Question

Consider the forward interest rates defined by the following equation: fk = 0.09 + 0.002k − 0.002k^2 for k = 0, 1, 2, 3, 4.

1) Find the 4-year spot rate

2) ) Find the 2 year deferred 3-year forward rate.

Answer 1

fk = 0.09 + 0.002k − 0.002k^2

This means the interest rate for 1-year k years from now is fk

f(0,1) = f0 = 0.09

f(1,1) = f1 =0.09+0.002-0.002 = 0.09

f(2,1) = f2 = 0.09+0.002*2-0.002*2*2 = 0.086

f(3,1) = f3 = 0.09 + 0.002*3 − 0.002*3*3 = 0.078

f(4,1) = f4 = 0.09 + 0.002*4 − 0.002*4*4 = 0.066

1) Find the 4-year spot rate f(0,4)

f(0,4) = ((1+f(0,1))*((1+f(1,1))*(1+f(2,1))*((1+f(3,1)) ^(1/4))-1

f(0,4) = ((1.09*1.09*1.086*1.078)^(1/4))-1 = 0.086

4-year spot rate f(0,4) = 0.086 = 8.6%

2) ) Find the 2 year deferred 3-year forward rate.

This means the interest rate from t=2 to t=5

2 year deferred 3-year forward rate f(2,3)

((1+f(0,1))*((1+f(1,1))*(1+f(2,3))^3) = (1+f(0,4))^4 * (1+f(4,1))

1.09*1.09*(1+f(2,3)^3) = 1.086^4 * 1.066

(1+f(2,3))^3 = 1.2480256458

f(2,3) = 0.0766 = 7.66%