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Arthur CHARPENTIER - ´ econom´ etrie de la finance (2008/2009) ´ Econom´ etrie

Arthur CHARPENTIER - ´ econom´ etrie de la finance (2008/2009) ´ Econom´ etrie de la ﬁnance Partie 1 Mesurer les risques Arthur Charpentier http ://perso.univ-rennes1.fr/arthur.charpentier/ Master 1, Universit´ e Rennes 1 1 Arthur CHARPENTIER - ´ econom´ etrie de la finance (2008/2009) 2 Arthur CHARPENTIER - ´ econom´ etrie de la finance (2008/2009) Premier ﬁl rouge du cours : la VaR 3 Arthur CHARPENTIER - ´ econom´ etrie de la finance (2008/2009) Premier ﬁl rouge du cours : la VaR 4 Arthur CHARPENTIER - ´ econom´ etrie de la finance (2008/2009) Premier ﬁl rouge du cours : la VaR 5 Arthur CHARPENTIER - ´ econom´ etrie de la finance (2008/2009) Second ﬁl rouge du cours : RiskMetrics 6 Arthur CHARPENTIER - ´ econom´ etrie de la finance (2008/2009) Second ﬁl rouge du cours : RiskMetrics 7 Arthur CHARPENTIER - ´ econom´ etrie de la finance (2008/2009) Plan du cours ◦Introduction g´ en´ erale • Mesurer les risques, une introduction au Risk Manageemnt ◦Mesurer les “risques” ? ◦Value-at-Risk ◦Contexte et cadre r´ egelementaire, Bˆ ale II ◦Un (tout petit) peu d’´ economie de l’incertain • Mod´ eliser des rendements boursiers ◦Que cherche-t-on ` a mod´ eliser ? ◦Processus ARCH et GARCH ◦Processus ` a volatilit´ e stochastique ◦Du rendement d’un titre au rendement d’un portefeuille • Retour ` a la VaR : les probl` emes d’estimation ◦Estimation de la Value-at-Risk, un mot de th´ eorie des extrˆ emes ◦Estimation de la Value-at-Risk pour des processus GARCH 8 Arthur CHARPENTIER - ´ econom´ etrie de la finance (2008/2009) La gestion des risques ` A la ﬁn des ann´ ees 90, les r´ eglementations prudentielles convergent vers l’adoption de la VaR comme mesure de risque de r´ ef´ erence. 9 Arthur CHARPENTIER - ´ econom´ etrie de la finance (2008/2009) History of risk measures The evolution of (analytical) Risk Management Tools (from Jorion (2007)) 1938 bond duration 1952 Markowitz mean-variance framework 1963 Sharpe’s single beta model 1973 Black & Scholes option pricing formula 1983 RAROC, Risk Adjusted Return 1992 Stress testing 1993 Value-at-Risk (VaR) 1994 RiskMetrics 1997 CreditMetrics 1998 integration of credit and market risk 1999 coherent risk measures 10 Arthur CHARPENTIER - ´ econom´ etrie de la finance (2008/2009) Market risks Classical models for stock prices, • dynamic models (Bachelier (1900), Black & Scholes (1973)), Brownian geometric dSt = µStdt | {z } drift + √ V StdWt | {z } random part , where (Wt)t≥0 is a standard brownian motion, • more advanced dynamic models (Heston (1993)) have stochastic volatility    dSt = µStdt + √VtdW S t dVt = κ(θ −Vt)dt + ξ√VtdW V t , where (W S t )t≥0 and (W V t )t≥0 are two standard brownian motions (possibly correlated). 11 Arthur CHARPENTIER - ´ econom´ etrie de la finance (2008/2009) 0.0 0.2 0.4 0.6 0.8 1.0 50 100 150 200 Stock price over 1 year, large volatility Time 0.0 0.2 0.4 0.6 0.8 1.0 50 100 150 200 Stock price over 1 year, large volatility Time Fig. 1 – Random generation of a stock price, dSt = µStdt + σStdWt. 12 Arthur CHARPENTIER - ´ econom´ etrie de la finance (2008/2009) How to quantify market risks : volatility All the information about uncertainty is summarized by the volatiliy - or variance - parameter. Note that this is one of the reason for the use of the Gaussian distribution, i.e. X ∼N(µ, σ2) having density f(x) = 1 σ √ 2π exp −1 2 x −µ σ 2! Then µ = E(X) and σ2 = V ar(X). 13 Arthur CHARPENTIER - ´ econom´ etrie de la finance (2008/2009) Market risks • the capital asset pricing model (Markowitz (1970) or the Sharpe index are based on the mean-variance framework, 0 5 #0 #5 !#.0 !0.5 0.0 0.5 #.0 #.5 %.0 %.5 &ca)t!type &sp/)ance 0 " #0 #" !#.0 !0." 0.0 0." #.0 #." %.0 %." &ca)t!t+pe &sp/)ance Fig. 2 – Capital asset pricing model, the mean-variance framework. 