Technique

A loan repaid in equal instalments under compound interest

Arithmetic • Interest • Compound Interest

Formula

P = \sum_{k = 1}^{n} \frac{X}{( 1 + \frac{r}{100} ) ^{k}}

If a sum

P

borrowed at

r %

per annum compounded annually is repaid in

n

equal annual instalments of

X

, each paid at the end of a year, then discounting every instalment back to the present gives

P = \frac{X}{( 1 + \frac{r}{100} )} + \frac{X}{( 1 + \frac{r}{100} ) ^{2}} + \dots + \frac{X}{( 1 + \frac{r}{100} ) ^{n}}

How to use it

1Keep the growth factor as a fraction — at $10%$ use $\frac{11}{10}$ , so each discount factor is $\frac{10}{11}$ , $\frac{100}{121}$ and so on — and the arithmetic stays exact.
2Factor $X$ out, add the discount factors over a common denominator, and divide.
3Check by running the loan forward: grow the balance by the factor, subtract the instalment, and the last subtraction should clear it exactly.

A sum of Rs.

1, 050

is borrowed at

5%

per annum compounded annually and repaid in two equal annual instalments. Find each instalment.

Step-by-step

1
$\frac{X}{\frac{21}{20}} + \frac{X}{( \frac{21}{20} ) ^{2}} = 1050$
Why: Discount both instalments back to the present.
2
$X (\frac{20}{21} + \frac{400}{441}) = X \times \frac{820}{441} = 1050$
3
$X = 1050 \times \frac{441}{820} = \frac{11025}{19} \approx 580.26$

Answer

Rs.

\frac{11025}{19} \approx 580.26

Watch out

Each instalment is discounted by a DIFFERENT number of years, so the instalment is not the amount divided by

n

, and the interest is not simply added on once. This assumes instalments fall at the end of each year; a payment made today is not discounted at all.

Builds on

Also known as

instalment, EMI, present value, discounting, loan repaid in equal instalments, annuity