Martha leaves an estate of 500,000. Interest on this estate is paid to John for the first X years at the end of each year. Karen receives annual interest payments from the end of year X+1 forever. At an annual effective interest rate of 5%, the present value of Karen’s interest payments is 1.59 times the present value of John’s.

Calculate X.

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

Solution: E

Let P be the annual interest paid. The present value of John’s payments is [math]Pa_{\overline{X}|0.05}[/math]. The present value of Karen’s payments is

[[math]] P(1.05)^{-X}\,a_{\underline{\infty}|0.05}=P(1.05)^{-X}\,/\,0.05. [[/math]]

Then,

[[math]] \begin{array}{l}{{P(1.05)^{-X}/{0.05=1.59P a_{\overline{x}|0.05}}}}\\ {{0.05=1.59-1.05^{-X}}}\\ {{\mathrm{~ln~39=2.59(1.05)^{-X}}}}\\ {{X=10.}}\end{array} [[/math]]

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