How the tax court’s risk distribution analysis misapplies the law of large numbers

The US tax court’s approach to risk distribution is rooted in a flawed reliance on the Law of Large Numbers, argue F. Hale Stewart, JD, LL.M., attorney with Higginbotham Insurance, and Sol Feinberg, FCAS, MAAA, consulting actuary at SIGMA Actuarial Consulting Group

“Historically, and commonly, insurance involves risk shifting and risk distribution.” This oft-cited Supreme Court sentence introduced risk distribution as a common law element of insurance for federal income tax purposes. A worse precedential foundation could not exist. Helvering v. LeGierse is an anti-abuse and estate tax case. It does not meaningfully discuss an insurance company’s internal intricacies, save for a single sentence. Later jurisprudence compounded this error by strictly applying the Law of Large Numbers to property and casualty insurers. This stands in stark contrast to insurance scholarship, which, as early as 1923, recognised that the Law of Large Numbers does not apply to certain coverages. Numerous other insurance texts have observed that this statistical concept has limited applicability to property and casualty underwriting.

This essay will first analyse current captive insurance safe harbors, explaining they are not only arbitrary but also fall outside the boundaries of “insurance in its commonly accepted sense” making them judicially suspect. The origins of risk distribution will next be discussed. This is followed by an examination of how the insurance industry first believed the Law of Large Numbers was central to risk pooling only to conclude it has limited application to insurers. The article concludes with a contribution by Sol Feinberg that not only explains the mathematical underpinnings of the Law of Large Numbers but also explains how this concept is frequently misapplied in insurance pricing.

Current IRS ‘safe harbors’

Revenue Ruling 2002-89 focuses on the amount of non-parent premium needed for the captive to be recognised for federal income tax purposes, concluding “less than 50 per cent” is required. This means a captive receiving 50.00 per cent of its premium from third parties is not an insurer, but one receiving 50.01 per cent is. The ruling offers no mathematical, actuarial, or underwriting justification for this distinction. Revenue Ruling 2002-90 focuses on the number of entities needed for a group captive to be valid for federal tax purposes, concluding that 12 are required. This means a group with 11 members is not an insurance company, but one with 12 is. The ruling offers no mathematical, actuarial, nor underwriting justification for the distinction between 11 and 12 companies.

Both rulings are based on a synthesis of the then-decided court decisions, none of which relied on a formula, calculation, or commonly used underwriting methodology. All decisions were based on the court’s subjective opinion based on a case’s facts. On the one hand, this is to be expected. But on the other hand, it overlooks the fact that insurance is deeply rooted in mathematical and actuarial principles that should be included in the court’s reasoning.

Recent decisions have focused on the number of individual exposure units the captive insures. Here, the court counts the number of autos covered by commercial auto, employees insured by workers’ compensation coverage, or the number of locations under a commercial general liability policy. Here, the precedents are equally suspect as the above-discussed Revenue Rulings. Dicta from the Swift case states that 350-531 individual units is insufficient but numbers from Rent-A-Center and Securitas indicate tens of thousands of specific units will suffice for the tax court. As with the Revenue Ruling ‘safe harbors’, no court offers mathematical, actuarial, nor underwriting underpinnings for its decisions. Instead, each court simply counts the number of exposures and offers a thumbs-up or thumbs-down assessment.

The current safe harbors all suffer from the same fatal flaw: none are based on recognisable insurance or mathematical principles. Compare this to the underwriting process, which is deeply rooted in higher-level math and statistics. The approaches could not be more pronounced nor disjointed, creating an untenable public policy situation.

The following example illustrates how this dichotomy creates an unworkable public policy situation. ABC company wants to form a captive that underwrites commercial auto, general liability, and workers’ compensation. The company hires a fronting carrier, paying it an annual fee of US$250,000. Additionally, ABC agrees to set aside US$300,000 in collateral for the programme’s first three years. ABC hires a captive insurance manager, a third-party claims administrator, and an investment adviser for the programme. Due to the uncertain nature of the law, no party can definitively state the captive has sufficient risk distribution. However, a consulting actuary hired by ABC analysed the captive’s projected losses and finances and determined it is financially viable, even in an adverse scenario.

A second actuary hired by the jurisdiction reviews this work. The captive files an application, containing an actuarial analysis that is reviewed by a second actuary hired by the jurisdiction. The IRS audits the captive after five years of operations. determining the programme has insufficient risk distribution but offering no explanation for this determination. After paying more than US$1 million in fees, hiring industry experts, and operating a programme in good faith, ABC now faces a tax liability with remarkably little clarity as to why the company owes the money. A company that has spent over US$1 million, signed several large and complex service-provider contracts, attended annual meetings, and submitted detailed reports to a regulator for review deserves far more legal clarity, especially considering the importance of adequate insurance coverage to its daily operations.

What is risk distribution?

The concept of risk distribution originated in Allan Willet’s 'Economic Theory of Risk and Insurance', which stated: “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 risk of many individuals in a group.” This combination has multiple goals, the most important of which is to stabilise the group’s statistical outcomes. For example, if a company insures a single employee’s workers’ compensation exposure, the possible financial outcomes range from no claim to the largest claim possible. Adding a single employee to this pool can noticeably lower volatility, a development furthered by the addition of more risks. While smaller pools can still show a significant drop in variability, they can also transfer portions of the pool’s risk to other insurers via reinsurance. As the pool grows, the risk per unit declines. Finally, and most importantly, uncertainty is never eliminated.

Increasing the size of the pool also diversifies the source of funds for claims funding. Assume the same company paid US$1,000 per employee for 10 employees workers’ compensation coverage. A claim for one employee would be 90 per cent sourced from funds contributed on behalf of the other nine workers’ individual risk. The increased number of pool participants also adds temporal distribution; of the 10 workers in the preceding example, not all will file a claim in a single policy year. This allows the time value of money to increase the premium contribution paid for these employees, increasing the captive’s capital and surplus.

Mathematical explanation of declining volatility

Originally, insurance academics believed the Law of Large Numbers caused the decreased volatility of larger pools. In 1923, Hardy wrote: “For [insurers], the chief hazard is not the hazard insured against, but the risk of failing to get sufficient business to furnish a working basis for the law of large numbers.” Several other texts agreed. In ‘Insurance Principles and Practices’, Riegel observed: “The Law of Large Numbers is the principle that the reliance to be placed upon a given probability is increased when the number of trials is increased.”

However, insurance authors and academics quickly realised the Law of Large Numbers was not the foundational mathematical principle of insurance pooling as documented in the following citations:

• The LLN didn’t apply to coverage purchased from Lloyd’s. (Charles O. Hardy, Risk and Risk Bearing, 1923.)


• “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.” (Pfeffer, Insurance and Economic Theory, p. 43.)


• “Neither the insurer nor the self-insurer is capable of collecting exactly homogenous risks in sufficient numbers to make theoretical predictions of absolute certainty.” (Robert Goshay, Corporate Self-Insurance and Risk Retention Plans, S.S. Huebner Foundation for Insurance Education, 1965, p. 17.)


• “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 the loss ratio.” (Donald L. MacDonald, Corporate Risk Control, 1966, p. 60.)


• “Even if we properly evaluate the mean frequency and the mean severity, the actual observed results will generally vary from the underlying mean.” (Foundations of Casualty Actuarial Science, third edition, 1996, p. 61.)


• “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. (Michael Elliot, George Head, and James D. Blinn, Risk Financing, American Institute for Chartered Property Casualty Underwriters, Inc., 2005, p. 231.)