Actuarial Analysis

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The origins of the actuarial profession can be traced to the late-seventeenth and early-eighteenth centuries when leading mathematicians were prevailed upon to compute the cost of annuities and life insurances. Many of the early great names of mathematics contributed in this way. The first professional body (the Institute of Actuaries) was established in London in 1848. Since that time, the professional interests of actuaries have widened to include pensions, general (property and casualty) insurance, health insurance, finance, and a wide range of "non traditional" problems, for which their quantitative skills and understanding of risk are readily applicable (e.g., pricing electricity supplied to a national grid).


Most very early life tables were used and/or prepared in connection with life annuities and life insurance. The Equitable Assurance Society, which established long-term life insurance on a scientific basis in 1762, for example, used James Dobson's life table (based on London Bills of Mortality between 1728 and 1750) and Richard Price's table of 1783 (based on death records for a parish in Northampton). Price later constructed a life table from the population and deaths in Sweden, the first national life table ever made. The standard life table symbols still used in the twenty-first century were adopted as part of the International Actuarial Notation as early as 1898. Government actuaries continue to prepare the official national life tables of many countries, including Australia, the United Kingdom, and the United States.

Within a national population there is a considerable degree of mortality heterogeneity. Persons accepted for life insurance tend to have mortality that is lower than that of the national population over much of the age span because they are generally better educated, more affluent, and subject to medical scrutiny by the insurer. Purchasers of life annuities have even lower mortality as no one expecting to live only a relatively short time would purchase a life annuity. Because of these and other differences between the mortalities of the various subpopulations, many different types of life tables are regularly prepared, covering, for example, nonsmoker insured lives, smoker insured lives, super-select insured lives, annuitants, members of pension funds, actively employed persons, age retirees, and persons who have retired because of ill health. Large insurance companies often can prepare their own life tables on the basis of their own experience, and those tables reflect their own standards of underwriting. Only a small proportion of life tables are ever published.

Standard tables based on confidential data collected from groups of insurers are prepared and reviewed regularly by the various actuarial professional bodies. More recent standard tables tend to be published on the Internet.

Finding a suitable life table for use in a developing market is a problem faced by many actuaries of the twenty-first century and requires considerable judgment. Actuaries usually have to rely on insurance tables prepared for similar products in another market that is believed to have similar characteristics. If national life tables are available, they may be used as collateral information. The collection of local insurance mortality data is a high priority.

Temporary Initial Selection

The mortality of persons recently selected for life insurance is normally lower than that of other insured lives of the same attained age who were selected in earlier years. For this reason, since the mid-nineteenth century, when the first life tables based on the combined mortality experience of several insurers were constructed, actuaries usually have estimated mortality rates that take account of both age at selection and duration since selection. The mortality rate of persons selected at age x who have been insured for t years and are now aged x + is denoted by q[x]+t (the +0 is suppressed when t = 0).

In theory, therefore, separate life tables are required for each age at selection. The effects of temporary initial selection tend to disappear after several years, however, so that lives the same attained age that are selected at different ages eventually develop mortality rates that are indistinguishable. When the effect of temporary initial selection has worn off, the insured lives are said to be "ultimate lives" and their mortality is given by the "ultimate life table" with mortality rates [qy], where y is the attained age. In other words, once the temporary initial selection has disappeared (the duration t is greater than or equal to the select period), q[x]+t = q[x-1]+t+1 = q[x-2]+t+2 =… = qy where y = x + t.

For pragmatic reasons, British actuaries have tended to use very short select periods, whereas their North American colleagues have used longer periods (up to 15 years). If one uses common ultimate (ly) values for the latter part of all the distinct life tables (corresponding to various ages at selection and durations in excess of the select period) and chooses appropriate radices (l[x]), survivorship values can be represented concisely as in Table 1. Based on this table, for example, the probability that a select life aged 47 will die before age 50 is 1 - 32,670/32,975 = 0.00925, the probability that a life now aged 47 who was selected at age 46 will die before age 50 is 1 - 32,670/33,020 = 0.01060, and the probability that a life now aged 47 who was selected on or before his or her forty-fifth birthday will die before age 50 is 1 - 32,670/33,045 = 0.01135.

