Question

A man borrows Rs. $20,000$ at $12\%$ per annum, compounded semi-annually and agrees to pay it in $10$ equal semi-annual installments. Find the value of each installment, if the first payment is due at the end of two years.

Answer 1

This is a case of deferred annuity. The segment depicts $10$ semi-annuals of $5$ years. The first installment is paid at the end of two years.
Let the value of each instalment be $a$.
The formula for deferred annuity is
$P=\frac{a}{i}.\frac{{(1+i)}^n-1}{{(1+i)}^{m+n}}$
Here, $m=3,n=7\Rightarrow$ $m+n=10,P=$Rs.$20,000$
$\frac{12}{100}\times \frac{1}{2}=0.06$
$\therefore$ $20,000=\frac{a}{0.06}.\frac{{(1+0.06)}^7-1}{{(1+0.06)}^{10}}$
$20,000=\frac{a}{0.06}.\frac{{\left(1.06\right)}^7-1}{{(1+0.06)}^{10}}$
$20,000=\frac{a}{0.06}.\frac{1.503-1}{1.791}$
$20,000=\frac{a}{0.06}\times \frac{0.503}{1.791}$
$a=\frac{20,000\times 0.06\times 1.791}{0.503}$
$a=\frac{2149.2}{0.503}$
$\therefore$ $a=427.76$
Hence, each instalment is of Rs. $427.67$