Ron is repaying a loan with payments of 1 at the end of each year for n years. The annual effective interest rate on the loan is i. The amount of interest paid in year t plus the amount of principal repaid in year t + 1 equals X.

Determine which of the following is equal to X.

• [math]1 + \frac{v^{n-t}}{i}[/math]
• [math]1 + \frac{v^{n-t}}{d}[/math]
• [math]1 + v^{n-t}i[/math]
• [math]1 + v^{n-t}d[/math]
• [math]1 + v^{n-t}[/math]

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

Solution: D

Year [math]t[/math] principal repaid is [math]i a_{\overline{n-t+1}|i}=1-\nu^{n-t+1}.[/math]

Year [math]t+1[/math] interest is [math]1-\bigl(1-{y^{n-t}}\bigr)={\nu}^{n-t} [/math]

[math]X=1-\nu^{n-t+1}+\nu^{n-t}=1+\nu^{n-t}(1-\nu)=1+\nu^{n-t}d.[/math]

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