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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 $P(1+r)^{t}$

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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.

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