Can we diversify extremal events?

This post was originaly written in French and translated below.

In a financial context, diversifying risks means investing in a variety of assets, sectors, or geographic regions to avoid having the poor performance of a single investment significantly affect the overall portfolio. Diversification allows for risk reduction, or, in its mathematical formulation, the reduction of variance. But what happens when we encounter large risks, infinite variance? Or worse, infinite expectation?

Extreme Risks and Infinite Expectation?

Formalizing quantities related to random and uncertain quantities is a complex exercise. Probabilities, in the sense the word is often understood, are defined as the limits of frequencies observed through repeated events. The probability of rolling a 3 on a die is 1/6 because, by rolling a die a million times [1], a billion times, the probability will be as close as desired to 1/6. This is what the law of large numbers states, in its weakest form. Saying that the probability it will rain today is 1/6 is entirely different because it is a unique event. If I get drenched by a shower today, it will not prove that the probability was not 1/6, nor will it disprove the meteorological model. This is just a reminder that when modeling, we try to imagine small values for rare events, and it is unfortunately very difficult to validate them.

When modeling large risks, very large risks, it is not uncommon to suggest that the risks have infinite variance or expectation. The notion of infinite expectation is both strange and probably counterintuitive [2]. If we consider a positive random variable $X$ (for simplicity), and let $S(x)=\mathbb{P}(X>x)$ be the survival function, and $f(x)$ the density function (corresponding to the opposite of the derivative of $S$), we can show that the empirical mean of a million or a billion draws of this variable will approach a value, called the mathematical expectation:$$\mathbb{E}(X) =\int_0^\infty S(x)dx= \int_0^\infty xf(x)dx$$There is nothing surprising here; this is still the law of large numbers, stated as early as 1713 by Jacob Bernoulli (the “golden theorem” of Raper (2018)) and especially by Pierre-Simon Laplace in 1814. However, this integral must be finite, which is not guaranteed. For example, the Pareto distribution with index $a$ satisfies $S(x)=\mathbb{P}(X>x)=x^{-a}$. As early as 1925, Karl-Gustaf Hagstroem noted that this distribution seemed particularly suited for modeling large risks, and thus in reinsurance [3]. For a variable following a Pareto distribution with index 1, its expectation is, mathematically, infinite.

What does this infinite expectation mean? There will be no “claim of infinite cost,” and it will always be possible to calculate an empirical average over $n$ observations. However, this average will tend toward infinity as $n$ increases. Louis Bachelier, in discussing the St. Petersburg paradox (a game with infinite expected gain), reminds us that “a paradoxical result in mathematical sciences necessarily stems from a flaw in our understanding, incapable of deciphering a too complex whole, unable to represent the infinitely large. Common sense cannot be invoked in delicate matters; it does not allow us to recognize whether the area between a curve and its asymptote is finite or not, whether a series is convergent or divergent.” This average will tend toward infinity as $n$ increases, meaning that we can be sure the average will always exceed any value we can imagine. This can be visualized at the top of Figure 1 with 10 simulations of 100,000 values. On the left, the case where variance is infinite, and expectation is finite; on the right, both are infinite.

Figure 1: Evolution of the average $n\mapsto (x_1+\cdots+x_n)/n$ for generated samples from a distribution with finite expected value (and infinite mean) on the left, and infinite expected value on the right.

Another interesting measure is the ratio of the maximum over $n$ observations to the sum. For variables with infinite expectation, this ratio does not tend towards 0. It is possible that if the $x$ variables represent claim costs, with 100,000 claims of infinite expectation, the largest claim could represent more than 90% of the total burden.

Figure 2 : Evolution of the ratio $n\mapsto \max\{x_1,\cdots,x_n\}/(x_1+\cdots+x_n)$ with a distribution of finite expectation (and infinite variance) on the left, and a distribution of infinite expectation on the right.

As we can see, this property is important, but it is difficult to identify because it is a fundamental property of the underlying model, related to the distribution of observations, since it is always possible to calculate the average. For example, the following sequence corresponds to eight values obtained by randomly drawing from a Pareto distribution with index 1 (and thus theoretically of infinite expectation):

1.657442 || 4.138543 || 15.592108 || 1.429090

1.684843 || 1.186745 || 1.341435 || 3.308316

How can we tell if a set of claim costs follows a distribution of finite expectation or not? The classic approach, presented for example in Zajdenweber (1996, 2000), is to use the so-called Pareto plot, with the logarithm of costs on the x-axis, and the logarithm of the survival probability on the y-axis. If the points are aligned along a straight line with slope $-a$, then the Pareto distribution with parameter $a$ is perfectly adapted. Indeed, if $\mathbb{P}(X>x)=x^{-a}$, then, taking the logarithm of both quantities, and ordering the sample ($x_1\leq x_2\leq\cdots\leq x_n$), we have
$$\log\left(\frac{n-i}{n}\right)=-a\cdot \log(x_i)$$And if the slope is too moderate, greater than -1, then the costs have infinite expectation.

