Revision as of 22:12, 18 November 2023 by Admin (Created page with "You are given the following information about a loan of L that is to be repaid with a series of 16 annual payments: *The first payment of 2000 is due one year from now. *The next seven payments are each 3% larger than the preceding payment *From the 9th to the 16th payment, each payment will be 3% less than the preceding payment. *The loan has an annual effective interest rate of 7%. Calculate L. <ul class="mw-excansopts"><li>20,689</li><li>20,716</li><li>20,775</li><...")
ABy Admin
Nov 18'23
Exercise
You are given the following information about a loan of L that is to be repaid with a series of 16 annual payments:
- The first payment of 2000 is due one year from now.
- The next seven payments are each 3% larger than the preceding payment
- From the 9th to the 16th payment, each payment will be 3% less than the preceding payment.
- The loan has an annual effective interest rate of 7%.
Calculate L.
- 20,689
- 20,716
- 20,775
- 21,147
- 22,137
Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.
ABy Admin
Nov 18'23
Solution: A
The present value of the first eight payments is:
[[math]]
2000v+2000(1.03)v^{2}+...+2000(1.03)^{7}v^{8}={\frac{2000v-20000(1.03)^{8}v^{7}}{1-1.03v}} = 13,136.41.
[[/math]]
The present value of the last eight payments is
[[math]]
\begin{split}
2000(1.03)^{7}0.97v^{9}+2000(1.03)^{7}(0.97)^{2}v^{10}+\cdots+2000(1.03)^{7}(0.97^{9})v^{96} \\
=\frac{2000(1.03)^{7}0.97v^{9}-2000(1.03)^{7}(0.97)^{9}v^{17}}{1-0.97v}=7,552.22.
\end{split}
[[/math]]
Therefore, the total loan amount is L = 20,688.63.
Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.