HUNT'S

MERCHANTS' MAGAZINE

AND

COMMERCIAL REVIEW.

AUGUST, 1860.

Art. I.-VALUATION OF LIFE INSURANCE POLICIES.

NUMBER IV.

IN former numbers we have insisted that the valuation of gross premiums ignores future expenses and other contingencies, and anticipates future profits that may never be received; that the method of net premiums reserves every farthing that has been earned; and even more, for it only provides for the net cost of the future hazard as far as it is met by the past premiums, besides omitting all reference to the deterioration of life, or the diminished mortality of the early years of the policy. But in order to anticipate the correct value of the future risk, it is necessary to know the true rate of mortality at every age for the whole duration of life. If this cannot be known, we must obtain the nearest approximation to it that is possible. With this we can determine what portion of the past payments belong to the future hazard not yet incurred; and what amount must be reserved out of past profits to make up any deficiencies in the future premiums towards paying their share of the future risk and future contingencies. With a false or defective table both these results will be more or less erroneous.

To learn the true mortality that shall hereafter occur in any company's experience is of course impossible. But, as the future may be judged by the past, in life and death as well as other natural phenomena, it is only necessary to obtain the past mortality at every period of life, and to add to it a proper margin for fluctuations, deteriorations of life, and the probable excess of American deaths over the experience of other countries. We have no vital statistics in the United States that are sufficient for this purpose, nor are we likely to have for years to come. The United States census, and the mortuary reports in our cities, and the imperfect registrations in the several States, do not furnish even an approximation to the true mortality. An extensive collection of materials has

been begun by our life companies, which will, in the course of time, be very valuable, and supply exactly what is wanted. But our offices are all so recent, the withdrawals are so numerous, and the yearly additions of new members so large, that the duration of membership cannot average now more than three or four years. A long time must therefore elapse before this collection, however extensively made and carefully continued, will enable us to know the average mortality among assured lives, much less the mortality that may be expected in the later years of insurance. Our dependence is therefore on the English and other European tables, and, with the proper additions, we can rely on these with safety.

Among these tables there is much accordance. If the best and most recent be selected, their substantial agreement is remarkable, when we consider the different sources from which they have been derived. In none does the expectation of life differ more than a year or two at any age, as appears by the following comparison of some of the best tables:

Although these tables agree so well with one another, every one of them is known to have defects, and the same may be said of the best that have been published. Some of these deficiencies are in the observations on which they are based, some in the small numbers from which they are derived, and some from the defective mode of construction. The Carlisle has a very small basis, and is badly adjusted. Davies' is from the experience of a single life company. Farr's tables are dependent upon the census returns of population in England, which are probably very imperfect, and on the official registrations of deaths, in which a great many of the ages are given in round numbers, and with more or less errors. The actuaries' is founded partly on policies and not on lives; and these or other defects belong to all the published tables.

These imperfections result in an excessive mortality at one age and a deficient rate at another, even when there is a general agreement in the whole table. Thus while the Carlisle gives the rate of mortality from 15 to 20 higher than Davies', from 21 to 29 it is lower; from 30 to 31 it is again higher, while from 32 to 39 it is lower; from 40 to 46 it again exceeds Davies', while from 47 to 59 it falls below; then rising again, only to be succeeded by another depression near the end of life. If the actuaries' be compared with Davies', we find it higher from 15 to 25, lower from 26 to 59, and then generally lower to the close of the table. If the actuaries' be compared with Farr's, we find it lower up to 50, and higher up to 100.

Even in the expectation of life some of these oscillations occur. Thus Farr's is generally lower than Davies', but near the close of life it is less. Farr's at all the earlier and later ages is above the Northampton, but from 65 to 74 it is less. Davies' is below the Carlisle from 15 to 50, above it from 55 to 75, and below it at the older ages.

