An insurer enters into a four-year contract today. The contract requires the insured to deposit 500 into a fund that earns an annual effective rate of 5.0%, and from which all claims will be paid. The insurer expects that 100 in claims will be paid at the end of each year, for the next four years. At the end of the fourth year, after all claims are paid, the insurer is required to return 75% of the remaining fund balance to the insured. To issue this policy, the insurer incurs 100 in expenses today. It also collects a fee of 125 at the end of two years.

Calculate the insurer’s yield rate.

• 9%
• 24%
• 39%
• 54%
• 69%

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

Solution: B

The fund will have

[[math]]500(1.05)^{4}-100s_{\overline{4}|0.05}=176.74 [[/math]]

after four years. After returning 75% to the insured, the insurer receives 0.25(176.74) = 44.19. So the insurer’s cash flows are to pay 100 at time 0, receive 125 at time 2, and receive 44.19 at time four. The equation of value and the solution are:

[[math]] \begin{align*} 100(1+i)^{4}-125(1+i)^{2}-44.19=0 \\ (1+i)^{2}=\frac{125\pm\sqrt{(-125)^{2}-4(100)(-44.19)}}{200}=1.5374 \\ 1+i = 1.2399 \\ i = 24\% \end{align*} [[/math]]

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