Question

1. Consider the following two bonds. One bond with a coupon rate of 4%, semi-annual coupons, and 10 years until maturity. The second bond has 5 years until maturity but is otherwise the same.
   1. What is the most you should pay for each asset if current yields are 6%?
   2. Do the bonds sell at a premium or a discount?
   3. Suppose current yields increase to 7%, what are the new bond prices?
   4. Which bond is more sensitive to yield changes? Why?

Answer 1

Number of periods to maturity = n

Yield to Maturity = r

Semi Annual Coupon Payment P

Face Value = FV

Hence, Present Value of bond = PV = P/(1+r) + P/(1+r)² + .... + P/(1+r)ⁿ + FV/(1+r)ⁿ

= P[1 - (1+r)^-n]/r + FV/(1+r)ⁿ

(a)

Bond 1
n = 10*2 = 20 semiannual periods
r = 0.06/2 = 0.03
FV = $100
P = 4%*100/2 = $2
Hence, PV = 2[1 - (1+0.03)^-20]/0.03 + 100/(1+0.03)²⁰ = $85.12

Bond 2
n = 5*2 = 10 semiannual periods
r = 0.06/2 = 0.03
FV = $100
P = 4%*100/2 = $2
Hence, PV = 2[1 - (1+0.03)^-10]/0.03 + 100/(1+0.03)¹⁰ = $91.47

(b) Since the present value of both the bonds is less than the par value (FV) of the bond, both sell at discount

(c)

Bond 1
n = 10*2 = 20 semiannual periods
r = 0.07/2 = 0.035
FV = $100
P = 4%*100/2 = $2
Hence, PV = 2[1 - (1+0.035)^-20]/0.035 + 100/(1+0.035)²⁰ = $78.68

Bond 2
n = 5*2 = 10 semiannual periods
r = 0.07/2 = 0.035
FV = $100
P = 4%*100/2 = $2
Hence, PV = 2[1 - (1+0.03)^-10]/0.03 + 100/(1+0.03)¹⁰ = $87.53

(d) The Bond with 10 years to maturity is more sensitive to yield changes. The longer the maturity of a bond, the more sensitive it is to yield changes.