Annuity, a yearly payment, subject to various conditions. The payment may be stipulated without regard to any contingency, in which case it is called an annuity certain. If limited in time, it is called a term or temporary annuity certain. If no limit is fixed, it is called a perpetuity. A contingent annuity is one of which the payment is limited by the occurrence of some future event, uncertain as to time, but more or less probable. It is temporary when it must cease at a fixed time, provided it has not already ceased by the previous occurrence of the contingent event. The consideration for insurance, as generally paid, is an example of contingent annuity, but is called premium. Annuities paid as reward for meritorious services are called pensions; and those paid for the use of real estate are called rent. - Though the term annuity implies a year as the interval between the payments, yet in practice it is made to include any series of equal or uniformly increasing or decreasing payments at equal intervals, as annual, semi-annual, quarterly, or monthly; and in mathematical theory the intervals may be infinitely small, when the annuity is said to be payable momently. - The most important contingency ever introduced into annuity contracts is that of death.

A fixed sum which is payable at equal intervals during the entire life of a person is called a life annuity. If it depends on two or more lives, and is to cease on the death of either, it is called a joint life annuity. A survivorship annuity is one which so depends upon two or more lives, that it is to commence only when one or more begin to be survivors. These annuities may be temporary, or for the whole life, immediate or deferred; that is, the first payment may take place in advance or immediately after the occurrence of the contingency, or it may be deferred one or more of the equal intervals. The most important question in regard to any such series of payments is its present value. This would be easy to answer in regard to annuities certain, but for the interest of money. For if money earned no interest, the present value would be the sum of all the future payments, which in case of a perpetuity would be infinite. If we assume any perpetual rate of interest, the present value of a perpetuity at that rate is obviously the principal that will yield that interest; and this principal is always less in regard to a given interest as the rate is higher. Any term annuity certain may be considered as the early portion of a perpetuity.

Hence the difference between the principal, P, which is the present value of the perpetuity, and the same discounted at compound interest for the intervals of the term n, at the assumed rate i (P - 1/(1+inP), is the present value of the term annuity for n intervals. The subtrac-tive quantity (1/1+i)n P, as it is usually written) is called the reversionary value of the term annuity, or of the estate whose income is absorbed by its payment. Any annuity is said to be worth as many "years' purchase " as the times it is contained in its present value. - The present value of any life annuity involves a medical as well as a mathematical question. The mathematical solution, which comes first, is founded on an assumed rate of mortality, more or less worthy of confidence, according to its agreement with observed facts when taken in large numbers. This gives the value supposing the life in question possessed of the average vitality due to its age. Medical science will modify this result so far as it can determine a variation of the individual from the assumed average, though it has no means of fixing a definite or numerical variation.

It is a common mistake to suppose that the present value of a life annuity can be found from the "'expectation of life," or average after-lifetime at the given age, by finding the present value of an annuity certain for the term of that expectation. This can be true only when the assumed interest is zero. This popular error has been much fostered by life insurance companies publishing tables of "expectation," which can have no possible application to their business except by this erroneous method, and which, so applied, only prove their premiums too high. The only correct method of applying the rates of mortality and interest to ascertain the present value of any series of payments contingent on life, is to apply them separately to each and every possible payment. Each future payment must, in effect, be multiplied by the present value at compound interest of a dollar, or monetary unit, payable certain at that time; and this product must again be multiplied by the fraction, derived from the table of mortality, expressing the probability of the party being then alive to pay it.

The sura of as many such products as there are possible payments is the present value of the life annuity. "When only one or two lives are concerned, there are tables which abridge the operation to a narrow compass; but when there are three or more, the combinations become too numerous to admit of exhaustive tables, and mathematicians content themselves with methods of approximation to solve particular problems. - After the mathematical solution, which can only be as correct as the assumptions on which it is founded, comes the medical, weighing the special facts by which the individual case differs from the average or general type. The reason why the business of selling annuities to commence in a year or less is always unprofitable to an honest company, and why it is unprofitable to a government as a means of borrowing money, is that the medical selection is in favor of the buyer. On the contrary, companies dealing in policies of insurance, or selling annuities long deferred, succeed by a medical selection in their own favor. - The most valuable recent contributions to the basis for calculating life annuities are contained in the works of Chisholm, and the writings of Dr. Farr, in connection with the reports of the registrar general of Great Britain. Very valuable observations have also been made by Mr. Meech on the United States census of 1860, and by Mr. Elliott on the population returns of Massachusetts.