At an annual effective interest rate of 10.9%, each of the following are equal to X:

• The accumulated value at the end of n years of an n-year annuity-immediate paying 21.80 per year.
• The present value of a perpetuity-immediate paying 19,208 at the end of each n-year period.

Calculate X.

• 1555
• 1750
• 1960
• 2174
• There is not enough information given to calculate X

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

Solution: C

From the first annuity,

[[math]] X = 21.8 s_{\overline{n}|0.109} = 21.8\cdot{\frac{1.109^{n}-1}{0.109}}=200[1.109^{n}-1]. [[/math]]

From the second annuity,

[[math]] X = 19,208( v^{n}+ v^{2n}+\cdots)=19,208{\frac{ v^{n}}{1- v^{n}}}=19,208{\frac{1}{1.109^{n}-1}} [[/math]]

Hence,

[[math]] \begin{array}{l}{{200[1.109^{n}-1]=19,208\frac{1}{1.109^{n}-1}}}\\ {{\mathrm{~[1.109^{n}-1]^{2}=19,208/200=96.04}}}\\ {{{ X=200(9.8)=1960.}}}\end{array} [[/math]]

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