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Risk Distribution: A History and the Law of Large Numbers Fallacy

Hale Stewart | May 2, 2024

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A scattered deck of cards.

This two-part article first looks at the academic and case law history of risk distribution, followed by an explanation of correlated, semicorrelated, and noncorrelated risk and the effect each of these has on the portfolio's total risk. The second part, "The Law of Large Numbers," is written by Gordon C. Thompson. It outlines the foundations of the law of large numbers and the underlying risk theories that support risk diversification by volume and by time, providing numerical examples.

"Historically and commonly insurance involves risk-shifting and risk-distributing."

Source: Helvering v. LeGierse, 312 U.S. 531, 539 (1941).

This sentence is one of the most cited in the captive insurance case law. Unfortunately, the case contains no discussion of either term. Later decisions would hold that risk shifting requires a valid contract 1 and underwriting risk, 2 giving planners some degree of certainty. 3 However, risk distribution is a different story.

The Tax Court—and the Internal Revenue Service (IRS) via revenue rulings—originally ruled that, for tax purposes, the insurer either needed to receive a certain percentage of premium from a nonparent or insure a certain number of entities in specific percentages. 4 Recent cases state that insurers must insure the risk of a certain number of statistically independent risk units, but any meaningful guidance stops there.

This article seeks to eliminate some of this uncertainty. It first explains the academic origins of risk distribution, followed by a brief discussion of the case law. Next is an explanation of independent and dependent risks and their respective level of risk for an insurer. This is followed by an explanation of why the law of large numbers doesn't apply to insurance in the manner understood by the courts. This is followed by an analysis of coefficient of variance using an example derived from the book by Scott E. Harrington and Greg Niehaus, Risk Management and Insurance. 5 The example illustrates that, as the number of independent risk exposures increase, the risk per exposure unit decreases. Unfortunately, the article offers no bright line rule, as none exists.

Historical and Case Law Development

The origins of addressing the statistical advantages of risk diversification 6 in insurance methods can be traced directly to Allan H. Willet's The Economic Theory of Risk and Insurance. Chapter VII first notes that, without insurance, individual businesses would have to raise prices to account for risk. He uses two examples: a fruit dealer (who would have to add to the price of his goods due to decay) and a shipowner (who would have to raise his prices for the time his ships were in port). 7 The increased price due to the cost of risk explains the need for capital to pay for future losses, which is Mr. Willet's first requirement of insurance.

To this, he adds the transfer of risk, 8 where the person exposed to a potential harm moves the economic cost of the risk to a third party. He then adds the need to combine risks:

To form a complete conception of insurance, it is necessary to add to the notions of accumulation of capital and transfer of risks the idea of the combination of the risks of many individuals in a group … Only when … joined in combination of risks in a group is the insurance complete.

Source: Allan H. Willett, The Economic Theory of Risk and Insurance, p. 106.

Mr. Willet provides an example of 10 men who each contribute $1,000 to a hypothetical company, creating a capital pool of $10,000. Each individual would only suffer one-tenth of the total loss.

A later author noted that increasing the number of risks increases the predictability of potential outcomes:

In the third class of cases the reduction of risk arises from the facts that the specialist who assumes the specific risk combines it with a great number of others and so reduces the amount of uncertainty, and hence the amount of risk, through the operations of the law of large numbers.

Source: Charles Oscar Hardy, Risk and Risk Bearing, University of Chicago Press, 1931, pp. 66–67.

Roger Cooley's Briefs on the Law of Insurance contains the best summary of the then-emerging definition of insurance:

There must, in order that there may be successful insurance, be a sufficiently large number exposed to the same risk to make it practicable and advantageous to distribute the loss falling upon a few. As indemnity against loss is at the foundation of insurance, the business must be regarded as a system of distributing losses upon the many who are exposed to the common hazard.

Source: Roger W. Cooley, Briefs on the Law of Insurance, Volume 1, West Publishing, 1960, p. 81.

The same concepts (capital, risk transfer, and risk distribution) are scattered throughout the captive insurance case law.

When an insurer has a sufficiently large number of risks such that great variations in aggregate losses are unlikely, and the premiums received plus its capital make it a viable risk bearer, one can say that risk distribution is present regardless of the number of insureds covered.

