10

INTRODUCTION.

THREE distinct Mortality Tables are, in the present work, made bases of computation. They are designated respectively by the suggestive symbols, HM, HF, and HM(5). The first two, HM and HF, have been deduced, by a highly scientific process of graduation, devised and applied by Mr. Woolhouse,* from the two, similarly designated, on pages 273 to 276 of the Mortality Experience of Life Assurance Companies, collected by the Institute of Actuaries, published in 1869. The original tables commence at age o, with a radix of 10,000. The new tables, on the other hand, commence with a radix of 100,000 at age 10, the numbers observed upon, between ages o and 10, being considered too small to afford trustworthy results; and therefore the numbers-living, in corresponding tables, do not admit of being directly compared. It will be shown hereafter how closely the results of the graduated tables are assimilated to those of the tables from which they have been respectively deduced.

As regards HM(5), the third table which is here made a basis of computation, there is no table in the Experience volume to which it holds a relation corresponding to that held by the two HM and HF, to the tables similarly designated in the volume referred to. It has been formed from the same data as HM, when modified by the exclusion from them of the experience of the first five years of assurance. Comparison of the results of this table with the corresponding results of HM serve to show the effect of recency of selection.

The arrangement that has been adopted, in regard to the Single-Life Tables, has been to group together, under each fundamental table, all the tables deduced from it. It is

* Much attention has been given by Mr. Woolhouse to the subject of the graduation of tables. A full description of his method, as applied to the Table HM, follows this Introduction. In the application of this method every fact in the table operated upon has its due weight accorded to it; and the effect is, that in the resulting table, while asperities are softened down, every well-pronounced characteristic in the original is faithfully reproduced.

b

believed that this arrangement will be found to be more conducive to facility in the use of the tables than any other that could be suggested.

The Two-Life Tables, which have been deduced from HM, follow the Single-Life Tables.

I. OF THE SINGLE-LIFE TABLES.

The Single-Life Tables, deduced from the several fundamental tables, may be classified as follows:

1. An Elementary Table. This contains the Mortality Table and such deductions from it as are requisite for the construction of the succeeding tables, and of any others that may be proposed.

2. Results involving the rate of mortality only. These are x, x and ex.

3. A Commutation Table.

4. Logarithms and Cologarithms of the principal columns of the Commutation Table.

5. Results deduced from the Commutation Table. These are ax, Ar and @x'

The monetary tables, comprising 3, 4 and 5 of the above enumeration, are given at various rates of interest. For each of the tables HM and HF they are given at the following six rates, viz., 3, 3, 4, 4, 5 and 6 per cent.; and for HM(5) they are given at the three rates 3, 3 and 4 per cent.

The tables as above classified will be now more particularly described; and attention will be directed to any specialties in their form, or in the methods employed in their construction, that may seem to deserve or to demand notice.

The illustrative references will be solely or chiefly to the table HM and the deductions from it, with interest, where this element enters, at 3 per cent.

1. The Elementary Table.

The functions here tabulated for each age are l, and d, with their logarithms and cologarithms; and also the logarithms and cologarithms of pa, the logarithmic functions being in all

usual, for what reason is not apparent, to tabulate log d; nor have the differences been heretofore tabulated. These find their uses, as will hereafter appear, in the construction of tables.

The bases of the logarithmic portion of the Elementary Table are log and log dz. The manner in which the remaining columns are deduced from the columns containing these functions is sufficiently obvious; and it is therefore necessary to explain only the methods employed for their verification.

Since, u, being any function of x,

Uz + Aux+ AU x+1+... +AUx+n−1=Ux+n

it appears that if in any column of differences we insert at intervals the proper values of u, they will in each case be the sum of all the terms which precede them. Thus, referring to p. 2, if in the blank spaces in the column headed log/x (which contains the differences of log ) we insert logo, log /15, log /20, &c., the column will be verified by continuous addition. But the requisite additions can very well be made without the actual transference of the terms from the adjoining column. All the columns of differences were checked in this way.

Again, since the sum of a logarithm and its complement is o, so the sum of any number of logarithms and their complements is also o. The complementary columns were verified therefore by adding together, as they stand, the corresponding groups of five in the two columns.

2. Results involving the Rate of Mortality only.

These results, as respects HM, are on p. 6. They consist of Pr, qz and ex, for each age. The first two are complements of each other to unity; and they might have been formed by taking out the numbers corresponding to log in the Elementary Table, and subtracting the results from unity. But it was preferred to employ for this purpose a method which brings into use certain of the tabulated functions; and as it is typical of other operations to follow, it will be described at some length.

log 9x+1=log 9x+Alog d+colog ĺx.

That is, we shall pass from log to log 1 by adding to the former the differences of its components; and the logarithms of the series 9 will consequently be formed by continuous addition.

The formation of the first few terms of the series is here exhibited.

Col. is adopted in the illustrative examples as an abbreviation for Colog, in order to

Log do is set down on the first line, and on the fourth, seventh, and every third succeeding line the differences of log d, in order, commencing with A log do; colog 40 is set down on the second line, and the differences of cology, in other words the successive terms of colog, commencing with colog/10, on the fifth, eighth, &c., that is, on every third line as before. The third, sixth, ninth, &c., lines thus left vacant, are intended to receive the results of the final additions, which results are the logarithms of the values sought. Before proceeding to the final additions the terms set down should be added continuously in groups; and the successive sums, which should be previously formed from the Elementary Table, and inserted in their places, will serve, as they are reached in order, to verify also the final additions.

In most, or all, of the logarithmic series with which wer have to do, the increase (or decrease) in the successive terms is so gradual that it is not necessary, in computation, to write the index of more than the first term. We have sufficient intimation of a change in the index by that which takes place at the same time in the mantissa of the logarithm. It is thus unnecessary to set down the indexes of the differences, and they are not inserted in the tables.

It is frequently desirable to form a series in reverse instead of direct order; and the Elementary Table furnishes the means of doing so. Thus:

Since log 9x+1=log 9x+A log d2+colog₤x, therefore,

log 9x=log 9x+1-A log dr-colog pr

=log 2+1+A colog d2+ log

(A colog de being the negative difference of log dr, marked -A in the Elementary Table); and the series may be formed by continuous addition, as before.

The following is the construction of the last six terms; and the results of this and the preceding formation may be compared with the complete table on p. 6.