I will be in Chicago next week, invited by 冯润桓 (Runhuan Feng). On Monday, with Jan Dhaene and Marco Mirabella, we will give some short courses “on decentralized insurance and risk sharing“.
Then, we will attend the IME Congress 2024 (that will be held between July 9th and 11th).
In September, I will be in Warsaw, in Poland, to give a course on Insurance, Biases, Discrimination and Fairness (based on the textbook recently published). More to come after the summer break
Notre article, avec Ewen Gallic, “Croissance, décroissance, de quoi parle-t-on ?” a été remis en forme pour être publié dans le prochain numéro de la revue (du magazine ?) Risques.
After defending his PhD last week, Samuel just arrived in 横浜市, at the International Joint Conference on Neural Networks (IJCNN’24), that will take place at the IEEE World Congress on Computational Intelligence (WCCI).
He will present our recent work on Boarding for ISS: Imbalanced Self-Supervised Discovery of a Scaled Autoencoder for Mixed Tabular Datasets,
The field of imbalanced self-supervised learning, especially in the context of tabular data, has not been extensively studied. Existing research has predominantly focused on image datasets. This paper aims to fill this gap by examining the specific challenges posed by data imbalance in self-supervised learning in the domain of tabular data, with a primary focus on autoencoders. Autoencoders are widely employed for learning and constructing a new representation of a dataset, particularly for dimensionality reduction. They are also often used for generative model learning, as seen in variational autoencoders. When dealing with mixed tabular data, qualitative variables are often encoded using a one-hot encoder with a standard loss function (MSE or Cross Entropy). In this paper, we analyze the drawbacks of this approach, especially when categorical variables are imbalanced. We propose a novel metric to balance learning: a Multi-Supervised Balanced MSE. This approach reduces the reconstruction error by balancing the influence of variables. Finally, we empirically demonstrate that this new metric, compared to the standard MSE: i) outperforms when the dataset is imbalanced, especially when the learning process is insufficient, and ii) provides similar results in the opposite case.
This morning (Montréal time), Samuel Stocksieker defended his PhD thesis entitled “contribution of machine learning in modeling rare values and imbalanced data“. Cécile Capponi, Marianne Clausel, Julie Josse, Frédéric Planchet and Anne Sabourin, Christian-Yann Robert and Stéphane Loisel were in the jury,
the work is structured around major axes: Imbalanced Features and Imbalanced Regression. The first axis addresses the issue of feature imbalance, that is, when it concerns the attributes and not the variable to be explained. The first solution involves adjusting the distribution of a continuous covariate relative to a given target distribution. It proposes to combine weighted resampling and synthetic data generators. This strategy notably allows to deal with selection bias: when the distribution of the covariate in the training sample is significantly different from that of the population. A second solution is proposed in the context of multi-supervised learning, particularly with autoencoders. It relies on a new metric aimed at balancing the influence of variables during learning and is applicable not only to supervised and unsupervised models, but also to generative models such as variational autoencoders. The second part deals with the issue of regression from imbalanced data. Various preprocessing solutions, including synthetic data generation, are proposed. Initially, we propose to explore the initial data space by introducing new generators and methodologies to address the specific case of regression. We then propose to immerse the data in a latent space in order to provide a more conducive framework for synthetic data generation.
Le dernier numéro (137) de la revue Risques est sorti (il y a plusieurs semaines, certes, mais je viens tout juste de le recevoir par la poste)
On y avait écrit un petit article avec Laurence Barry.
Jeudi, je vais donner un exposé (et animer un atelier) au centre de recherche sur l’intelligence2 en gestion de systèmes complexes (CRI2GS), à l’UQAM, que je rejoins tout juste.
J’ai mis quelques slides en ligne.
Our paper, Generalized Oversampling for Learning from Imbalanced datasets and Associated Theory: Application in Regression, written with Samuel Stocksieker and Denys Pommeret, has been accepted for publication in TMLR (Transactions on Machine Learning Research)
In supervised learning, it is quite frequent to be confronted with real imbalanced datasets. This situation leads to a learning difficulty for standard algorithms. Research and solutions in imbalanced learning have mainly focused on classification tasks. Despite its importance, very few solutions exist for imbalanced regression. In this paper, we propose a data augmentation procedure, the GOLIATH algorithm, based on kernel density estimates and especially dedicated to the problem of imbalanced data. This general approach encompasses two large families of synthetic oversampling: those based on perturbations, such as Gaussian Noise, and those based on interpolations, such as SMOTE. It also provides an explicit form of such machine learning algorithms. New synthetic data generators are deduced. We apply GOLIATH in imbalanced regression combining such generator procedures with a new wild-bootstrap resampling technique for the target values. We evaluate the performance of the GOLIATH algorithm in imbalanced regression where we compare our approach with state-of-the-art techniques.
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?
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)dxThere 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:
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) ↑
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 ?
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 météorologique s’est trompé. 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)dxRien 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,
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.
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)
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Ibragimov, R., Jaffee, D., & Walden, J. (2011). Diversification disasters. Journal of financial economics, 99(2), 333-348.
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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) ↑
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.
Great time at the IDSC’24, Insurance Data Science Conference, in Stockholm, those two days…
I am glad to see so many people using the datasets of the CASdatasets package… good news, we are working with Christophe Dutang, Julien Siharath and Ewen Gallic this summer to enrich it, with fresh and new data, and with vignettes ! more about it this Fall !
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.
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).
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.