Source: Malone & Hyde, Inc. v. Commissioner, 1993 T.C. Memo. 585, 66 T.C.M. 1551, 1993 Tax Ct. Memo LEXIS 601.

The Tax Court in Gulf Oil Corp. v. Commissioner, 89 T.C. 1010 (1987), noted that a large pool of insureds "diffuses the risks through a mass of separate risk shifting contracts." Both the Malone and Gulf definitions imply that pooling lowers volatility of the pool. 9 The court in Humana Inc. v. Commissioner, 881 F.2d 247, 251 (6th Cir. 1989), noted the relationship between an individual policyholder and the pool: "The focus is broader and looks more to the insurer as to whether the risk insured against can be distributed over a larger group."

Case Law and IRS Guidance on Achieving Risk Distribution

Prior to the Rent-A-Center, Inc. v. Commissioner, 142 T.C. No. 1 (2014), decision, case law and IRS rulings provided two general methods to create risk distribution. The first requires the captive to receive a certain percentage of premium from a nonparent. For example, a captive formed for Acme Corporation receives 75 percent of its premium from Acme and 25 percent from a third-party (nonparent). The case law provided minimal guidance on the percentage required to achieve risk distribution. 10 The IRS eventually stated in Rev. Rul. 2002-89 that "more than 50 percent" was required.

The second method to achieve risk distribution is by selling insurance to a certain number of subsidiary companies in a specific proportion. This concept was loosely based on two IRS decisions: Crawford Fitting Co. v. United States, 606 F. Supp 136 (N.D. Ohio 1985), and Humana Inc. & Subsidiaries v. Commissioner, T.C. Memo 1985-426. The IRS eventually stated in Rev. Rul. 2002-90 that this method required at least 12 companies with no company accounting for less than 5 percent nor more than 15 percent of the total risk and premium received by the captive.

Neither the Tax Court nor the US Treasury provide an actuarial, mathematical, or statistical basis for either method. The rulings and decisions simply wrote approvingly about a specific percentage of third-party premium or a number of insured entities.

Rent-A-Center created a new and far superior methodology to achieve risk distribution. According to the decision, the captive must insure a sufficient number of "statistically independent risks"—a far better metric, as this concept clearly translates into the underwriting concept of "exposure," which is "the basic unit that measures a policy's exposure to loss." 11 Not only must there be a sufficient number of units, but each must be independent, meaning no unit's loss can have a causal relationship with another unit's loss.

For example, assume that 100 trucks are tethered so that each would move in the same direction. Each truck would be a "unit," but their risks would be dependent: One truck's accident would cause several others. Untethering the trucks makes them independent risks. An accident caused by truck number 3 on the west side of town at the beginning of the week would have no causal relationship to an accident caused by truck number 46 that occurred 3 days later on the east side of town.

Regarding the number of units needed, the Rent-A-Center decision stated, "Risk distribution occurs when an insurer pools a large enough collection of unrelated risks (i.e., risks that are generally unaffected by the same event or circumstance)." This sentence is regrettably imprecise and statistically wrong. A small number of unrelated risks can significantly lower the portfolio's total risk. 12 Semicorrelated risks also lower the standard deviation but not to the same degree as uncorrelated exposures. Purely correlated risks have no impact on the portfolio's comprehensive risk exposure. 13

The Fallacy of the Law of Large Numbers Regarding Risk Distribution

According to the Tax Court's captive decisions, the law of large numbers is an absolute requirement for all insurance companies.

Risk distribution is one of the common characteristics of insurance identified by the Supreme Court in Le Gierse, and occurs when the insurer pools a large enough collection of unrelated risks. The idea is based on the law of large numbers—a statistical concept that theorizes that the average of a large number of independent losses will be close to the expected loss. As the Ninth Circuit explained in Clougherty, "[b]y assuming numerous relatively small, independent risks that occur randomly over time, the insurer smoothes out losses to match more closely its receipt of premiums."

Source: Avrahami v. Commissioner, 149 T.C. 144, 181 (2017).

