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Encyclopedia > 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. Finance theory is the field that deals with investment making decisions and the concept of the time value of money. ... The time value of money (TVM) is a way of calculating the value of a sum of money, at any time in the present or future. ...

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 present value of an ordinary annuity can be calculated through the formula

n | r

In the limit as n increases,

Thus even an infinite series of finite payments with a non-zero discount rate has a finite Present Value.

The future value of an ordinary annuity can be calculated through the formula

$FV(A) ,=,pcdotfrac{left(1+rright)^n-1}{r} ,=,pcdot S$n | r

In each of these formulae, p is the periodic amount of the annuity, r is the period interest rate, and n is the number of periods.

## Proof

Let A be the constant value of the payments to be paid at the end of the next n periods at effective rate of interest r. The next payment is to be paid in one period. Thus, the present value, noted PV, is computed to be

We notice that the second term is a geometric progression of scale factor 1 and of common ratio . We can write Diagram showing the geometric series 1 + 1/2 + 1/4 + 1/8 + ... which converges to 2. ...

$PV , = , frac{A}{1+r} times frac{1 - frac{1}{(1+r)^n}}{1-frac{1}{1+r}}.$

Finally, after simplifications, we obtain Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic. ...

$PV , = , A cdot frac{1 - frac{1}{(1+r)^n}}{r}.$

## Annuity Due

An annuity-due is an annuity whose payments are made at the beginning of each period.

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+r). Thus, the present value of an annuity-due can be calculated through the formula

$PV = A cdot { 1 - { 1 over (1+r)^n } over r } cdot (1+r)$

The future value of an of annuity-due can be calculated through the formula

$FV = A cdot { (1+r)^n - 1 over r } cdot (1+r)$

Thus

Another intuitive way to interpret an annuity-due is as the sum of one annuity payment now (at time = 0) and an ordinary annuity without an annuity payment at the end of the last period (e.g. n-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 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.

The Securities and Exchange Commission, commonly referred to as the SEC, is the United States governing body which has primary responsibility for overseeing the regulation of the securities industry. ... An equity index annuity in the United States is a type of tax-deferred annuity whose return is indexed to an equity index, typically the S&P 500 but which also guarantees a minimum interest rate (typically about 3% as of 2007) and against a loss of all or most...

## Finding Annuity Values with a Financial Calculator

Texas Instruments BA II Plus Professional[1]

To calculate present value of an ordinary annuity, with an annual payment of \$2000 for 10 years and an interest rate of 5%

To Press Display
Set all variables to defaults [2nd] [RESET] [ENTER] RST 0.00
Enter number of payments 10 [N] N= 10.00<
Enter interest rate per payment period 5 [I/Y] I/Y= 5.00<
Enter payment 2000 [PMT] PMT= 2,000.00<
Compute present value [CPT] [PV] PV= 15443.47

note: Press [CPT] [FV] in the last step instead of [CPT] [PV] to calculate the future value

To calculate present value of an annuity due, with an annual payment of \$2000 for 10 years and an interest rate of 5%

To Press Display
Set all variables to defaults [2nd] [RESET] [ENTER] RST 0.00
Enter number of payments 10 [N] N= 10.00<
Enter interest rate per payment period 5 [I/Y] I/Y= 5.00<
Enter payment 2000 [PMT] PMT= 2,000.00<
Set beginning-of-period payments [2nd] [BGN] [2nd] [SET] BGN
Compute present value [CPT] [PV] PV= 16215.64

note: Press [CPT] [FV] in the last step instead of [CPT] [PV] to calculate the future value(1)

## References

1. ^ "Texas Instruments BA II Plus Guide Book", Texas Instruments

A perpetuity is an annuity in which the periodic payments begin on a fixed date and continue indefinitely. ... The life annuity (also known as a single-payment annuity) is a financial instrument that allows for a seller (issuer), typically a financial institution such as a life insurance company, to provide a series of future payments to a buyer (annuitant) for a known sum with a net present value...

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