Question

If the market interest rate rises from 8% to 9% this year, which of the following bonds will show the highest price sensitivity?

- A 30-year bond, paying 8% annual coupon

- A 30-year bond, deep discount bond (you may assume a zero-coupon bond)

(Assume the face values are $1,000 for both coupon bond and the zero-coupon bond)

Show that a deep discount bond will give higher risk than a coupon bond with same maturity.

Answer 1

When interest rate is 8%:

Price of the coupon paying bond = -PV (Rate, Nper, PMT, FV) = - PV (8%, 30, 8% x 1000, 1000) = 1,000

Price of the deep discount bond = FV / (1 + r)ⁿ = 1,000 / (1 + 8%)³⁰ = 99.38

When interest rate is 9%:

Price of the coupon paying bond = -PV (Rate, Nper, PMT, FV) = - PV (9%, 30, 8% x 1000, 1000) = 897.26

Price of the deep discount bond = FV / (1 + r)ⁿ = 1,000 / (1 + 9%)³⁰ = 75.37

Hence, %age change in price of

Coupon bond = 897.26/1000 - 1 = -10.27%

Deep discount bond = 75.37 / 99.38 - 1 = -24.16%

Thus, deep discount bond has higher decline in prices and it therefore has higher sensitivity of prices to the interest rate. It has higher interest rate risk than the coupon paying bond.