Art. 186. An annuity in reversion is an annuity which does not commence until a specified time has elapsed, or some particular event has taken place.

Suppose we wish to find the present worth of an annuity of $100, which is to commence at the end of 2 years, and then to continue 4 years, allowing 6 per cent. compound interest.

The present worth is evidently such a sum as would, in 2 years, at 6 per cent. compound interest, produce an amount equal to the present worth of the annuity, were it to commence immediately.

We must therefore find the present worth of an annuity of $100 to commence immediately and to continue 4 years, which is $346.5106. We wish now to obtain a sum whose amount in 2 years will equal this present worth. This may be found by dividing $346.5106 by the amount of $1 for 2 years; thus, $346.51061.1236-$308.394+.

Hence, to find the present worth of an annuity in reversion, at compound interest, we have the following rule.

RULE. Find the present worth of the given annuity for the time of its continuance, supposing it to commence immediately; then divide this present worth by the amount of $1 for the time in reversion; the quotient will be the present worth of the annuity in reversion.

Or, find the present worth of the annuity from the present time to the end of the time of its continuance, by Table IV.; find also, by Table IV., its present worth before it commences; the difference of these two results will be the present worth.

1. What is the present worth of an annuity of $200, which is to continue 5 years, but not to commence until the end of 2 years, allowing compound interest at 6 per cent. ? Ans. $749.7976.

2. What amount of ready money will purchase the reversion of an estate which rents for $60 a year, to continue 6 years, but not to commence until the end of 3 years, allowing compound interest at 6 per cent. ? Ans. $247.72.

3. Suppose a parent leaves to his son an annual rent of $310 for 8 years, and the reversion of the same rent to his daughter for the next succeeding 14 years. What is the present worth of the legacy of each, allowing compound interest at 6 per cent.?

Son's, $1925.036+.
Daughter's, $1807.854+.

LIFE INSURANCE.

Art. 187. LIFE INSURANCE is a contract whereby one or more, or a company of persons, in consideration of the payment of an amount called the premium, promise, either on the decease of a person named, whenever it may happen, or on his decease within a given time, to pay a certain sum to his heirs, or others for whose benefit the contract is made. This contract is now more frequently made by Mutual Life Insurance Companies, all the members of which contribute, in certain agreed proportions, to a common fund, out of which the sums due on the decease of a member, that is, the losses, are to be paid; the surplus, if any, being distributable, from time to time, among the members.

In the early stages of life insurance in England, in the seventeenth century, a premium of five per cent. on the amount insured was charged upon each life, without discrimination on account of difference of age. It, however, early became obvious that, all other things being equal, the younger of two persons has a better chance, or probability, or, as it is usually termed, expectation of life, than the elder; and consequently, if the two, or any greater number, by the annual payment of equal sums during their whole lives, make up a common fund for the payment of a given sum to the heirs of each, it is plain that the younger associates will pay a greater amount into this joint stock than the older, so that the partnership will be a very unequal one. The question then arises, what different yearly amounts the contributors of different ages ought to pay into the common fund, out of which the respective heirs of each are to be entitled to a given sum on his decease. To answer this question, it is necessary to determine upon some law of mortality, that is, how many out of a given number will die annually at each age. Registers of deaths have been kept at various places in Europe at different periods, for the purpose of ascertaining this law in those places, and the experience of life insurance companies has also been resorted to for a similar purpose. Life insurance has hitherto been little practised in the United States, and the records of population and registers of deaths have been very imperfect. Some such registers have been kept, which, with the censuses of the population of the United States taken every ten years, afford some materials for making an estimate of the rate of mortality. At the time of the New England Mutual Life Insurance Company going into operation, in 1843, in Boston, very considerable investigation was made, in order to ascertain a rate of mortality which could be safely adopted as the basis of the premiums to be charged by the company. A very material inquiry was, whether the European registers and experience would be a safe guide, and it appeared from the registers kept and the censuses of population, that in the older and long-settled parts of New England and the Middle States, at least, the_rate of mortality approximated very nearly to those assumed in Europe as the basis of calculations of premiums of insurance of lives, and the value of life annuities. From the various materials thus offered, the following table of mortality was formed for the ages of ten and upwards, as the basis of the computation of the premiums to be paid by the members of that

company. This table starts with 10000 lives at the age of ten, and notes the annual decrements by deaths, until the whole number is exhausted.