Temporary initial selection also is observed in other situations. Persons who have retired more recently because of ill health, for example, tend to have mortality that is higher than that of the survivors of those who retired from ill health earlier, but again, the effect wears off with duration since retirement.

The technique is a convenient one that could be applied in a number of demographic situations, such as immigrant mortality, where the mortality of recent immigrants differs from that of the host population but gradually approaches the same level. Other possible applications include the study of the mortality of divorced and widowed persons, with the age at selection being the age at which the person became divorced or widowed.

Effects of Lifestyle and Medical Conditions

A number of life insurance companies formed in the nineteenth century distinguished between persons who abstained from alcohol and nonabstainers. Actuary Roderick Mckenzie Moore (1904), for example, was able to produce separate life tables for the two groups and investigate the effects of transitions between the two classes. Such a distinction normally would not be made in the twenty-first century, although "excessive" consumption might be taken into account at the underwriting stage.

Although the standard insurance life tables referred to above are usually for lives insured on normal terms, persons in less than perfect health can often obtain insurance on special terms. In determining the terms, company actuaries work alongside experienced medical officers, making use of a wealth of international data on the effect on mortality of many different medical conditions, personal habits (tobacco, alcohol, and drug consumption, exercise, etc.), and fitness, including weight to height measures. The data come from a wide range of sources: clinical trials, longitudinal studies of whole communities, special longitudinal studies for particular diseases, surveys, and cancer registries. A two-volume reference work entitled Medical Risks–Trends in Mortality by Age and Time Elapsed (Lew and Gajewski 1990), for example, provides an extensive description of many different conditions and advice on the relative mortality of persons suffering from those conditions. The major international life reinsurance companies produce their own electronic rating manuals to advise client insurers on the rating of impaired lives, and special investigations are undertaken from time to time by actuarial professional organizations.


Improvements in Mortality

Improvements in mortality can undermine the financial viability of companies that sell life annuities. For this reason, actuaries have long been interested in measuring mortality improvements and estimating future mortality. Projected generation life tables are required, as annuities will be taken out at different ages in the same calendar year. The simplest commonly used approach has been to observe the annual rates of improvement in q values over time and then to extrapolate the q rates by using improvement factors at each age, although other approaches are also adopted. In most cases the actuaries' assumptions have led to underestimates of improvement.

Variation of Mortality with Age

Since the eighteenth century mathematical "laws" of mortality have been explored in an attempt to facilitate the otherwise very tedious life contingencies calculations essential for pricing and valuing life assurance and annuity contracts. The mathematician Abraham de Moivre was possibly the first to do this (in 1725), but the most celebrated early development was that of mathematician and Fellow of the Royal Society Benjamin Gompertz (1825), modifications of which have been proposed ever since. The model allows many quick approximate calculations that are remarkably accurate even with mortality tables that are not strictly of the Gompertz shape. Actuary T. N. Thiele, in 1872 proposed a model applicable over the whole age span, as did demographer Larry Heligman and actuary John Pollard (1980). Actuaries David Forfar and David Smith (1987) applied the latter model in 1987 to all 26 English Life Tables to project the English Life Tables for 1991. The projected mortality rates for females turned out to be very good, but those for the males were less satisfactory.

A variety of models were studied by actuary Wilfred Perks in 1932, who noted the effects of heterogeneity, and variants of his models were used to graduate (smooth) British standard tables in the 1950s and 1960s. Other more generalized formulas that have been used in more recent British standard tables are discussed in Forfar et al. (1988).

Mortality Heterogeneity

In recent years some life insurance companies have begun marketing policies to super-select lives, persons with characteristics that tend to make their mortality extremely low even compared with those accepted for life insurance under normal conditions. In doing so, the companies are attempting to exploit the considerable mortality heterogeneity that exists in any national population. Actuaries are therefore becoming very interested in measuring heterogeneity and understanding its underlying causes.


Before the development of the welfare state in the early twentieth century there was little financial security for those who were sick and unable to work; they had to depend on charity or small payments to the destitute from the local parish. "Friendly Societies" began to proliferate, providing small benefits in times of need to members in return for small weekly contributions. In this way workers in particular occupations and regions were able to support each other. Actuaries were soon required to ensure that these mutual institutions were financially viable and, as a result, became involved in sickness investigations. The largest and most thorough of these studies was the Manchester Unity investigation of 1893–1897, and the tables derived by actuary Alfred Watson (1901) showing age-specific proportions sick were used extensively (with adjustments) well into the twentieth century.

