Annuity

Annuities are fixed cash flows that are generated each period for a fixed number of periods.

A [[#Regular Annuity]] produces the first payment at the end of the period, while [[#Annuity due]] produces its first payment at the start of the period.

Assumptions

All cash flows are separated by exactly one period.
All time periods are of equal length.
The interest rate is the same for each time period.
The value of each cash flow is the same.

Present Values

Present Value of Annuity at $t = 1$

P V A = P M T \times (\frac{1}{r} - \frac{1}{r (1 + r)^{T}})

Present Value of Annuity Due at $t = 0$

P V A_{D u e} = (1 + r) \cdot P M T \times (\frac{1}{r} - \frac{1}{r (1 + r)^{T}})

$r$ is the interest rate
$T$ is the number of payments
$P M T$ is the periodic payment

Future Values

F V A = P M T \times (\frac{(1 + r)^{T}}{r} - \frac{1}{r})

F V A_{D u e} = P M T \times (1 + r) \times (\frac{(1 + r)^{T}}{r} - \frac{1}{r})

Perpetuity

A perpetuity is an annuity with an infinite number of payments.

P V = \frac{P M T}{r}

P V A_D u e = \frac{P M T \times (1 + r)}{r}