However, the law of large numbers doesn't apply to insurance in the manner the Tax Court believes. Irving Pfeffer observed:

Strictly speaking, there is no line of insurance written which is able to meet the complete set of tests implied by any of the Law of Large Numbers because the universe of insurance experience is constantly changing with the economic and social environment. This means that the results gleaned from the past no longer have the same measure of relevance for the present—much less the future."

Source: Irving Pfeffer, Insurance and Economic Theory, S.S. Huebner Foundation for Insurance Education, 1956, p. 43.

In Corporate Self-Insurance and Risk Retention Plans, Robert C. Goshay noted, "Neither the insurer nor the self-insurer is capable of collecting exactly homogeneous risks in sufficient numbers to make theoretical predictions with absolute certainty." 14 Donald L. MacDonald had a similar observation in Corporate Risk Control: "No matter how many risks it has pooled, however, an insurer almost always will experience a deviation from its expected loss ratio if the risks—collectively—differ substantially in capacity for producing loss from those in the group which was the source of that loss ratio." Finally, as noted in Essentials of Risk Financing:

The law of large numbers has two significant limitations. First, no matter how many past losses have occurred and have been used to compute expected losses for a current insurance policy period, there is some chance that the actual losses for that period will not follow past patterns. Second, and more important, underlying conditions affecting loss experience might change, so the future no longer comes close to repeating the past. 15

These shortcomings have been incorporated into mathematical insurance teachings and are now part of the standard insurance curriculum. 16

For more explanation of the law of large numbers, its foundations, current insurance practices, and some numerical examples, continue to part two of this article.

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1 Humana Inc. v. C.I.R., 881 F.2d 247, 251 (6th Cir. 1989). ("Risk shifting involves the shifting of an identifiable risk of the insured to the insurer. The focus is on the individual contract between the insured and the insurer.")
2 R.V.I. Guar. Co. Ltd. and Subsidiaries v. C.I.R., (145 T.C. 209) (2015). ("Underwriting risk is the difference between the premium received and the face amount of the policy. For example, if a $1 million insurance policy is sold for $250,000, the insurance company faces $750,000 of underwriting risk.")
3 Outside of case law, there is FASB 113 and Risk Transfer Testing in Reinsurance Contracts by David L. Ruhm and Paul J. Brehm, published by the Casualty Actuarial Society.
4 As noted below, neither concept is rooted in statistics.
5 Scott E. Harrington and Greg Niehaus, Risk Management and Insurance, McGraw-Hill, 2003.
6 The use of "distribution" to represent "diversification" is a common mistake in the statistical discussions of insurance methods. When referring to the frequency of an event, distribution refers to the probability of an event's occurrence. When referring to cost of a loss given the occurrence of an event, distribution reflects the probability distribution of claim costs. When referring to the aggregate loss costs for a risk portfolio, distribution reflects the probability of the aggregate claim costs. Diversification refers to the aggregation of separate risk units.
7 Allan H. Willett, The Economic Theory of Risk and Insurance, Leopold Classic Library, New York, 1901.
8 Willett, p. 106. ("Risk" is defined as uncertainty as to timing of an occurrence of an event and the associated economic costs given the occurrence of such an event)
9 Great variations are unlikely; the pooling "diffuses" the risk.
10 The lowest percentage in a taxpayer victory was 29 percent. (Harper Group & Includible Subsidiaries v. C.I.R., 96 T.C. 45 (1991).)
11 Geoff Werner, Basic Ratemaking, Casualty Actuarial Society, 2016, p. 49.
12 This concept was first observed by Harry M. Markowitz in Portfolio Selection, published in March 1952.
13 Harrington and Niehaus, p. 57.
14 Robert C. Goshay, Corporate Self-Insurance and Risk Retention Plans, Studies/S.S. Huebner Foundation for Insurance Education, 1964, p. 17.
15 George L. Head, Michael W. Elliot, and James D. Blinn, Essentials of Risk Financing, Insurance Institute of America, 3rd edition, July 1, 1996, p. 231.
16 I.B. Hossack, Introductory Statistics with Applications in General Insurance, Cambridge University Press, 1983, Chapter 11; Hans Bühlmann, Mathematical Methods in Risk Theory, Springer-Verlag Berlin, 1970.