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 | 1/1.08^{2} | 857.34
March | 1000 | 1/1.08^{3} | 793.83
April | 1000 | 1/1.08^{4} | 735.03
May | 1000 | 1/1.08^{5} | 680.58
June | 1000 | 1/1.08^{6} | 630.17
July | 1000 | 1/1.08^{7} | 583.49
August | 1000 | 1/1.08^{8} | 540.27
September| 1000 | 1/1.08^{9} | 500.25
October | 1000 | 1/1.08^{10} | 463.19
November | 1000 | 1/1.08^{11} | 428.88
December | 1000 | 1/1.08^{12} | 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 *a*_{n} 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 - v^{n}
a_{n} = -------
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.