A borrower takes out a 50-year loan, to be repaid with payments at the end of each year. The loan payment is 2500 for each of the first 26 years. Thereafter, the payments decrease by 100 per year. Interest on the loan is charged at an annual effective rate of i (0% < i < 10%). The principal repaid in year 26 is X.

Determine the amount of interest paid in the first year.

• [math]Xv^{25}[/math]
• [math]2500v^{25} - Xv^{25}[/math]
• [math]2500-X[/math]
• [math]2500-Xv^{25}[/math]
• [math]25Xi[/math]

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

Solution: D

The outstanding balance at time 25 is

[[math]] 100(D a)_{\overline{{{25}}|}}=100\frac{25-a_{\overline{{{25}}}|}}{i}. [[/math]]

The principle repaid in the 26th payment is

[[math]] X=2500-i(100){\frac{25-a_{\overline{{{25}}}|}}{i}}=2500-2500+100a_{\overline{{{25}}|}}=100a_{\overline{{{35}}|}}. [[/math]]

The amount borrowed is the present value of all 50 payments,

[[math]] 2500a_{\overline{{{25}}}|}+{{\nu}^{25}}100(D a)_{\overline{{{25}}}|} [[/math]]

Interest paid in the first payment is then

[[math]] \begin{array}{l}{{i\Big[2500a_{\overline{{{25}}}|}+v^{25}100(D a)_{\overline{{{25}}}|}]}}\\ {{=2500(1-v^{25})+100v^{25}(25-a_{\overline{{{25}}}|})}}\\ {{=2500-2500v^{25}+2500v^{25}-v^{25}100a_{\overline{{{25}}}|}}}\\ {{=2500-Xv^{25}.}}\end{array} [[/math]]

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