Insurance and mathematics

Summary: Society has long used mathematical methods to quantify risk and protect against loss, and professionals like actuaries help make these decisions.

Insurance involves the exchange of a fixed amount of money or sequence of payments (called premiums) by the insured to an entity or group for indemnification of the insured from specified losses. Thus, insurance involves trading a small but certain cost (the premium) for payment of a potentially large but uncertain loss in the future.

It is used to manage risk of loss in uncertain situations by hedging the risk (for example, by pooling money with others and sharing losses) or transferring it to some entity, like an insurer, for a price. Because the price paid today must cover future costs and future uncertain indemnification payments, the insurance industry employs many mathematicians to calculate and predict expected future costs and payments.

Importance

Risk transfer and risk pooling via insurance are very important. Following the government, insurance is probably the second most important mechanism available to alleviate social upheaval and to reduce risks to citizens. Social upheaval is reduced by supplying a financial safety net in times of loss. Risk reduction is achieved since insurance establishes risk reduction incentives, such as lowering the cost of insurance, for those who undertake risk reduction behaviors. Examples of risk reduction behavior include premium reduction in automobile insurance for defensive driving classes or having air bags; lowering premiums and providing loss control consulting to business firms concerning risk exposures; and lobbying governments for stronger safety standards.

Insurance allows entrepreneurs to create new products, explore new energy alternatives, and engage in selective risk-taking beneficial to society, such as creating new pharmaceuticals, which might be too uncertain or create potential liability exposure consequences too great to be undertake if not insured. Through insurance, cash flows of firms are stabilized, bankruptcy likelihood is reduced, and the cost of capital to firms is lowered.

History

Because of the individual and societal benefits of insurance, it is no wonder that the rudiments of insurance date back millennia—although the modern approach to insurance awaited the development of mathematical tools to create the logical underpinning of the industry. The Code of Hammurabi (c. 1750 b.c.e.) details how early Babylonian merchants who had a loan on cargos or vessels could pay a little extra so that if the ship were lost at sea, the loan would be forgiven—an early example of risk transfer. Early civilizations also had arrangements wherein members pooled resources, and if one suffered a loss, such as a building burning down, others would pitch in and furnish materials and labor to rebuild the member’s lost structure—an example of risk pooling. Before formal life insurance companies were developed, people in England in the seventeenth century would band together in groups called friendly societies, each contributing a small sum such that if an emergency or death occurred, the group would pay medical expenses, funeral costs, and sometimes give a stipend to the widow. Some of these friendly societies later developed into insurance companies.

Mathematics of Premiums

A crucial element in insurance is determining the insurance premium. The premium is the amount of money to be paid by the insured whose risk of loss is being indemnified, but needs to be an amount sufficient for the insurer selling the insurance to both cover potential loss costs and make a profit. Indeed, many early insurance-type organizations failed from the lack of correct assessments of risk and potential exposures to financial loss by the group furnishing the insurance—an incorrect quantification of risk. Without quantification of risk, the expected lost costs cannot be formalized and monitored. It is in this area of risk quantification that mathematics of insurance arises, mostly in the area of probability and statistics, which deal with the quantification of uncertainty.

The mathematics of insurance, known as “actuarial science,” had its birth amid the incredible growth in mathematics in the seventeenth century. Most major mathematicians of the seventeenth and eighteenth centuries contributed to insurance mathematics in a variety of ways, such as calculating annuity tables based on interest rates and tables listing the probability of death at each age (called “life tables”). Some, such as Abraham DeMoivre, made a living, in part, by consulting on the calculation of annuity values. The first life table was constructed in 1694 by mathematician and astronomer Edmund Halley, now most famous for identifying Halley’s Comet.

The development of modern probability theory—an essential element of the quantification of risk needed to price insurance—is usually attributed to French mathematicians Blaise Pascal and Pierre Fermat from a series of letters from 1654 concerning games of chance left unfinished. Using this new mathematical theory, the fair price of insurance could be rationally developed for the first time. For example, if, in the case of the occurrence of an event having a probability p, a benefit B is to be paid at some future time T, then the fair price today is pBv^T where v is the “discount rate” accounting for interest available on money invested today and paid at time T, expressed algebraically as the following:

In this formula, i denotes the annual interest rate on invested money. Subsequent developments in mathematics have allowed for uncertainty in B, v, and T, enabling one to obtain the fair value of the insurance in more-complex risk transfer situations.

A mathematical foundation for insurance lies in the Law of Large Numbers (LLN), developed by mathematician Jacob Bernouli, and the Central Limit Theorem (CLT), developed by Abraham de Moivre and extended by mathematician Pierre Simon Laplace. The LLN is fundamental to insurance since it proves that the empirical relative frequency with which an event occurs in a risk pool will, as the size of the sample increases, approach the “true” probability of the event.

This allows insurance companies to objectively obtain the likelihood of loss-producing events from their experience in large collections of policyholders. The CLT proves that the average of a sample of homogeneous independent observations, such as losses within a pool of risks, will be well approximated by the bell-shaped Gaussian distribution as the number in the pool increases. From this idea, the setting of premiums for insurers who are appropriately confident of remaining solvent can be calculated.

Bibliography

Baranoff, Etti G., Patrick Brockett, and Yehuda Kahane. “Risk Management for the Enterprise and Individuals.” Flatworld Knowledge (2009). http://www.flatworldknowledge.com/printed-book/1635.

Pearson, Egon. The History of Statistics in the Seventeenth and Eighteenth Centuries Against the Changing Background of Intellectual, Scientific and Religious Thought. New York: Macmillan, 1978.

Trieschmann, James S., Robert Hoyt, and David Sommer. Risk Management and Insurance. Cincinatti, OH: South-Western College Publishing, 2004.