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)

Peut-on diversifier des risques extrêmes ?

Dans un contexte financier, diversifier les risques signifie investir dans une variété d’actifs, secteurs ou régions géographiques pour éviter que la mauvaise performance d’un investissement n’affecte trop l’ensemble du portefeuille. La diversification permet de réduire le risque, ou, dans sa formulation mathématique, de réduire la variance. Mais que se passe-t-il quand on est en présence de grands risques, de variance infinie ? Ou pire encore, d’espérance infinie ?

Risques extrêmes, et espérance infinie ?

Formaliser des grandeurs en lien avec des quantités aléatoires et incertaines est un exercice compliqué. Les probabilités, au sens où le mot est souvent entendu, sont définies comme des limites de fréquences observées par répétitions d’événements. La probabilité d’avoir 3 en lançant un dé est ⅙ car en lançant un dé un million de fois [1], un milliard de fois, la probabilité sera aussi proche qu’on veut de ⅙. C’est ce que dit la loi des grands nombres, dans sa version la plus faible. Dire que la probabilité qu’il pleuvra aujourd’hui est de ⅙ est totalement différent, car c’est un évènement unique. Si je me fais tremper par une averse aujourd’hui, cela ne permettra aucunement de dire que la probabilité n’était pas, a priori, de ⅙, et que le modèle météorologie. Tout ça pour rappeler que lorsqu’on fait de la modélisation, on va essayer d’imaginer les valeurs petites d’évènements rares, et qu’il est malheureusement très difficile de les valider.

Et quand on modélise les grands risques, les très grands risques, il n’est pas rare d’avancer l’idée que les risques sont de variance ou d’espérance infinie. Or la notion d’espérance infinie est un à la fois étrange, et probablement contre-intuitive [2].. Si on considère une variable aléatoire $X$ positive (pour simplifier), et si on note $S(x)=\mathbb{P}(X>x)$ la fonction de survie, et $f(x)$ la densité (correspondant à l’opposé de la dérivée de $S$), on peut montrer que la moyenne empirique d’un million ou d’un milliards de tirage de cette variable va s’approcher autant qu’on veut d’une grandeur, appelée l’espérance mathématique$$\mathbb{E}(X) =\int_0^\infty S(x)dx= \int_0^\infty xf(x)dx$$Rien de bien étonnant ici, c’est encore la loi des grands nombres, énoncée dès 1713 par Jacob Bernoulli (le “golden theorem” de Raper (2018)) et surtout Pierre-Simon Laplace, en 1814. A condition toutefois que cette intégrale soit finie. Ce qui n’est pas garanti. Par exemple, la loi de Pareto d’indice $a$ vérifie $S(x)=\mathbb{P}(X>x)=x^{-a}$. Dès 1925, Karl-Gustaf Hagstroem avait noté que cette loi semblait particulièrement adaptée à la modélisation des grands risques, et donc en réassurance [3]. Et pour une variable qui suit une loi de Pareto d’indice 1, son espérance est, mathématiquement, infinie.

Que signifie cette espérance infinie ? Il n’y aura aucun “sinistre de coût infini”, et il sera toujours possible de calculer une moyenne empirique sur n observations. Mais cette moyenne va tendre vers l’infini quand n croit. Louis Bachelier, en évoquant le paradoxe de Saint Pétersbourg (qui est un jeu dont l’espérance de gain est infinie), rappelle que “un résultat paradoxal, dans les sciences mathématiques, provient nécessairement d’un défaut de notre intelligence inhabile à déchiffrer un ensemble trop complexe, inapte à se représenter l’infiniment grand (…) le bon sens ne peut être invoqué lorsqu’il s’agit de questions délicates ; il ne permet pas de reconnaître si l’aire comprise entre une courbe et son asymptote est finie ou non, si une série est convergente ou divergente”. Cette moyenne va tendre vers l’infini quand $n$ croit, c’est-à-dire qu’on peut être certain que la moyenne va toujours dépasser n’importe quelle valeur aussi grande qu’on puisse l’imaginer. On peut le visualiser en haut de la Figure 1 avec 10 simulations de 100,000 valeurs. À gauche, le cas où la variance et l’espérance sont finies; au centre, la variance est infinie, et l’espérance finie; à droite, les deux sont infinies.

