0
$\begingroup$

I'm struggling with this question

A 20-year loan of 20,000 may be repaid under the following two methods: i) amortization method with equal annual payments at an annual effective rate of 6.5% ii) sinking fund method in which the lender receives an annual effective rate of 8% and the sinking fund earns an annual effective rate of j. Both methods require a payment of X to be made at the end of each year for 20 years. Calculate j

My attempt:

I first tried to find the deposits of the amortization method.

$$20000 - X = P\cdot a_{20|0.08}$$

However, with the X variable, I can't figure it out.

$\endgroup$

2 Answers 2

0
$\begingroup$

$X = 20000/a_{20|0.065}$

$X = 20000/s_{20|j} + 0.08 \cdot 20000$

Solve for $s_{20|j}$ and then $j$.

$\endgroup$
0
$\begingroup$

From $i)$ you have that for a loan $L=20,000$ in $n=20$ years at an annual interest rate $r=6.5\%$ $$ L=X\,a_{\overline{n}|r}\quad\Longrightarrow \; X=\frac{L}{a_{\overline{20}|6.5\%}}\tag 1 $$ For $ii)$ we have for $i=8\%$ $$ X=L\left(i+\frac{1}{s_{\overline{n}|j}}\right)=\left(8\%+\frac{1}{s_{\overline{20}|j}}\right) \tag 2 $$ and then equating (1) and (2) $$ i+\frac{1}{s_{\overline{n}|j}}=\frac{1}{a_{\overline{n}|r}}\quad\Longrightarrow\;\frac{1}{s_{\overline{20}|j}}=\frac{1}{a_{\overline{20}|6.5\%}}-8\%=\frac{1}{11.0185 }-0.08= 0.0108\tag 3 $$ and then solve (3) numerically for $j$ finding $s_{\overline{20}|j}=92.97$ and $j=14.18\%$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.