Last updated 1996 Sep 29
Introduction to Actuarial Science
Applications
| History
| Notation
| Computations
| References
Provenence
| Insurance
| Pensions
| Policy Issues
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- Uncertain Events. Many events in the lives of individuals and businesses occur at times and in ways that are unknowable in advance.
- Predictable Patterns. People have always been aware there are patterns in such events. Actuarial science quantifies the probabilities of events and integrates those probabilities with information about costs and rates of return. Life insurance and pension plans are two applications of actuarial science that touch most individuals and businesses.
- Life. Individuals pool resources to care for the dependents or partners of someone who dies. The problem is to compute the cost, that is, the premium, for the death benefit to be paid.
- Health. The two tasks are to define insured events, such as covered illness, and to calculate costs.
- Casualty. The two tasks are to define insured events, such as fire and flood, and to calculate costs.
Life annuities are the principal example. A life annuity pays a constant sum at regular intervals, usually monthly, for as long as the person lives. See Pensions and Cost Methods for a further introduction.
- Definition of risks
- general categories of risks to be pooled
- casualty
- disease
- old age
- other
- operational details of recognizing instances of the risky events
- Definition of risk sharing pools
- criteria for membership
- age
- gender
- location
- occupation
- other
- size of pools
- double law of large numbers. Large groups are necessary to
- accurately estimate probabilities
- reliably share risks
- collecting data about patterns
- spreading future risk
- necessity for uncertainty at the individual level
- self selection
Source: O'Donnell, History of Life Insurance, 1936.
Antiquity
| Middle Ages
| 17th and 18th Centuries
| 19th Century
| 20th Century
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In Rome before 230AD the value at age x of a life annuity of 1 was set at
- 30 for x < 31
- 60 - x for x > 30
In 230AD the praetorian prefect Domitius Ulpianus produced a table of annuity values that was the best table in Europe until almost 1700. It was used until 1814.
In England there are records going back to 1512 of annuities sold by the crown.
Pascal (1623-1662) started the modern theory of probability, including the law of large numbers.
1671 John DeWitt (Holland) made the first conscious application of modern mathematics and probability to annuities.
1692 Until this time the English crown sold annuities with the yearly payment equal to 14% of the premium, that is, of the selling price. In Holland, too, the general practice was for government annuity debt to pay double the straight loan interest rate. [In essence such practice assumes a death rate that is constant with age.]
1693 Edmund Halley (England) combined birth and death records from the city Breslau [page 124] with the new theory of probability and created the first modern life table [page 12]. This is the Halley of Halley's comet.
1719 First stock life insurance company in England. The eighteenth century saw rapid development of actuarial science in England and France and of insurance companies
1725 Abraham DeMoivre (Huguenot from France who fled to England) based a theory on Dx = constant
1740-1752 Thomas Simpson constructed life tables based on observation. He introduced whole life insurance with level annual payments.
M. Boffon (France) computed many values of mqx, for example,
- 8q0 = .5
- 33q1 = .5
- 22q40 = .5
1747 James Hodgson (England) constructed life tables from London mortality records.
1779 William Morgan was actuary at the Equitable (? precursor to Prudential)
Condorcet (1743-1794, France) connected mathematics with social goals.
1806 M. Duvillard (France) constructed life tables [pages 213-214]
distinct male and female life tables
computers
Actuaries
Actuaries study demographic statistics and prepare the tables for mortality, morbidity and other events that happen unpredictably to individuals but with solemn regularity to large groups of people. Actuaries also work on pricing insurance products and on planning reserves.
Approximate number of actuaries in the United States in 1984
- Members of the Society of Actuaries
- working for insurance companies 5,000
- in private consulting practice 3,000
- in government, colleges or retired 1,000
- Enrolled Actuaries under the rules of federal agencies 2,500
Every field of expertise develops abbreviations and special ways of describing the elements of its work. Such jargon and notation are useful and convenient to the expert, but they are a barrier to outsiders, just like a foreign language. Keep in mind that actuarial notation does help those who must do the work of actuaries. The many different symbols reflect the different concepts with which an actuary must deal on a regular basis. It's like Eskimos having seven different words for all the things that English calls snow.
Acturial notation is a shorthand for quantities such as actuarial present values, future values, probabilities, and premiums. For example, there are special notations for
- present value of an
- annuity of $1 for n years
- annuity due of $1 for n years
- value in n years of an
- annuity of $1 for n years
- annuity due of $1 for n years
- probability of living
- from age x to age x+1
- from age x to age x+n
- Net Single Premium at age x for a
- whole life annuity
- whole life annuity due
- life annuity due for n years
- whole life annuity due beginning at age x + m
- pure endowment at age x + n
- whole life insurance
- n years of term life insurance
There is also extensive notation for parameters such as interest rates and survival rates, together with intermediate quantities that facilitate computation. See Actuarial Terms and Algebra for more information, including definitions of terms such as whole life annuity, Net Single Premium, and pure endowment.
How much should members of a pool put aside from time to time to pay a specified benefit to those who experience a specified future event? The answer depends on how many members will share in the proceeds of the fund and on what the amounts set aside can earn. See Discounting and Examples for the details of actual computations.
Panjer, H.
Actuarial Mathematics
American Mathematical Society, 1986
QA1 A5217 v.35
Mathematical Model of an Insurance Company
f HC107.W6 W68 v.4 no.3
North Wing, third floor, row 101
History of Mutual
HG 8963.M872 B35
Chiang, Chin Long
The Life Table and Its Applications
R. E. Krieger Pub. Co.
1984
HB 1322.C47 1984
Beard, Robert E. et al.
Risk Theory
Methuen
1969
HG 8781.B
Studies in Insurance and Acturial Science
University of Texas at Austin
HG 8018.S8
Nesbitt, Cecil J. and Butcher, Majorie V.
Mathematics of Compound Interest
Ulrich's Books
1971
HF 5695.B83
O'Donnell
History of Life Insurance
1936
HG 8761 O25
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