An annuity-immediate provides annual payments of 10 for 20 years. Immediately following the 11th payment, the annuity is exchanged for a perpetuity-immediate of equal value with semi- annual payments. The present values at the time of the exchange are based on an annual effective interest rate of 6%. The first payment of the perpetuity is K and each subsequent payment is 0.5% larger than the previous payment.

Calculate K.

• 1.53
• 1.67
• 2.37
• 3.42
• 3.74

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

Solution: B

The PV of the annuity following the [math]11^{\text {th }}[/math] payment is: [math]10 a_{\left.9\right|_{0.06}}=68.0169[/math].

The effective semi-annual rate is [math]j=\frac{i^{(2)}}{2}=1.06^{1 / 2}-1=0.02956301[/math]. Next,

[[math]] \begin{aligned} & P V=K\left[\frac{1}{0.02956301-0.005}\right]=68.0169 \\ & K=1.67 \end{aligned} [[/math]]

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