Seth repays a 30-year loan with a payment at the end of each year. Each of the first 20 payments is 1200, and each of the last 10 payments is 900. Interest on the loan is at an annual effective rate of i, i > 0. The interest portion of the 11th payment is twice the interest portion of the 21st payment.

Calculate the interest portion of the 21st payment.

• 250
• 275
• 300
• 325
• There is not enough information to calculate the interest portion of the 21st payment.

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

Solution: C

[[math]] \begin{array}{l} {{\mathrm{Int}_{11} = i \cdot [ 900 \cdot a_{\overline{20}|i} + 300a_{\overline{10}|i}] =900(1-{ v}^{20})+300(1-{ v}^{10})=1200-300{ v}^{10}-900{ v}^{20}}} \\ {{\displaystyle{\mathrm{Int}_{21}=i\left[900\cdot a_{\overline{{{10}}}|i}\right]=900(1- v^{{\mathrm{i0}}})}}}\\ {{\displaystyle{\mathrm{Int}_{11=}2\mathrm{Int}_{21}\displaystyle\Rightarrow1200-300 v^{{\mathrm{i0}}}-900 v^{{\mathrm{i0}}}=1800-1800 v^{{\mathrm{i0}}}}=1800-1800 v^{{\mathrm{i0}}}}}\\ {{\displaystyle{\mathrm{Int}}_{21}=900(1- v^{10})=300}}\end{array} [[/math]]

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