Besides these irregularities, the general rate in some tables is higher than in others. But the excess or diminution is in all cases small. Thus

the actuaries' is ten per cent lower than Farr's, from 20 to 25, fifteen per cent up to 35, twenty per cent to 40, fifteen to 45, equal to it at 50, and exceeding it two or three per cent to 80, with a small excess to the end of life.

The near agreement between all the good tables indicates most clearly the remedy to be applied to correct the errors of each. It is a combination of all, giving different weights to each in proportion to its reliability and value. This will eliminate the excesses and deficiencies of each at every age, and give a mean between the highest and the lowest, more worthy of confidence than any single table.

In the most accurate sciences, this reduction to a mean result is continually resorted to, and what we may do with safety and propriety in astronomy, geodesy, and every department of natural philosophy, may, and ought to be, applied to vital statistics. We must, indeed, make a selection of these, and adopt only the good and reliable. Just as we reject defective observations in astronomy, so we must here exclude what is conjectural and unworthy of confidence.

Such a procedure is warranted by the very nature of the case. As it would be improper to determine the law of increase in the population of our whole country from a single city or State, so it is wrong to anticipate our general mortality from that of Carlisle, or Northampton, or Glasgow. As widely extended or long continued observations are needed to determine the proportions of the two sexes, the ratio of births to marriages, and all vital phenomena subject to law, so are they necessary for obtaining the true mortality at every period of life.

This is especially true for the necessities of our American life offices. As they cannot know, before experience, which of the European tables will best suit them, their proper course is to adopt an average of all. Even if they have reason to believe that the American mortality is likely to correspond with the higher European tables, it is best for them to take an average table and add such a percentage as they may think best suited to their wants; since whatever depressing causes are here operating, their intensity is alike active at all ages.

If any one should say we have reason to believe from past experience, that the actuaries' table corresponds best with our mortality, it is easy to reply, that it is in the highest degree improbable that the future experience of our life companies will correspond, either in its total amount or in its comparative rate of mortality at different ages, with the past when its lives were fresh and its numbers constantly recruited with new accessions. The past agreement with any European table is therefore no guide for the future.

If any one should contend that the actuaries' table is best suited for our calculations, because it was made up from the same class as ours, the answer is ready, that the actuaries' table is more or less defective on account of its use of policies, and not lives, and that the law and rate of mortality which our offices wish to know, is for the future or later years of a company, and not for its whole experience.

So if any one should prefer the equitable, or Farr's, or Neison's, or any other table, the objection is, that all have their defects, and that it is best to use an average of all, adding a proper margin to meet the peculiar necessities of any company.

In making this average,

it is not contended that all tables should be

used. Some have been made on insufficient observations which have not been recorded and preserved. Various defects are known to exist in many, and it is only proposed to use those whose merits have secured for them general esteem and confidence. To each of these, different weights ought to be allowed in proportion to their value, so that the mean will approximate to the true mortality at every period of life.

Of the earlier tables published by Dr. Price, or his contemporaries or predecessors, few are of any value, because for the most part they are founded on the deaths only, with some hypothesis as to the living. Halley's, for Breslau, is worth something; but the Switzerland, the Vienna, the London, the Norwich, the Brandenburg, the Berlin, the Warrington, the Chester, the Shrewsbury, the Stockholm, the Montpelier, and those of Kerseboom, Des Parcieux, Duvillard, and De Moivre are of very little value. The Northampton has been more esteemed, and at the older ages it is so nearly correct that Mr. Morgan, who is high authority, preferred it to any of the recent tables for the purposes of life valuations. At the younger ages it gives the mortality too high. But as the hypothesis on which it was founded, of a stationary population without emigration or immigration, was certainly incorrect, we shall exclude it from the combination which we propose to make. Mr. Farr has shown how erroneous are its results and the hypothesis on which it is based, and taken away from it all the estimation which remained to it. The Swedish table in Dr. Price's book, being founded on a large number of observations, and based on correct principles, is the earliest table on which any reliance can be placed.

