rapid growth in the writing of this form of policy is an indication that it is to be an ever increasing factor in the insurance business of the nation, there is another and broader aspect of the subject which has constantly forced itself upon my attention.

The growth of the "income idea" is not only an indication of the increased thought of life insurance investors, but it is a guage and a sign of the increasing growth of thought and sense in the American people. It is an almost certain indication of the solid, sure and inevitable development of the character of the people of this country toward care and thrift. It is one of the great evidences of the passing of the pioneer, the carelessness for the future, the instability of aim which mark all new countries. It is the growth of the desire to make life more certain, comfort more sure, reward more definite, which is the underlying desire of civilized man and which is the measure, in its presence and force, of his civilization. It is not the only evidence of this national development, but it is one of the evidences which is most easily discernible and lends itself to accurate measurement. It is not necessary for me to add that this increasing tendency toward thrift is something which does, and indeed ought to, give rise to a sentiment which may be described only as patriotic.

Not only in its revelation of the growing thrift of the people does this growth of income insurance give material for wider thoughts and open up great vistas of economic speculation. It is a further indication, it seems to me, of the inherent purpose of the American people to solve problems in a democratic way. It is a step toward the solution of such vexing questions as "old age pensions" and "compensation" which have for some years occupied the minds of European governments and resulted in advanced and debated legislation. It would be a remarkable thing indeed if the American nation, of itself, within its own homes, by the might of the massed consciousness of a sane and thinking people, should solve these problems without the aid of laws by which the thrifty majority force thrift upon the thriftless minority. It is possible to believe that the people of this nation may in such increasing numbers subtract investment from wages in sufficient amount to safeguard old age and disability that the government shall never have to do it for them. That this is not Utopian may be learned from a glimpse at the statistics of life insurance; that it is democratic may be main

tained by the statement that the mutual life insurance company is one of the most voluntary and democratic institutions ever developed in society.

It is in consideration of these wider aspects of the relation between life insurance as an institution and the nation as a whole that the income policy takes on a singularly significant, almost prophetic aspect. Nor can the man who reflects upon this aspect, with its great and ever widening growth, its increasing weaving of itself into the very heart of the thrift and personal prosperity of a vast people, fail to reflect also upon the equally enormous growth of life insurance institutions which must inevitably go with it. It is a reflection which brings with it a solemn feeling of responsibility— but not of doubt. For I firmly believe that the same desire to protect the home which has been the root and stock of American life insurance, the same growing sense of thrift and responsibility which has been the main spring of all its developments, such as the income policy, will keep pace with material growth, and that the same spirit which has increased insurance savings to the billions will walk hand-in-hand with the spirit which converted insurance companies from private corporations into great mutual democracies. It is of a vision such as this, not only possible but probable, that the rapid growth and development of the income policy is a hope and a sign.

LIFE ANNUITIES

BY M. ALBERT LINTON, F.I.A., F.A.S.,

Vice-President and Associate Actuary, The Provident Life and Trust Company of Philadelphia.

The primary function of the life annuity is to insure that a given sum of money will produce a life income larger in amount than could be safely secured through the channels of ordinary investment. The regular life annuity contract is a promise to pay, in consideration of a single cash sum, a fixed amount periodically during the lifetime of a designated person, called the annuitant. Annuities in practice are paid yearly, half-yearly, quarterly or monthly.

The periodical payment yielded by a given sum invested in an annuity is larger than the return through the channels of regular investment, for the reason that each annuity payment consists of interest and of a portion of the principal of the invested fund. If the length of life of the individual could be exactly foretold, he could establish his own annuity fund. He would be able to use a definite portion of the principal each year with assurance that the fund would not be exhausted during his lifetime. For example, if a man knew that he would live exactly ten years from a given moment and if he knew that during that period he could realize exactly $4.50 for each $100 invested in the regular manner, he would find by a simple computation that each $100 applied as an annuity would yield $12.64 at the end of each year for ten years. Each yearly payment would contain a portion of the principal as well as interest. At the expiration of the ten years the last cent of principal would have been withdrawn.

