{"id":10075,"date":"2023-02-01T21:51:53","date_gmt":"2023-02-01T20:51:53","guid":{"rendered":"https:\/\/www.simtrade.fr\/blog_simtrade\/?p=10075"},"modified":"2026-02-17T11:15:32","modified_gmt":"2026-02-17T11:15:32","slug":"mesures-risques","status":"publish","type":"post","link":"https:\/\/www.simtrade.fr\/blog_simtrade\/mesures-risques\/","title":{"rendered":"Mesures de risques"},"content":{"rendered":"

\"Shengyu<\/a><\/p>\n

Dans cet article, Shengyu ZHENG<\/a> (ESSEC Business School, Grande Ecole<\/i> Program – Master in Management, 2020-2023) pr\u00e9sente les mesures de risques bas\u00e9es sur la distribution statistique des rentabilit\u00e9s d’une position de march\u00e9, ce qui est une approche possible pour mesurer les risques (comme expliqu\u00e9 dans mon article Cat\u00e9gorie de mesures de risques<\/a>).<\/p>\n

Les mesures de risques bas\u00e9es sur la distribution statistique sont des outils largement utilis\u00e9s pour la gestion des risques par de nombreux de participants du march\u00e9, dont les traders, les teneurs de march\u00e9, les gestionnaires d\u2019actifs, les assureurs, les institutions r\u00e9glementaires et les investisseurs.<\/p>\n

Ecart-type \/ Variance<\/h2>\n

La variance (moment d\u2019ordre deux de la distribution statistique) est une mesure de la dispersion des valeurs par rapport \u00e0 la moyenne. La variance est d\u00e9finie par<\/p>\n

Var(X) = \u03c3 2<\/sup> = \ud835\udd3c[(X-\u03bc)2<\/sup>]<\/p>\n

Par construction, la variance est toujours positive (ou nulle pour une variable al\u00e9atoire constante).<\/p>\n

En finance, l’\u00e9cart-type (racine carr\u00e9e de la variance) mesure la volatilit\u00e9 des actifs financiers. Un \u00e9cart-type (ou une variance \u00e9lev\u00e9e) indique une dispersion plus importante, et donc un risque plus important, ce n\u2019est pas appr\u00e9ci\u00e9 par les investisseurs qui ont de l\u2019aversion au risque. L’\u00e9cart-type (ou la variance) est un param\u00e8tre clef dans la th\u00e9orie moderne du portefeuille de Markowitz.<\/p>\n

La variance a un estimateur non biais\u00e9 donn\u00e9 par<\/p>\n

\u015c2<\/sup> = (\u2211n<\/sup>i=1<\/sub>(xi<\/sub> – X\u0304)2<\/sup>)\/(n-1)<\/p>\n

Value at Risque (VaR)<\/h2>\n

La Value at Risque (VaR, parfois traduite comme valeur en enjeu) est une notion classique pour mesurer les risques de perte d\u2019un actif. Elle correspond au montant de perte d\u2019une position qui ne devrait \u00eatre d\u00e9pass\u00e9 qu\u2019avec une probabilit\u00e9 donn\u00e9e sur un horizon pr\u00e9cis\u00e9, ou autrement dit, au montant de la pire perte attendue sur un horizon de temps pour un certain niveau de confiance. Elle est essentiellement le quantile de la probabilit\u00e9 donn\u00e9e de la distribution de perte (rendement n\u00e9gatif).<\/p>\n

Dans le langage math\u00e9matique, la VaR est d\u00e9finie comme :<\/p>\n

VaR\u03b1<\/sub> = inf{y \u2208 : \u2119[L>y] \u2264 1 – \u03b1} = inf{ y \u2208 : \u2119[L \u2264 y] \u2265 \u03b1 }<\/p>\n

VaR\u03b1<\/sub> = q\u03b1<\/sub>(F) \u2254 F\u2190<\/sup>(\u03b1)<\/p>\n

\u03b1 est la probabilit\u00e9 donn\u00e9e ; L est une variable al\u00e9atoire de montant de perte ; F est la distribution cumulative de perte (rendement n\u00e9gatif), ce qui est continue et strictement croissante ; F\u2190<\/sup> est l\u2019inverse de F.<\/p>\n

Les organismes financiers se servent assez souvent de cette mesure pour la rapidit\u00e9 et la simplicit\u00e9 des calculs. Toutefois, elle pr\u00e9sente certaines lacunes. Elle n\u2019est pas une mesure coh\u00e9rente. Cela dit, l\u2019addition des VaRs de 2 portefeuilles aurait aucun sens. \u00c0 part cela, bas\u00e9e sur une hypoth\u00e8se gaussienne, elle ne tient pas compte de la gravit\u00e9 et la possibilit\u00e9 des \u00e9v\u00e8nements extr\u00eames, tant que les distributions du march\u00e9 financier sont, pour la plupart, leptokurtiques.<\/p>\n

Expected Shortfall (ES)<\/h2>\n

L\u2019Expected shortfall (ES) est la perte esp\u00e9r\u00e9e pendant N jours conditionnellement au fait de se situer dans la queue (1 – \u03b1) de la distribution des gains ou des pertes (N est l\u2019horizon temporel et \u03b1 est le niveau de confiance). Autrement dit, elle est la moyenne des pertes lors d\u2019un choc qui est pire que \u03b1% cas. L\u2019ES est donc toujours sup\u00e9rieure \u00e0 la VaR. Elle est souvent appel\u00e9e VaR conditionnelle (CVaR).<\/p>\n

ES\u03b1<\/sub> = \u222b 1<\/sup>\u03b1<\/sub> (VaR\u03b2<\/sub>(L) d\u03b2)\/(1 – \u03b1)<\/p>\n

