Question

Consider the following? bonds:

                                               

Bond	Coupon Rate? (annual payments)	Maturity? (years)
A	?0%	14
B	?0%	10
C	2?%	14
D	7?%	10

a. What is the percentage change in the price of each bond if its yield to maturity falls from

5%

to

4?%?

b. Which of the bonds

Aminus?D

is most sensitive to a? 1% drop in interest rates from

5?%

to

4?%

and? why? Which bond is least? sensitive? Provide an intuitive explanation for your answer.

Note?:

Assume annual compounding.

Answer 1

Price of a bond is present value of all cashflows associated with the bond (namely coupons and maturoty value) discounted at Yield to Maturity.

It is mathematically represented as:

Now, we need to calculate the value of P for each bond at YTM 5% and 4% and change in price.

Bond A, C = $0, n = 14, M = $100

At i = 5%

P = $50.51

At i = 4%

P = $57.75

Change in price = (57.75/50.51) - 1 = 14.33%

Bond B, C = $0, n = 10, M = $100

At i = 5%

P = $61.39

At i = 4%

P = $57.75

Change in price = (67.5564/61.3913) - 1 = 10.04%

Bond C, C = $2, n = 14, M = $100

At i = 5%

P = $70.30

At i = 4%

P = $78.87

Change in price = (78.87/70.30) - 1 = 12.19%

Bond D, C = $7, n = 10, M = $100

At i = 5%

P = $115.44

At i = 4%

P = $124.33

Change in price = (124.33/115.44) - 1 = 7.70%

Looking at the above price changes, Bond A was most sensitive for 1% change in interest rates, showing the maximum change in price. We could have also ascertained this from the theoretical concept of duration. Duration measures sensitivity of price of a bond to change in interest rates. Lower the coupon rate or higher the term to maturity, greater is the duration of bond. It is maximum (and equal to years to maturity) for a zero coupon bond.

In our question, Bond A has lowest coupon (0%) and highest time to maturity (14 Years) and hence that has the maximum sensitivity to interest rate change.

Contrary to this, Bond D has highest coupon (7%) and takes less time to mature (10 years) and hence is least sensitive to interest rate change.