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

Geometric series

${\displaystyle \sum _{i=m}^{n}x_{i}=x_{m}+x_{m+1}+x_{m+2}+\cdots +x_{n-1}+x_{n}.}$

Level payment annuities

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

Contributions	0	${\displaystyle C_{1}}$	${\displaystyle C_{2}}$	${\displaystyle C_{3}}$	${\displaystyle \ldots }$	${\displaystyle C_{n}}$
Time	0	1	2	3	${\displaystyle \ldots }$	n

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

Contributions	${\displaystyle C_{0}}$	${\displaystyle C_{1}}$	${\displaystyle C_{2}}$	${\displaystyle C_{3}}$	${\displaystyle \ldots }$	${\displaystyle C_{n}}$	0
Time	0	1	2	3	${\displaystyle \ldots }$	${\displaystyle n-1}$	n

An annuity is said to have level payments if all payments ${\displaystyle C_{j}}$ are equal. An annuity is said to have non-level payments if some payments ${\displaystyle C_{j}}$ 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.

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

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

${\displaystyle s_{{\overline {n}}|}={\frac {a(n)}{a(1)}}+{\frac {a(n)}{a(2)}}+{\frac {a(n)}{a(3)}}+\cdots +{\frac {a(n)}{a(n)}}=\sum _{j=1}^{n}{\frac {a(n)}{a(j)}}}$

${\displaystyle a_{{\overline {n}}|}=v+v^{2}+v^{3}+\cdots +v^{n}=\sum _{j=1}^{n}{v^{j}}={\frac {1-v^{n}}{i}}}$

${\displaystyle s_{{\overline {n}}|}=(1+i)^{n-1}+(1+i)^{n-2}+\cdots +(1+i)+1={\frac {(1+i)^{n}-1}{i}}}$

Level payment perpetuities

Payable m-thly, or Payable continuously

Arithmetic increasing/decreasing payment annuity

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

Geometric increasing/decreasing payment annuity

Term of annuity