Probabilities for life insurance are represented in a mortality table. Conceptually it is much like drawing cards from a deck. Table 13-2 shows portions of the 1979�81 U.S. Life Table that we can use to illustrate these ideas.

Column (2) in the table shows the number of survivors on each birthday out of 100,000 newborns. We call the number at the youngest age, here 100,000, the radix of the table. The number surviving at age one is 98,740. As with a deck of cards, if we draw at random one of the 100,000 members at age zero, the probability is 98,740/100,000 that the newborn will be a survivor at age one. Then the probability of not being a survivor at age one is

aged 45 will live until age 80?" Using the number from column (2) of people still living at age 80 (43,180) the answer is the same�0.46133.

Let�s summarize the pattern of the reasoning in the symbols shown at the top of the columns of table 13-2.

An additional column in a mortality table may show the expectation of life or life expectancy at each age. Column (5) in table 13-2 shows this value. The life expectancy at any age is the average number of years remaining once a person has attained that same age. Life expectancy is an average future lifetime for a representative group of persons at the same age. The probable future lifetime of any individual, of course, depends on his or her state of health, among other things, and may be much longer or much shorter than the average.

For most mortality tables, life expectancy is greatest at age one. It is usually less at age zero due to the high mortality rate in the first year of life. On the other hand, the average age at death, which is the sum of the life expectancy and the attained age (or number of years already lived), increases with age. The excerpt in table 13-3 from the 1979�81 U.S. Life Table illustrates this.

Probabilities for coin tossing, card drawing, and mortality tables all depend on the law of large numbers, known more formally as Bernoulli�s Law. This law asserts that in a series of trials, the ratio of the number of occurrences of an event to the number of trials approaches the actual underlying probability of the event as the number of trials increases. In other words, in n tosses of a coin, the number of heads observed divided by n, will be closer to � more often when n is large than when n is small. In the extreme, for only one toss, the result will be either zero or one�both of which are far from �. When n is 2, the ratio will be 0, �, or one, and it will be zero or one only half the time. Table 13-4 illustrates the idea by showing the pattern for just a few tosses.

The law of large numbers asserts that the probability of the number of heads being near one-half increases to one as the number of tosses increases. In the table you can see that the probability increases from zero to nearly 1.000 as the number of tosses increases from one to 256. This mathematical law forms the basis of the actuarial application of probability to insurance.

While the application of mortality tables is similar to applications of probabilities in coin tossing and card drawing, the building of the probabilities in mortality tables has a different basis than does the determination of probabilities for coins and cards. For cards and coins we can theorize the probability by physically counting the number of favorable outcomes compared to the number of total outcomes. In the simple world of coins and cards this is possible.

In the complex and changing world of life and death, actuaries use statistical methods to estimate the probabilities. The first step is to select a sample population representative of the population where the resulting probabilities are to be used. In life insurance this means that both populations should have the same smoking habits and gender, should be buying the same type of contract (insurance or annuity, individual or group, permanent insurance versus term insurance), and that similar underwriting information is available�that is, that medically examined applicants are grouped separate from nonmedical applicants.

The next step is to observe the mortality experience of the representative population for a period of time. The observation period should be as recent as possible, sufficiently long to give adequate data, and free of nonrepresentative events such as war and epidemics.

After following the sample population during the observation period, the data are processed into the number of life-years and the number of deaths observed in each age interval. We will use a one-year interval for our examples here, although in practice the age intervals can be different from one year. Suppose analysis has produced the values in table 13-5.

From these figures, death rates may be computed for the respective ages as shown in table 13-6. Death rates at the other ages to be included in the table are obtained in like manner.

This illustration assumes the observation of many lives at each age. In practice, such a large "exposure" is not likely to be obtained during any one year. The results of the investigation, if limited to a single year, even in a very large life insurance company, will be subject to considerable distortion because of the small number of lives involved. Therefore actuaries usually study the experience of several years in deriving the rates to be incorporated in a mortality table. Rates derived for a 5-year period are the ratios for the number of deaths that occur at each age during the 5 years to the total exposure at that age. Each person who

survived the 5-year period would have contributed five years� exposure�one year in each of the 5 one-year intervals. Most mortality tables in use today include the combined experience of several large companies over a period of years. The 1980 CSO (Commissioners Standard Ordinary) Table is based on the experience of several life insurers during the period of 1970 to 1975.

