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 economies.

Viewpoints

Insurance companies are profit-making organizations. Therefore, it is essential that they structure their finances and set insurance premiums in a manner that will allow them to pay claims and also make a profit.

This section describes the differences between higher group insurance risk and higher individual insurance risk; examines some of the ethical and economic issues inherent in setting insurance premiums based on statistical methods and tools; and presents strategies for mitigating the cost of insurance risks.

Higher insurance risk usually falls into one of two categories (a person could fall into both categories if his or her situation shows characteristics of both):

• Higher group insurance risk.
• Higher individual insurance risk.

Higher Group Insurance Risk

When the higher risk is attributed to a group of insured persons — often due to a common geographic or economic factor — each insured person in the group bears the risk.

The following two examples describe situations of higher group insurance risk:

• A group of people who live in a flood zone will be considered higher risk for property insurance claims. The insured persons would be required to carry extra property insurance called "flood insurance."
• Motor vehicle owners who live in a neighborhood with historically high vehicle theft rates will be considered to be in a higher risk group. This fact would be reflected in higher vehicle insurance premiums for each vehicle.

Higher Individual Insurance Risk

When an individual is considered a higher insurance risk, it is due to factors unique to that individual's situation. The individual's higher risk is determined based on statistical tools and models that consider the individual's situation as well as historical statistics developed through probability tables.

The following four examples describe situations of higher individual insurance risk:

• Individuals who make a higher than average number of claims for motor vehicle damage present a higher risk for motor vehicle insurance claims.
• Individuals who make a higher than average number of claims for property arson damage present a higher risk for property insurance claims.
• Individuals who smoke or have a serious, chronic health problem present a higher risk for health insurance claims and/or an earlier risk for disability and life insurance claims. (Although these individuals are considered to be at a higher risk for health insurance claims and for earlier life insurance or disability claims, the cost of these risks is mitigated for the insurance company if the individuals are enrolled in group health or life insurance programs because the risk is spread around among a large number of persons. This is the reason that health insurance policies are so much more expensive if the individual is not part of a group plan.)
• Individuals who default on their mortgage payments present a higher risk to companies that fund and insure mortgages.

Ethical Issues: A Question of Fairness

The major ethical issue that surrounds the setting of insurance premiums based on statistical methods and tools is the concept of fairness to all insured persons. Generally, the higher the insurance risk — or the greater the likelihood that a claim will result — the higher the insurance premium will be. This seems like a reasonable business practice; however, the risks are greater for certain people or groups of people who often can't change their situation.

For example, the following sets of people are generally considered to either present a higher risk for claims or do not qualify to have their premium rates averaged out among a large group:

• People who are forced to live in high crime areas due to their economic circumstances.
• People who have lost their jobs.
• People who do not have access to group health insurance.
• Minorities and low-income populations. Kabler (2004) notes that insurance credit scores for minorities and low-income individuals are lower than for other populations. A lower insurance credit score equates to a higher insurance risk.

So, the complicated situation for an insurance company becomes one of setting up a financial and premium structure that allows the insurance company to insure people; charge fair premiums; and reap a profit.

Strategies for Mitigating the Cost of Insurance Risks

Strategies for mitigating the cost of insurance risks may be employed by insurance companies and by society. (For purposes of this article, we will consider "society" to include any person or organization.)

Insurance Companies

Insurance companies employ a number of strategies to mitigate their cost of insurance risks, including the following four strategies:

• The establishment of group insurance pools, such as those for employers and professional societies, to spread out the total cost of claims to the insurance company and the premium rates to insured members. (See #3 under "Higher Individual Insurance Risk" above.)
• The establishment of Bonus-Malus practices for property or motor vehicle insurance: Whenever an insured individual makes a claim for property or motor vehicle damage, his future premium is raised. According to Moreno, Vazquez, & Watt (2006), the institution of Bonus-Malus greatly reduces the likelihood of false claims.
• The establishment of an investment strategy by the insurance companies that reflects a balanced investment portfolio in order to reduce dependency upon any one type of investment.
• The purchasing of reinsurance, which is insurance for insurers. A reinsurer assumes part of the risk and part of the premium originally taken by the insurance company, and reimburses the insurance company for claims paid (Insurance Information Institute, n.d.).

