Question

The yield curve for Government-guaranteed zero-coupon bonds is based as follows:

Term to maturity (years) Yield to maturity (% per annum)

1 8%

2 9%

3 10%

REQUIRED:

i. What are the implied one-year forward rates for years 1, 2 and 3 respectively?

ii. If the expectations hypothesis of the term structure of interest rates is correct, in one year’s time, what will be the yield to maturity on a one-year zero-coupon bond?

iii. Based on the same hypothesis as in ii. above, in one year’s time, what will be the yield to maturity on a two-year zero-coupon bond?

Answer 1

i.

1-year spot rate, s1 = 1 year yield to maturity of the zero coupon bond = 8%

2-year spot rate, s2 = 2 year yield to maturity of the zero coupon bond = 9%

3-year spot rate, s3 = 3 year yield to maturity of the zero coupon bond = 10%

 

Therefore, implied 1 year forward rate for year 1, f0,1 = s1 = 8%

 

Implied 1 year forward rate for year 2, f1,1 = (1+s2)2/(1+s1)-1 

                                                                             = (1+9%)2/(1+8%)-1 

                                                                             = 10.0093%

 

Implied 1 year forward rate for year 3, f2,1 = (1+s3)3/(1+s2)2-1 

                                                                             = (1+10%)3/(1+9%)2-1 

                                                                             = 12.0276%

 

ii.

Let 1 year yield to maturity one year from now = y

 

Then according to expectation theory, (1+s2)2 = (1+s1)*(1+y)

or, y = (1+s2)2/(1+s1)-1 = (1+9%)2/(1+8%)-1 = 10.0093%

Therefore, yield to maturity on a one-year zero-coupon bond one year from now = 10.0093%

 

iii.

Let 2 year yield to maturity one year from now = r

Then according to expectation theory, (1+s3)3 = (1+s1)*(1+r)2

or, r = [(1+s3)3/(1+s1)]1/2-1 = [(1+10%)3/(1+8%)]1/2-1 = 11.0138%

 

Therefore, yield to maturity on a two-year zero-coupon bond one year from now = 11.0138%

i. Implied 1 year forward rate for year 1, f0,1 = s1 = 8%

Implied 1 year forward rate for year 2, f2,1 = 12.0276%

 

Implied 1 year forward rate for year 3, f2,1 = 10.0093%