Amortization vs Sinking Fund Financial Mathematics

Question

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.

Answer 1

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

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

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

Answer 2

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 $j=14.18\%$.