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The Law of Large Numbers

Gordon Thompson | May 3, 2024

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This is a two-part article. The first article, "Risk Distribution: A History and the Law of Large Numbers Fallacy," is written by F. Hale Stewart, JD, LLM. It 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.

This second part outlines the foundations of the law of large numbers (LLN), and the underlying risk theories that support risk diversification by volume and by time, providing numerical examples.

Basic Requirements of the Law of Large Numbers

An understanding of the LLN's two basic requirements is required to understand its shortcomings relative to risk diversification. First, the risks must be independent. Second, the risk must be identically distributed (i.e., homogeneous). These requirements are commonly referred to as independent and identically distributed.

Statistical independence requires no correlation between events. Identical distribution requires all risk units to have the same risk characteristics. A common way of interpreting this requirement is the withdrawal of one risk unit can be replaced with another risk unit without a change to the risk portfolio.

Neither of these statistical requirements is completely met under insurance operations. There is some recognition of this fact given the rise of underwriting methods for all insurance perils. One method given the lack of risk homogeneity was the introduction of risk classifications. Insurance underwriters assign individual risks to separate groups in an attempt to reflect common risk characteristics within a group.

It is recognized that membership in any group does not represent true risk homogeneity. While the individuals in any specific group are heterogeneous, there is an assumption that underwriting can recognize "similar" risk characteristics. That is, while some members in the group will have lower loss frequency/costs, and other members may have higher loss frequency/costs, the average loss frequency/costs for the group can be used to represent the risk characteristics of all members within the group.

This underwriting method is typically used where risk perils, displaying common characteristics and of uniform size, can be approximately identified (e.g., life insurance, health insurance, personal automobile liability, and automobile physical damage). In agreement with Hans Bühlmann's 1 Stability Theorem from Mathematical Methods in Risk Theory, the goal of risk classification is to achieve operational stability through volume.

To account for these shortcomings, underwriters utilize other methods of risk selection and classification. These include individual risk rating, facultative underwriting, risk pools (e.g., nuclear pool), Lloyd's of London, derivative and options (financial risks), etc. Under these methods, the underwriters evaluate each risk separately and, based on this appraisal, select a premium for which the insurer is willing to assume the risk perils to be transferred. Again, in agreement with Mr. Bühlmann's Stability Theorem, the goal of these alternative methods is to achieve operational stability across time for the individual insured risk.

Determining the Existence of Risk Diversification

Within the insurance literature, there are several proposed methods to determine the existence of this phenomenon. Here, two notes of caution are in order.

First, no method offers a bright-line rule such as, "above level X, risk diversification exists; below level X, risk diversification doesn't exist." The existence of—or lack of—sufficient risk diversification is dependent on the risk aversion of the insurer. Quite simply, an insurer with lower risk aversion may represent a "speculator" value system. While an insurer with higher risk aversion represents a "hedger" value system. Both value systems are observed in the general insurance industry.

Second, Hale Stewart, author of part one of this article, stated, "No court has offered an opinion on any of these proposed methods. That should not deter the use of any of them since they are derived from an appropriate source. But it should also be understood that these methods are not officially sanctioned."

Risk Stability Using Volume

There are specific methods of potentially determining sufficiency in risk diversification under an insurer's risk reduction strategy of gaining volume. Each section cites a source and includes a very basic explanation of the method. The reader is strongly encouraged to investigate each concept in more detail.

Simple Probability Analysis

One method of determining risk diversification sufficiency involves a simple probability analysis along with the calculation of the standard deviation of a portfolio of risks.

For example, Emily and Samantha each face the same risk. Each has a 20 percent probability of a $2,500 loss and an 80 percent probability of no loss. Each faces the following expected $500 [ = (0.8 x $0) + (0.2 x $2,500)] cost without pooling. Individually, each also has a standard deviation of $1,000 [ = ((0.8 x (0 – 500)2 + (0.2 x (2,500 – 500)2)0.5].

The coefficient of variation (CoV or standard deviation over mean) is 2.000 [= $1,000/$500]. Their individual loss distribution ranges from $0 to $2,500. At the end of the period, each has an 80 percent chance that they can recover their $500 average expected loss costs. However, they each have a 20 percent probability that they will need access to an additional $2,000 if a loss occurs.

