Question

What is the price of a 15-year, $1000 par value bond with a 7% coupon that pays interest seminannually if we assume that its yield to maturity is 8%? What would be the price of the bond if its YTM were 9%? Compute the percentage change in price: (new price - initial price) / initial price. Repeat the exercise for a 10-year, $1000 bond with a 7% coupon paying interest semiannually using the same two yields. What do you notice about the percentage change in price for the 10-year bond versus that for the 15-year bond?

Answer 1

Let us first find the price of 15 year, $ 1000 par value of bond with 7% coupon rate and YTM of 8% (SEMI ANNUAL INTEREST)

Price of bond = (Coupon/2) [1-(1+YTM/2)^(-2*15)/(YTM/2)] + 1000/(1+YTM/2)^(2*15)

Price of the bond = 35 * [1-(1.08^-30/0.04)] + 1000/1.04^30 = 913.5398

Thus the price of this bond at 8% YTM would be $ 913.54

If the YTM were 9%, then the price of the bond would be,

Price of the bond = 35 * [1-(1.09^-30/0.045)] + 1000/1.045^30 = $ 837.1111

Thus the price of this bond at 9% YTM would be $ 837.11

Percentage change in price (formula given in question) = (837.11-913.54)/913.54 = -8.3664%

Now repeating the same exercise for 10 year $1000 bond with 7% coupon (semi annual) with 8% YTM,

Price of bond = (Coupon/2) [1-(1+YTM/2)^(-2*10)/(YTM/2)] + 1000/(1+YTM/2)^(2*10)

Price of the bond = 35 * [1-(1.08^-20/0.04)] + 1000/1.04^20 = $ 932.0484

Thus the price of this bond at 8% YTM would be $ 932.05

If the YTM were 9%, then the price of the bond would be,

Price of the bond = 35 * [1-(1.09^-20/0.045)] + 1000/1.045^20 = $ 869.9206

Thus the price of this bond at 9% YTM would be $ 869.92

here we notice that the price of 8% YTM bond with 15 year maturity was 913.54 and that with 10 yr maturity was 932.05 (% change = 913.54-932.05/932.05*100) = 1.99%) The bond with lower maturity here has the highest price as against the bond with higher maturity of 15 years (all other factors are same between the 2 bonds)

Similarly for 9% YTM, we see that price of 15 yr maturity bond is $ 837.11 and that of 10 yr maturity bond is $ 869.92, thus here we again see that bond with lower maturity has higher price and vice versa.