Simple & Compound Interest

Formula sheet

Everything for this chapter, in one place. $P$ is the principal, $r$ the annual rate in per cent, $t$ the time in years.

Simple interest — the same amount every year


Interest	$\text{SI} = \dfrac{Prt}{100}$
Amount	$A = P\left(1+\dfrac{rt}{100}\right)$
Shape	an arithmetic progression — equal steps

Compound interest — the same factor every year


Amount	$A = P\left(1+\dfrac{r}{100}\right)^{t}$
Interest	$\text{CI} = A - P$
Shape	a geometric progression — equal multipliers

Compounding more often


$k$ times a year	$A = P\left(1+\dfrac{r}{100k}\right)^{kt}$
Half-yearly	rate halves, periods double: $\left(1+\dfrac{r}{200}\right)^{2t}$
Effective annual rate, half-yearly	$\left(1+\dfrac{r}{200}\right)^2 - 1$
6% half-yearly is really	$6.09\%$ a year

The gap between them


Over 2 years	$\text{CI} - \text{SI} = P\left(\dfrac{r}{100}\right)^2$
Over 3 years	$\text{CI} - \text{SI} = P\left(\dfrac{r}{100}\right)^2\left(3+\dfrac{r}{100}\right)$
Over 1 year	they are equal

Year-on-year interest under CI


Each year's interest	is $\left(1+\dfrac{r}{100}\right)$ times the previous year's
So two consecutive years	give the rate directly, by division
And two amounts $n$ years apart	give $\left(1+\dfrac{r}{100}\right)^n$ by division

Repaying by instalments


Each instalment $I$ paid after $n$ years is worth	$\dfrac{I}{\left(1+\frac{r}{100}\right)^{n}}$ today
So a loan $P$ repaid in two equal instalments	$P = \dfrac{I}{1+\frac{r}{100}} + \dfrac{I}{\left(1+\frac{r}{100}\right)^2}$
A part-payment mid-loan	grow the balance to that date, subtract the payment, then grow what is left