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

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