Table 1. Table One
Members: 1 Claims Claim Amount $2,500 Claim Amount – Mean (Claim Amount – Mean)2 Probability x (Claim Amount – Mean)2
Probability: 0.2
Combinations Frequency
(1) (2) Binomial Distribution (3) (4) = (3) x $2,500 (5) = (4) – $500 (6) = (5)2 (7) = (2) x (6)
1 0.800 0 $0 ($500) $250,000 $200,000
1 0.200 1 $2,500 2,000 4,000,000 800,000
Totals/Avgs 1.000 4,250,000 1,000,000
Mean 0.200 $500 Stdev: 1,000
CoV 2.000 Stdev/1: 1,000
Reduction in Stdev: 0.00%

But, if they combine their respective risks and agree to share costs equally, two things happen. First, there is a decrease in the probability of no losses [(0.8 x 0.8) = 0.64]. This is because there's an increased probability of one experiencing a maximum loss [2 x (0.2 x 0.8) = 0.32]. In addition, there's also a possibility of both experiencing a large loss [(0.2 x 0.2) = 0.04].

Second, there is also a reduction in the CoV under the two-person collective. The mean of this distribution is $1,000 or twice the individual average of $500. So, the average expected losses don't change per individual. The standard deviation also increases to $1,414, an increase from the single individual of $414.

At first, there doesn't appear to be an advantage. However, the CoV now decreases by 29 percent from 2.000 to 1.414. The standard deviation has a tighter fit around the expected losses even though the aggregate loss distribution is now wider, going from $0 to $5,000.

Under the collective, each individual's expected pure loss costs remain at $500 (portfolio mean of $1,000 divided by 2 members). At the end of the period, each has a 64 percent chance of receiving a return of their $500 expected costs. However, they still had a 32 percent probability of needing an additional $750 [= (2,500 – $1,000)/2], but only a 4 percent chance of needing an additional $2,000 [ = (5,000 – 1,000)/2]. Each has reduced their financial risk when each looks at her downside for additional payments.

Table 2. Table Two
Members: 2 Claims Claim Amount $2,500 Claim Amount – Mean (Claim Amount – Mean)2 Probability x (Claim Amount – Mean)2
Probability: 0.2
Combinations Frequency
(1) (2) Binomial Distribution (3) (4) = (3) x $2,500 (5) = (4) – $ (6) = (5)2 (7) = (2) x (6)
1 0.640 0 $0 ($1,000) $1,000,000 $640,000
2 0.320 1 $2,500 1,500 2,250,000 720,000
1 0.040 2 $5,000 4,000 16,000,000 640,000
Total/Avgs 1.000 19,250,000 2,000,000
Mean 0.400 $1,000 Stdev: 1,414
CoV 1.414 Stdev/2: 707
Reduction in Stdev: 29.29%

In Risk Management and Insurance, 2 author Scott E. Harrington notes that the pooling arrangement changes the probability distribution of aggregate loss costs facing the two-person collective. For either Emily or Samantha to pay the maximum amount of damages ($2,500), both would have to experience a catastrophic loss. This development lowers the portfolio's CoV, or the standard deviation divided by the mean—a standard measure of risk. In this example, the standard deviation decreased from $1,000 to $707 per individual. This 29.29 percent [ = 1.00 – 707/1000] decrease in the volatility of outcomes accomplishes a key objective of developing risk diversification to reduce the risk to the collective.

If Emily and Samantha continue to add other participants to their risk collective, they achieve additional risk reduction benefits. But the risk reduction benefits come at a price. While the CoV decreases in size, the range of the aggregate loss distributions widen. Both the mean and the standard deviation increase, but at a reducing rate as indicated by the CoV.

Table 3. Table Three
Member Count Claims Frequency Average Aggregate Loss Distribution
Probable Minimum Mean Probable Maximum Range Stdev CoV CoVMC CoV1 CoVMC – CoV1 CoVMC – CoVMC-1
(1) (2) (3) (4) (5) = (4) – (2) (6) (7) (8) (9) (10)
1 0.2 $0 500 2,500 2,500 1,000 2.000 1.000 0.0% XXX
2 0.4 $0 1,000 5,000 5,000 1,414 1.414 0.707 -29.3% -29.3%
3 0.6 $0 1,500 7,500 7,500 1,732 1.155 0.577 -42.3% -13.0%
4 0.8 $0 2,000 10,000 10,000 2,000 1.000 0.500 -50.0% -7.7%
5 1.0 $0 2,500 12,500 12,500 2,229 0.892 0.446 -55.4% -5.4%
6 1.2 $0 3,000 15,000 15,000 2,449 0.816 0.408 -59.2% -3.8%
7 1.4 $0 3,500 17,500 17,500 2,646 0.756 0.378 -62.2% -3.0%
8 1.6 $0 4,000 20,000 20,000 2,828 0.707 0.354 -64.6% -2.4%
9 1.8 $0 4,500 22,500 22,500 3,000 0.667 0.333 -66.7% -2.0%
10 2.0 $0 5,000 25,000 25,000 3,162 0.632 0.316 -68.4% -1.7%
15 3.0 $0 7,500 35,000 35,000 3,873 0.516 0.258 -74.2% -5.8%
25 5.0 $0 12,500 50,000 50,000 5,000 0.400 0.200 -80.0% -5.8%

As noted above, there is no bright-line rule as to how much of a drop in CoV is required. Generally speaking, the benefit in the CoV reductions is smaller with each additional member. What is important to note is that it's possible to demonstrate in a simple way the decline in risk associated with a specific portfolio of coverages.