Adjustments of several types occur in converting raw mortality rates into a usable mortality table. Among the most important of these adjustments are smoothing, projections, and safety margins.

For theoretical and practical reasons actuaries want the schedule of mortality rates to be smooth with respect to age. First, resistance to disease declines and the degeneration of the body system increases in continuous and minute gradations. Thus sharp changes in the death rates by age should not occur. Second, if the mortality rates are irregular by age, then the premiums and reserves based on them will also be irregular. Such a pattern is not desirable.

In most mortality studies a smooth set of mortality rates cannot be obtained by the simple procedure described above because sufficient data are not available. Actuaries have developed methods that produce smooth sets of rates from the initial nonsmooth sets. These methods, called graduation techniques, are based on mathematics and statistics and their use requires a high degree of specialized training.

Projections reflect changes that have occurred in mortality since the observation period. At least 5 years usually pass between the observation period and the publication of a public mortality table. For the table�s users, who want current rates for pricing, mortality rates published with projections fill this need. A table of projection factors may be provided as well so that users can adjust the published rates for future use.

Another adjustment provides a margin of safety. The appropriate adjustment depends on the purpose of the table. In a life insurance mortality table, including a safety margin means increasing the mortality rates above those anticipated. In an annuity table, including a safety margin means lowering the rates of mortality below those expected.

In early life insurance tables, safety margins occurred implicitly through the use of data from an earlier period known to have had rates of mortality higher than those currently being experienced. In some mortality tables, the margins are provided explicitly. For example, the observed rates for the 1941 CSO Table were increased by use of a mathematical formula. While the adjustment may appear arbitrary, it had a definite objective. Insurance commissioners use this table to monitor solvency of life insurance companies. The adjustment was made in such a way that the table would be appropriate for at least 95 percent of the ordinary insurance business under regulation. On the other hand, the rates entering the GA 1951 Table, an annuity table described in table 13-10, were reduced by 10 percent at all ages for males and 12.5 percent for females. All subsequent Group Annuity Tables including the GA 1983 are sex distinct.

The security behind insurance contracts depends on the existence of safety margins in the contract premiums. Savings developed by the use of conservative actuarial assumptions can be returned as dividends to owners of participating policies. For nonparticipating policies, the safety margins must be smaller in order for premium rates to be competitive.

The previous section describes statistical procedures that transform observations into a smooth series of mortality rates with the needed margins. From that point it is an arithmetic procedure to prepare the other columns of the mortality tables. The first step is a choice of radix for the number of lives at the first age shown. This number is arbitrary and usually is chosen so the number dying at the end of the life span will not be less than a whole number.

To illustrate we reproduce the construction of table 13-2. We have derived the q_x column values by the statistical procedures and chosen arbitrarily to set

l₀ = 100,000. Having determined the radix and the rates of death, the partially complete table appears as table 13.7.

Now d₀ = l₀ x q₀ = 100,000 x (0.01260) = 1,260 and l₁ = l₀ � d₀ = 100,000 � 1,260 = 98,740. With these two values entered in the proper places in the table, we repeat the process at each age. That is, d₁ = l₁ x q₁ = 98,740 x (0.00093) = 92 and l₂ = l₁�d₁ = 98,740�92= 98,648.

Despite the amount of work involved in developing a mortality table, it still falls far short of fitting every situation where mortality rates are needed. As explained below, variations in the published rates are needed to reflect mortality for a group of individuals who are more or less healthy than average, to adjust for the amount of time that has passed since a group of life insurance policies was issued, and to make it easier to consider what events short of death may trigger a policy benefit or other important change.

Mortality varies among different types of lives. A life insurance company uses the mortality table that seems most appropriate for its business. For example, a company may use one table for annuitants and another for lives insured, one for men and another for women, and one for smokers and another for nonsmokers. Different tables or adjustments to standard tables are used to estimate the lower mortality of preferred risks, those whose personal characteristics suggest better-than-average mortality. Similarly developed tables or adjustments that reflect higher mortality are used for substandard risks, those whose personal characteristics suggest worse-than-average mortality.

