Jack inherited a perpetuity-due, with annual payments of 15,000. He immediately exchanged the perpetuity for a 25-year annuity-due having the same present value. The annuity-due has annual payments of X. All the present values are based on an annual effective interest rate of 10% for the first 10 years and 8% thereafter.

Calculate X.

• 16,942
• 17,384
• 17,434
• 17,520
• 18,989

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

Solution: B

[[math]] \begin{align*} \mathrm{PV}_{\mathrm{perp.}}=\left[\frac{1}{0.1}+\frac{\frac{1}{0.08}-\frac{1}{0.1}}{1.1^{10}}\right](15,000)+15,000 \\ 164, 457.87 + 15, 000 = 179, 457.87 \\ X\left(\ddot{a}_{\overline{{{10}}}|0.10}+\frac{\ddot{a}_{\overline{{{15}}}|0.08}}{1.10^{10}}\right)=179,458 \\ X\biggl(6.759+\frac{9.244}{1.10^{10}}\biggr)=179,458 \\ X = 17,384. \end{align*} [[/math]]

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