Figure 1: Évolution de la moyenne $n\mapsto (x_1+\cdots+x_n)/n$ avec une loi d’espérance finie et de variance infinie à gauche, et une loi d’espérance infinie à droite.

Une autre grandeur intéressante est le ratio du maximum sur $n$ observations sur la somme. Pour des variables d’espérance infinie, ce ratio ne tend pas vers 0. Il est alors possible, si les variables x désignent les coûts de sinistres, avec 100,000 sinistres d’espérance infinie, il est possible que le plus gros des sinistres représente plus de 90% de la charge totale.

Figure 2 : Évolution du ratio $n\mapsto \max\{x_1,\cdots,x_n\}/(x_1+\cdots+x_n)$ avec une loi d’espérance finie et de variance infinie à gauche, et une loi d’espérance infinie à droite.

On le voit, cette propriété est importante, mais elle est difficile à identifier car il s’agit d’une propriété fondamentale du modèle sous-jacent, en lien avec la distribution des observations, puisqu’il est toujours possible de calculer la moyenne. Par exemple, la suite suivante correspond à huit valeurs obtenues en tirant au hasard une loi de Pareto d’indice 1 (et donc théoriquement d’espérance infinie)

1.657442 || 4.138543 || 15.592108 || 1.429090

1.684843 || 1.186745 || 1.341435 || 3.308316

Comment savoir si un ensemble de coûts de sinistres suit un loi d’espérance finie, ou pas ? L’approche classique, présentée par exemple dans Zajdenweber (1996, 2000), consiste à utiliser le graphique dit de Pareto, avec le logarithme des coûts sur l’axe des abscisses, et le logarithme de la probabilité de survie en ordonnées. Si les points sont alignés suivant une droite de pente -a, alors la loi de Pareto de paramètre a est parfaitement adaptée. En effet, si $\mathbb{P}(X>x)=x^{-a}$, alors, en prenant le logarithme des deux grandeurs, et si on ordonne l’échantillon ($x_1\leq x_2\leq\cdots\leq x_n$), alors $$\log\left(\frac{n-i}{n}\right)=-a\cdot \log(x_i)$$Et si la pente est trop modérée, plus grande que -1, alors les coûts sont d’espérance infinie.

Figure 3: Graphique de Pareto, avec $\log((n-i)/n)$ sur l’axe des ordonnées et $\log(x_i)$ sur l’axe des abscisses. Les points sont alignés suivant une droite, de pente -a, correspondant à une loi de Pareto d’indice $a$. $a\leq 1$ signifie que les risques sont d’espérance infinie.

Cette hypothèse d’indice de Pareto proche de 1 n’est pas irréaliste quand on parle de catastrophes naturelles, ou industrielles,

• ouragans, Hsieh (1999), $a\sim1.5$
• incendies entreprises, Biffis et al. (2014), $a\sim1.25$
• tremblement de terre, Sornette et al. (1996), $a\sim1$
• tsunamis, Embrechts et al. (2024), $a\sim1$
• risque opérationnel, Moscadelli (2004) et Chavez-Demoulin et al. (2006) $a\sim1$
• risque cyber, Eling et al. (2019) $a\sim1$
• risque nucléaires, Hofert et al. (2012), $a\in(0.6;0.7)$

De la diversification des grands risques

Au lieu de travailler par type de risque, on peut envisager l’agrégation de ces risques, tous ensemble. Heuristiquement, avoir des portefeuilles avec des risques d’inondation, de tremblement de terre, ou de sécheresse pourrait offrir un peu de “diversification”. Le concept de “diversification” peut être introduit avec la loi des grands nombres, dont on parlait auparavant, et il sera très proche de l’idée même d’assurance, de mutualisation des risques. Smith & Kane (1994), par exemple, rappellent que la contribution d’un $n+1$-ième risque, indépendant, dans un groupe de $n$ risques, tarifés de manière actuariellement juste, permet généralement de faire marginalement diminuer le risque, ce qui renforce la mutualisation des risques par l’assureur. Cet effet de diversification marche encore, même si les risques sont corrélés (mais pas parfaitement corrélés, et les gains de diversification diminuent avec la corrélation, comme le rappelait Charpentier (2011)).

