Annuity

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.

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

Failed to parse (syntax error): {\displaystyle PV \,=\,A\cdot\frac{1-\frac{1}{\left(1+r\right)^n}{r}}

In the limit as $n$ increases,

$\lim _{n\,\rightarrow \,\infty }\,PV\,=\,{\frac {A}{r}}$

Thus even an inifinite series of 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

$FV\,=\,A\cdot {\frac {\left(1+r\right)^{n}-1}{r}}$

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

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

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

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).

Finding Annuity Values with a Financial Calculator

Texas Instruments BA II Plus Professional

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
Return to calculator mode	[2nd] [QUIT]	0.00
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

See also

• Annuity formula
• Perpetuity
• Variable Annuity
• Deferred Annuity

External links

• Annuities: Ordinary? Due? What Do I Do? -- Annuity tutorial (with quiz) from Prof. John Wachowicz at the University of Tennessee.
• Edmond Halley, An Estimate of the Degrees of the Mortality of Mankind, drawn from curious Tables of the Births and Funerals at the City of Breslaw; with an Attempt to ascertain the Price of Annuities upon Lives, Philosophical Transactions, 196 (London, 1693), p.596-610.
• Non-commercial webform to calculate annuities
• Annuity Problems -- Practice time-value-of-money annuity problems with answers and detailed solutions.
• Growing Annuities Article explains how to determine the future/present value of a growing annuity (i.e., an annuity in which the payments/receipts increase each period at a constant percentage).
• Immediate annuity calculator for commercial annuity present values Annuity-immediate
• calculate above formulas with your own values to understand