CHAPTER VI ANNUITIES 30. Definitions and notation. An annuity is a series of pay- ments, usually equal in amount, made at equal intervals of time, called intervals of payment. Unless the contrary is specifically stated, the first payment is made at the end of the first interval. If the first payment is made at the beginning instead of at the end of the 'first interval, the annuity is called an annuity due. An annuity certain is one for which the payments begin and end at fixed dates. The time to elapse between the beginning of the first interval of payment and the end of the last is called the term of the annuity certain. If the date either of the first or of the last payment depends upon some event, the time of whose occurrence cannot be fore- told, the annuity is called a contingent annuity. An annuity whose payments begin or end with the death of an individual would be a contingent annuity. A deferred annuity is one for which a certain specified number of intervals must elapse before the first payment is made. A de- ferred annuity becomes an ordinary annuity after the lapse of the specified number of intervals. To be exact, if the annuity is deferred m intervals, the first payment is made at the end of the (m+1) st interval. A perpetuity is an annuity whose payments are supposed to continue forever. When the payments are equal in amount, the sum of the pay- ments made in one year is called the annual rent. If the annual rent be -R, we speak of an annuity of R per annum. If the payments of an annuity are allowed to accumulate to the end of the term during which they are made, the annuity is said to be forborne, and the total sum due at the end of this term \s called the amount of the annuity. 77 78 MATHEMATICAL THEORY OF INVESTMENT The present value, or the cash equivalent, of an annuity is the sum of the present values of all the payments of the annuity. One has only to cite a few examples to see the wide range of application of the theory of annuities. The rental of a house, the interest payments on a mortgage note, the annual premiums on a life-insurance policy, the stated return from an interest- bearing bond, a pension, are all examples of annuities. In fact, any process involving the stated payment of a sum of money may be looked upon as giving rise to an annuity. The following notation is standard among English-speaking writers on the subject of annuities: 8yi denotes the amount of an annuity of 1 per annum, payable annually for n years. s^ denotes the amount of an annuity of 1 per annum, payable va.p installments at equal intervals throughout the year for n years. o^ denotes the present value of 1 per annum, payable annually for n years. a^ denotes the present value of 1 per annum, payable p times a year for n years. ^a^ denotes the present value of a deferred annuity of 1 per annum, payable p times a year for n years, the first payment to be made after the lapse of m + — years. If the annuity is due, the roman " fullface " a is used for the present value. Thus, a^ and a^ are used to denote the present values of the annuities due, the first of 1 per annum, payable annually for n years, and the second of 1 per annum, payable p times a year for n years. When interest is convertible oftener than once a year, the rate of interest may be given as either effective or nominal. All formulas involving the rate of interest may be expressed in tlie one or the other of these at will by means of the fundamental relation / \m (i+.)=(i^). All theoretical computations are made for an annuity of 1 per annum.