Society

The individuals and organizations that make up society can employ a variety of strategies to mitigate the cost of insurance risks. Before we examine some of these strategies, it is useful to consider that by mitigating the cost of insurance risks, society as a whole benefits through lower cost insurance premiums and better and more widespread insurance coverage for individuals.

Society may mitigate the cost of insurance risks by studying the statistics and probability tables to identify the riskiest groups and individuals and then leveling the playing field for those high risk groups and individuals by tackling the following four initiatives:

• Expand healthcare coverage options for uninsured persons.
• Reduce the environmental factors that lead to illness and premature death.
• Improve access to fire and police protection in order to reduce the number and severity of safety-related issues that lead to insurance claims.
• Increase the number of health and safety education programs in order to promote well-being and reduce insurance claims.

Conclusion

Actuaries perform vital services for insurance companies. They utilize statistical methods and tools — including probability tables, scenario analyses, and risk models — to predict insurance claim risks and costs, and to advise insurance companies on risk management strategies. While statistical methods and tools aim to present an accurate, objective chart of risks and costs, they can pigeonhole certain individuals and groups into higher risk groups. Insurance companies can employ a number of strategies to mitigate the cost of higher insurance risks including the establishment of group insurance pools, Bonus-Malus practices, balanced investment portfolios, and reinsurance policies. In addition to the insurance companies, individuals and other organizations can help level the playing field for the higher risk groups by tackling initiatives that expand healthcare coverage, reduce environmental contamination, improve access to fire and police protection, and expand health and safety education.

Terms & Concepts

Actuary: "Actuaries assemble and analyze data to estimate the probability and likely cost of the occurrence of an event such as death, sickness, injury, disability, or loss of property. Actuaries also address financial questions, including those involving the level of pension contributions required to produce a certain retirement income and the way in which a company should invest resources to maximize its return on investments in light of potential risk. Using their broad knowledge of statistics, finance, and business, actuaries help design insurance policies, pension plans, and other financial strategies in a manner that will help ensure that the plans are maintained on a sound financial basis. Most actuaries are employed in the insurance industry, specializing in life and health insurance, or property and casualty insurance. Actuaries produce probability tables that determine the likelihood that a potential future event will generate a claim. From these tables, they estimate the amount a company can expect to pay in claims" (United States Bureau of Labor Statistics, 2006-07, p. 24).

Bonus-Malus: "An insurance contract in which premiums are tied to the number of claims; each time an insured person places a claim, his future premiums are increased. This practice can be used to discourage frivolous claims and combat fraudulent claims and penalizes the individual but not the whole group of insured persons" (Moreno, Vasquez & Watt, 2006, p. 125).

Collective Risk Model: Premium determination based on the total claim amount in a fixed period in a portfolio of insurance contracts.

Individual Risk Model: Premium determination based on a sum of the claims of many insured individuals.

Lee-Carter Mortality Model: Based upon long-term data, this model forecasts mortality rates by age group (Friedberg, L., & Webb, A., 2007).

Premium: The consideration paid for a contract of insurance (Merriam-Webster's collegiate dictionary, 2000). Laymen usually refer to a premium as their "rate."

Probability: The chance that a given event will occur (Merriam-Webster's collegiate dictionary, 2000).

Reinsurance: "Insurance bought by insurers. A reinsurer assumes part of the risk and part of the premium originally taken by the insurer, known as the primary company. Reinsurance effectively increases an insurer's capital and therefore its capacity to sell more coverage. The business is global and some of the largest reinsurers are based abroad. Reinsurers have their own reinsurers, called retrocessionaires. Reinsurers don't pay policyholder claims. Instead, they reimburse insurers for claims paid" (Insurance Information Institute, n.d., Glossary of Terms).

Risk: In the insurance industry, risk is a situation where the probability distribution of a variable (such as the burning down of a building) is known but its mode of occurrence or actual value (whether the fire will occur at a particular property) is not (Business Dictionary, 2007).

