# Calculating fair value of a bond

Philadelphia Electric has many bonds trading on the New York Stock Exchange. Suppose PhilElÃ‚'s bonds have identical coupon rates of 9.125% but that one issue matures in 1 year, one in 7 years, and the third in 15 years. Assume that a coupon payment was made yesterday.

a. If the yield to maturity for all three bonds is 8%, what is the fair price of each bond?

b. Suppose that the yield to maturity for all of these bonds changed instantaneously to 7%. What is the fair price of each bond now?

c. Suppose that the yield to maturity for all of these bonds changed instantaneously again, this time to 9%. Now what is the fair price of each bond?

d. Based on the fair prices at the various yields to maturity, is interest-rate risk the same, higher, or lower for longer- versus shorter-maturity bonds?

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#### Solution Preview

Solution:

a.If the yield to maturity for all three bonds is 8%, what is the fair price of each bond?

Let us assume face value of each bond=$1000

For bond with one year maturity, investor will get face value (i.e. $1000) only as coupon payment is already made. Its fair price=$1000 (if maturity amount is not paid so far)

For bond with 7 year maturity,

Number of coupon payment left=n=6

Coupon payment=C=1000*9.125%=$91.25

Maturity amount =$1000

Required rate of return=YTM=r=8%

Fair Price of a bond= C/r*(1-1/(1+r)^n)+M/(1+r)^n

=91.25/8%*(1-1/(1+8%)^6)+1000/(1+8%)^6=$1052.007

For bond with 15 year maturity,

Number of coupon payment left=n=14

Coupon payment=C=1000*9.125%=$91.25

Maturity amount =$1000

Required rate of return=YTM=r=8%

Fair Price of a bond= C/r*(1-1/(1+r)^n)+M/(1+r)^n

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#### Solution Summary

Solution describes the steps to calculate fair value of bond with different maturity periods and different yields to maturity.