14 Arthur CHARPENTIER - ´ econom´ etrie de la finance (2008/2009) A (very) short word on diversiﬁcation Naturally, in higher dimension (when dealing with multiple stocks), Gaussian vectors are considered X =         X1 X2 . . . Xd         ∼N                 µ1 µ2 . . . µd         ,         σ2 1 ρ1,2σ1σ2 · · · ρ1,dσ1σd ρ2,1σ2σ1 σ2 2 · · · ρ2,dσ2σd . . . . . . . . . ρd,1σdσ1 ρd,2σdσ2 · · · σ2 d                 All the information about marginal risks is in the variances (σ2 i ) while all the information on the dependence is in the correlation coeﬃcients (ρi,j). 15 Arthur CHARPENTIER - ´ econom´ etrie de la finance (2008/2009) On the Gaussian distribution The Gaussian distribution is very important for many reasons, • it is a stable distribution, i.e. it appears as a limiting distribution in the central limit theorem : for i.i.d. Xi’s with ﬁnite variance, √nX −E(X) √ V X L →N(0, 1). • it is an elliptic distribution, i.e. X = µ + AX0 where A′A = Σ, and where X0 has a spheric distribution, i.e. f(x0) is a function of x′ 0x0 (spherical level curves), 16 Arthur CHARPENTIER - ´ econom´ etrie de la finance (2008/2009) −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 Level curves of a spherical distribution −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 Level curves of a elliptical distribution Fig. 3 – The Gaussian distribution. 17 Arthur CHARPENTIER - ´ econom´ etrie de la finance (2008/2009) On the Gaussian distribution As a consequence, if X ∼N(µ, Σ), and if X =  X1 X2  ∼N    µ1 µ2  ,  Σ11 Σ12 Σ21 Σ22     • Xi ∼N(µi, Σi), for all i = 1, · · · , d, • α′X = α1X1 + · · · + αdXd ∼N(α′µ, α′Σα), • X1|X2 = x2 ∼N(µ1 + Σ12Σ−1 2,2(x2 −µ2), Σ1,1 −Σ12Σ−1 2,2Σ21) 18 Arthur CHARPENTIER - ´ econom´ etrie de la finance (2008/2009) −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 0.00 0.05 0.10 0.15 0.20 Density of the Gaussian distribution −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 Level curves of a elliptical distribution Fig. 4 – The Gaussian distribution. 19 Arthur CHARPENTIER - ´ econom´ etrie de la finance (2008/2009) Value-at-Risk The expression is quite recent and its origin is uncertain : in the 80’s, some papers introduced dollars-at-risk, capital-at-risk, income-at-risk, earning-at-risk and ﬁnally value-at-risk Denomination has been stabilized after the publication of RiskMetrics Technical Document in 1994, by JPMorgan. Note that the work accomplished by JPMorgan was more a pulic relation campaign than an advanced technical study : VaR is more a practice than a theory. “VaR summarizes the worst loss ever on a target horizon that will not be exceeded with a given level of conﬁdence”, i.e. formaly it is a quantile of the projected distibution of gains and losses over the target horizon Till Guldimann (1992) created the term value-at-risk while head of global research at JP Morgan in the late 80’s. It appeared in the G30 report (group of thirty) in July 1993. 20 Arthur CHARPENTIER - ´ econom´ etrie de la finance (2008/2009) The Basel II accord (2004) June 2004, the Basel Committee ﬁnalized the Basel Accords, based on three pillars • minimum regulatory requierements, i.e. some risk-based capital requirements : set capital charges against credit risk (internal rating based), market risk (internal model approach) and operational risk. the goal is to keep constant the level of capital in the global banking syste : 8% of risk weighted assets, • supervisorv review, i.e. expanded role for bank regulartors, to ensure that banks operate above the minimum regulatory capital ratios, that banks have appropriate processes for assessing their risks, and appropriate corrective actions • market discipline, i.e. set of disclosure recommendations, encouraging to publish informations about exposures, risk proﬁles, capital cushion... 21 Arthur CHARPENTIER - ´ econom´ etrie de la finance (2008/2009) The Basel II accord (2004) From the ﬁrst pillar, there should be a credit risk charge (CRC), a market risk charge (MRC) and an operationnal risk charge (ORC), and the bank’s total capital must exceed the total-risk charge (TRC) Capital > TRC = CRC + MRC + ORC. 22 Arthur CHARPENTIER - ´ econom´ etrie de la finance (2008/2009) Why using VaR as a risk measure ? Markowitz (1952) claimed that standard deviation should be an intuitive and appropriate risk measure (leading to the mean-variance trade-oﬀ). The same year, Roy (1952) claimed uploads/Finance/ cours-econometrie-finance-r1-part-1 1 .pdf