Risk Management: Policies, procedures, and practices involved in identification, analysis, assessment, and control, avoidance, minimization, or elimination of unacceptable risks (Business Dictionary, 2007).

Scenario Analyses: A method of determining insurance or actuarial risk that involves: Specifying a portfolio (such as property, casualty or life, country, or type of product); identifying general economic and underwriting risks; constructing a loss distribution model based on past data and expert judgment; specifying the likelihood of exposure to the various risk factors; and formulating the impact of the scenarios considered (Dowd & Blake, 2006, p. 216).

Bibliography

Business Dictionary. (2007). Retrieved October 15, 2007 from http://www.businessdictionary.com.

Dowd, K., & Blake, D. (2006). After VaR: The theory, estimation, and insurance applications of quantile-based risk measures. Journal of Risk & Insurance, 73, 193-229. Retrieved October 15, 2007, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=20967407&site=ehost-live

Ergashev, B. (2012). A theoretical framework for incorporating scenarios into operational risk modeling. Journal of Financial Services Research, 41 , 145-161. Retrieved November 17, 2013, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=74467298&site=ehost-live

Friedberg, L., & Webb, A. (2007). Life is cheap: Using mortality bonds to hedge aggregate mortality risk. B.E. Journal of Economic Analysis & Policy, 7, 1-33. Retrieved October 17, 2007, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=26040324&site=ehost-live

Gomez-Deniz, E., & Vazquez-Polo, F.J. (2005). Modeling uncertainty in insurance Bonus-Malus premium principles by using a Bayesian robustness approach. Journal of Applied Statistics, 32, 771-784. Retrieved October 23, 2007, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=18517953&site=ehost-live

Ibiwoye, A., Adeleke, I.A., & Aduloju, S.A. (2011). Quest for optimal Bonus-Malus in automobile insurance in developing economies: An actuarial perspective. International Business Research, 4, 74-83. Retrieved November 17, 2013, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=69615581&site=ehost-live

Insurance Information Institute. Glossary. Retrieved October 29, 2007, from Insurance Information Institute Web site: http://www.iii.org/static/site/tools/glossary%5ffrset.htm

Iwanik, J., & Nowicka-Zagrajek, J. (2005). Premiums in the individual and collective risk models. In Statistical Tools in Finance & Insurance (pp. 407-426). Netherlands: Springer Science & Business Media. Retrieved October 17, 2007, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=18968807&site=ehost-live

Kabler, B. (2004). Insurance-based credit scores: Impact on minority and low-income populations. Journal of Insurance Regulation, 22, 77-90. Retrieved October 15, 2007, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=12983041&site=ehost-live

Merriam-Webster's collegiate dictionary (10th ed.). (2000) Springfield, MA: Merriam- Webster.

Moreno, I., Vazquez, F.J., & Watt, R. (2006). Can bonus-malus alleviate insurance fraud? Journal of Risk & Insurance, 73, 123-151. Retrieved October 15, 2007, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=20458676&site=ehost-live

United States Bureau of Labor Statistics. (2006-07). Occupational Outlook Handbook. Retrieved October 15, 2007, from, http://www.bls.gov/oco/ocos041.htm.

Zhao, B. (2012). A modified Lee-Carter model for analysing short-base-period data. Population Studies, 66, 39-52. Retrieved November 17, 2013, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=71346972&site=ehost-live

Suggested Reading

Kipling, M. (2007, May 17). The rate composers. Money Marketing, 55-56. Retrieved October 15, 2007, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=25331058&site=ehost-live

Lescourret, L., & Robert, C.Y. (2006). Extreme dependence of multivariate catastrophic losses. Scandinavian Actuarial Journal, 2006, 203-225. Retrieved October 15, 2007, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=22306608&site=ehost-live

Mitchell, O.S., & Phillips, J.W.R. (2006). Social Security replacement rates for alternative earnings benchmarks. Benefits Quarterly, 22, 37-47. Retrieved October 15, 2007, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=23131010&site=ehost-live

Risk hazard assessment, not only actuary skill, add non-traditional areas as well. (2000). Insurance Advocate, 111, 7. Retrieved October 15, 2007, from EBSCO Online Database Business Source Complete. http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=4012032&site=ehost-live