Mais bien souvent, quand on parle de “diversification” on pense aux travaux d’Harry Markowitz ou d’Arthur Roy en finance dans les années 1950, comme base de la théorie du portefeuille. Cette théorie montre comment des investisseurs rationnels peuvent utiliser la diversification, correspondant à la corrélation entre actifs, afin d’optimiser leur portefeuille financier. Dans cette approche, on suppose généralement que la préférence des investisseurs pour un couple risque / rendement peut être décrite par une fonction d’utilité quadratique. Autrement dit, seuls le rendement attendu (l’espérance de gain) et la volatilité (l’écart type) ou la variance, sont les paramètres considérés par l’investisseur. Ce que montre cette littérature, c’est qu’un investisseur peut réduire le risque de son portefeuille simplement en détenant des actifs qui ne sont pas (ou peu) corrélés, donc en diversifiant ses placements. Il peut alors obtenir la même espérance de rendement en diminuant la variabilité de son portefeuille.

Mais que se passe-t-il si la variance n’existe plus ? Cette question revient à questionner l’utilisation de la loi normale pour modéliser les rendements financiers. La loi normale était intéressante en partie parce qu’elle vérifie une propriété de stabilité par sommation[4]. Garder cette propriété tout en considérant une loi ayant davantage d’extrêmes que la loi normale, revient à utiliser les lois “stables” étudiées par Paul Lévy, comme le proposait[5] Benoit Mandelbrot dans les années 1960.

Dans le cas où la variance est infinie, il convient d’utiliser une mesure de risque plus générale que l’écart-type, et heuristiquement, la “diversification” est liée à la sous-additivité de la mesure de risque : un portefeuille contenant la moyenne des avoirs de deux autres portefeuilles a un risque plus faible de la moyenne des risques des deux autres portefeuilles. Daníelsson et al. (2013) rappellent qu’en présence de grands risques (d’espérance infinie), la diversification ne fonctionne plus. Cette propriété avait été décrite et discutée par Paul Samuelson dès 1967, Stephen Ross en 1976, ou plus récemment Rustam Ibragimov,  Dwight Jaffee, Johan Walden, Paul Embrech ou Ruodo Wang, entre autres. L’introduction de Ibragimov et al. (2015) l’explique bien, “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 ”. Ces propriétés, largement discutées d’un point de vue mathématiques, sont compliquées à faire admettre car elles sont théoriques, et contre-intuitives. De plus, il est souvent difficile de savoir pour qui la diversification devient dangereuse, puisqu’il y a plusieurs acteurs, les assurés, les assureurs, les réassureurs, l’état. Ibragimov et al. (2011) donnent des éléments de réponse, “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”.

Depuis une vingtaine d’années, les exemples où la diversification ne fonctionne pas sont nombreux, et connus des praticiens. Fabozzi et al. (2014), évoquant la crise financière le rappellent “The financial crisis has clearly shown that when you need diversification most, it may not work.” Quand on s’intéresse aux risques liés aux catastrophes climatiques, on voit que ces risques sont extrêmes, potentiellement inassurables car d’espérance potentiellement infinie. L’inassurabilité signifie surtout qu’un mécanisme de marché n’a pas de sens, sans intervention de l’État. On pourrait aussi penser qu’il pourrait être intéressant de diversifier les risques, en offrant une couverture multi-périls (comme le propose le mécanisme cat-nat actuel), ou bien en envisageant une diversification géographique, par exemple au niveau européen, comme le suggéraient récemment Carlo Cimbri, Thierry Derez et Philippe Lallemand. Mais la littérature scientifique nous rappelle que cette diversification est dangereuse, en tous cas inenvisageable sans une intervention forte et claire des États.

