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There Is a Var Beyond Usual Approximations. Towards a Toolkit to Compute Risk Measures of Aggregated Heavy Tailed Risks

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There Is a Var Beyond Usual Approximations. Towards a Toolkit to Compute Risk Measures of Aggregated Heavy Tailed Risks There is a VaR beyond usual approximations. Towards a toolkit to compute risk measures of aggregated heavy tailed risks Marie Kratz September 2013 SWISS FINANCIAL MARKET SUPERVISORY AUTHORITY FINMA Bern - Z¨urich 1 Acknowledgement. This work has been carried out during my internship (July-December 2012) at the Swiss Financial Market Supervisory Authority (FINMA). I would like to thank FINMA for its hospitality and rich environment, and ISFA (Univ. Claude Bernard, Lyon 1) for having given me the opportunity to learn more about actuarial sciences and finance, in view of becoming an IA - actuary. A special thanks to Dr. Hansj¨orgFurrer (Head of the QRM department, FINMA) for having made this stay possible and for bringing to my attention this interesting practical issue. Thanks to all of my colleagues from the Quantitative Risk Management (QRM) - Fi- nancial Risk SST department, for making my stay at FINMA a rich, pleasant and fruitful experience. Besides the study presented in this report, it was very interesting to get the chance to participate in the review process, discussion and report of some company, to learn more about the Swiss Solvency Test (SST) through discussions with colleagues and the revision of the french translation of SST. My warm thanks also to Dr. Michel Dacorogna, for stimulating and interesting discus- sions on this study. 2 Abstract Basel II and Solvency 2 both use the Value-at-Risk (VaR) as the risk measure to compute the Capital Requirements. In practice, to calibrate the VaR, a normal approximation is often chosen for the unknown distribution of the yearly log re- turns of financial assets. This is usually justified by the use of the Central Limit Theorem (CLT), when assuming aggregation of independent and identically dis- tributed (iid) observations in the portfolio model. Such a choice of modeling, in particular using light tail distributions, has proven during the crisis of 2008/2009 to be an inadequate approximation when dealing with the presence of extreme returns; as a consequence, it leads to a gross underestimation of the risks. The main objective of our study is to obtain the most accurate evaluations of the aggregated risks distribution and risk measures when working on financial or insurance data under the presence of heavy tail and to provide practical solutions for accurately estimating high quantiles of aggregated risks. We explore new ap- proaches to handle this problem, numerically as well as theoretically, based on properties of upper order statistics. We compare them with existing methods, for instance with one based on the Generalized Central Limit Theorem. R´esum´e Les r`eglesde B^aleII et de Solvabilit´e2 sont toutes deux bas´eessur le choix de la Value-at-Risk (VaR) comme mesure de risque pour calculer le capital de solvabilit´e. En pratique, il est souvent d'usage de choisir une approximation normale pour estimer la distribution sous-jacente des log-returns annuels d'actifs financiers, en vue de l'´evaluation de la VaR. Cela est justifi´epar l'utilisation du TLC lorsqu'on suppose le mod`eledu portefeuille constitu´ede l'aggr´egation d'observations iid. Un tel choix de mod´elisation,reposant sur l'utilisation d'une distribution `aqueue dite 'l´eg`ere',s'est av´er´etotalement inappropri´elors de la crise financi`erede 2008/09 durant laquelle la pr´esencede returns extr^emesest devenue manifeste. Cette erreur de mod`elea conduit `aune sous-estimation cons´equente des risques. L'objectif principal de cette ´etudeest d'obtenir une ´evaluation de la distribution des risques aggr´eg´eset des mesures de risque associ´ees,la meilleure possible, lorsqu'on traite des donn´eesfinanci`eresou d'assurance `aqueue de distribution lourde, et de proposer des solutions pratiques pour estimer de fa¸conla plus op- timale possible les quantiles extr^emesdes risques aggr´eg´es. Nous explorons de nouvelles approches pour appr´ehenderce probl`eme,d'un point de vue th´eorique comme num´erique,bas´eessur les propri´et´esdes plus grandes statistiques d'ordre. Nous comparons ensuite ces approches avec les m´ethodes existantes, telle celle reposant sur le TLC g´en´eralis´e. Keywords: aggregated risk, (refined) Berry-Ess´eeninequality, (generalized) central limit theorem, conditional (Pareto) distribution, conditional (Pareto) moment, convo- lution, expected shortfall, extreme values, financial data, high frequency data, market risk, order statistics, Pareto distribution, rate of convergence, risk measures, stable distribution, Value-at-Risk 3 Synopsis • Motivation Toute institution financi`ere,banque ou assurance, doit g`ererun portefeuille de risques. Ces risques agr´eg´es,mod´elis´espar des variables al´eatoires,constituent la base de tout mod`eleinterne. On constate qu'il est