THE CONSTRUCTION OF MORTALITY AND SICKNESS TABLES CHAPTER I INTRODUCTORY IT is well known that the practical conclusions of an actuary are based on mortality, sickness, and other tables, which are prepared from the statistics given in the census returns or collected by life assurance offices and friendly societies. The study of the methods by which these tables are constructed is therefore of great importance, and it would naturally be anticipated that the funda- mental character of the subject would make it attractive; but, as a matter of fact, there is no other part of actuarial work that appears to the average student so troublesome and uninteresting. This may be because he leams about several tables of mortality which are never used, some of which never have been used, and many of which were constructed by more or less unsuitable methods, and he is left with the erroneous impression that the whole subject is a mass of bewildering detail. He may even go so far as to think that he would employ his time more profitably by evolving methods to suit special circum- 2 MORTALITY AND SICKNESS TABLES stances, than in learning the way other people have solved, or failed to solve, similar problems. And here he is right. Thinking out such solutions is the best way to understand the subject; and when it has been mastered the history is easy enough and less uninteresting. The real difficulty for most people is that the amount of detail obscures not only the importance of the subject but even the problem that has to be solved. This is, of course, fatal; when we are trying to solve any problem we must be clear about its nature, and if, as is nearly always the case in statistical work, an approximate result is all that can be hoped for, we must try to see where the approximation falls short of accuracy. Let us begin then by saying that the object of our investigation is to find the rate of mortality at any age from data obtained from censuses and death registers of the general population, or from the books of insurance offices, where the " rate of mortality" at a specified age may be defined as the ratio of the number of persons in a particular population dying within one year from the attainment of that age, to the number who were under observation for one year from the attainment of that age, or until death if occurring within the year. If, for example, there are 10,000 people aged 30 exactly, each of whom is kept in sight until he attains age 31 or dies (if death occurs before that age), and if it is found that there are 104 deaths among the 10,000, then the rate of mortality, or chance of dying in a year at age 30, is -0104. INTRODUCTORY 3 Now let us see how this definition can help us to appreciate some of the difficulties of our problem and the ways these difficulties can be met. In the first place, we want a large number of people (in our numerical illustration, 10,000) who are exactly 30 years of age. We cannot expect to trace many people who were all born on the same day, but we might trace 10,000 people all of whom attained age 30 in the same year (or some other period) and observe them from their thirtieth to their thirty-first birthdays, or we might observe for a year all those who are between 29^ and 30^ at a particular moment. In the second place, we must know the number of deaths among the people observed; and in the third place, our definition suggests that we must observe each case for a year. Here we are faced with the difficulty that people emigrate or if we are dealing with particulars from the registers of an assurance office, some policies lapse or are surrendered and the lives are consequently lost sight of. If there were 10,200 people all aged 30, of whom 400 withdrew (by emigration or their policies lapsing) at the end of exactly half a year, we ought to consider the 400 people withdrawing as equivalent to 200 under observation throughout the year, because we do not know whether they died in the second half of the year or not; in other words, within our experience these 400 who withdrew only had half the chance of dying in the year that was given to the rest of the lives under observation. The consequence is that the 10,200 people in the circumstances described are equivalent to 10000 4 MORTALITY AND SICKNESS TABLES people observed for the whole year. If the 400 left at various times during the year a similar argument will hold, and we shall see later how these points are dealt with in practice. As we have already indicated, we can use either the data obtained from the general population, or the particulars that can be found in the registers of a life assurance company, to reach the rates of mortality at various ages; but more accurate results can be obtained from the latter, owing to the more detailed information available. We shall therefore deal with the life office methods first, and refer afterwards to the methods to be adopted in the case of the general population. The course that is usually followed in order to obtain rates of mortality, is first to decide exactly how many years of the assurance company's experi- ence are to be taken into account and whether any cases are to be excluded. It might, for instance, be thought well to exclude those who effected policies many years ago or those who were charged an extra premium, while for some purposes the separation of with-profit policies from without-profit policies might be deemed advisable. Males and females should be dealt with separately whenever possible. If it is decided to neglect certain cases, great care must be taken to see that their exclusion will not vitiate the results of the investigation. Thus deaths arising from some specified cause, such as cancer, could not be excluded, because each life has undergone the risk of dying from this disease, and if we exclude those persons whose deaths have actually taken place from INTRODUCTORY
this cause the general rate of mortality would be under-estimated. There would, however, be no fallacy
Life Assured . . ADAM SMITH Policy Number . 1001
Date of Exit . Date of Entry . Duration . Date of Birth . Age at Entry . Made of Exit .
