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 equivalently, 3.75% every six months for the next 10 six-month periods). Also suppose that the payment 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 insurance company seeks to make on the in- vested 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 circumstances will the investment in bond X fail to satisfy the target accumulated value?

TABLE A Accumulated Value and Total Return after Five Five-Year 7.5% Bond Selling to Yield 7.5% Investment horizon (years): 5 Coupon rate: 7.50% Maturity (years): 5 Yield to maturity: 7.50% Price: 100.00000 Par value purchased: $9,642,899 Purchase price: $9,642,899 Target accumulated value: $13,934,413 After Five Years New Total Yield (%) Coupon Interest on Interest Price of Bond Accumulated Return Value (%) 11.00 $3,616,087 $1,039,753 $9,642,899 $14,298,739 8.04 10.00 3,616,087 9.00 3,616,087 8.00 3,616,087 7.50 3,616,087 7.00 3,616,087 6.00 3,616,087 5.00 3,616,087 4.00 3,616,087 343,427 9,642,899 13,602,414 7.00

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 circumstances will the investment in bond Y fail to satisfy the target accumulated value?

f. 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 circumstances 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%: Bond Five-year, 7.5% coupon, selling at par 12-year, 7.5% coupon, selling at par Six-year, 6.75% coupon, selling for 96.42899 Modified Duration 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?

TABLE B Accumulated Value and Total Return after Five Years 12-Year 7.5% Bond Selling to Yield 7.5% Investment horizon (years): 5 Coupon rate: 7.5% Maturity (years): 12 Yield to maturity: 7.5% Price: 100.00000 Par value purchased: $9,642,899 Purchase price: $9,642,899 Target accumulated value: $13,934,413 After Five Years New Yield (%) Total Coupon Interest on Interest Price of Accumulated Return Bond Value (%) 11.00 $3,616,087 $1,039,753 $8,204,639 $12,680,479 5.55 10.00 3,616,087 9.00 3,616,087 8.00 3,616,087 7.50 3,616,087 7.00 3,616,087 6.00 3,616,087 5.00 3,616,087 4.00 3,616,087 343,427 11,685,837 15,645,352 9.92 TABLE C Accumulated Value and Total Return after Five Yea Six-Year 6.75% Bond Selling to Yield 7.5% Investment horizon (years): 5 Coupon rate: 6.75% Maturity (years): 6 Yield to maturity: 7.5% Price: 96.42899 Par value purchased: $10,000,000 Purchase price: $9,642,899 Target accumulated value: $13,934,413 After Five Years New Yield Total (%) Coupon Interest on Interest Price of Bond Accumulated Return Value (%) 11.00 $3,375,000 $970,432 $ 9,607,657 $13,953,089 7.53 10.00 3,375,000 9.00 3,375,000 8.00 3,375,000 7.50 3,375,000 7.00 3,375,000 6.00 3,375,000 5.00 3,375,000 4,00 3,375,000 320,531 10,266,965 13,962,495 7.54.

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