Références

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.

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1. Le cas des dés est un peu particulier car la géométrie du cube, en particularité sa régularité (on parle d’hexaèdre régulier, à 6 faces), permet d’inférer la probabilité sans faire la moindre expérience

2. La littérature théorique des probabilités s’est largement construite sur l’idée de variables d’espérances finies, et il est très dur de s’en passer (tout raisonnement “en moyenne” devenant impossible)

3. Il faudra attendre les années 1970, et les travaux de Guus Balkema ou Laurens de Haan, pour avoir une preuve mathématique de ce résultat. L’école néerlandais de statistique a fait des avancées majeures sur l’analyse des évènements extrêmes suite au raz-de-marée de 1953 en mer du Nord, qui a eu des conséquences majeures et désastreuses aux Pays-Bas, comme le rappellent Embrechts et al. (2024)

4. La somme (ou la moyenne) de variables normales indépendantes suit aussi une loi normale

5. Il appelle ces lois Pareto-Lévy pour souligner la forme des queues de distributions, correspondant à des lois de type Pareto, sur les pertes extrêmes (à gauche) et les gains extrêmes (à droite)

Prix des Sciences du Risque de la Fondation Optimind – Accenture

Bravo à Franklin Feukam Kouhoue, qui a obtenu le prix des sciences du risques pour son mémoire de maîtrise en actuariat, à l’ENSAE, sur le thème de l’interprétabilié des modèles d’actuariat en tarification, sous la supervision de Laurence Barry, et moi-même.

Talk in Stockholm, Sweden, at the Insurance Data Science Conference

This week, I will attend the Insurance Data Science conference in Sweeden. It has been a while… I was a keynote speaker at the one in London, ten years ago (to give a talk I still have feedbacks about – Getting into Bayesian Wizardry… (with the eyes of a muggle actuary) – by that time, the conference was “R in Insurance”), and then, we organized the one in Paris, back in 2017. Then we had the online events, but it was… different.

This time, I will get back to our recent paper A Sequentially Fair Mechanism for Multiple Sensitive Attributes, with François Hu and Philipp Ratz, and the equipy package, wrote with Agathe Fernandes-Machado and Suzie Grondin. The slides are available online.

Talk in København

Tomorrow, I will give a talk at København Universiteit, on Using optimal transport to quantify and mitigate unfair insurance predictions. It is based on recent work with François Hu and Philipp Ratz (2310.20508, 2309.06627, 2306.12912 and 2306.10155).

The insurance industry is heavily reliant on predictions of risks based on characteristics of potential customers. Although the use of said models is common, researchers have long pointed out that such practices perpetuate discrimination based on sensitive features such as gender or race. Given that such discrimination can often be attributed to historical data biases, an elimination or at least mitigation is desirable. With the shift from more traditional models to machine-learning based predictions, calls for greater mitigation have grown anew, as simply excluding sensitive variables in the pricing process can be shown to be ineffective. In this talk, we first investigate why predictions are a necessity within the industry and why correcting biases is not as straightforward as simply identifying a sensitive variable. We then propose to ease the biases through the use of Wasserstein barycenters instead of simple scaling. To demonstrate the effects and effectiveness of the approach we employ it on real data and discuss its implications. The talk will be based on a recent textbook (978-3-031-49782-7) as well as work with François Hu and Philipp Ratz (2310.20508, 2309.06627, 2306.12912 and 2306.10155).

From Contemplative to Predictive Modeling

As mentioned yesterday, I gave a talk, this afternoon, entitled From Contemplative to Predictive Modeling (in actuarial science and risk management). Slides are available online, but maybe I can take some time to explain what I talked about…

It is usually claimed that actuaries build ‘predictive models’ but most of the time, what they consider would be simply ‘contemplative modeling’, in the sense that they use past information and hope that the future will be more or less the same (corresponding to the idea of generalization in machine learning). In the context of climate change (but also when modeling insurance market competition) it is not the case, data used to train models do not have the same distribution as the one we will have in the future.

