Annuity (finance theory)

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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.

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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[1]:

Let:

r = the annual 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{r}{m}
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 \,=\,R\left[\frac{\left(1+i\right)^n-1}{i}\right] \,=\,R\cdot s_{\overline{n}|r} (Actuarial_notation#Annuities)

Also:

P \,=\,R\left[\frac{1-\frac{1}{\left(1+i\right)^n}}{i}\right] = R\cdot a_{\overline{n}|i}

Clearly, in the limit as n increases,

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

Thus even an infinite series of finite payments with a non-zero discount rate has a finite Present Value (q.v. Perpetuity).

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

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

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

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

Finally, after simplifications, we obtain

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

Hence:

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

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+i). Thus, the future value of an annuity-due can be calculated through the formula (variables named as above)[2]:

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

  • 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.

Annuity due is useful for lease payment calculatios

Texas Instruments BA II Plus Professional[3]


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)

  1. ^ Finite Mathematics, Eighth Edition, by Margaret L. Lial, Raymond N. Greenwell, and Nathan P. Ritchey. Published by Addison Wesley. ISBN 032122826X
  2. ^ ibid.
  3. ^ "Texas Instruments BA II Plus Guide Book", Texas Instruments

Retrieved from "http://en.wikipedia.org/wiki/Annuity_%28finance_theory%29"
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