REMAKKS
30 March 1886 jo Dec. i8y<) 6 i Jan. 1850 }0 Lapse
in working only on people resident in a certain district or following a certain occupation. Having decided on these points, the usual practice 6 MORTALITY AND SICKNESS TABLES is to fill up a card (see specimen) for every case to be included, giving the date of birth, date of entry, the date of withdrawal and the mode of withdrawal, i.e. death, lapse, surrender, or if the life was still living at the close of the observations. The above specimen card is suitable, the corner is cut off for convenience in sorting, and the particulars are arranged so that the calculations can be made readily. As the easiest way to appreciate what these particulars give and how they can be used, is by taking one or two examples, Table I has been pre- pared so that the first four lines give the particulars furnished by the office, and the later lines the calcul- ated ages and durations to be used in the subsequent work. The particulars in the first four lines are easy to follow, and can be left to explain themselves while we consider the rest of the table. It will be seen in the first place that the nearest age at entry has been calculated in each case. This is used as a convenient and accurate approximation to the exact age; it means that we group together all persons on whose lives policies have been granted between the ages 29^ and 30^, and assume the exact age at entry to have been 30. The next two lines of the table show the method of calculating the time during which a policy is in force in various circumstances. Taking the first example, we see that the duration from the date of entry to the day when the policy lapsed was six years and three months. The life assured comes under our TABLE I
8 MORTALITY AND SICKNESS TABLES observation for that time, and no longer; he might, within our experience, have died during that period, but as a matter of fact he did not do so; or in more technical language, he was exposed to risk of death for six years and three months. In the same way, the life assured in the second case was exposed to risk of death for twenty years and seven months. If we go back to our definition of the rate of mortality, we see that these cases do not fit in conveniently, because we ought to observe every case until the end of the year. The first six years in the first example fit well enough, but the life assured had a chance of dying in only three months of the seventh. We ought, strictly speaking, to count him for one quarter of that year, and the assured in the second example for twenty years and seven-twelfths of the twenty-first year. In practice it is convenient to avoid fractional durations, so we take the nearest duration in a similar way to that followed in connection with the age at entry. We assume, therefore, that Example 1 was exposed to risk for six years, and Example 2 for twenty- one years. We balance the understatement of some cases with the overstatement of others. The next two examples relate to deaths, and have been dealt with differently from those that have just been considered. We have not taken the nearest duration but the number of complete years lived, and neglected the odd months altogether. To understand this we must go back again to our definition of the rate of mortality. In finding the rate of mortality at age 30, for instance, we want to obtain the ratio of the deaths between ages 30 and 31 (which in INTRODUCTORY 9 practice we call the deaths at age 30) to the corre- sponding number living at age 30, so that if a person dies at any time between 30 and 31 we must assume he was exposed to risk for the whole of that year of age. What we have to find is the chance of a man dying in a year, and it does not matter in what portion of the year death occurs. If we adopted any other method, such as the nearest duration, we should reach some other function than the rate of mortality as we have defined it. The one remaining example relates to a policy which is still in existence when the observations end. The usual practice is to assume that the experience ends not on a fixed date, but on the anniversaries in a particular calendar year (e.g. 1911) of the dates when the policies were effected. This fits in well with our definition of the rate of mortality, because it enables us to observe these cases for an exact number of years. Those who are studying the subject of the con- struction of mortality tables for the first time fre- quently have considerable difficulty in following the meaning of the term " those existing at the close of observations," and seem to find it hard to see why they have to be taken into account. It must be borne in mind that the experience covers a fixed period of years. Each policy is observed from the date of entry until its cessation by surrender, lapse, or death, or until its anniversary in the final year of the period. There will necessarily be several of the latter cases, some effected one year before the final year, some two years, and so on. These are termed 10 MORTALITY AND SICKNESS TABLES the " Existing at the close of Observations," and since the policy-holders might have died during the time covered by the experience, they must be included for each year of their existence up to the date when the observations end. This may be made clearer if we consider four policies effected ten years ago. Let us assume that in one case the life assured died; in another the policy was allowed to lapse; a third was surrendered ; and the remaining one is still being maintained. This last one would be termed " Existing." It must be taken into account for each of the ten years, because the life assured might have died during that period, even though he did not do so. If it was excluded altogether we should exaggerate the mortality, as can easily be seen by considering the extreme case of ten persons observed for ten years, of whom only one died and none withdrew. If we did not take the nine existing people into account for the whole of the ten years, we should assume that a person was certain to die within that period, whereas as a matter of fact only one out of ten had done so. Having thus provided ourselves with the necessary data, we have to consider how it can best be mani- pulated, but before doing so it will be useful to recapitulate what has been said in the following way, which shows the procedure to be adopted:— 1. Decide the limits of the experience. 2. Write a card for each case, giving—(a) date of birth; (6) date of entry; (c) date of exit; (d) cause of exit, or if existing. INTRODUCTORY 11 3. Calculate the nearest age at entry in each case. 4. Calculate the nearest duration for with- drawals. 5. Calculate the curtate duration (i.e. integral number of years) for deaths. 6. Calculate the exact duration for those existing at the close of the observations by taking them up to the anniversaries of the policies in a chosen calendar year. In this connection a word of caution must be added. If the calendar year which ends the period is 1911 and the anni- versary of the policy is 30th June, the policy passes out of observation on that date. This means that if the life assured died, or if the policy was surrendered between the 30th June and the end of the year, we should count the case as existing, and not as a death or surrender.