A perpetuity-due with semi-annual payments consists of two level payments of 300, followed by a series of increasing payments. Beginning with the third payment, each payment is 200 larger than the preceding payment.

Using an annual effective interest rate of i, the present value of the perpetuity is 475,000.

Calculate i.

• 4.05%
• 4.09%
• 4.13%
• 4.17%
• 4.21%

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

Solution: E

Let j be the semi-annual interest rate. Then,

[[math]] \begin{array}{c} {{475,000 = 300 + 300a_{\overline{\infty}|j} + (1+j)^{-1}200(Ia)_{\overline{\infty}|j} = 300 + 300/j + 300/j^2}} \\ {{474,700j^{2}-300j-200=0}}\\ {{j=\frac{300+\sqrt{300^{2}-4(474,700)(-200)}}{2(474, 700)} = 0.02084}}\\ {{i=(1+j)^{2}-1=0.04212 = 4.21\%}}\end{array} [[/math]]

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