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*** The above paper will no doubt prove practically useful to the managers of the companies which adopt the method of charging for impaired health therein discussed; but it appears to us to be even more valuable to the student of the theory of life contingencies, as suggesting the course which future investigations into the mortality among under-average lives should follow. Mr. Makeham's investigations clearly demonstrate that the point to be ascertained is whether the increased mortality among such lives is more apparent immediately after the grant of the policy or in later years; and any future investigation into such mortality must be considered incomplete that does not give special attention to this point.—ED. J. I. A.
On the Construction of a Graduated Table of Mortality from a
Limited Experience. By S. C. CHANDLER, JR., Actuary of the Continental Life Insurance Company, of New York.
[Reprinted from the Spectator of New York and Chicago.] THE discussion of the methods of dealing with the mortuary statistics of a life insurance company has been hitherto almost entirely neglected by American actuarial writers. This is not a little singular, considering the attention bestowed on other subjects of not greater difficulty or importance. Viewing the large and rapidly-growing body of data already accumulated by American companies, the problem is not of less concern to us than to our English and German brethren, to whom, notwithstanding, we are indebted for the principal contributions to this branch of actuarial science.
In bringing to notice, with great diffidence, the methods which have suggested themselves to me as the most practicable and efficient for the treatment of a limited mortuary experience, I venture to express the hope that the way may be opened for a general discussion of the subject.
There is at present a very great diversity of opinion among actuaries in regard to what is commonly termed the “graduation” of the results of observations of human mortality. Many object to any interference with the naked results of observations. Prominent among these may be mentioned the late Professor de Morgan, and, if I mistake not, Mr. David Chisholm. The language of the former on this point,* though decided, had greater force at the date it was written than it has now, and I am not sure that his views were not subsequently modified.
* Treatise on Probabilities, p. 162.
In a second class may be placed those actuaries who, while admitting the desirableness of removing the grosser deviations of the observations at individual ages, due to the limited data employed, prefer to effect that object by processes which are independent of any hypothetical assumption as to the nature of the law of mortality.
Finally, there is a third class who deem the approximate knowledge we possess of the nature of the law of mortality sufficient to justify the use of processes based, to a greater or less degree, upon an assumed relation between the mortality at different ages, for the purpose of securing the required adjustment.
It is not intended here to enter upon an examination of the subject in its general relations. I will only say, with reference to the object directly in view, that when the data are very limited, the raw results of observation are unfit for use in many of the comparisons and deductions which it is desirable to make; while the methods which exclude considerations of the character of the law of mortality, of necessity more or less desultory in their nature, become too uncertain in their application to afford trustworthy results.
If there be a general law governing human mortality, and if the variations from this law that obtain among any given body of individuals, to whatever circumstances due, are such as do not affect the form of the mathematical expression of the general law, but merely the values of the constant elements of that expression, -it is not essential for the purpose of eliminating the errors in a limited body of observations, that the true form of the general expression should be known. Any expression, however arbitrary, will be of service, if we possess the means of estimating the accuracy with which it represents the observations, and of determining the limits within which the deviations from the true law, being inappreciable, may be neglected.
To apply this principle to the case of a life insurance company, we observe that the ages at which lives are subject to observation are ordinarily comprised within a period of forty-five or fifty years. In most of our American companies the limits of age for the acceptance of risks lie between 15 and 25, and between 60 and 70. The experience at ages above the latter limit, pertaining to risks which have continued in force from the younger ages, is, even in the older companies, exceedingly scanty. So that, even if we were in possession of the true formula, we might not have sufficient material for fixing the values of the constants required for its numerical application outside the limits named.
Now, in the assumption that the force of mortality is the resultant of two single forces, one constant at all ages, the other increasing in geometrical progression with the age, we have a relation which, though it requires considerable modification to represent the mortality in infancy and childhood, and in extreme old age, conforms very closely to the observations on which our best life tables are founded, within the limits of age above stated. It seems, therefore, to be admirably adapted for the present purpose, being at once simple, and so far as the application here to be made of it is concerned, as accurate in its results as the true law of which it is but the approximate expression.
