A life insurance company sells a two-year immediate annuity with annual payments of 1000 for a price of 1817. The investment actuary invests the 1817 in two zero-coupon bonds

• The first bond matures in one year and earns an annual effective interest rate of 6%. The second bond matures in two years and earns an annual effective interest rate of 7%.
• 999.35 is invested in the first bond and 817.65 is invested in the second bond.
• The two bonds are held to maturity

As long as the effective annual one-year reinvestment rate is at least X% one year from now, the principal and interest earned will be sufficient to make the two annuity payments.

Calculate X.

• 6.0
• 6.6
• 7.0
• 7.3
• 7.7

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

Solution: E

999.35 x 1.06 = 1059.31 will be available to make the first payment of 1000, leaving 59.31 to be reinvested at X%. 817.65 x 1.072 = 936.13 will be available from the second bond to make the second payment of 1000, leaving 63.87 to come from the reinvestment of 59.31. X = 100(63.87 / 59.31 – 1) = 7.69.

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