The Carlisle is later than the time of Dr. Price, and has been highly esteemed. The facts on which it depends were carefully observed by Dr. Heysham, and the general results obtained from it correspond with those of the best tables. At the age of ten its expectation of life corresponds with the Equitable; up to fifty it is less than half a year above, from fifty to eighty it is about the same amount below, and from eighty upwards it is again slightly in excess. It needs adjustment, however, very much, and although prepared by a skillful mathematician, the interpolations for each decade of years were made in total neglect of all mathematical rules, the graphic or ocular method having been used instead of any arithmetical formula or principle. As the value of an annuity by this table, as well as the expectation of life, agrees very nearly with the best tables, we shall use it in the combination we propose to make. Before doing so we will, however, adjust it to some extent, and also construct a new table from Dr. Heysham's observations.

The number of the living, as given by Mr. Milne, at every age, from fifteen upwards, in a stationary population following the same law of mortality as at Carlisle, is inserted in the second column of the table at the end of this article. It is found to be true in most tables of mortality, and in all the best ones, that there is a regular increase in the rate of mortality from the age of ten to the end of life. At the earlier years the increase is slow and almost in arithmetical progression. After sixty it is more rapid, and seems to follow very nearly a geometrical progression. Thus, in the actuaries' table the mortality or ratio of the dying to the living at

15 20 25 30 35 40 60 65 70 75 80 85 .0069 73 78 84 93 104 303 441 649 956 140 205 And their ratio is 1.06 1.07 1.08 1.10 1.12 1.45 1.47 1.47 1.47 1.47

Is...

....

At all ages, for short periods of four or five years, the rate, whether increasing or decreasing, may be supposed to be in geometrical progression, and though this is not true for long periods, nor exactly true for short periods, especially at the middle time of life, it is sufficiently cor rect for all ages to be used to adjust the irregularities that will be found in constructing any table of mortality within the limit here proposed. Mr. Finlaison has used this principle for adjusting his tables, and even extended it. By applying this to Mr. Milne's Carlisle it is relieved of most of its anomalies. In the third column of the table below we have placed the rate of mortality at Carlisle, or the ratio of the dying to the living at every age, omitting the decimal cyphers, which can easily be supplied, and in the fourth the adjusted rates or the geometrical mean of five consecutive ratios. As it is the rates of mortality that we shall use in making our combination of the several tables, we will not continue this work and construct an adjusted table for all ages. The harmony and regularity introduced by the proposed mode of adjustment can be seen by a glance along the two columns.

To reconstruct the Carlisle table by some mathematical rule, various methods might be tried. The plan proposed by Mr. Farr is, perhaps, the most simple; but, in this case, we have preferred the interpolation of the living and dying at each age by the method of differences. This method represents so nearly the true population and the deaths at every age that the rate of mortality which it gives must approximate the reality very closely. By adjusting this rate we shall approach more nearly the

true rate.

The following are Dr. Heysham's observations of the deaths and the living at Carlisle, from which Mr. Milne constructed his table:—

The deaths being for the nine years from 1779 to 1787, the increase of the people between the two enumerations is used to obtain the living at the middle period of the time for which the deaths were recorded, and thence the whole number of the inhabitants for the whole time is deduced. These numbers of the living and dying are interpolated by the method of differences, and form columns fifth and sixth in the table below, the deaths being multiplied tenfold on account of their small numbers. From these we get the ratio of the dying to the living, which is inserted in the seventh column. This ratio is for the middle of the year at each age, since all who are over twenty and under twenty-one are counted at twenty in taking the census, or reporting the deaths. Making a correction for this half year, by Farr's or Milne's formula, first adjusting the ratios by the geometrical method used before, we obtain column eighth, or the adjusted rate of mortality at the beginning of each year of life. And hence the corrected table of Carlisle mortality, which

forms the last column of the table.