Definitions

# Annuity (finance theory)

The term annuity is used in finance theory to refer to any terminating stream of fixed payments over a specified period of time. This usage is most commonly seen in academic discussions of finance, usually in connection with the valuation of the stream of payments, taking into account time value of money concepts such as interest rate and future value.

Examples of annuities are regular deposits to a savings account, monthly home mortgage payments and monthly insurance payments. Annuities are classified by payment dates. The payments (deposits) may be made weekly, monthly, quarterly, yearly, or at any other interval of time.

## Ordinary annuity

An ordinary annuity (also referred as annuity-immediate) is an annuity whose payments are made at the end of each period (e.g. a month, a year). The values of an ordinary annuity can be calculated through the following:

### Formulae

Let:
$r$ = the yearly nominal interest rate.
$t$ = the number of years.
$m$ = the number of periods per year.
$i$ = the interest rate per period.
$n$ = the number of periods.

Note:

$i=frac\left\{r\right\}\left\{m\right\}$
$n=tm$

Also let:

$P$ = the principal (or present value).
$S$ = the future value of an annuity.
$R$ = the periodic payment in an annuity (the amortized payment).

$S ,=,Rleft\left[frac\left\{left\left(1+iright\right)^n-1\right\}\left\{i\right\}right\right] ,=,Rcdot s_\left\{overline\left\{n\right\}|i\right\}$ (annuity notation)

Also:

$P ,=,Rleft\left[frac\left\{1-frac\left\{1\right\}\left\{left\left(1+iright\right)^n\right\}\right\}\left\{i\right\}right\right] = Rcdot a_\left\{overline\left\{n\right\}|i\right\}$

Clearly, in the limit as $n$ increases,

$lim_\left\{n,rightarrow,infty\right\},P,=,frac\left\{R\right\}\left\{i\right\}$

Thus even an infinite series of finite payments (perpetuity) with a non-zero discount rate has a finite present value.

### Proof

The next payment is to be paid in one period. Thus, the present value is computed to be:

$P , = , frac\left\{R\right\}\left\{1+i\right\} + frac\left\{R\right\}\left\{\left(1+i\right)^2\right\} + dots + frac\left\{R\right\}\left\{\left(1+i\right)^n\right\} = frac\left\{R\right\}\left\{1+i\right\} left\left[1 + frac\left\{1\right\}\left\{1+i\right\} + frac\left\{1\right\}\left\{\left(1+i\right)^2\right\} + dots + frac\left\{1\right\}\left\{\left(1+i\right)^\left\{n-1\right\}\right\}right\right].$

We notice that the second term is a geometric progression of scale factor $1$ and of common ratio $frac\left\{1\right\}\left\{1+i\right\}$. We can write

$P , = , frac\left\{R\right\}\left\{1+i\right\} times frac\left\{1 - frac\left\{1\right\}\left\{\left(1+i\right)^n\right\}\right\}\left\{1-frac\left\{1\right\}\left\{1+i\right\}\right\}.$

Finally, after simplifications, we obtain

$P , = , frac\left\{R\right\}\left\{i\right\} left\left[1 - frac\left\{1\right\}\left\{\left(1+i\right)^n\right\} right\right] = frac\left\{Rm\right\}\left\{r\right\} left\left[1 - frac\left\{1\right\}\left\{\left(1+frac\left\{r\right\}\left\{m\right\}\right)^\left\{\left(tm\right)\right\}\right\} right\right].$

Similarly, we can prove the formula for the future value. The payment made at the end of the last year would accumulate no interest and the payment made at the end of the first year would accumulate interest for a total of (n-1) years. Therefore,

$S , = , R + R\left(1+i\right) + R\left(1+i\right)^2 + dots + R\left(1+i\right)^\left\{n-1\right\} = R left\left[1 + \left(1+i\right) + \left(1+i\right)^2 + dots + \left(1+i\right)^\left\{n-1\right\}right\right].$

Hence:

$S , = , R left\left[frac\left\{\left(1+i\right)^n-1\right\}\left\{i\right\} right\right].$

If an annuity is for repaying a debt P with interest, the amount owed after n payments is:
$frac\left\{S\right\}\left\{i\right\}-\left(1+i\right)^n\left(frac\left\{S\right\}\left\{i\right\}-P\right)$
because the scheme is equivalent with lending an amount $frac\left\{S\right\}\left\{i\right\}$ and putting part of that, an amount $frac\left\{S\right\}\left\{i\right\}-P$, in the bank to grow due to interest. See also fixed rate mortgage.

## Annuity-due

An annuity-due is an annuity whose payments are made at the beginning of each period. Deposits in savings, rent payments, and insurance premiums are examples of annuities due.

Because each annuity payment is allowed to compound for one extra period, the value of an annuity-due is equal to the value of the corresponding ordinary annuity multiplied by (1+i). Thus, the future value of an annuity-due can be calculated through the formula (variables named as above):

$S , = , R left\left[\left\{ \left(1+i\right)^\left\{n+1\right\} - 1 over i \right\} right\right] - R,=,Rcdot ddot\left\{s\right\}_\left\{overline\left\{n|\right\}i\right\}$ (annuity notation)
It can also be written as

$S ,=,Rleft\left[frac\left\{left\left(1+iright\right)^n-1\right\}\left\{i\right\}right\right]\left(1 + i \right)$
,=,Rcdot s_{overline{n}|r}$\left(1 + i\right)$

An annuity-due with n payments is the sum of one annuity payment now and an ordinary annuity with one payment less, and also equal, with a time shift, to an ordinary annuity with one payment more, minus the last payment.

Thus we have:

$ddot\left\{a\right\}_\left\{overline\left\{n|\right\}i\right\}=a_\left\{overline\left\{n\right\}|i\right\}\left(1 + i\right)=a_\left\{overline\left\{n-1|\right\}i\right\}+1$ (value at the time of the first of n payments of 1)
$ddot\left\{s\right\}_\left\{overline\left\{n|\right\}i\right\}=s_\left\{overline\left\{n\right\}|i\right\}\left(1 + i\right)=s_\left\{overline\left\{n+1|\right\}i\right\}-1$ (value one period after the time of the last of n payments of 1)

## Other types of annuities

• Fixed annuities - These are annuities with fixed payments. They are primarily used for low risk investments like government securities or corporate bonds. Fixed annuities offer a fixed rate up to ten years but are not regulated by the Securities and Exchange Commission.
• Variable annuities - Unlike fixed annuities, these are regulated by the SEC. They allow you to invest in portions of money markets.
• Equity-indexed annuities - Lump sum payments are made to an insurance company.

Annuity due is useful for lease payment calculations