According to the hypothetical principle just enunciated, the force of mortality at the age x may be represented by an equation of the form Fr=a+bq*
(1) The credit of this formula is due to Mr. Makeham, to whose labours in this department the actuarial world owes so much. In the Journal of the Institute of Actuaries for October 1871, Mr. Makeham has given a mode of employing it, with a numerical application to a case where the number of deaths is but 27, and the years of life exposed, for all ages, less than 3000. The method of Mr. Makeham, though exceedingly simple, is, however, open to the grave objection that the values of the constants a and b depend entirely upon the age t, which is chosen arbitrarily. There would thus be as many values for the mortality at any age as there are possible combinations of the equations of condition into two groups. Moreover, it furnishes no criterion of the accuracy of the determination. In requiring that the sum of the deaths in each of the arbitrarily-chosen groups shall be the same for the adjusted as for the unadjusted numbers, Mr. Makeham assumes, in effect, two rigorous equations of condition, which, when q is known, of course define a and b.
The best mode of procedure, it seems to me, even where the data are as scanty as in the case exemplified by Mr. Makeham, is to form equations of condition assigning appropriate weights at each age, and then deduce from the whole series the most probable values of the constants. I propose in this paper to give the necessary formulas for accomplishing this, with such explanation and illustration as shall enable computers not familiar with the calculus of errors to make a practical application of them.
Representing the true annual rate of mortality at the age x by Mx, we shall have from (1), since Mx=Fx+}, very nearly, a+bqu+i=mx .
(2) If in any series of observations we denote the years of life exposed and the actual number of deaths at age x by E, and d'x,
d' respectively, and the corresponding ratio* by m'r, we shall have for every age under observation an approximate equation of condition of the form a+bqx+1-m'x=0
(3) from which a, b, and q are to be found.
In most cases that arise in practice the experience will be insufficient to determine the constant q with any accuracy. But the value of this element is nearly the same in all tables of mortality. Its logarithm may be taken =:04 without appreciable
This reduces the number of unknown quantities to two. The weight of the observed probability of dying in a year found from E, observations is proportional to
(4) m.(1 -mx) Multiplying the equations of the form of (3) each by the square root of its weight, we have
VWzoa + Wrqx+36-vw.m'. =0
Wx+1a + VWx+19+ im'x+1=0 (5)
(6) Ewrq*+sa+Ew=q2+1b-Ewrqu+im's=0 from which the most probable values of a and b may be obtained by elimination.
If it is desired to determine q, we may proceed as follows. Calling m's the hypothetical value of the rate of mortality which results from the substitution of the approximate values a',6',a', in (2), and substracting the resulting equation
a' +b'q'x+=m's from (2), we obtain
a-a'+bqx+} – b'q'*+=m'z-m's.
and neglecting the terms involving the squares and higher powers of Z in the development, we find
X+q'*+1Y+B'(x + )q*+-Z-Mo=0
&c., whence we derive the normal equations for determining the corrections to the assumed values of the constants, Ew.X+Ewxq'x+1Y+Ew b'(x++)g'r-12-Ew.zfo=0 Ewxq'x++X+ Ewxq%2x+1Y+Ewdb' (x + })q'2Z-Ewxq'*+p=0
(10) Ewxb'(x + })q'--1X+Ewxb' (x + })q'2Y+Ew.,b'?(x+ })?q%20–1Z
-Ew b'(=+ })q-te=0 and from these the most probable values of a, b, and q, by (7). Unless Z is very small, it may be necessary to make another approximation, using the results of this solution for the assumed values.
For the calculation of the numerical values of the weights, we may employ provisional values of a and b found by taking the definite sums of the equations of the form given in (3), from the youngest age n to an arbitrarily-chosen age t, and from t to the oldest age w, or
να +ΟΣ- gt +1=Σ.-m's
(11) where v and v' are the numbers of equations in each group. Then, from (2) and (4) the values of w can be calculated. If the results deduced from (6) differ much from the approximate values employed in finding the weights, the latter may be computed anew and applied as before.
To exemplify the method just developed, I propose to apply it to the data* in columns (2) and (3) of the accompanying table. Column (4) contains the actual ratios of mortality deduced from these.
Our first business will be to compute the weights. Taking log 9=:04 and t=30, we find
v=16, 38q+++= 132:61, $m'x=•1283,
v'=41, x2qx+=7336.79, szom's=•8381; which, substituted in (11), give
a= .00732, b= 0000728.
* The published experience of the New York] Mutual Life for the first fifteen years.