Two immediate annuities have the following characteristics:

X: [math]\quad[/math] Pays [math]1 / m, m[/math] times per year, for 10 years

Y: [math]\quad[/math] Pays [math]P[/math] at the end of years 2, 4, 6, 8, and 10

You are given

1. The accumulated value at time 1 of [math]1 / m[/math] paid at times [math]1 / m, 2 / m, \ldots, 1[/math] is 1.0331 .
2. [math]\quad s_2=2.075[/math].
3. The present value of [math]X[/math] equals the present value of [math]Y[/math].

Calculate [math]P[/math].

• 1.94
• 2.01
• 2.03
• 2.07
• 2.14

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

Solution: E

The present value of annuity [math]\mathrm{X}[/math] is [math]1.0331 a_{10}=1.0331 \frac{1-v^{10}}{i}[/math]. The present value of annuity [math]\mathrm{Y}[/math] is [math]P\left(v^2+\cdots+v^{10}\right)=P \frac{v^2-v^{12}}{1-v^2}=P \frac{1-v^{10}}{(1+i)^2-1}[/math]. Equating the present values and solving,

[[math]] P=1.0331 \frac{(1+i)^2-1}{i}=1.0331 s_{\overline{2} \mid}=1.0331(2.075)=2.14 [[/math]]

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