Studies show that the rate of mortality among recently insured lives is lower, age for age, than that among those insured for some years. For example, the mortality rate for a group of 35-year-olds insured 5 years ago will be higher than the mortality rate for a group of newly insured 35-year-olds. Sometimes underwriting�the process of deciding whether to issue a policy and on what terms�includes extensive medical screening. Underwriting screens out applicants suffering from a disease or physical condition likely to prove fatal in the near future. This disparity in death rates is greatest during the first year of insurance, and it diminishes gradually after that but never completely disappears. The difference is measurable for at least 15 years, but for practical purposes actuaries generally assume that the effect of selection "wears off" after approximately 5 years.

To illustrate, the death rate of policyowners insured at age 25 will be substantially lower than that of policyowners now 25 who were insured at age 20. One year later, when these policyowners are age 26, the difference in the death rates will be somewhat smaller. At age 30, death rates for the two groups will be almost the same.

One way to recognize the reduced mortality in the early years of insurance is by a mortality table that shows the rate of mortality not only by age but also by duration of insurance. This recognizes the time passed since an applicant was approved to receive a policy. The result is called a select mortality table. While in theory actuaries could create an entire series of mortality tables (one for each age of issue), such extensive effort is not usually necessary. Since the effect of selection is negligible after 5 years, different rates are needed only for durations below 6 years. Select mortality tables usually are presented in the form shown in table 13-8.

Table 13-8 gives insight into the mortality reduction due to selection. Column (1) shows the death rate at the various ages during the first year of insurance when the effect of selection is greatest. Column (6) shows the rate after the effect of selection has worn off. Contrast the various death rates for attained age 41 by comparing the six values. All six of these rates apply to a life

aged 41; however, the rates apply to six different periods since issue of insurance.

Companies use select tables in developing gross premiums for both participating and nonparticipating insurance and for testing�through asset share

calculations described later in this book�the appropriateness of existing or proposed schedules of dividends and surrender values. Companies also base

profit projections for new blocks of nonparticipating business on select tables. Another important use is for tabulating mortality experience for analysis and comparison. Comparisons among companies, or of the experience of different periods within the same company, would be far less valuable unless the comparisons considered the relative proportion of new business and the lower rates of mortality experienced on those recent issues.

If the construction of a mortality table excludes the effects of selection, the table is called an ultimate mortality table. That is, it reflects only the rates of mortality that can be expected after the influence of selection has worn off. Standing alone, column (6) of table 13-8 is an ultimate mortality table. Using an ultimate table during the early years of a life insurance contract provides companies with a source of extra mortality savings. This in turn helps to offset the heavy expenses incurred in placing the business on the books.

A mortality table may be constructed from the experience of insured lives, without regard to the duration of insurance. Such a table is an aggregate mortality table. An aggregate table blends in the experience of recently selected lives without segregating it by duration.

In the construction process discussed above for mortality tables, death is the only factor that decreases the number of persons living at each age. Actuaries call such downward reductions decrements. For calculation of statutory reserves� those required by state laws or statutes�and cash values, companies use mortality tables that recognize one decrement only. However, operating an insurance company requires more than the basic information required to comply with state laws. Often the actuary combines this additional information with death rates to produce a table that considers more than one decrement.

Several important decrements other than death can be included. For example, the table used to calculate gross premium needs to recognize that policies will lapse before maturity. Calculating the cost of a pension plan needs to include projections for both the number of employees who will die and the number who will leave employment before reaching retirement. If disability benefits are to be provided, accurate projections are needed for the number of covered lives who will become disabled as defined in the policy and the length of time they will remain so disabled. Some pension plans, including the federal Old Age, Survivors, and Disability Insurance (OASDI) program, need to include the probabilities of remarriage in the estimates of the cost of surviving spouse benefits.

Tables that include rates for more than one decrement are called multiple- decrement tables. The construction of multiple decrement rates is complicated and is beyond the scope of this book. However, once the rates are available, the construction of the multiple-decrement table is very similar to the procedure described above for the single-decrement mortality table.