Question

The following is a list of prices for zero-coupon bonds of various maturities. Calculate the yields to maturity of each bond and the implied sequence of forward rates. Maturity (Years) Price of Bond

1 $917.43 YTM= IFR=

2 $826.51 YTM= IFR=

3 $737.96 YTM= IFR=

4 $653.06 YTM= IFR=

Answer 1

Solution :

Zero coupon bond is traded at discount based on the prevailing yield and

price of a zero coupon bond = Par Value / (1+ yield)^maturity

Bond having 1-year maturity is priced at 917.43. Par value = 1000

917.43 = 1000 / (1+ YTM)^1

1 + YTM = 1000/917.43 = 1.090001

YTM = 9.01% , Implied forward rate = 9.01% ( It is same for 1 year )

Bond having 2-year maturity is priced at 826.51 Par value = 1000

826.51 = 1000 / (1+ YTM)^2

(1 + YTM)^2 = 1000/826.51 = 1.209907

1+ YTM = (1.209907)^(1/2) = 1.1000

YTM = 10% ,

Implied forward rate :

(1+ YTM1) ( 1+ FR) = (1+ YTM2)^2

Implied forward rate = (1+YTM2)^2 / (1+ YTM1)^1 - 1 = 1.10^2/1.09^1 - 1 = 11.00%

Bond having 3-year maturity is priced at 737.96 Par value = 1000

737.96 = 1000 / (1+ YTM)^3

(1 + YTM)^3 = 1000/737.96 = 1.355087

1+ YTM = (1.355087)^(1/3) = 1.1066

YTM = 10.66% ,

Implied forward rate = (1+YTM3)^3 / (1+ YTM2)^2 - 1 = 1.1066^3/1.10^2 - 1 = 12.00%

Bond having 4-year maturity is priced at 653.06 Par value = 1000

653.06 = 1000 / (1+ YTM)^4

(1 + YTM)^4 = 1000/653.06 = 1.531253

1+ YTM = (1.531253)^(1/4) = 1.1124

YTM = 11.24% ,

Implied forward rate = (1+YTM4)^4 / (1+ YTM3)^3 - 1 = 1.1124^4/1.1066^3 - 1 = 13.00%

1 $917.43 YTM= 9.00% IFR= 9.00%

2 $826.51 YTM= 10.00% IFR= 11.00%

3 $737.96 YTM= 10.66% IFR= 12.00%

4 $653.06 YTM= 11.24% IFR= 13.00%