuss exchange-rate conversion between currencies, credit adjustments, or alternative curve conventions that may be needed in a full market valuation.
Key ideas
- A swap’s value is the present value of its remaining future cash flows.
- Discounting divides each future amount by one plus the relevant yield raised to its time to payment.
- Compounding a cash flow forward does not calculate its present value.
- Each currency leg uses its own yield and cash flow schedule in the example.
- Coupon and principal payments are discounted according to their remaining maturities.
Tags
Full text
# discounting currency swap
# discounting currency swap
I need help with a currency swap problem:
Remaining life: 15 months. Exchanging interest at 10% on £20 million in Sterlings for interest 6% on $30 million in Dollars. If swap were negotiated today, the interest rates exchanged would be 4% in dollars and 7% in sterling. All are quoted with annual compounding.
What is the value of the swap to the party paying sterling?
What is the value of the swap to the party paying dollars?
The solutions guide states the answer to be:
$\frac{2}{(1.07)^{1 / 4}}+\frac{22}{(1.07)^{5 / 4}}=22.182$
and
$\frac{1.8}{(1.04)^{1 / 4}}+\frac{31.8}{(1.04)^{5 / 4}}=\$ 32.061$
When I try to use my normal: $P(1+r)^{t}$ (because it is annually), I do not get these results. Can anybody explain to me, how $\frac{P}{(1+r)^{t}}$ goes to $P(1+r)^{t}$
## Answer by teoeme139 (score 1)
https://quant.stackexchange.com/a/51679
The value of the swap is the present value of all future payments. In order to find the present value, flows must be discounted at the current yield.
So:
$PV = FV ⁄ (1+r)^t$
In this case, for the GBP receiver, 10% coupon is calculated over the 10 million face value. This coupon is to be received in 3 months. The first term in the first equation, represents the present value of this coupon as it is discounting (expressing in today's Sterlings) the value of the future coupon. It's raised to the power 1/4 or 0.25 since 3 months is one quarter of a year and is what's left to receive the payment.
The second term in the first equation represents the present value of the coupon that accrues in the last 12 months of the swap life plus the principal payment of 20 million. Now the exponential factor is 5/4, which represents the 15 months that remains to maturity. If you multiply in the form $P(1+r)^t$ you would be compounding your flows and finding what the value of your flows would be at time 't' yielding rate 'r'.
The same reasoning applies to the Dollar receiver.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.