A bank agrees to lend 10,000 now and X three years from now in exchange for a single repayment of 75,000 at the end of 10 years. The bank charges interest at an annual effective rate of 6% for the first 5 years and at a force of interest

[[math]] \delta_t = \frac{1}{t+1}, \quad \textrm{for} \, t \geq 5.[[/math]]

Calculate X

• 23,500
• 24,000
• 24,500
• 25,000
• 25,500

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

Solution: D

The effective annual interest rate is

[[math]]\dot{l}=(1-d)^{-1}-1=(1-0.055)^{-1}-1=5.829 \%. [[/math]]

The balance on the loan at time 2 is 15, 000, 000(1.0582)² =16, 796,809.

The number of payments is given by

[[math]]1,200,000 a_{\overline{n}|} = 16, 796,809[[/math]]

which gives [math]n = 29.795 \Rightarrow 29[/math]

payments of 1,200,000. The final equation of value is

[[math]] \begin{align*} 1.200,000a_{\overline{{{29}}}|}+X(1.0582)^{-30}=16,796,809 \\ X=(16,796,809-16,621,012)(5.45799)=959,490. \end{align*} [[/math]]

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