This table gives at the age of eleven, that is, after one year, 9935

lives surviving, 65 out of this society of 10000 having died during

the year. Let it be proposed that the heirs of each one of the society of 10000, of the age of ten, who shall die within the year, shall receive 100 dollars out of a fund to be raised by a contribution from the whole 10000. This fund must be 6500 dollars, and the assessment or premium for each to pay will be 65 cents. Looking to the age of twenty in the above table, we find the number of surviving lives to be 9388, of whom, according to the chance or probability presented by this table, the number of deaths or decrement during a year will be 71, leaving 9317 survivors at the age of twenty-one. The number of deaths or decrement during a year is, as we see, greater than at the age of ten to eleven; and this greater number is out of a less number of lives; the ratio at ten to eleven being To, and that at twenty to twenty-one being. Therefore, the fund requisite for the payment of 100 dollars to the heirs of each member of the society who shall die during a year at this age, is 7100 dollars; which assessed upon 9388 members of the society at the beginning of their twenty-first year, gives a premium of 75 cents to be paid by each.

65

So if the society is for more than one year, -either for a term of years or for life, it is to be ascertained what sum must be paid out each year, which is readily ascertained by the above table, and a sufficient assessment, payable annually, or at other fixed periods, or all at once, or payable in less or diminishing amounts, or greater or increasing amounts, at the beginning of each successive year or other period, must be made upon the members to meet the payments.

But when the association is for more than a year, and the fund is to be raised by the same assessment or premium paid by a member every year as long as he continues to be a member, it is plain that a part of the payment made the first year, and so on to the expiration of about one half of the period of his membership, is paid to meet the increasing amounts of the losses during the latter part of the period; the consequence is that the company must accumulate a fund to be held over into the subsequent part of the period.

This brings a new element into the calculation, namely, the interest to be gained on this accumulated fund, which the company is supposed to invest in mortgages, stocks or otherwise, so that it may yield an income. Accordingly, in the computation of the premium to be paid by each member whose insurance is for more than one year, the interest to be derived upon a part of his early payments must be taken into the account. If the company is formed for a period of a few years only, it may be presumed that the existing rate of interest will continue without any material change; but such companies are usually formed for an indefinite period, and must be so if they insure for the whole life of a member, for the chance is, as we see by the table, and know very well otherwise, that some of the members may survive to a very old age; and it is absurd for a company, which is by its constitution to subsist but twenty years, to agree to pay a sum on the decease of a member who may live ninety or a hundred; a member of such a company, who is insured for his whole life, ought to get insurance in another that he shall die within twenty years. The probable rate of interest for a long series of years to come must be

assumed in calculating the premium. That assumed by the New England Mutual Life Company was four per cent. per annum.

Other expenses and contingencies besides the payment of losses resulting from the ordinary rate of mortality, must be taken into the account in calculating the premium, such as the chance of a greater mortality than that assumed, incidental expenses, loss on investments, which may take place notwithstanding the utmost prudence. To provide against those contingencies and expenses, that company added twenty per cent. to the rate of premium requisite for merely meeting the losses at the assumed rate of mortality; and the calculation thus made results in the following rates of annual premiums to be paid for insurance on the lives of persons of different ages, from ten to sixty. In determining the rate for a mutual company, the premium should be put at a point which will be certainly high enough; and if it should prove higher than is necessary, the excess is distributed back, whereas if it be too low, the company will sooner or later become insolvent.

TABLE OF PREMIUMS.

Annual payment for each hundred dollars insured.

It is apparent that if the business of a mutual company is prudently managed, there is a probability amounting to almost a cer tainty of a handsome return to the members on the premiums paid