Virtual Historical Simulation for estimating the conditional VaR of large portfolios

In order to estimate the conditional risk of a portfolio's return, two strategies can be advocated. A multivariate strategy requires estimating a dynamic model for the vector of risk factors, which is often challenging, when at all possible, for large portfolios. A univariate approach based on a dynamic model for the portfolio's return seems more attractive. However, when the combination of the individual returns is time varying, the portfolio's return series is typically non stationary which may invalidate statistical inference. An alternative approach consists in reconstituting a "virtual portfolio", whose returns are built using the current composition of the portfolio and for which a stationary dynamic model can be estimated. This paper establishes the asymptotic properties of this method, that we call Virtual Historical Simulation. Numerical illustrations on simulated and real data are provided.

[1]  Christian Gourieroux,et al.  ESTIMATION-ADJUSTED VAR , 2013, Econometric Theory.

[2]  J. Laurent,et al.  Market Risk and Volatility Weighted Historical Simulation after Basel III , 2017 .

[3]  G. Barone-Adesi.  VaR Without Correlations for Nonlinear Portfolios , 1998 .

[4]  W. Newey,et al.  A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelationconsistent Covariance Matrix , 1986 .

[5]  Dongming Zhu,et al.  Properties and Estimation of Asymmetric Exponential Power Distribution , 2007 .

[6]  L. Bauwens,et al.  Multivariate GARCH Models: A Survey , 2003 .

[7]  Robert F. Engle,et al.  Large Dynamic Covariance Matrices , 2017 .

[8]  Marno Verbeek,et al.  Evaluating portfolio Value-at-Risk using semi-parametric GARCH models , 2009 .

[9]  Luc BAUWENS,et al.  Handbook of Volatility Models and Their Applications , 2012 .

[10]  Glyn A. Holton.  Value at Risk: Theory and Practice , 2003 .

[11]  R. Koenker,et al.  Unit Root Quantile Autoregression Inference , 2004 .

[12]  C. Francq,et al.  Optimal predictions of powers of conditionally heteroscedastic processes , 2013 .

[13]  Ivana Komunjer,et al.  Evaluation and Combination of Conditional Quantile Forecasts , 2002 .

[14]  G. P. Aielli,et al.  Dynamic Conditional Correlation: On Properties and Estimation , 2011 .

[15]  Richard A. Davis,et al.  M-estimation for autoregressions with infinite variance , 1992 .

[16]  F. Diebold,et al.  Comparing Predictive Accuracy , 1994, Business Cycles.

[17]  E. Beutner,et al.  A residual bootstrap for conditional Value-at-Risk , 2018, Journal of Econometrics.

[18]  J. Zakoian,et al.  GARCH Models: Structure, Statistical Inference and Financial Applications , 2010 .

[19]  Christian Francq,et al.  Estimating multivariate volatility models equation by equation , 2016 .

[20]  Kris Boudt,et al.  Positive Semidefinite Integrated Covariance Estimation, Factorizations and Asynchronicity , 2016 .

[21]  J. Carlos Escanciano,et al.  Backtesting Parametric Value-at-Risk With Estimation Risk , 2008 .

[22]  Fabio Trojani,et al.  Robust Value at Risk Prediction , 2007 .

[23]  Laura Spierdijk,et al.  Confidence Intervals for ARMA-GARCH Value-at-Risk: The Case of Heavy Tails and Skewness , 2013, Comput. Stat. Data Anal..

[24]  YUKUN ZHANG.  Comparing univariate and multivariate models to forecast portfolio value-at-risk , 2011 .

[25]  Peter Christoffersen,et al.  Série Scientifique Scientific Series Estimation Risk in Financial Risk Management , 2022 .

[26]  J. Zakoian,et al.  Estimation risk for the VaR of portfolios driven by semi-parametric multivariate models , 2018, Journal of Econometrics.

[27]  P. Billingsley,et al.  Convergence of Probability Measures , 1970, The Mathematical Gazette.

[28]  A CENTRAL LIMIT THEOREM FOR MIXING TRIANGULAR ARRAYS OF VARIABLES WHOSE DEPENDENCE IS ALLOWED TO GROW WITH THE SAMPLE SIZE , 2005, Econometric Theory.

[29]  R. Bass,et al.  Review: P. Billingsley, Convergence of probability measures , 1971 .

[30]  Stephen P. Boyd,et al.  Introduction to Applied Linear Algebra , 2018 .

[31]  D. Andrews.  Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation , 1991 .

[32]  Franz C. Palm,et al.  Testing for jumps in conditionally Gaussian ARMA-GARCH models, a robust approach , 2016, Comput. Stat. Data Anal..

[33]  Y. Gong,et al.  Empirical likelihood intervals for conditional Value‐at‐Risk in ARCH/GARCH models , 2009 .

[34]  Esther Ruiz,et al.  Frontiers in VaR forecasting and backtesting , 2016 .

[35]  C. Francq.  Estimating multivariate GARCH models equation by equation , 2015 .

[36]  Keith Knight,et al.  Limiting distributions for $L\sb 1$ regression estimators under general conditions , 1998 .

[37]  Christian Francq,et al.  Risk-parameter estimation in volatility models , 2012 .

[38]  Jianqing Fan,et al.  Quantile autoregression. Commentary , 2006 .

[39]  Stephan Smeekes,et al.  Risk Measure Inference , 2015 .

[40]  T. Gneiting.  Making and Evaluating Point Forecasts , 2009, 0912.0902.

[41]  Stephan Smeekes,et al.  A justification of conditional confidence intervals , 2017, Electronic Journal of Statistics.

[42]  Peter Christoffersen,et al.  Elements of Financial Risk Management , 2003 .

[43]  J. Carlos Escanciano,et al.  Robust Backtesting Tests for Value-at-Risk Models , 2008 .

[44]  D. Hunter.  Estimation risk in financial risk management , 2005 .

[45]  Christophe Pérignon,et al.  Do Banks Overstate Their Value-at-Risk? , 2007 .