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 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 $C_{1}$ 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)