We need to determine the fixed rate, which is the rate that makes the present value of the payments made equal to the present value of the payments received.
Obviously for the last type of currency swap, in which both sides pay floating, we do not need to price the swap because both sides pay a floating rate.
Note the relationship between interest rate and currency swaps (c1: currency 1; c2: currency 2).
We price currency swaps in the same way that we learned how to price interest rate swaps.
Reston Technology enters into currency swap with GSI. Reston will pay Euros at 4.35% based on NP of €10 million semi-annually for two years. GSI will pay dollars at 6.1% based on NP of $9.804 million semi-annually for two years. Notional principals will be exchanged.
The following term structures, discount bond prices, and the resulting swap fixed rates are given below:
Let dollar notional principal be NP$. Then Euro notional principal is NP€ = 1/S0 for every dollar notional principal. Here Euro notional principal will be €10 million. With S0 = $0.9804, NP$ = $9,804,000.
During the life of the swap, we value it by finding the difference in the present values of the two streams of payments, adjusting for the notional principals, and converting to a common currency. Assume new exchange rate is $0.9790 three months later.
Three months into the Reston-GSI swap, the new term structures and zero coupon bond prices for dollars are:
The present value of the dollar fixed payments of 0.061 (180/360) plus a $1 notional principal is: 0.061 (180/360) (0.9860 + 0.9563 + 0.9259 + 0.8965) + 1 (0.8965) = 1.01132335.
If the swap had been designed with floating payments, the present value of the dollar floating payments would be found by discounting the next floating payment, which is at the original 180-day floating rate of 5.5%, plus the market value of the floating-rate bond on the next payment day: [1 + 0.055 (180/360)] 0.9860 = 1.013115.
The new term structure and discount bond prices for the euro are:
The present value of the Euro fixed payments of 0.0435 (180/360) plus a €1 notional principal is: 0.0435 (180/360) (0.9903 + 0.9688 + 0.9467 + 0.9255) + 1 (0.9255) = 1.00883078.
If the payments were floating, the present value of the euro floating payments would be found by discounting the next floating payment, which is at the original 180-day floating rate of 3.8%, plus the market value of the floating-rate bond on the next payment date: [1.0 + 0.038 (180/360)] 0.9903 = 1.0091157.
Valuation of Currency Swaps
What is the annualized fixed rate in Euros?
First we compute the discount factors:
B0euro(180) = 1 / (1 + 0.06 x 180/360) = 0.9709.
B0euro(360) = 1 / (1 + 0.066 x 360/360) = 0.9381.
The fixed rate in Euro is: (1/0.5) x (1 - 0.9381) / (0.9709 + 0.9381) = 0.0648.
The annualized fixed rates in US$ and Euros are 7.84% and 6.48%, respectively.
Ninety days after, the term structure is as follows:
If the current exchange rate is 0.7, what is the market value of the swap to pay dollar fixed and receive euro fixed?
First we compute the dollar discount factors:
B90(180) = 1 / (1 + 0.071 x 90/360) = 0.9826.
B90(360) = 1 / (1 + 0.074 x 270/360) = 0.9474.
Fixed dollar payments:
Floating dollar payments:
Then we compute the Euro discount factors:
Fixed Euro payments:
Floating Euro payments:
The Euro notional principal, established at the start of the swap, is 1/0.75 = 1.3333 euro.
Now we obtain the following values for the four swaps: