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An annuity is a regular series of payments. There are two types of annuities :

Annuity Immediate where the payments are at the end of the period
Annuity Due where the payments are at the beginning of the period

The future value of an annuity is the accumulated value at $t=n$ of the annuity paying a certain value every period for n years.

Arrears

This is when the payments are at the end of each period i.e. annuity immediate.

${S}_{n|}^{(p)}$ is the actuarial notation for the accumulated value of an annuity paying p times a year for n years with payments of ${1 \\over p}$ at the end of each period.

The formula is given by:

${S}_{n|}^{(p)}={(1+i)^n-1 \\over i^{(p)}}$

where:

$i$= annual effective rate
$i^{(p)}$= nominal rate (see Numbas- Nominal Rates)
$p$= number of periods in a year
$n$= number of years

In order to get the future value of an annuity paying a certain value p times a year at the end of every period, multiply ${S}_{n|}^{(p)}$ by both the value and p.

So the future value is given by:

$px{S}_{n|}^{(p)}$

where:

$x$ is the amount paid every period.

Advance

This is when the payments are at the beginning of each period i.e. annuity due.

$\\ddot{S}_{n|}^{(p)}$ is the actuarial notation for the accumulated value of an annuity paying ${1 \\over p}$ at the start of each period for n years.

The formula is given by:

$\\ddot{S}_{n|}^{(p)}={(1+i)^n-1 \\over d^{(p)}}$

where:

$i$= annual effective rate
$p$= number of periods in a year
$n$= number of years
$d^{(p)}$=nominal discount rate

The relationship between the nominal and effective discount rate is given by:

$1-d=(1-{d^{(p)} \\over p})^p$

Re-arranging this and making $d^{(p)}$ gives the nominal discount rate.

In order to get the future value of an annuity paying a certain value p times a year at the beginning of every period, multiply $\\ddot{S}_{n|}^{(p)}$ by both the value and p.

So the future value is given by:

$px\\ddot{S}_{n|}^{(p)}$

where:

$x$ is the amount paid every period.

Substituting in

$x$=£{value}
$p$={p}
$n$={n} years
$i$={interest}%

into the above formulas will give the future value of an annuity that is payed in arrears and in advance.

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the payments are made in arrears?

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the payments are made in advance?

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Compute the {time} value (to two decimal places) of an annuity paying £{value} {p} times a year for {n} years at an interest rate of {interest}% if:

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