# Financial Math FM/Annuities

NOTE: This chapter assumes knowledge of capital-sigma notation for summations and some basic properties of summations and series, particular geometric series.

## Geometric series[edit]

## Level payment annuities[edit]

An **annuity** is a sequence of payments made at equal intervals of time. We have n periods of times . These periods could be days, months, years, fortnights, etc but they are of equal length. An **annuity-immediate** (also referred to as an ordinary annuity or simply an annuity) has each payment made at the end of each interval of time. That is to say, a payment of at the end of the first period, , a payment of at the end of the second period, etc.

Contributions | 0 | |||||

Time | 0 | 1 | 2 | 3 | n |

An **annuity-due** has each payment made at the beginning of each interval of time.

Contributions | 0 | ||||||

Time | 0 | 1 | 2 | 3 | n |

An annuity is said to have **level payments** if all payments are equal. An annuity is said to have **non-level payments** if some payments are different from other payments. Whether an annuity has level or non-level payments is independent of whether an annuity is an annuity-due or annuity-immediate. First we'll look at the present value of an annuity-immediate with level annual payments of one using accumulation function notation.

The accumulated value of an annuity-immediate with level annual payments of one is

## Level payment perpetuities[edit]

## Payable m-thly, or Payable continuously[edit]

## Arithmetic increasing/decreasing payment annuity[edit]

A(t) = (P-Q)s(nbox) + Q(Ds)(nbox)