A perpetuity-immediate with annual payments consists of ten level payments of k, followed by a series of increasing payments. Beginning with the eleventh payment, each payment is 200 larger than the preceding payment. Based on an annual effective interest rate of 5.2%, the present value of the perpetuity is 50,000.

Calculate k.

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

Solution: C

[[math]]\begin{aligned} & 50,000=k a_{\overline{10} \mid 0.052}+v^{10}\left[(k+200)\left(\frac{1}{0.052}\right)+200\left(\frac{1}{0.052^2}\right)\right] \\ & 50,000=k(7.647284)+(0.602341)[19.230769 k+3846.153846+73964.50] \\ & 50,000=19.230765 k+46,868.54 \\ & k=162.83\end{aligned}[[/math]]

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