Figure 3: Pareto plot, with $\log((n-i)/n)$ on the y-axis and $\log(x_i)$ on the x-axis. The points are aligned along a line with slope -a, corresponding to a Pareto distribution with index aa. a≤1a≤1 means that the risks have infinite expectation.

This hypothesis of a Pareto index close to 1 is not unrealistic when we talk about natural or industrial disasters:

• hurricanes, Hsieh (1999), $a\sim1.5$
• company fires, Biffis et al. (2014), $a\sim1.25$
• business interruption, Zajdenweber (1996), $a\sim1$
• earthquakes, Sornette et al. (1996), $a\sim1$
• tsunamis, Embrechts et al. (2024),  $a\sim1$
• operational risk, Moscadelli (2004) and Chavez-Demoulin et al. (2006) $a\sim1$
• cyber risk, Eling et al. (2019) $a\sim1$
• nuclear risk, Hofert et al. (2012), $a\in (0.6;0.7)$

On the Diversification of Large Risks

Instead of working by risk type, we can consider the aggregation of these risks together. Heuristically, having portfolios with flood, earthquake, or drought risks could offer some “diversification.” The concept of “diversification” can be introduced with the law of large numbers, as previously mentioned, and it will be very close to the idea of insurance, of risk pooling. Smith & Kane (1994), for example, remind us that the contribution of an $n+1$-th independent risk in a group of $n$ risks, fairly priced, generally allows for a marginal reduction in risk, which reinforces the insurer’s risk pooling. This diversification effect still works even if the risks are correlated (but not perfectly correlated, and the diversification gains decrease with correlation, as Charpentier (2011) pointed out).

Often, when we talk about “diversification,” we think of the work of Harry Markowitz or Arthur Roy in finance in the 1950s, which laid the foundation for portfolio theory. This theory shows how rational investors can use diversification, corresponding to the correlation between assets, to optimize their financial portfolio. In this approach, it is generally assumed that investors’ preference for a risk/return trade-off can be described by a quadratic utility function. In other words, only the expected return (the expected gain) and the volatility (the standard deviation) or variance are the parameters considered by the investor. This literature shows that an investor can reduce the risk of their portfolio simply by holding assets that are not (or only slightly) correlated, thus diversifying their investments. They can then achieve the same expected return while reducing the variability of their portfolio.

But what happens if the variance no longer exists? This question challenges the use of the normal distribution to model financial returns. The normal distribution was interesting partly because it satisfies a property of stability by summation[4]. Keeping this property while considering a distribution with more extremes than the normal distribution amounts to using “stable” distributions studied by Paul Lévy, as proposed[5] by Benoit Mandelbrot in the 1960s.

In cases where the variance is infinite, it is necessary to use a more general risk measure than the standard deviation, and heuristically, “diversification” is related to the sub-additivity of the risk measure: a portfolio containing the average of the holdings of two other portfolios has a lower risk than the average of the risks of the two other portfolios. Daníelsson et al. (2013) remind us that in the presence of large risks (infinite expectation), diversification no longer works. This property was described and discussed by Paul Samuelson as early as 1967, Stephen Ross in 1976, and more recently by Rustam Ibragimov, Dwight Jaffee, Johan Walden, Paul Embrech, or Ruodu Wang, among others. The introduction by Ibragimov et al. (2015) explains it well, “there are limitations to diversification with such risk distributions [heavy-tailed distribution]. Specifically, whereas diversification is preferred by risk-averse agents when risks are thin-tailed (the traditional case that has been extensively studied), it may actually be hurtful for agents to diversify when risks are heavy-tailed […] nondiversification traps may arise when risk distributions have heavy left tails and insurance providers have limited liability.” These properties, widely discussed from a mathematical perspective, are difficult to accept because they are theoretical and counter-intuitive. Moreover, it is often difficult to determine for whom diversification becomes dangerous, since there are several stakeholders: the insured, insurers, reinsurers, and the state. Ibragimov et al. (2011) provide some answers, “when these risks are thin-tailed, risk-sharing is always optimal for both individual intermediaries and society. But, with moderately heavy-tailed risks, risk-sharing may be suboptimal for society, although individual intermediaries still benefit from it […] and it is well-known that diversification may be suboptimal in the extremely heavy-tailed case.

Over the past twenty years, there have been many examples where diversification does not work, and practitioners are aware of them. Fabozzi et al. (2014), discussing the financial crisis, remind us, “the financial crisis has clearly shown that when you need diversification most, it may not work.” When considering risks related to climate disasters, we see that these risks are extreme, potentially uninsurable because of potentially infinite expectation. Uninsurability mainly means that a market mechanism does not make sense without state intervention. One might also think that it could be interesting to diversify risks by offering multi-peril coverage (as proposed by the current cat-nat mechanism), or by considering geographical diversification, for example at the European level, as recently suggested by Carlo Cimbri, Thierry Derez, and Philippe Lallemand. But the scientific literature reminds us that this diversification is dangerous, in any case, unimaginable without strong and clear state intervention.

References

Biffis, E., & Chavez, E. (2014). Tail risk in commercial property insurance. Risks, 2(4), 393–410.

