Estimation risk for the VaR of portfolios driven by semi-parametric - PowerPoint PPT Presentation

General framework Estimating the conditional VaR Numerical comparison of the different VaR estimators Estimation risk for the VaR of portfolios driven by semi-parametric multivariate models Christian Francq Jean-Michel Zakoïan CREST and University of Lille, France Troisièmes Journées d’Econométrie de la Finance, JEF’2016 Rabat, November 18-19, 2016 Francq, Zakoian Conditional VaR of a portfolio

General framework Estimating the conditional VaR Numerical comparison of the different VaR estimators Objectives Estimate the conditional risk of a portfolio of assets (market risk) Setup: the portfolio’s composition is time-varying. The vector of individual returns follows a general dynamic model. Aims: Evaluate the accuracy of the estimation: ⇒ quantify simultaneously the market and estimation risks. Compare univariate and multivariate approaches. Crystallized portfolios; Optimal (conditional) mean-variance portfolios; Minimal VaR porfolios. Francq, Zakoian Conditional VaR of a portfolio

General framework Risk factors Estimating the conditional VaR Dynamic model Numerical comparison of the different VaR estimators Conditional VaR parameter Risk factors p t = ( p 1 t ,..., p mt ) ′ vector of prices of m assets y t = ( y 1 t ,..., y mt ) ′ vector of log-returns, y it = log ( p it / p i , t − 1 ) V t value of a portfolio composed of µ i , t − 1 units of asset i , for i = 1 ,..., m : � m V t = µ i , t − 1 p it , i = 1 where the µ i , t − 1 are measurable functions of the past prices. Francq, Zakoian Conditional VaR of a portfolio

General framework Risk factors Estimating the conditional VaR Dynamic model Numerical comparison of the different VaR estimators Conditional VaR parameter Self-financing constraint At date t , the investor may rebalance his portfolio in such a way that � m i = 1 µ i , t − 1 p it = � m SF: i = 1 µ i , t p it . The value at time t of the portfolio bought at time t − 1 serves to buy the new portfolio at time t . Francq, Zakoian Conditional VaR of a portfolio

General framework Risk factors Estimating the conditional VaR Dynamic model Numerical comparison of the different VaR estimators Conditional VaR parameter Return of the portfolio Under SF , the return of the portfolio over the period [ t − 1 , t ] , assuming V t − 1 �= 0 , is � m V t − 1 = a i , t − 1 exp ( y it ) − 1 ≈ r t V t − 1 i = 1 where � m a i , t − 1 y it = a ′ r t = t − 1 y t , i = 1 with µ i , t − 1 p i , t − 1 � m i = 1 ,..., m , a i , t − 1 = , j = 1 µ j , t − 1 p j , t − 1 y t = ( y 1 t ,..., y mt ) ′ . and a t − 1 = ( a 1 , t − 1 ,..., a m , t − 1 ) ′ , Francq, Zakoian Conditional VaR of a portfolio

General framework Risk factors Estimating the conditional VaR Dynamic model Numerical comparison of the different VaR estimators Conditional VaR parameter Conditional VaR of the portfolio’s return The conditional VaR of the portfolio’s return r t at risk level α ∈ ( 0 , 1 ) is defined by � � r t < − VaR ( α ) P t − 1 t − 1 ( r t ) = α , where P t − 1 denotes the historical distribution conditional on � p u , u < t � . Consequence The evaluation of the portfolio’s conditional VaR requires either a dynamic model for the vector of risk factors y t , or a dynamic univariate model for the portfolio’s return r t . Remark: The univariate approach seems simpler, but is likely to be inefficient, or even inconsistent when a t is time-varying. Francq, Zakoian Conditional VaR of a portfolio

General framework Risk factors Estimating the conditional VaR Dynamic model Numerical comparison of the different VaR estimators Conditional VaR parameter Dynamic model for the vector of log-returns Let ( y t ) be a strictly stationary and non anticipative solution of the multivariate model with conditional mean and GARCH-type errors: y t = m t ( θ 0 ) + ǫ t , ǫ t = Σ t ( θ 0 ) η t θ 0 ∈ R d and iid where η t ∼ ( 0 , I m ), m t ( θ 0 ) = m ( y t − 1 , y t − 2 ,..., θ 0 ), Σ t ( θ 0 ) = Σ ( y t − 1 , y t − 2 ...., θ 0 ). Examples of MGARCH Thus, the portfolio’s return satisfies r t = a ′ t − 1 m t ( θ 0 ) + a ′ t − 1 Σ t ( θ 0 ) η t , and its conditional VaR at level α is � a ′ � VaR ( α ) t − 1 m t ( θ 0 ) + VaR ( α ) t − 1 ( r t ) = − a ′ t − 1 Σ t ( θ 0 ) η t . t − 1 Francq, Zakoian Conditional VaR of a portfolio