encore tr`esfr´equent, en pratique, d'utiliser une approximation normale pour estimer la distribution (in- connue) des rendements logarithmiques annuels d'actifs financiers. Cela est jus- tifi´epar l'application du Th´eor`eme de la Limite Centrale (TLC) sous l'hypoth`ese d'observations ind´ependantes et identiquement distribu´ees(iid) `avariance finie. Un tel choix de mod´elisation,utilisant une loi `aqueue de distribution dite l´eg`ere, s'est av´er´einappropri´epour l'´evaluation de mesures de risque, car elle sous estime le risque. Deux probl`emesessentiels se posent lors de l'application du TLC lorsqu'on con- sid`erece type de donn´ees `aqueue de distribution relativement lourde (par exem- ple de type Pareto ayant un param`etrede forme sup´erieur`a2). Le premier con- cerne la qualit´ede l'approximation li´eeaux moments de la variable parente, le sec- ond la pertinence d'une m´ethode graphique pour d´ecelerune distribution `aqueue ´epaisselorsqu'on agr`egeles donn´eeset que l'on obtient ainsi des ´echantillons de taille plus r´eduite. ,! Revenons sur la 1`erequestion soulev´ee. Il ne faut pas perdre de vue que le TLC est un th´eor`emepermettant d'´evaluer le comportement moyen d'un ph´enom`eneet non celui des extr^emes(localis´esen queue de distribution). Il a ´et´ed´emontr´e,depuis les ´etudesdes ann´ees80, en particulier celle de Hall ([27]), qu'enlever les extr^emesde l'´echantillon peut am´eliorerla vitesse de convergence (variance plus faible) de la moyenne lors de l'application du TLC. De plus, m^emesi l'on ne s'int´eressaitqu'au comportement moyen, on sait qu'il s'agit d'un th´eor`emeasymptotique fonction de la taille de l'´echantillon et que l'on peut obtenir une mauvaise approximation pour des ´echantillons de taille r´eduite. Pour am´eliorerla qualit´ede l'approximation, l'existence de moments d'ordre sup´erieur`a2 s'av`eren´ecessaire,comme l'atteste le d´eveloppement de Edgeworth. ,! Concernant le second probl`emeli´e`al'agr´egation, notons que l'existence d'une distribution sous-jacente `aqueue lourde peut se d´ecelerpar des m´ethodes empiriques/graphiques, telles le QQ-plot, - clairement sur des donn´ees`ahaute fr´equence (par exemple des donn´ees journali`eres) - et non plus sur des donn´eesagr´eg´ees,par exemple des donn´eesannuelles (´echantillons de petite taille) 4 alors qu'il est bien connu, depuis le th´eor`emede Fisher, que l'indice de queue de la distribution sous-jacente reste invariant apr`esagr´egation.C'est un ph´enom`enesur lequel ont insist´ebien des auteurs (cf. par exemple [13]). Ce second probl`emes'illustre tr`esbien par les QQ-plots obtenus pour les ren- dements S&P 500 figurant dans la section Introduction. En effet, alors que le QQ-plot effectu´esur des donn´eesjournali`eresde 1987 `a2007 permet de d´etecter une distribution `aqueue lourde, il n'en est pas de m^emelorsqu'on agr`egeces donn´eesmensuellement. Dans ce cas, le QQ-plot appara^ıtplut^otcomme normal, et inciterait alors `aconsid´ererles crises financi`eresde 1998 et 1987 comme des valeurs aberrantes. M^emeen consid´erant les donn´eesagr´eg´eesmensuellement sur une p´eriode un peu plus longue, `asavoir de 1987 `a2013, le QQ-plot reste plus ou moins inchang´e, i.e. normal, sauf que cette fois un autre point 'aberrant' appara^ıt, avec la date de ... Octobre 2008 ! Sans revenir aux donn´eesjournali`eresmais en augmentant sensiblement la taille de l'´echantillon, avec des donn´eesmensuelles de 1791 `a2013, il devient `anouveau manifeste que la distribution sous-jacente est `aqueue lourde. La crise financi`erede 2008 est un ´ev´enement apparaissant dans la queue de distribution et ne peut alors plus ^etre´ecart´ede l'analyse en tant que point aberrant. Ces diff´erentes figures illustrent bien l'importance de la taille d'´echantillon, lorsque nous devons ´evaluer des risques en pr´esencede distri- bution `aqueue relativement lourde. Il s'av`eredonc fondamental de proposer une m´ethode qui ne d´epende pas de la taille de l'´echantillon pour d´ecelerla forme de la queue de distribution. • Objectif La recherche de telle(s) approche(s) afin d'obtenir les ´evaluations les plus fines possibles des mesures de risque lorsque nous analysons des donn´eesfinanci`eres en pr´esencede queues de distribution ´epaissesconstitue notre principal objectif. Nous allons explorer diverses m´ethodes, existantes et nouvelles, pour r´esoudrece probl`eme,th´eoriquement et num´eriquement. Ayant en vue des applications financi`eres ou actuarielles, nous utiliserons des mod`eles de loi puissance pour les marginales des risques, telle la loi de Pareto de param`etre α d´efiniepar F (x) := 1 − F (x) = x−α; α > 0; x ≥ 1 (1) Nous d´efinironsles rendements selon (n) Xi := ln Pi − ln Pi−n; n ≥ 1 o`u Pi est le prix journalier et n le facteur d'agr´egation. Notons que nous pouvons aussi ´ecrire n (n) X (1) Xi = Xi i=1 5 (1) Nous simplifierons la notation de Xi en Xi et consid´ereronsalors un n-´echantillon (Xi; i = 1; : : : ; n) de variable parente X suivant une loi de Pareto, de statistiques d'ordre associ´ees X(1) 6 ··· 6 X(n).
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