Charpentier, A. (2011). La loi des grands nombres et le théorème central limite comme base de l’assurabilité ? Risques, 86.

Chavez-Demoulin, V., Embrechts, P., & Nešlehová, J. (2006). Quantitative models for operational risk: extremes, dependence and aggregation. Journal of Banking & Finance, 30(10), 2635-2658.

Chen, Y., Embrechts, P., & Wang, R. (2024). An unexpected stochastic dominance: Pareto distributions, dependence, and diversification. Operations Research.

Cimbri, C., Derez, T. & Lallemand, P. (2024). Mutualisons l’assurance pour offrir aux Européens une protection à la hauteur des risques actuels ! La Tribune, 23 mai.

Daníelsson, J., Jorgensen, B. N., Samorodnitsky, G., Sarma, M., & de Vries, C. G. (2013). Fat tails, VaR and subadditivity. Journal of econometrics, 172(2), 283-291

Eling, M., & Wirfs, J. (2019). What are the actual costs of cyber risk events? European Journal of Operational Research, 272(3), 1109–1119.

Embrechts, P., Hofert, M., & Chavez-Demoulin, V. (2024). Risk Revealed: Cautionary Tales, Understanding and Communication. Cambridge University Press.

Fabozzi, F. J., Focardi, S. M., Jonas, C.: Investment Management: A Science to Teach or an Art to Learn?. CFA Institute Research Foundation (2014)

Fama, E. F. (1965). Portfolio analysis in a stable Paretian market. Management science, 11(3), 404-419.

Hagstroem, K.-G. (1925). Pareto and reinsurance. Scandinavian Actuarial Journal, 216–248

Hofert, M., & Wüthrich, M. V. (2012). Statistical review of nuclear power accidents. Asia-Pacific Journal of Risk and Insurance, 7(1).

Hsieh, P.-H. (1999). Robustness of tail index estimation. Journal of Computational and Graphical Statistics, 8(2), 318–332.

Ibragimov, R., & Walden, J. (2007). The limits of diversification when losses may be large. Journal of banking & finance, 31(8), 2551-2569.

Ibragimov, R., Jaffee, D., & Walden, J. (2011). Diversification disasters. Journal of financial economics, 99(2), 333-348.

Ibragimov, M., Ibragimov, R., & Walden, J. (2015). Heavy-tailed distributions and robustness in economics and finance (Vol. 214). Springer.

Lévy, Paul (1925). Calcul des probabilités. Paris: Gauthier-Villars.

Mandelbrot, B. (1960). The Pareto–Lévy Law and the Distribution of Income. International Economic Review. 1 (2): 79–106.

Markowitz, H. (1952). Portfolio Selection, Journal of Finance, 7 (1), 77-91.

Markowitz, H. (1971). Portfolio selection : efficient diversification of investments. Yale University Press.

Moscadelli, M. (2004). The modelling of operational risk: experience with the analysis of the data collected by the Basel committee. Technical Report 517, Banca d’Italia

Raper, S. (2018). Turning points: Bernoulli’s golden theorem. Significance, 15(4), 26-29.

Ross, S. A. (1976). A note on a paradox in portfolio theory. Unpublished Mimeo, University of Pennsylvania.

Roy, A. D. (1952). Safety first and the holding of assets. Econometrica, 431-449.

Samuelson, P. A. (1967). Efficient portfolio selection for Pareto-Lévy investments. Journal of financial and quantitative analysis, 2(2), 107-122.

Sornette, D., Knopoff, L., Kagan, Y. Y., & Vanneste, C. (1996). Rank‐ordering statistics of extreme events: Application to the distribution of large earthquakes. Journal of Geophysical Research: Solid Earth, 101(B6), 13883-13893.

Smith, M. L., & Kane, S. A. (1994). The law of large numbers and the strength of insurance. In Insurance, Risk Management, and Public Policy: Essays in Memory of Robert I. Mehr (pp. 1-27). Dordrecht: Springer Netherlands.

Zajdenweber, D. (1996). Extreme values in business interruption insurance. Journal of Risk and Insurance, 95-110.

Zajdenweber, D. (2000). Économie des extrêmes. Flammarion.

1. The case of dice is somewhat peculiar because the geometry of the cube, particularly its regularity (we refer to it as a regular hexahedron with six faces), allows us to infer the probability without any experimentation

2. The theoretical literature on probabilities is largely built on the idea of finite expectation variables, and it is very hard to do without them (making any reasoning “on average” impossible)

3. It was not until the 1970s, with the work of Guus Balkema and Laurens de Haan, that we had a mathematical proof of this result. The Dutch school of statistics made significant advances in the analysis of extreme events following the 1953 North Sea flood, which had major and disastrous consequences in the Netherlands, as recalled by Embrechts et al. (2024)

4. The sum (or average) of independent normal variables also follows a normal distribution

5. He calls these laws Pareto-Lévy to emphasize the shape of the distribution tails, corresponding to Pareto-type laws, on extreme losses (on the left) and extreme gains (on the right)