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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

P V = A ⋅ 1 − 1 ( 1 + r ) n r {\displaystyle PV\,=\,A\cdot {\frac {1-{\frac {1}{\left(1+r\right)^{n}}}}{r}}}

In the limit as n {\displaystyle n} increases,

lim n → ∞ P V = A r {\displaystyle \lim _{n\,\rightarrow \,\infty }\,PV\,=\,{\frac {A}{r}}}

Thus even an infinite 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

F V = A ⋅ ( 1 + r ) n − 1 r {\displaystyle FV\,=\,A\cdot {\frac {\left(1+r\right)^{n}-1}{r}}}

In each of these formulae, A {\displaystyle A} is the periodic amount of the annuity, r {\displaystyle r} is the period interest rate, and n {\displaystyle 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

P V   =   A ⋅ 1 − 1 ( 1 + r ) n r ⋅ ( 1 + r ) {\displaystyle 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

F V   =   A ⋅ ( 1 + r ) n − 1 r ⋅ ( 1 + r ) {\displaystyle 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 RST 0.00
Enter number of payments 10 N= 10.00<
Enter interest rate per payment period 5 I/Y= 5.00<
Enter payment 2000 PMT= 2,000.00<
Compute present value PV= 15443.47

note: Press in the last step instead of 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 RST 0.00
Enter number of payments 10 N= 10.00<
Enter interest rate per payment period 5 I/Y= 5.00<
Enter payment 2000 PMT= 2,000.00<
Set beginning-of-period payments BGN
Return to calculator mode 0.00
Compute present value PV= 16215.64

note: Press in the last step instead of to calculate the future value(1)

References

See also

External links

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