## Actuarial Statistics (Research Starters)

Actuarial science, and specifically probability and statistics, deal with the concepts of uncertainty and risk. Many of the decisions that we make everyday involve uncertainty; actuarial statistics are used by actuaries to assess the financial risk that is inherent in insurance, pensions, or investment plans. Statistics are used by actuaries to determine the best way to collect data and analyze data. As actuaries analyze data, outcomes are studied that can reveal patterns related to risk and human behavior. Actuaries are experts at modeling risk and applying probabilistic decision-making. It is critical for actuaries to understand the conditions and processes under which historical data was obtained as well as the evaluation and quality assessment of available data. This article discusses the application of random variables that actuaries apply when building risk models. A discussion of data mining, predictive analytics and the identification of new data correlations follow as a means of understanding the constantly changing methods used by actuaries in applying statistical methods to risk analysis.

Keywords Correlation-Statistics; Credit Rating Insurance; Insurance Statistics; Predictive Analysis; Random Variables; Statistics and Probability

### Actuarial Science: Actuarial Statistics

OverviewThe world is an uncertain place and human beings have always been uncomfortable with the realization that they must live with risk. Risk, like death and taxes is a foregone conclusion and cannot be avoided. Since humans are a risk adverse species, it is easy to see why insurance is seen as a necessary evil. Uncertainty is the reason that insurance companies exist. People and organizations pay premiums to insurers because the insurer will assume the majority of financial risk for the insured. Insurance companies are able to assume risks by pooling large numbers of policy holders together and estimating that only a small number will make a claim.

“Insurance works through the magic of the Law of Large Numbers. This law assures that when a large number of people face a low-probability event, the proportion experiencing the event will be close to the expected proportion. For instance, with a pool of 100,000 people who each face a 1 percent risk, the law of large numbers dictates that 1,100 people or more will have losses only one time in 1,000” (Zeckhauser, 2003, ¶6).

When determining ratings, insurance companies must manage a large number of unknowns, including:

- Potential number of claims to be paid to insured (future claims payments).
- Uncertainty about how much to charge for premiums (premium rating projections).
- Predictive modeling of future events and trends (risk projections).
- Volatility in financial markets that affect the assets of an insurance company.

Insurers rely on the analysis of historical data to project the likelihood of future events; the data that is analyzed by actuaries is compiled by insurers, government agencies and private companies. Today, data mining has become commonplace due to the great availability of electronic data available from a variety of sources. The more data points that can be collected about a policy holder, the greater the number of variables an insurer or actuary has to describe an entity (individual policy holder or fund.)

An additional challenge for actuaries is that they must determine premium rates that will cover the costs of operations of an insurer without knowing how many claims will be paid out in the future. The competitive nature of insurance today requires that insurers keep rates competitive, so the job of the actuary is very challenging indeed. Actuaries balance the asset reserves of insurance companies while at the same time hedging the risks associated with setting rates. "Insurance companies pay claims out of premiums that were set some time before the claim arose and that were based on information drawn from an even earlier period. Obviously, in determining the premium it should charge, the insurer must forecast its expected claims and expenses." In short, an insurance company cannot be certain of its future assets, profits or its solvency (Hart, Buchanan, & Howe, 1996); all this risk makes being an insurer-very risky indeed.

A “fundamental task of the actuary is to use historical observations to draw conclusions about future outcomes. This is similar to the work of the statistician; it is the context that defines the work of the actuary. Therefore, it is appropriate that the initial principles be taken from probability and statistics” (“Principles,” 1999, p.5).

Insurers face a risk of ruin or insolvency if premiums are charged that are equal to the expected costs of operation. An insurer can reduce the risk of insolvency through the following means (Hart, Buchanan, & Howe, 1996):

- Increase its capital.
- Increase its profit margin
- Reduce exposure on risk (re-insurance).
- Increase the numbers of risks (risk pool).
- Reduce correlations between risks.

"Claims are not settled as soon as they occur and may take many years to be finalized. The capital requirements of an insurer, particularly one writing long tail classes of business, are quite complex as care is required not just for the period of policy exposure but until the last claim has been settled" (Hart, Buchanan, & Howe, 1996).

Variables"Insurance relies on pooling to reduce relative variability. If all of the risks in a pool are identical, then each should contribute an equal amount to the pool. In practice, however, the risks in the pool are often different and these differences need to be taken into account. The purpose of risk classification is to find variables, called rating variables, that can be used to distinguish between different levels of risk and to quantify the differences on the basis of these variables" (Hart, Buchanan, & Howe, 1996).

There are "usually a large number of potential rating variables, each of which could reasonably be expected to affect the risk. It is seldom practical to use all possible rating variables. Some, such as ethnic origin, are barred under anti-discrimination legislation or are socially unacceptable. Some, such as mileage, are difficult to collect, unreliable, or both. Some have only a small effect. Too many rating variables would create an unwieldy rating structure. The idea is to find a small number of variables that explain as much of the variation between risks as possible" (Hart, Buchanan, & Howe, 1996).

Probability & StatisticsThe theory of probability has its foundations in mathematics of the seventeenth and eighteenth centuries, and introduced the study of random variables. Early study of probability emphasized games of chance where the number of possible outcomes was finite. The physical characteristics of card or dice give strong clues to the evaluation of underlying probabilities; there are a finite number of numbers on a die, or cards in a deck. A later concept of probability theory introduced the concept of the continuous random variable. In the use of continuous random variables, probabilities need to be obtained empirically (via experimentation or observation) (Trowbridge, 1989).

The use of random variables leads only to estimates in outcomes. For example, if the height of 100 men is sampled...

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