The target risk reduction for a portfolio is based on many factors relating to business strategy, risk aversion of owners, premium risk loads, and available capital. An arbitrary selection as to some predetermined minimum number of members within a collective cannot designate a specific level of risk reduction. There are simply too many variables. The necessary level of risk reduction can only be determined by the entity assuming the risk of adverse loss positions within the risk collective. In insurance, the level of acceptable risk reduction has historically been made by the insurer's domiciliary regulator.

Risk Stability over Time

The following section identifies specific methods of determining sufficiency in risk diversification under an insurer's risk reduction strategy of time. Each section cites a source and includes a very basic explanation of the method. The reader is strongly encouraged to investigate each concept in more detail.

For commercial insureds, peril risks relate to the uncertainty as to the timing of a loss event, not whether such event will or will not occur. It is assumed the event will occur, but the timing and costs associated with such a loss are unknown. This concept is more similar to mortality risk; the event will occur, but the timing is uncertain.

Under such a view of risk, underwriters review historical experience for a potential insured. This experience provides a limited time view of types of historical losses observed for the entity. Under this approach, the underwriter is looking for the type of loss, source of loss, time of loss events, and size of loss given a loss event of the individual entity.

An advantage of such an approach is that there is a clearer appreciation for both the systematic and idiosyncratic risks associated with the entity. By reviewing the loss history, there is an improved view as to the risks of the entity.

From a statistical perspective, the premium analysis is based on reviewing the entity's individual losses and its aggregated loss experience across exposure periods. Individual company losses only provide small sample sizes. To account for this, underwriting considers employing assumptions to gather more data (e.g., assuming the data is homogeneous and stable from year to year and assuming the exposure and loss results consistently reflect the individual entity's operations and management). When tested, these assumptions are often found to be inappropriate/inaccurate.


Ultimately, insurance literature and basic statistical concepts lead to some conclusions regarding the existence of risk diversification. Generally, risk diversification is thought to provide a statistical benefit through the reduction in the CoV, the standard deviation divided by the mean. 3 While adding additional risks within an insurance portfolio can reduce the CoV, the size and amounts of the standard deviation and the mean of the aggregate loss distribution increases. The smaller CoV for the aggregate loss distribution allows for an assumption of smaller variations around the mean, hence smaller risk loads to achieve the same confidence levels.

However, adding additional risks also increases the spread of the aggregate loss distribution. Although there is a smaller risk premium needed from each insured, both the total premium volumes and capital support requirement increase due to increased volume.

While there is no exact number, risk diversification leads to a minimum level of exposure such that the variability of projected losses meets the risk tolerance levels required by the insurer. For insurers with higher (lower) risk tolerance, the number of insured risk exposures may be small (high). Note the insured risks within an insurance collective can represent a group of insureds representing a single peril or a group of risks representing multiple differing perils. Again, as the number of insured risks increase, the CoV may be reduced, but there are higher capital costs to maintain solvency levels.

Also, the statistical benefit for adding one additional risk represents a declining reduction in the standard deviation. This means each additional risk added to the portfolio has a smaller impact on the risk reduction within the portfolio.

For additional information on this topic, please see the following books.

  • Neil A. Doherty, Corporate Risk Management, McGraw-Hill College, January 1, 1985.
  • Charles M. Nyce, Foundations of Risk Management and Insurance, American Institute for Chartered Property Casualty Underwriters/Insurance Institute of America, 2006.
  • C. Arthur C. Williams, Jr., Peter C. Young, and Michael L. Smith, Risk Management and Insurance, McGraw-Hill/Irwin, 8th edition, August 1, 1997.

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1 Hans Bühlmann, Mathematical Methods in Risk Theory, Springer Berlin, Heidelberg, 1970.
2 Scott E. Harrington and Greg Niehaus, Risk Management and Insurance, McGraw-Hill, 2nd edition, 2003.
3 This is true if the risk units can be considered to be "independent." It may or may not be true under conditions of correlation.