Question

Consider the following​ bonds:

                                               

Bond	Coupon Rate​ (annual payments)	Maturity​ (years)
A	​0%	15
B	​0%	12
C	5​%	15
D	10​%	12

a. What is the percentage change in the price of each bond if its yield to maturity falls from 7​% to 6​%?

b. Which of the bonds A−D is most sensitive to a​ 1% drop in interest rates from 7​% to 6​% and​ why? Which bond is least​ sensitive? Provide an intuitive explanation for your answer.

Note​: Assume annual compounding.

Answer 1

a: Price of each bond = present value of all future cash flows expected from the bond.

Using the YTM rate of 7% and then 6% we get the following prices and following % changes in the prices:

	Bond A	Bond B	Bond C	Bond D
Price at 7%	362.45	444.01	817.84	1,238.28
Price at 6%	417.27	496.97	902.88	1,335.35
% change in price from 7% to 6%	13.14%	10.66%	9.42%	7.27%

(b): Bond A is the most sensitive to the 1% drop in interest rates from 7% to 6%. This is because Bond A has the highest duration of 15 years. While Bond C also has a duration of 15 years it is less sensitive than Bond A. This is because Bond A is not paying any coupons and hence its maturity value is the only cash flow and hence entirely subjected to the change in interest rate.

The bond that is least sensitive is Bond D. This is because Bond D has the lowest duration of 12 years. While bond B also has a duration of 12 years it is bond D that has lower degree of sensitivity because it is paying a coupon unlike bond B which is not paying any coupons.

Note that price of bond A at 7% = 1000/1.07^15 = $362.45. At 6% = 1000/1.06^15 = 417.27

Price of Bond B at 7% = 1000/1.07^12 = 444.01 and at 6% = 1000/1.06^12 = 496.97

Price of Bond C at 7% = 50/1.07+50/1.07^2.....+50/1.07^15+1000/1.07^15 = 817.84. At 6% the price will be computed using the same equation except that 1.07 will be substituted by 1.06.

Price of Bond D at 7% = 100/1.07+100/1.07^2....+100/1.07^12+1000/1.07^12 = 1238.28. At 6% the price will be computed using the same equation except that 1.07 will be substituted by 1.06.