Question $5$

A loan of $L,$ paid annually, is made for a term of $n$ years at an effective interest

rate of $i.$ The lender allows only annual year$-$end interest payments made until at

the end of $n$ years. In order to pay the loan of $L$ at the end of $n$ years, the

borrower will make level annual year$-$end payments into a sinking fund earning $j$

interest rate per year.

$($a$)$ Show that the annual payment made is $iL+\frac{L}{s_{\frac{?}{\text{b}\text{a}\text{r}}(n)j}}.$

$($b$)$ If $j=\frac{i}{2},$

$($i$)$ Determine the level sinking fund deposit and the balance in the

sinking fund at $k-$th payment.

$($ii$)$ Show that the total principal repayment for the $k-$th payment is$,$

$\frac{2L(1+\frac{i}{2}{)}^k}{(2+i)s_{\frac{?}{\text{b}\text{a}\text{r}}(n)|\frac{i}{2}}}.$

$($c$)$ For $L=RM50,000,i=5\%,j=2\mathrm{.}5\%$ and $n=15,$ find,

$($i$)$ the annual payment under this scheme

$($ii$)$ the total principal repayment for the $5-$th payment.