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Recall the four types of currency swaps: (1) pay one currency fixed, receive the other fixed, (2) pay one currency fixed, receive the other floating, (3) pay one currency floating, receive the other fixed, and (4) pay one currency floating, receive the other floating. In determining the fixed rate on a swap, we must keep in mind one major point: The fixed rate is the rate that makes the present value of the payments made equal the present value of the payments received. In the fourth type of currency swap mentioned here, both sides pay floating so there is no need to find a fixed rate. But all currency swaps have two notional principals, one in each currency. We can arbitrarily set the notional principal in the domestic currency at one unit. We then must determine the equivalent notional principal in the other currency. This task is straightforward: We simply convert the one unit of domestic currency to the equivalent amount of foreign currency, dividing 1.0 by the exchange rate.
Consider the first type of currency swap, in which we pay the foreign currency at a fixed rate and receive the domestic currency at a fixed rate. What are the two fixed rates? We will see that they are the fixed rates on plain vanilla interest rate swaps in the respective countries.
Because we know that the value of a floating-rate security with $1 face value is $1, we know that the fixed rate on a plain vanilla interest rate swap is the rate on a $1 par bond in the domestic currency. That rate results in the present value of the interest payments and the hypothetical notional principal being equal to 1.0 unit of the domestic currency. Moreover, for a currency swap, the notional principal is typically paid, so we do not even have to call it hypothetical. We know that the fixed rate on the domestic leg of an interest rate swap is the appropriate domestic fixed rate for a currency swap in which the domestic notional principal is 1.0 unit of the domestic currency.
What about the fixed rate for the foreign payments on the currency swap? To answer that question, let us assume the point of view of a resident of the foreign country. Given the term structure in the foreign country, we might be interested in first pricing plain vanilla interest rate swaps in that country. So, we know that the fixed rate on interest rate swaps in that country would make the present value of the interest and principal payments equal 1.0 unit of that currency.
Now let us return to our domestic setting. We know that the fixed rate on interest rate swaps in the foreign currency makes the present value of the foreign interest and principal payments equal to 1.0 unit of the foreign currency. We multiply by the spot rate, So, to obtain the value of those payments in our domestic currency: 1.0 times So equals So, which is now in terms of the domestic currency. This amount does not equal the present value of the domestic payments, but if we set the notional principal on the foreign side of the swap equal to l/So, then the present value of the foreign payments will be So(l/So) = 1.0 unit of our domestic currency, which is what we want.