Employers in most developed countries of the twenty-first century offer some level of income maintenance for short periods of sickness, for example, a certain number of days of full pay while sick, with the number of allowable days generally increasing with length of service. National sickness schemes also may pay basic income benefits. A need for private sickness and disability insurance remains, particularly for the self-employed, and insurers offer a wide range of products designed for specific markets. As with mortality, the actuarial professional bodies coordinate the collection and analysis of morbidity data and the preparation of standard tables of incidence and recovery. More recent standard tables tend to be published on the Internet. The major international reinsurers also provide underwriting manuals for their clients.

Competing Risks

The first detailed study of competing risks was done by the British actuary William Makeham (1874), although some of the ideas can be traced to eighteenth-century Swiss mathematician Daniel Bernoulli, who attempted to estimate the effect on a population of the eradication of smallpox. Makeham's approach was to extend the concept of the "force of mortality" (which was well known to actuaries of that time) to more than one decrement, and he noted the essential independence between the different decrements implied by his analysis.

Actuaries who have used multiple decrement tables ever since have almost invariably assumed independence between the "competing risks." Important applications include pension schemes, where active employees may be depleted by a number of different decrements (death, resignation, termination, ill-health retirement, and age retirement), and mortality analysis, where mortality rates for certain causes may be changed to take account of trends or to answer "what if" questions about possible future changes in mortality.

The formulas relating the decrement rates in a multiple decrement table to those in the associated single decrement tables or with other multiple decrement tables (e.g., tables with fewer decrements) depend on the manner in which the decrements operate. In cause of death analyses, for example, decrements in the related single cause tables often are assumed to be spread evenly over the year of age. Formulas derived under this assumption may not necessarily be transferable to other situations, such as pension funds, where certain events may be concentrated at birthdays. There is an extensive literature on this topic, and attempts have been made to deal with dependence between decrements.

Multiple decrement tables belong to a very special class of the Markov process, and more general Markov chain processes are often required in morbidity studies, because persons can recover from their illnesses.

Population Modeling: HIV/AIDS

The HIV/AIDS epidemic that started in the 1980s caused considerable alarm in the insurance industry, particularly in respect to policies providing death benefits, those providing income replacement during illness, and medical and health policies. Actuaries in various countries therefore began modeling the development of the disease in the community at large and the numbers at risk or already HIV-positive in the insuring subpopulation.

Crucial to the modeling of the insurance process were assumptions concerning the numbers of existing policyholders at risk and the numbers already infected and the numbers and sizes of new policies that would be issued to persons in those categories once the community and the insurers reacted to the epidemic. A major concern was the possibility of high-risk groups and those already HIV-positive selecting against the insurers (taking out a disproportionate amount of insurance). Because the diffusion of the disease differed from country to country and because legislation controlled the extent to which insurers were permitted to discriminate between different groups in their underwriting, a model developed in one country was not necessarily immediately transferable to another.

Improved community awareness and safer sexual practices in developed countries ultimately caused the spread of HIV/AIDS and the effects on insurers to be less serious than had been projected.

Population Modeling: Genetic Testing

Almost since the dawn of life insurance, insurance companies and their actuarial advisers have sought genetic information from those applying for life insurance by asking details about survivorship and cause of death of family members. With the recent rapid developments in genetics considerably more information about the likely survivorship and morbidity of an individual can be provided by a genetic test. A person who has taken a test may be aware that he or she is more likely to die younger or be subject to increased ill health. Serious ethical questions ensue. Should insurers be permitted to demand genetic tests? If not, should an individual who has taken a genetic test be required to reveal the results to the insurer under the basic insurance principle of utmost good faith (uberrima fides)? If such information is available only to the proposer, there is a serious risk of selection against the insurer, to the detriment of the company and others insured with it.

There are also serious privacy issues. Genetic information about an individual also provides information about that individual's relatives. Such indirect genetic information also can be used to select against an insurer. For example, a person may submit to a genetic test and learn that he or she bears an undesirable gene. Knowing this, that person might advise a sibling to take out insurance, and the sibling could justifiably claim not to have undergone a test.

