Question

The 5-year, 8-year, and 10-year zero rates are 5%, 7%, and 8%. The rates are given per annum with annual compounding.

a) What is the forward rate for an investment initiated 5 years from today and maturing 10 years from today? (Give your answer per annum with annual compounding)?

b) What is the forward rate for an investment initiated 5 years from today and maturing 8 years from today? (Give your answer per annum with continuous compounding)?

Answer 1

As per expectation hypothesis long-term interest rates contain an implicit prediction of short-term interest rates. Hence, we can compute forward rates as:

(1+rk) k = (1+rk-n) k-n (1+fn) n

rk = Current rate for k periods

rk-n = Current rate for k-n periods

fn = Forward rate for n periods

a)

(1 + r10)10 = (1+r5)5 x (1+f5)5

(1 + 0.08)10 = (1+0.05)5 x (1+f5)5

(1.08)10 = (1.05)5 x (1+f5)5

2.15892499727279 = 1.2762815625 x (1+f5)5

(1+f5)5 = 2.15892499727279 /1.2762815625

(1+f5)5 = 1.69157422680608

1+f5 = (1.69157422680608)1/5

       = (1.69157422680608)0.2

          = 1.11085714285714

f5 = 1.11085714285714 – 1 = 0.11085714285714 or 11.09 %

Forward rate for the investment initiating 5 years from today and maturing in 10 years is 11.09 %

b)

(1 + r8)8 = (1+r5)5 x (1+f3)3

(1 + 0.07)8 = (1+0.05)5 x (1+f3)3

(1.07)8 = (1.05)5 x (1+f3)3

1.71818617983192 = 1.2762815625 x (1+f3)3

(1+f3)3 = 1.71818617983192/1.2762815625

(1+f3)3 = 1.34624383076279

1+f3= (1.34624383076279)1/3

       = (1.69157422680608) 0.333333333

        = 1.10418347173766

F3 = 1.10418347173766 – 1 = 0.10418347173766 or 10.42 %

Forward rate for the investment initiating 5 years from today and maturing in 8 years is 10.42 %