## Statistical Methods for Actuaries

The insurance industry employs actuaries to collect and analyze data that will provide the basis for setting insurance premiums to customers and developing risk management strategies. This article describes two statistical tools and four statistical models utilized by actuaries; examines the characteristics of higher group insurance risks and higher individual insurance risks; discusses ethical and economic considerations for insurance companies; and presents strategies for mitigating the costs of insurance risks. A glossary of related terms is included.

Keywords Actuary; Bonus-Malus; Collective Risk Model; Individual Risk Model; Lee-Carter Mortality Model; Premium; Probability; Reinsurance; Risk; Risk Management; Scenario Analyses

### Actuarial Science: Statistical Methods for Actuaries

### Overview

The insurance industry employs actuaries to collect and analyze data that will provide the basis for setting insurance premiums for customers. Actuaries estimate the probability and cost of the occurrence of claims due to death, sickness, injury, disability, or loss of property. The goal of an actuary is to predict the likelihood, frequency, and cost of claims placed to the insurance company. In order to arrive at their estimates, actuaries employ a variety of tools to analyze historical data, economic conditions, and any other relevant factors such as natural disaster patterns and predictions. Because of the need for a high level of accuracy in their analyses, actuaries rely upon methods that utilize statistical tools and models to collect and analyze data, predict claim activity and costs, and advise insurance companies on how to manage their risks. The insurance companies base many of their practices on data and recommendations from actuaries, including rate-setting for premiums and strategies for mitigating their costs for claims paid to higher risk groups and individuals.

### Applications

This section defines some of the statistical tools and models that are utilized by actuaries to analyze data, forecast claim activity and costs, and recommend premium-pricing strategies for insurance companies.

### Statistical Tools Utilized by Actuaries

Actuaries frequently utilize the following two statistical tools in their work for insurance companies:

- Probability Table
- Scenario Analyses

### Probability Table

The first statistical tool utilized by actuaries is the probability table. A probability table plots the likelihood or risk that an event will occur. For example, a probability table may indicate the likelihood that a hurricane will occur within a certain timeframe in a target geographic location, and that this situation will result in an estimated number of claims at an estimated cost to an insurance company. The validity of such a probability table will depend upon the careful collection and statistical analysis of data, including the following seven factors:

- Historical hurricane patterns in the target location.
- Future hurricane predictions for the target location within a target time period.
- The number of potential claimants in the target location.
- Historical costs to pay out claims for the target location due to hurricanes.
- The revenue generated by the potential claimants over the target time period.
- The predicted cost to pay out claims for future hurricanes.
- The suggested cost for future premiums.

Every time one or more of the factors in the probability table are changed or manipulated, the estimates and recommendations may change.

### Scenario Analyses

The second statistical tool utilized by actuaries is called scenario analysis. In scenario analyses, the actuary determines the degrees of insurance risk for a particular insurance portfolio by examining multiple risk situations.

To set up the scenario analyses, the actuary first identifies the economic and underwriting risks. Armed with this information, the actuary can then draw upon historical data and his or her own expertise to construct a loss distribution model that specifies the insurance company's risks according to the various scenarios (Dowd & Blake, 2006).

Ergashev (2012) introduced a theoretically justified framework that incorporates scenario analysis into operational risk modeling. The basis for the framework is that only worst-case scenarios contain valuable information about the tail behavior of operational losses. Worst-case scenarios also introduce a natural order among scenarios that makes possible a "comparison of the ordered scenario losses with the corresponding quantiles of the severity distribution that research derives from historical losses."

### Statistical Models Utilized by Actuaries

Depending upon the particular portfolio or goal, actuaries may utilize a variety of statistical models to determine premium rate recommendations to insurance companies.

The following four models are briefly described:

- Collective Risk Model
- Individual Risk Model
- Lee Carter Mortality Model
- Bonus-Malus

### Collective Risk Model

The first statistical model utilized by actuaries to determine premium rate recommendations is the collective risk model. In a collective risk model, the premium determination is based upon the total claim amount in a fixed period in a portfolio of insurance contracts (Iwanik & Nowicka-Zagrajek, 2005, p. 416).

### Individual Risk Model

The second statistical model utilized by actuaries to determine premium rate recommendations is the individual risk model. In an individual risk model, the premium determination is based upon a sum of the claims of many insured individuals (Iwanik & Nowicka-Zagrajek, 2005, p. 412).

### Lee-Carter Mortality Model

The third statistical model utilized by actuaries to determine premium rate recommendations is the Lee-Carter mortality model. The Lee-Carter mortality model, which is based upon long-term data, charts and projects mortality rates by age group. Actuaries and insurance companies may consult the Lee-Carter model then make adjustments in premiums that reflect longer life spans (Friedberg, L., & Webb, A., 2007).

Zhao (2012) presented a modified Lee-Carter model for analyzing short-base-period mortality data, for which the original Lee-Carter model produces "severely fluctuating predicted age-specific mortality." Approximating the unknown parameters in the modified model by "linearized cubic splines and other additive functions," the model can be simplified into a logistic regression when used with binomial data. The expected death rate estimated using the modified model is "smooth" over both ages and years.

### Bonus-Malus

The last statistical model utilized by actuaries to determine premium rate recommendations is Bonus-Malus. Bonus-Malus refers to the practice of raising an insured person's premiums each time that person makes a claim (Moreno & Watt, 2006).

Strictly speaking, Bonus-Malus may be considered more of a practice than a model. It is however, a statistically based method of determining premium rates and may be recommended as a rate-setting strategy by an actuary.

Evaluating the Bonus-Malus system in practice in the Nigerian motor insurance industry, Ibiwoye, Adeleke, & Aduloju (2011) constructed an alternative Bonus-Malus scale that issues reasonable penalties and yet is "commercially feasible." The authors assert that the model can be replicated for other...

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