Financial Math FM/Annuities

From Wikibooks, open books for an open world
Jump to: navigation, search

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 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 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)

Geometric increasing/decreasing payment annuity[edit]

Term of annuity[edit]