Company Q invests X at the end of each year for 25 years at an annual effective interest rate of 9%. Company R invests 100 at the end of each year for 25 years at an annual effective interest rate of 9%, but, at the end of each year, the interest earned is reinvested at an annual effective interest rate of 8%. Immediately after the 25th payment, Company R’s total investment, including the reinvested interest, has the same value as Company Q’s investment.

Calculate X.

• 91.22
• 91.93
• 92.67
• 93.41
• 94.03

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

Solution: D

For [math]\mathrm{Q}[/math] the accumulated value is [math]Xs_{25 \mid 0.09}=84.7009 X[/math]. For [math]\mathrm{R}[/math] the accumulated value is

[[math]]100(25)+9(Is)_{\overline{24}|0.08}=2500+9 \frac{\ddot{s}_{\overline{24} | 0.08}-24}{0.08}=2500+9 \frac{72.1059-24}{0.08}=7911.91[[/math]]

. Then [math]X=7911.91 / 84.7009=93.41[/math].

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