SCOR Foundation – Scope and limits of Artificial intelligence

On May 15, 2024, the SCOR Foundation for Science hosted a webinar titled “Scope and limits of Artificial intelligence”, delivered by Arthur Charpentier. A professor in the Department of Mathematics at the University of Quebec in Montreal and a member of the Institute of Actuaries, Arthur Charpentier is an internationally recognized expert in actuarial science and the author of numerous academic articles published in the best actuarial academic journals worldwide.

During the webinar, Arthur Charpentier discussed the research project “Fairness of predictive models: an application to insurance markets”, which is supported by the SCOR Foundation for Science. This project addresses biases within the automatic artificial intelligence algorithms utilized to determine optimal pricing in individual policies. Its aim is to mitigate or eliminate such biases, which could lead to inequities or discriminatory practices based on factors such as gender, race, religion, or origin in the coverage provided by insurers or reinsurers to policyholders.

Trip in (Northern) Europe

The next two weeks, in will be in (Northern) Europe, with a first stop in Brussels (to visit colleagues), then in Leuven (I will give a talk on Monday at KU Leuven), then in København (I will give a talk on Friday at Københavns Universitet), and finally in Stockholm (at Stockholm University, for the Insurance Data Science conference).

In the Fall, I will be in Europe, with Lisbon (European Actuarial Journal conference), in France (Cerisy Colloques) and in Warsaw in Poland. In Poland, I will give a two day cours on Insurance, Biases, Discrimination and Fairness

More to come soon…

Quantifying Fairness and Discrimination in Predictive Models

The article Quantifying Fairness and Discrimination in Predictive Models was just published in Machine Learning for Econometrics and Related Topics, Springer.

The analysis of discrimination has long interested economists and lawyers. In recent years, the literature in computer science and machine learning has become interested in the subject, offering an interesting re-reading of the topic. These questions are the consequences of numerous criticisms of algorithms used to translate texts or to identify people in images. With the arrival of massive data, and the use of increasingly opaque algorithms, it is not surprising to have discriminatory algorithms, because it has become easy to have a proxy of a sensitive variable, by enriching the data indefinitely. According to [69], “technology is neither good nor bad, nor is it neutral”, and therefore, “machine learning won’t give you anything like gender neutrality ‘for free’ that you didn’t explicitely ask for”, as claimed by [61]. In this article, we will come back to the general context, for predictive models in classification. We will present the main concepts of fairness, called group fairness, based on independence between the sensitive variable and the prediction, possibly conditioned on this or that information. We will finish by going further, by presenting the concepts of individual fairness. Finally, we will see how to correct a potential discrimination, in order to guarantee that a model is more ethical.

Assurabilité, vers de nouveaux partages de risque, Congrès des Actuaires

Jeudi, je vais participer, à distance, au 23e congrès de l’Institut des Actuaires, en France, avec Florence Picard et Laurence Barry.

Notre exposé a pour titre “assurabilité : vers de nouveaux partages de risques?”. Je parlerais un peu des catastrophes naturelles… et du risque de sécheresse, ou plutôt du risque “RGA“.

Workshop on Trustworthy AI, in Montreal

This Monday, a Workshop on Trustworthy AI will be held May 27, 2024 in Montreal.

We will be there with Agathe and Olivier, to chat with people who might have some interest

Here are our posters. I wil talk about discrimination and insurance

Agathe will explain why callibration of scores is important,

and finally, Olivier will talk about building (causal) graphs for fairness

Conference on New Developments in Probability, in September

The next iteration of the Conference on New Developments in Probability (CNDP) will take place from September 26-28, 2024 [Thursday morning-Saturday noon] in Montreal, Quebec, Canada. The event will be hosted by Women in Probability and the Centre de Recherches Mathématiques (CRM)…

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