Suppose that 1year 2year and 3year forward prices for the Br

Suppose that 1-year, 2-year, and 3-year forward prices for the British pound are $1.76/£, $1.67/£, and $1.37/£, respectively. The 1-year, 2-year, and 3-year effective annual interest rates in the U.S. are 5.3%, 4.9%, and 4.7%. What is the fixed exchange rate in a 3-year British pound swap? (In other words, what 3-year U.S. dollar annuity is equivalent to a 3-year annuity of £1?)

Answers: a. $1.97 b. $1.80 c. $1.46 d. $1.61 e. $1.86

Please show formulas and provide an explanation - thank you.

Solution

We first find sum of present value of all 3 forward prices,

PV =1.76/(1.053) + 1.67/(1.049)^2 + 1.37/(1.047)^3 = 4.382703

We then solve for the price X such that,( 3-year U.S. dollar annuity is equivalent to a 3-year annuity of £1)

(X/(1.053) + X/(1.049)^2 + X/(1.047)^3) = 4.382703

X * [1/(1.053) + 1/(1.049)^2 + 1/(1.047)^3] =4.382703

X = 4.382703 / [ 1/(1.053) + 1/(1.049)^2 + 1/(1.047)^3 ]

X = $1.6056

X = $1.61 (Rounding to two decimal point)