# 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

$\sum_{i=m}^n x_i = x_m + x_{m+1} + x_{m+2} +\cdots+ x_{n-1} + x_n.$

## Level payment annuities

An annuity is a sequence of payments $C_j$ made at equal intervals of time. We have n periods of times $[0, 1], [1, 2], [2, 3], . . . [n - 1, n]$. These periods could be days, months, years, fortnights, etc but they are of equal length. An annuity-immediate (also referred to an 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 $C_1$ at the end of the first period, $[0,1]$, a payment of $C_2$ at the end of the second period, $[1,2]$ etc.

 Contributions 0 $C_1$ $C_2$ $C_3$ $\ldots$ $C_n$ Time 0 1 2 3 $\ldots$ n

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

 Contributions $C_0$ $C_1$ $C_2$ $C_3$ $\ldots$ $C_n$ 0 Time 0 1 2 3 $\ldots$ $n - 1$ n

An annuity is said to have level payments if all payments $C_j$ are equal. An annuity is said to have non-level payments if some payments $C_j$ 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.

$a_{\overline{n}|}= \frac{1}{a(1)} + \frac{1}{a(2)} + \frac{1}{a(3)} + \cdots + \frac{1}{a(n)} = \sum_{j=1}^n {\frac{1}{a(j)}}$

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

$s_{\overline{n}|}= \frac{a(n)}{a(1)} + \frac{a(n)}{a(2)} + \frac{a(n)}{a(3)} + \cdots + \frac{a(n)}{a(n)} = \sum_{j=1}^n {\frac{a(n)}{a(j)}}$
$a_{\overline{n}|}= v + v^2 +v^3 + \cdots +v^n = \sum_{j=1}^n {v^j} = \frac{1-v^n}{i}$
$s_{\overline{n}|}= (1+i)^{n-1} + (1+i)^{n-2} + \cdots + (1+i) + 1 = \frac{(1+i)^n-1}{i}$

## Arithmetic increasing/decreasing payment annuity

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