More abstractly, in general interest theory an annuity is any payment that is made regularly over a certain length of time. With the exception of the Arabic banking system that does not make use of interest, annuities are not particular to any economic system and are fairly widespread.

Examples of annuities are mortgage payments on a house, repayment of a loan in monthly installments and monthly payment of bills. In the first two cases the annuity has a predetermined end date, while in the third example the annuity has no apparent end date (You'll be paying bills for the rest of your life. Get used to it.) Special annuities with no end date are known as perpetuities. Another example of a perpetuity is the regular payments of health insurance, which only end with the death of the insured.

For annuities with a final date of payment, by the term of the annuity we mean the length of time for which the annuity is to be paid. The regular time intervals in which the payments of the annuity are made as known as the period of the annuity. The word "annuity" reminds us of a time when the periods were always years, but now the word has acquired a more broad meaning with other possible periods. For example, the payment of a loan might have a term of one year and period of one month if paid on monthly installments. If the payments are made at the end of the period, then we speak of an annuity-immediate, while annuities with payment at the beginning are annuities-due. The difference is small and not terribly important.

Because of the time value of money, (money you have today is more valuable than money you have tomorrow), the computation of the present value of an annuity can be a little interesting. If we take time value and interest rates into account, an annuity of twelve monthly payments of 1000 roubles is not worth 12,000 roubles today, but must be discounted in accordance with interest rates. An example should be make this clear.

Suppose that Vronsky and Karenin have made an agreement for Vronsky to repay a debt in the form of twelve monthly payments of 1000 roubles, and that the payments are made at the end of the month (an annuity-immediate), with the first payment on January 31st, 1878, and suppose that interest is compounded on the same day that payments are due. Suppose that Karenin is compassionate towards Vronsky's pride and wants to salvage him from the embarassment and discomfort of coming each month to his manor to pay the amount. Karenin being the the gentleman he is instead offers Vronsky the opportunity to repay the entire debt in one sum today, January 1st 1878, taking into account an interest rate of 8%.

In short, at an effective interest rate of 8%, how much is an annuity of twelve payments of 1000 worth today?

Because Karenin would prefer to have 1000 roubles today than by the end of January, we must discount the value of 1000 roubles at end of January by 108%, so they are in fact only worth 1000/1.08 = 925 roubles and 93 kopeks to Karenin today. With compound interest, the payment at the end of February is worth less, only 1000/(1.08)2 = 857 roubles and 34 kopeks, and that at the end of March is even less, a mere 1000/(1.08)3 = 793 roubles and 83 kopeks. And so on. A table may help:

```
Month    |  Payment due | Discount factor | Present value
---------+--------------+-----------------+--------------
January  |  1000        |  1/1.08         |   925.93
February |  1000        |  1/1.082        |   857.34
March    |  1000        |  1/1.083        |   793.83
April    |  1000        |  1/1.084        |   735.03
May      |  1000        |  1/1.085        |   680.58
June     |  1000        |  1/1.086        |   630.17
July     |  1000        |  1/1.087        |   583.49
August   |  1000        |  1/1.088        |   540.27
September|  1000        |  1/1.089        |   500.25
October  |  1000        |  1/1.0810       |   463.19
November |  1000        |  1/1.0811       |   428.88
December |  1000        |  1/1.0812       |   397.11

TOTAL:  7536.07

```

So because Karenin is considering such an attractive interest rate for Vronsky, Vronksy has the choice to pay only 7536 roubles and 7 kopeks today and save himself some embarassment in the face of Society.

The mathematically inclined will have no problems in recognising the geometric series involved with annuities and in devising appropriate general formulae, perpetuities being the case of infinite geometric series (convergence is assured if we assume positive interest rates). If an denotes the discounted value of one dollar at the end of n periods, i the rate of interest, and v = (1 + i)-1 the rate of discount, then

```          1 - vn
an = -------
i
```

Annuities-due require similar computations which can be simplified by the introduction of the concept of discount rate instead of interest rate. I leave these generalisations to the industrious reader.

References:

Kellison, Stephen. The Theory of Interest McGraw-Hill, 1991.