Even in situations where no genetic test has been undertaken, an insured life may take one and, after learning that he or she does not have deleterious genes, discontinue the insurance, leaving the insurer with a higher than average proportion of policyholders with genes associated with increased morbidity and premature death.

Human rights supervisors, privacy officials, insurers, actuaries, insurance regulators and legislators are grappling with these issues, and actuaries are endeavoring to model the underlying genetic processes in the population and in the insuring subpopulation.

See also: AIDS; Genetic Testing; Gompertz, Benjamin; Life Tables; Lotka, Alfred; Mortality, Age Patterns of; Risk.


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Continuous Mortality Investigation Committee, Institute of Actuaries and Faculty of Actuaries.1998. "The Mortality of Holders of Permanent (Whole Life and Endowment) Policies of Assurance 1991–94." Continuous Mortality Investigation Reports 16: 1–34.

——. 1998. "The Mortality of Smokers and Non-Smokers 1991–94." Continuous Mortality Investigation Reports 16: 83–94.

——. 1998. "The Mortality of Impaired Insured Lives." Continuous Mortality Investigation Reports 16: 95–112.

——. 2000. "An Analysis of the PHI Experience of Individual Companies in the United Kingdom: I. Claim Inception Rates." Continuous Mortality Investigation Reports 18: 109–150.

——. 2000. "An Analysis of the PHI Experience of Individual Companies in the United Kingdom: II. Claim Termination Rates." Continuous Mortality Investigation Reports 18: 151–180.

Forfar, David O., and David M. Smith. 1987. "The Changing Shape of English Life Tables." Transactions of the Faculty of Actuaries 40: 98–134.

Forfar, David O., et al. 1988. "On Graduation by Mathematical Formulae." Journal of the Institute of Actuaries 115: 1–149.

Gompertz, Benjamin. 1825. "On the Nature of the Function of the Law of Human Mortality and on a New Mode of Determining the Value of Life Contingencies." Philosophical Transactions of the Royal Society 115: 513–585.

Heligman, Larry, and John H. Pollard. 1980. "The Age Pattern of Mortality." Journal of the Institute of Actuaries 107: 49–80.

Institute of Actuaries and Faculty of Actuaries. 1975. A1967-1970 Tables for Assured Lives. Oxford: Institute of Actuaries and Faculty of Actuaries.

——. 1979. A (90) Tables for Annuitants. Oxford: Institute of Actuaries and Faculty of Actuaries.

Lee, Ronald. 2000. "The Lee-Carter Method for Forecasting Mortality, with Various Extensions and Applications." North American Actuarial Journal 4: 80–93.

Lew, Edward A., and Jerzy Gajewski. 1990. Medical Risks: Trends in Mortality by Age and Time Elapsed. New York: Praeger.

Macdonald, A. S. 1999. "Modelling the Impact of Genetics on Insurance." North American Actuarial Journal 3(1): 83–101.

Makeham, William. 1874. "On an Application of the Theory of the Composition of Decremental Forces." Journal of the Institute of Actuaries 18:317–322.

Moore, Roderick M. 1904. "On the Comparative Mortality among Assured Lives of Abstainers and Non-Abstainers from Alcoholic Beverages." Journal of the Institute of Actuaries 38: 213–259.

Panjer, Harry H. 1988. "AIDS: Survival Analysis of Persons Testing HIV Positive." Transactions of the Society of Actuaries 40: 517–530.

Pollard, John H. 1987. "Projection of Age-specific Mortality Rates." Population Studies of the United Nations 21/22: 55–69.

——. 1991. "Fun with Gompertz." Genus 47: 1–20.

Subramanian, Krupa, Jean Lemaire, John C. Hershey, Mark V. Pauly, Katrina Armstrong, and David A. Asch. 1999. "Estimating Adverse Selection Costs from Genetic Testing for Breast and Ovarian Cancer: The Case of Life Insurance." Journal of Risk and Insurance 66: 531–550.

Watson, Alfred W. 1901. "An Account of an Investigation of the Sickness and Mortality Experience of the I.O.O.F. Manchester Unity Friendly Society, during the Years 1893–1897." Review (Editorial) 369. Journal of the Institute of Actuaries 38: 334.

internet resources.

Faculty and Institute of Actuaries. 2002. <>.

Society of Actuaries. 2002. <>.

John H. Pollard