Because of the uncertainty of life for the individual, and, on the other hand, of the comparative certainty of the average mortality rates that will be experienced by a large group of individuals collectively, annuities are made possible through the coöperative association of annuitants into groups, sufficiently large to insure the operation of the law of average. In practice, this coöperative association is made possible by the insurance companies which

establish annuity funds and sell annuities to the public. It is the purpose of the present article to discuss in a non-technical manner some of the problems that arise in theory and practice in connection with annuities and to portray some of the developments that have come about in recent years in the application of annuities to the needs of the public.

COMPUTATION OF ANNUITY RATES

To the layman, the calculation of the annuity premium is presumably based upon the so-called "Expectation of Life" of the annuitant. That is to say, if the mortality table which forms the basis of the calculation indicates that the expectation of life is a certain number of years, it is assumed that the true annuity premium is the present value of a series of annuity payments extending for exactly that number of years. A calculation of this nature will, it is true, yield a rough approximation to the true present value of the life annuity, but that is all. It can be demonstrated that the method always overstates the annuity value. In practice it is never employed for the scientific computation of rates. In fact, to the actuary, the expectation of life is hardly more than of academic interest. He may now and then employ it to compare the relative characteristics of different mortality tables, but for the scientific computation of monetary values, never.

The correct method of computing annuity premiums is essentially as follows: The mortality table, upon which the computation is based, consists fundamentally of a series of numbers, showing how many persons out of a given number alive at the youngest age in the table, survive to each age throughout the possible range of life. Given, therefore, a large group of persons all of the same age, the mortality table renders it possible to forecast how many of the group will be alive one year hence, two years hence, three years hence, and so on until all of the members of the group have passed away. If, therefore, a promise should be made to pay a yearly annuity of a dollar to each member of the original group, it could be foretold how much would have to be paid at the end of the first year to those surviving at that time, how much would have to be paid at the end of the second year and at the end of the third year, and so on throughout the number of years covering the possible span of life of any of the members of the group. This

series of payments may be compared to a serial bond issue maturing in definite amounts throughout a period of years. And just as the banking house computes the present value of the principal payments under the serial bond issue, so the actuary computes the present value of the series of annuity payments that will be made to the members of the annuity group. Dividing the present value of the complete series of future annuity payments, by the original number of members of the group, he arrives at the true present value of the life annuity on the basis of the mortality table employed and of the rate of interest assumed in determining the present value of the future payments.1

It will be of interest to compare the present value of the life annuity computed by the erroneous expectation-of-life method, with the true present value. Employing the Carlisle Table of mortality as the standard, we find that at age 35 the expectation of life is 31 years. On a 4 per cent basis the present value of an annuity of $1,000 payable yearly for 31 years, is $17,588. The true value of the life annuity of the same amount at age 35, on the basis of the Carlisle Table and 4 per cent interest, is $16,041. At age 75, where the expectation of life is 7 years by the Carlisle Table, the erroneous value of the 4 per cent life annuity of $1,000 payable yearly is $6,002 and the true value is $5,239. It is strange that in so many of our courts, the expectation-of-life method is still retained for the calculation of claims where life annuities are involved. One would have surmised that the error therein involved would not have remained unchallenged and that a demand would have arisen for the abandonment of a method which is so unfitted for scientific computation.

ANNUITY RATES IN AMERICA AND IN GREAT BRITAIN

At this point it will be instructive to set out in tabular form the percentage yearly return upon capital invested in an annuity at certain ages in America and in Great Britain. For the purpose of comparison, the fifteen largest United States' and Canadian companies which write annuities, and the fifteen largest British com

1 In practice, this extended method of computation is not actually required, since mathematical short-cuts have been developed which greatly facilitate the actuarial calculation. The final results of the short method, however, are identical with those obtained by the extended process described.