CHAPTER II SELECT TABLES THE mortality amongst assured lives depends not only upon the present ages of the lives but on the ages at which they were examined or selected for life assurance. People who have just been accepted for life assurance are far less likely to die within one year than those of the same age accepted twenty years ago, consequently it is best for most purposes to tabulate the cases according to age at entry. The natural procedure, therefore, is to sort all the cards according to the age at entry and count up the number of entrants at each age. Table II gives the particulars for age 30 as they are usually scheduled for each age at entry. In the experience from which this table was formed the number of cards in the bundle relating to age 3 0 was 1499. These cards were sorted into three groups— (1) existing, (2) withdrawals (i.e. lapses and surrenders), (3) deaths. Then each one of these groups was sorted according to the duration entered on each card, which had of course been previously calculated by the methods already described. The number of cards for each duration was then inserted in the table. We can now go through the table and see what the entries mean. Opposite duration 0 are entered 12 SELECT TABLES
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all the cards in which that duration appears on the cards in accordance with the practice described on p. 11; thus there were 30 withdrawals; this tells us that 30 people allowed their policies to be dis- continued within six months of the date of issue, the nearest duration in these cases being 0. Six people died within a year of the time when their policies TABLE II
were effected. There were no existing, because, as we observe up to policy anniversaries in a particular year, a case in order to be entered as existing must have been in force for at least a year. The exposed to risk is the equivalent of the "number of people observed for one year or until death " in our definition, i.e. the number of entrants making proper allowance for withdrawals and existing. The exposed to risk for dura-
14 MORTALITY AND SICKNESS TABLES tion 0 was 1469, i.e. 1499 entrants less the 30 with- drawals. The rate of mortality is 6—1469 == -00408. Considering duration 1, we find that 45 people who were still living and whose policies were still in force when the observation closed, had taken out their policies a -year before; 157 people had with- drawn at various times between six months and eighteen months (the nearest duration was one year); and 10 died between one and two years from the time when their policies were issued. Now since the 45 existing were observed up to their exact durations, none of them had a chance of dying in the second year of assurance 1—2, so they must be deducted in finding the exposed to risk for that year, and for similar reasons the 157 who withdrew must be deducted. We must also deduct the 6 people who died in the year 0—1; unless we do this we shall be giving them the chance of dying twice! The exposed to risk for duration 1 is therefore 1469 — 45 — 157 — 6 = 1261, and the rate of mortality is 10-1261 =-00793. Subsequent durations follow in the same way, until all the 1499 cards are accounted for. Similar tables are made for all other ages at entry, and finally the results can be collected in the form shown in Table III, which, if completed, would give the rates of mortality for all ages at entry, and all durations. Such tables are called " Select Tables of Mortality," because they show the mortality for each year subsequent to the date on which the life was " selected " for assurance by Medical Examination or otherwise. This supplies one solution of the problem of finding SELECT TABLES
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the rates of mortality from the books of insurance offices, but before turning to other methods it will be well to see where this method is open to criticism. The main points at which this can be directed are the use of the nearest age at entry and the nearest duration for withdrawals. The error in the former arises mainly because there is a tendency for people TABLE III.—RATES OF MORTALITY (SELECT)
to enter before a birthday rather than after it, so that they may obtain the assurance at a lower rate of premium, with the result that the true average age of each group is slightly less than the assumed age, and the rates of mortality for each year of assurance relate to slightly lower ages at entry than those that are assumed ; in other words, the morality is slightly understated. The use of the nearest duration is open to a little criticism through a practical difficulty that
16 MORTALITY AND SICKNESS TABLES arises. Withdrawals occur partly from lapses, which necessarily take place when a premium becomes pay- able. If premiums are payable yearly the nearest duration gives an exact result if we reckon durations only to the date when the premiums fall due. In practice, however, thirty days of grace are allowed, and if an insured person dies within the days of grace the sum assured is paid. In order to make a proper comparison between the " Deaths " and the " Exposed to Eisk" we should either ignore all such claims and treat every case as a withdrawal in which the premium is not paid on the day it falls due; or if we wish to include such claims among the " Deaths," we should treat the other lapses as withdrawing at the end of the days of grace, otherwise each lapse is given a month's exposure too little and the exposed to risk is underestimated. If, however, we reckon the durations up to the end of the days of grace, the following scheme shows the error that occurs in cer- tain circumstances by assuming the nearest duration:— Yearly Premiums . . Each withdrawal by lapse understated by a month. Half-yearly Premiums . Falling due on anniversary of policy—each withdrawal understated by a month. Half-yearly Premiums . Falling due 6 months after anniversary—each with- drawal overstated by 5 months. Quarterly Premiums . Falling due on anniversary of policy—each withdrawal understated by 1 month. SELECT TABLES 17 Quarterly Premiums . Falling due 3 months after anniversary of policy— each withdrawal under- stated by 4 months. „ „ . Falling due 6 months after anniversary of policy— each withdrawal over- stated by 5 months. „ „ . Falling due 9 months after anniversary of policy— each withdrawal over- stated by 2 months. If the proportionate distribution of cases is 12 yearly cases, 8 half-yearly, and 4 quarterly, the total understatement is 12 months in respect of yearly premium cases, 4 months in respect of half-yearly, 5 months in respect of quarterly, while the overstate- ments are 20 in respect of half-yearly, and 7 in respect of quarterly premiums, leaving a total over- statement of 6 months on 24 lapses. This is hardly a large error, and on the basis of Table II it would mean an error in the rate of mortality of (I)6 if every withdrawal is due to lapse, which is not the case. The error in any individual experience depends on the proportion of yearly, half-yearly, and quarterly cases, but it is unlikely that the error involved would be large enough to make it worth while to remodel the method. A short preliminary investigation can almost always be made to settle the point.