Question

A bond has a coupon rate of 6 percent, with payments semi-annually. It matures in 2.5 years and has a yield to maturity of 7 percent (15 points). a. Use the “long method” to determine the duration and modified duration of this bond? b. If the yield to maturity increases to 9 percent, what is the approximate percent change in price based on the modified duration calculated in ‘a?’ c. What is the actual percentage change in price if the yield to maturity increases to 9 percent? d. What is a reason for the difference in prices between ‘b’ and ‘c?

Answer 1

The price of the bond is given by:

3/(1.035^1) + 3/(1.035^2) + 3/(1.035^3) + 3/(1.035^4) + 103/(1.035^5) = 97.7425 = P

Macaulay Duration = (3/(1.035^1)/P)*.5 + (3/(1.035^2)/P)*1 +  (3/(1.035^3)/P)*1.5 + (3/(1.035^4)/P)*2 + (103/(1.035^5)/P)*2.5 = 2.3566

Modified duration = 2.3566 / (1.035) = 2.2769% (since Mod Dur = Mac Dur / (1 + y/k) where k is the frequency of compounding, in this case =2

Therefore by definition of mod duration, if the ytm rises by 2%, the price of this bond should fall by ~2.28*2% = 4.56%

Actual Price at ytm of 9% = 93.4150

Actual Percentage change = 4.43%

Duration always underestimates the price of the bond since it assumes a linear relationship between price and ytm. The actual relationship is convex. Therefore on account of convexity, the price of the bond falls less.