General framework Risk factors Estimating the conditional VaR Dynamic model Numerical comparison of the different VaR estimators Conditional VaR parameter Conditional VaR parameter The conditional VaR is a stochastic process � a ′ � VaR ( α ) t − 1 m t ( θ 0 ) + VaR ( α ) t − 1 ( r t ) = − a ′ t − 1 Σ t ( θ 0 ) η t . t − 1 Depends on i) the mean-volatility parameter θ 0 and ii) the law of η t . Under certain assumptions, VaR ( α ) t − 1 ( r t ) can be related to a so-called conditional VaR parameter θ ∗ 0 : VaR ( α ) t − 1 ( r t ) = − a ′ t − 1 m t ( θ ∗ 0 ) +� a ′ t − 1 Σ t ( θ ∗ 0 ) � Francq, Zakoian Conditional VaR of a portfolio

General framework Risk factors Estimating the conditional VaR Dynamic model Numerical comparison of the different VaR estimators Conditional VaR parameter A simplification for elliptic conditional distributions In the multivariate volatility model ǫ t = m t ( θ 0 ) + Σ t ( θ 0 ) η t , ( η t ) iid ( 0 , I m ), assume that the errors η t have a spherical distribution: d for any non-random vector λ ∈ R m , λ ′ η t A1: = � λ � η 1 t , where �·� is the euclidean norm on R m . Remark: This is equivalent to assuming that the conditional distribution of ǫ t given its past is elliptic. Under A1 we have � � � VaR ( α ) � � VaR ( α ) t − 1 ( r t ) = − a ′ � a ′ t − 1 m t ( θ 0 ) + t − 1 Σ t ( θ 0 ) η , where VaR ( α ) � � is the (marginal) VaR of η 1 t . η Example of spherical distributions Francq, Zakoian Conditional VaR of a portfolio

General framework Risk factors Estimating the conditional VaR Dynamic model Numerical comparison of the different VaR estimators Conditional VaR parameter Assumption on the conditional variance model B1: There exists a continuously differentiable function G : R d �→ R d such that for any θ ∈ Θ , any K > 0 , and any sequence ( x i ) i on R m , m ( x 1 , x 2 ,...; θ ) = m ( x 1 , x 2 ,...; θ ∗ ), and K Σ ( x 1 , x 2 ,...; θ ) = Σ ( x 1 , x 2 ,...; θ ∗ ), where θ ∗ = G ( θ , K ). Examples of the CCC and DCC-GARCH Francq, Zakoian Conditional VaR of a portfolio

General framework Risk factors Estimating the conditional VaR Dynamic model Numerical comparison of the different VaR estimators Conditional VaR parameter VaR parameter for an elliptic conditional distribution At the risk level α ∈ ( 0 , 0 . 5 ) , the conditional VaR of the portfolio’s return is � a ′ � VaR ( α ) t − 1 m t ( θ 0 ) + VaR ( α ) t − 1 ( r t ) = − a ′ t − 1 Σ t ( θ 0 ) η t t − 1 � � � a ′ � VaR ( α ) ( η ) = − a ′ t − 1 m t ( θ 0 ) + t − 1 Σ t ( θ 0 ) = − a ′ t − 1 m t ( θ ∗ 0 ) +� a ′ t − 1 Σ t ( θ ∗ 0 ) � , where, under B1 , � � θ ∗ θ 0 , VaR ( α ) ( η ) 0 = G . The parameter θ ∗ 0 can be called conditional VaR parameter. Remark: The conditional VaR parameter does not depend on the portfolio composition summarizes the risk at a given level Francq, Zakoian Conditional VaR of a portfolio

General framework Multivariate estimation under ellipticity Estimating the conditional VaR Relaxing the ellipticity assumption Numerical comparison of the different VaR estimators Univariate approaches General framework 1 Estimating the conditional VaR 2 Multivariate estimation under ellipticity Relaxing the ellipticity assumption Univariate approaches Numerical comparison of the different VaR estimators 3 Francq, Zakoian Conditional VaR of a portfolio

General framework Multivariate estimation under ellipticity Estimating the conditional VaR Relaxing the ellipticity assumption Numerical comparison of the different VaR estimators Univariate approaches Estimating the conditional VaR parameter Observations: y 1 ,..., y n (+ initial values � y − 1 ,... ). y 0 , � � θ n : estimator of θ 0 . m t ( θ ) = m ( y t − 1 ,..., y 1 , � � y 0 , � y − 1 ,..., θ ), � Σ t ( θ ) = Σ ( y t − 1 ,..., y 1 , � y 0 , � y − 1 ,..., θ ), for t ≥ 1 and θ ∈ Θ . − 1 t ( � m t ( � η t = � η mt ) ′ . Residuals: � θ n ){ y t − � θ n )}) = ( � η 1 t ,..., � Σ Under the conditional sphericity assumption, an estimator of the conditional VaR at level α is ( α ) ∗ ∗ � m t ( � Σ t ( � S , t − 1 ( r ) = − a ′ n ) +� a ′ t − 1 � VaR t − 1 � θ θ n ) � , where � �� � ( α ) ∗ � θ n , � � VaR θ n = G η , n � � ( α ) � VaR η = ξ n , 1 − 2 α : ( 1 − 2 α ) -quantile of { | � η it | , 1 ≤ i ≤ m , 1 ≤ t ≤ n }. n Francq, Zakoian Conditional VaR of a portfolio