A loan of 1500 is to be repaid with payments made at the end of each year for 20 years. There are two repayment options:

1. equal payments at an annual effective interest rate of 5%
2. nonlevel payments of interest on the unpaid balance at an annual effective interest rate of i, plus 75 of principal

The sum of the payments is the same under the two options.

Calculate i.

• 5.26%
• 5.51%
• 5.76%
• 6.01%
• 6.26%

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

Solution: C

Option 1

[[math]] \begin{aligned} & 1500=P a_{\overline{2000.05}} \\ & P=120.3639 \\ & \text { Total Payment }=120.3639(20)=2407.28 \end{aligned} [[/math]]

Option 2

[[math]] \begin{aligned} & \text { Total Payment }= 75 * 20+1500 i+(1500-75) i+ \\ &(1500-2 * 75) \cdot i+\cdots+(1500-19 * 75) i \\ &= 1500+1500 i(20)-75 i \cdot \sum_{k=0}^{19} k \\ &= 1500+30,000 i-75 i \cdot\left(\frac{19(20)}{2}\right) \\ &= 1500+15,750 i \\ & 2407.28=1500+15,750 i \\ & i=0.0576 \quad \end{aligned} [[/math]]

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