Question

3.There are three zero coupon bonds. Their prices are:1 year =$920; 2 year =$840; 3 year =$750.

Three zero coupon bonds have the face value of 1000.

To calculate YTM

Year 1   

Face value = Price (1+ YTM)

1000 = 920 ( 1+ YTM)

1+ YTM = 1000/920

YTM = 1.0896-1

YTM = 0.0869 = 8.70%

Year 2

Face value = Price (1+ YTM)²

1000 = 840 ( 1+ YTM)²

(1+ YTM)² = 1000/840

(1+YTM)² = 1.1905

1+ YTM = 1.0911

YTM = 1.0911-1

YTM = 0.0911 = 9.11%

Year 3

Face value = Price (1+ YTM)³

1000 = 750 ( 1+ YTM)³

(1+ YTM)³ = 1000/750

(1+YTM)^{3 =} 1.333333

1+ YTM = 1.1006

YTM = 1.1006-1

= 0.1006

= 10.06%

The answers are in the brackets. I want to know the process to get the answers.

***Question**** b. What are the 1 year forward rate at the end of year 1, and the annualized forward rate between end of years 1 and 3? [answers: 9.52%, and 10.75%]

Answer 1

One Year forward rate at the end of one year.

If you are investing for two years an amount of Dollar 100,

You can :

. 1.Invest directly for two years at 9.11%

2. Invest now for one year at 8.70% now and invest for one more year at one year forward rate at the end of one year

Assume one year forward rate at the end of one year=r

Amount available at the end of 2 years under alternative 1: 100*((1+0.0911)^2)

Amount available at the end of 2 years under alternative 2: 100*(1+0.0870)*(1+r)

100*(1+0.0870)*(1+r)=100*((1+0.0911)^2)

1.0870*(1+r)=(1.0911^2)=1.1904992

1+r=(1.1904992/1.0870)=1.0952155

r=0.0952155=9.52%

One year forward rate at the end of one year=9.52%

One Year forward rate at the end of one year

9.52%

Annualized Forward Rate from end of year 1 to end of year 3

Alternatives:

.1 Invest for 3 years at 10.06%

2.Invest now for one year at 8.7% and a two year forward annualized rate =r

Future value at end of 3 years of Alternative 1 =(1+0.1006)^3

Future value at end of 3 years of Alternative 2 =(1+0.0870)*((1+r)^2)

(1+0.0870)*((1+r)^2)=(1+0.1006)^3

1.0870*((1+r)^2)= 1.333179188

(1+r)^2=1.333179188/1.0870=1.226475794

1+r=1.226475794^(1/2)= 1.107463676

r=0.107463676=10.75%

Two year forward annualized rate at end of one year=10.75%

Annualized Forward Rate from end of year 1 to end of year 3

10.75%