An investment of 10,000 produces a series of 30 annual payments. The first payment of X is made one year after the investment is made. Each successive payment decreases by 5 from the previous payment.

At an annual effective interest rate of 5%, calculate X.

• 685
• 695
• 705
• 715
• 725

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

Solution: C

The equation of value is

[[math]] \begin{aligned} 10,000=(X+5)a_{\overline{{{30}}}|0.05}^{}-5(I a)_{\overline{{{30}}}|0.05}^{} = (X+5)a_{\overline{{{30}}}|0.05}^{} - 5\frac{\ddot{a}_{\overline{30}|0.05}-30v^{30}}{0.05}\\ =(X+5)(15.37245)-5{\frac{6.14107-6.94132}{0.05}}=15.37245X-843.11275 \\ X = (10, 000 843.11275) /15.37245 =705.36 \end{aligned} [[/math]]

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