A borrower takes out a 15-year loan at an annual effective interest rate of i with payments of 50 at the end of each year. The borrower decides to pay off the loan early by making extra payments of 30 with each of the sixth through tenth regularly scheduled payments. As a result, the loan will be paid off at the end of 10 years (instead of 15).

Determine which of the following equations of value is correct.

• [math]50a_{\overline{15}|}= 50a_{\overline{10}|} + 30a_{\overline{5}|}[/math]
• [math]50s_{\overline{15}|}= 50s_{\overline{10}|} + 30s_{\overline{5}|}[/math]
• [math]50v^5s_{\overline{15}|}= 50s_{\overline{10}|} + 30s_{\overline{5}|}[/math]
• [math]50s_{\overline{15}|}= 50s_{\overline{10}|} + 30v^5s_{\overline{5}|}[/math]
• [math]50s_{\overline{15}|}= 50s_{\overline{10}|} + 30(1+i)^5s_{\overline{5}|}[/math]

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

Solution: C

Let X be the original loan value. From the original loan terms, [math]X = 50a_{\overline{15}|}[/math] .Under the revised repayment plan, [math]X = 50a_{\overline{10}|} + 30v^5a_{\overline{5}|}[/math]. Equating the two gives

[[math]]50a_{\overline{15}|} = 50a_{\overline{10}|} + 30v^5a_{\overline{5}|}[[/math]]

which does not match answer A. All the other choices use s. Multiplying both sides by [math](1+i)^{10}[/math] gives

[[math]]50v^5s_{\overline{15}|} = 50s_{\overline{10}|} + 30s_{\overline{5}|}[[/math]]

, which is answer C. This can also be obtained by equating the values of the two payment streams at time 10 rather than time 0.

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