Tim takes out an n-year loan with equal annual payments at the end of each year. The interest portion of the payment at time (n − 1) is equal to 0.5250 of the interest portion of the payment at time (n − 3) and is also equal to 0.1427 of the interest portion of the first payment.

Calculate n.

• 18
• 20
• 22
• 24
• 26

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

Solution: C

[[math]] \begin{array}{l}{{1- v^{2}=0.52{\bar{s}}(1- v^{4})\Rightarrow1=0.52{\bar{s}}(1+ v^{2})\Rightarrow v^{2}=0.90476\Rightarrow v=0.95119}}\\ {{1- v^{2}=0.1427(1- v^{n})\Rightarrow1- v^n=(1-0.90476)/0.1427=0.667414\Rightarrow v^{n}=0.3325966}}\\ {{n=\ln(0.32596)/\ln(0.95119)=22}}\end{array} [[/math]]

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