A life insurance company invests two million in a 10-year zero-coupon bond and four million in a 30-year zero-coupon bond. The annual effective yield rate for both bonds is 8%. When the 10-year bond matures, the company reinvests the proceeds in another 10-year zero- coupon bond. At that time the bond yield rate is 12% annual effective. After 20 years from the initial investment, the 30-year bond is sold to yield an annual effective rate of 10% to the buyer. The maturity of the second 10-year bond and the sale of the 30-year bond result in a gain of X on the company’s initial investment of six million.

Calculate X.

• 23 million
• 29 million
• 32 million
• 34 million
• 42 million

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: A

All calculations are in millions. For the ten-year bond, at time ten it is redeemed for 2(1.08)¹⁰= 4.31785. After being reinvested at 12% it matures at time twenty for 4.31785(1.12)¹⁰ = 13.4106. The thirty-year bond has a redemption value of 30 4(1.08)³⁰ = 40.2506. For the buyer to earn 10%, it is sold for 10 40.2506(1.1)^-10 = 15.5184 . The gain is 13.4106 + 15.5184 – 6 = 22.9290.

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.