En comparaison de la VaR, ES est capable de montrer la gravit\u00e9 de perte dans des cas extr\u00eames. Ce point est primordial pour la gestion moderne de risques qui souligne la r\u00e9silience surtout en cas d\u2019extr\u00eame.<\/p>\n

La VaR a \u00e9t\u00e9 pr\u00e9f\u00e9r\u00e9e par les participants du march\u00e9 financier depuis longtemps, mais les d\u00e9fauts importants pr\u00e9sent\u00e9s ci-dessus ont occasionn\u00e9 des reproches, notamment face aux souvenances des crises majeures. L\u2019ES, rendant compte des \u00e9v\u00e8nements extr\u00eames, tend d\u00e9sormais \u00e0 s\u2019imposer.<\/p>\n

Stress Value (SV)<\/h2>\n

La Stress Value (SV) est un concept similaire \u00e0 la VaR. Comme la VaR, la SV est d\u00e9finie comme un quantile. Pour la SV, la probabilit\u00e9 associ\u00e9e au quantile est proche de 1 (par exemple, un quantile de 99.5% pour la SV, en comparaison d\u2019un quantile de 95% pour la VaR habituelle). La SV d\u00e9crit plus pr\u00e9cis\u00e9ment les pertes extr\u00eames.<\/p>\n

L’estimation param\u00e9trique de SV normalement s\u2019appuie sur la th\u00e9orie de valeurs extr\u00eames (EVT), alors que celle de VaR est bas\u00e9e sur une distribution gaussienne.<\/p>\n

Programme R pour calculer les mesures de risques<\/h2>\n

Vous pouvez t\u00e9l\u00e9charger ci-dessous un programme R qui permet de calculer les mesures de risques d’une position de march\u00e9 (construite \u00e0 partir d’indices d’actions ou d’autres actifs).<\/p>\n

\"Mesures_de_risque\"<\/a><\/p>\n

Voici est une liste des symboles d’actif (“tickers”<\/i>) que nous pouvons int\u00e9grer dans le programme R.
\n\"Download<\/a><\/p>\n

Example de calcul des mesures de risque de l\u2019indice S&P 500<\/h2>\n

Ce programme nous permet de calculer rapidement des mesures de risque pour des actifs financiers dont les donn\u00e9es historiques peuvent \u00eatre t\u00e9l\u00e9charg\u00e9es sur le site Yahoo! Finance. Je vous pr\u00e9sente une analyse de risque pour l\u2019indice S&P 500.<\/p>\n

En saisissant la date de d\u00e9but comme 01\/01\/2012 et la date d\u2019arr\u00eat\u00e9 comme 01\/01\/2022, ce programme est en mesure de calculer les mesures de risque pour toute la p\u00e9riode consid\u00e9r\u00e9e.<\/p>\n

Vous trouverez ci-dessous les mesures de risque calcul\u00e9es pour toute la p\u00e9riode : la volatilit\u00e9 historique, la volatilit\u00e9 conditionnelle sur les 3 derniers mois, VaR, ES et SV.<\/p>\n

\"risk<\/p>\n

Autres articles sur le blog SimTrade<\/h2>\n

\u25b6 Shengyu ZHENG Cat\u00e9gories de mesures de risques<\/a><\/p>\n

\u25b6 Shengyu ZHENG Moments de la distribution<\/a><\/p>\n

\u25b6 Shengyu ZHENG Extreme Value Theory: the Block-Maxima approach and the Peak-Over-Threshold approach<\/a><\/p>\n

\u25b6 Youssef LOURAOUI Markowitz Modern Portfolio Theory<\/a><\/p>\n

Ressources<\/h2>\n

Articles acad\u00e9miques<\/h3>\n

Merton R.C. (1980) On estimating the expected return on the market: An exploratory investigation, Journal of Financial Economics<\/i>, 8:4, 323-361.<\/p>\n

Hull J. (2010) Gestion des risques et institutions financi\u00e8res<\/i>, Pearson, Glossaire fran\u00e7ais-anglais<\/a>.<\/p>\n

Donn\u00e9es<\/h3>\n

Yahoo! Finance <\/a><\/p>\n

A propos de l’auteur<\/h2>\n

Cet article a \u00e9t\u00e9 \u00e9crit en f\u00e9vrier 2023 par Shengyu ZHENG<\/a> (ESSEC Business School, Grande Ecole<\/i> Program – Master in Management, 2020-2023).<\/p>\n","protected":false},"excerpt":{"rendered":"

Dans cet article, Shengyu ZHENG (ESSEC Business School, Grande Ecole Program – Master in Management, 2020-2023) pr\u00e9sente les mesures de risques bas\u00e9es sur la distribution statistique des rentabilit\u00e9s d’une position de march\u00e9, ce qui est une approche possible pour mesurer les risques (comme expliqu\u00e9 dans mon article Cat\u00e9gorie de mesures de risques). Les mesures de … Read more<\/a><\/p>\n","protected":false},"author":70,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[5,10],"tags":[231,387,567,607,612],"class_list":["post-10075","post","type-post","status-publish","format-standard","hentry","category-contributors","category-financial-techniques","tag-expected-shortfall","tag-mesure-de-risque","tag-stress-value","tag-value-at-risk","tag-variance"],"yoast_head":"\nMesures de risques - SimTrade blog<\/title>\n<meta name=\"description\" content=\"D\u00e9couvrez les mesures de risques en finance (VaR, ES et SV) - Insight sur l'analyse des risques financiers avec Shengyu ZHENG | SimTrade Blog\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.simtrade.fr\/blog_simtrade\/mesures-risques\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Mesures de risques\" \/>\n<meta property=\"og:description\" content=\"D\u00e9couvrez les mesures de risques en finance (VaR, ES et SV) - 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