Suppose that a life insurance company sells a five-year guaranteed investment contract that guarantees an interest rate of 7.5% per year on a bond-equivalent yield basis (or equiva- lently, 3.75% every six months for the next 10 six-month periods). Also suppose that the pay- ment made by the policyholder is $9,642,899. Consider the following three investments that can be made by the portfolio manager:

Bond X: Buy $9,642,899 par value of an option-free bond selling at par with a 7.5% yield to maturity that matures in five years. Bond Y: Buy $9,642,899 par value of an option-free bond selling at par with a 7.5% yield to maturity that matures in 12 years.

Bond Z: Buy $10,000,000 par value of a six- year 6.75% coupon option-free bond selling at 96.42899 to yield 7.5%.

a. Holding aside the spread that the insur- ance company seeks to make on the invested funds, demonstrate that the target accumulated value to meet the GIC obligation five years from now is $13,934,413.

b. Complete Table A assuming that the manager invests in bond X and immedi- ately following the purchase, yields change and stay the same for the five- year investment horizon.

c. Based on Table A, under what circum- stances will the investment in bond X fail to satisfy the target accumulated value?

d. Complete Table B, assuming that the manager invests in bond Y and immedi- ately following the purchase, yields change and stay the same for the five- year investment horizon.

e. Based on Table B, under what circum- stances will the investment in bond Y fail to satisfy the target accumulated value? . Complete Table C, assuming that the manager invests in bond Z and immedi- ately following the purchase, yields change and stay the same for the five- year investment horizon. g. Based on Table C, under what circum- stances will the investment in bond Z fail to satisfy the target accumulated value? h. What is the modified duration of the Liability? i. Complete the following table for the three bonds assuming that each bond is trading to yield 7.5%:AppendixLO1

Transcribed Image Text:

Modified Duration Bond 5-year, 7.5% coupon, selling at par 12-year, 7.5% coupon, selling at par 6-year, 6.75% coupon selling for 96.42899 j. For which bond is the modified duration equal to the duration of the liability? k. Why in this example can one focus on modified duration rather than effective duration?