## Financial maths

#### Simple interest

$I = Prn$
Where:

I = Interest
P = Principal (original) amount
r = Interest rate per period of time expressed as a decimal
n = Number of periods

#### Compound interest

$A = P(1 + r)^n$
Where:

A = Final amount
P = Principal (original) amount
r = Interest rate per period of time expressed as a decimal
n = Number of time periods

#### Annuities

An annuity is an investment where equal amounts are contributed to an account at regular intervals.

The present value of an annuity is the amount of money that could be invested now (at the same rate of a compound interest for the same length of time) to give the same result as an annuity with regular contributions, M.

Future value
$FV = M\left[\frac{(1 + r)^n - 1}{r} \right]$
Where:

FV = Future value of an annuity
M = Contribution per period (paid at the end of the period)
r = Interest rate per period of time expressed as a decimal
n = Number of periods

Present value
$\begin{eqnarray*} PV &=& \frac{FV}{(1+r)^n} \\\\ &=& M\left[\frac{(1 + r)^n - 1}{r(1+r)^n} \right] \\\\ &=& M\left[\frac{ 1 - (1 + r)^{-n}}{r} \right] \end{eqnarray*}$