An annuity has payments of 1000 at the beginning of every three months for six years. Another annuity has payments of X at the end of the first, third, and fifth years. At an annual effective rate of 8%, the present values of the two annuities are equal.

Calculate X.

• 7931
• 7981
• 8033
• 8085
• 8137

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

Solution: D

[[math]]\begin{aligned} & X=1000 \exp \left(\int_2^6 \frac{0.5}{5+0.5 t} d t\right) \\ & =1000 \exp \left[\left.\ln (5+0.5 t)\right|_2 ^6\right] \\ & =1000 \exp (\ln 8-\ln 6)=1000\left(\frac{8}{6}\right)=1333.33 \\ & 1333.33=Y\left[1-\frac{0.08}{4}\right]^{-4(2)} \\ & Y=1134.35\end{aligned}[[/math]]

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