Joint inference on market and estimation risks in dynamic portfolios

Francq, Christian and Zakoian, Jean-Michel (2015): Joint inference on market and estimation risks in dynamic portfolios.

Abstract

We study the estimation risk induced by univariate and multivariate methods for evaluating the conditional Value-at-Risk (VaR) of a portfolio of assets. The composition of the portfolio can be time-varying and the individual returns are assumed to follow a general multivariate dynamic model. Under sphericity of the innovations distribution, we introduce in the multivariate framework a concept of VaR parameter, and we establish the asymptotic distribution of its estimator. A multivariate Filtered Historical Simulation method, which does not rely on sphericity, is also studied. We derive asymptotic confidence intervals for the conditional VaR, which allow to quantify simultaneously the market and estimation risks. The particular case of minimal variance and minimal VaR portfolios is considered. Potential usefulness, feasibility and drawbacks of the different approaches are illustrated via Monte-Carlo experiments and an empirical study based on stock returns.

• Item Type: MPRA Paper
• Language: English
• Keywords: Confidence Intervals for VaR; DCC GARCH model, Estimation risk; Filtered Historical Simulation; Optimal Dynamic Portfolio
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C22 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C58 - Financial Econometrics

References

Aielli, G.P. (2013) Dynamic conditional correlation: on properties and estimation. Journal of Business and Economic Statistics 31, 282--299.

Alexander, G.J. and A. M. Baptista (2002) Economic implications of using a mean-VaR model for portfolio selection: a comparison with mean-variance analysis. Journal of Economic Dynamics and Control 26, 1159--1193.

Barone-Adesi, G., Giannopoulos, K., and L. Vosper (1999) VaR without correlations for nonlinear portfolios. Journal of Futures Markets 19, 583--602.

Bauwens, L., Hafner, C.M. and S. Laurent (2012) Handbook of Volatility Models and Their Applications. Wiley.

Bauwens, L., Laurent, S. and J.V.K. Rombouts (2006) Multivariate GARCH models: a survey. Journal of Applied Econometrics 21, 79--109.

Berkes, I., Horv\'ath, L. and P. Kokoszka (2003) GARCH processes: structure and estimation. Bernoulli 9, 201--227.

Billingsley, P.(1961) The Lindeberg-Levy theorem for martingales. Proceedings of the American Mathematical Society 12, 788--792.

Boswijk H.P., and R. van der Weide (2011) Method of moments estimation of GO-GARCH models. Journal of Econometrics 163, 118--126.

Campbell, R., Huisman, R. and K. Koedijk Optimal portfolio selection in a value-at-Risk framework. Journal of Banking \& Finance 25, 1789--1804.

Christoffersen, P.F. (2003) {\em Elements of financial risk management. Academic Press, London.

Christoffersen, P.F. and S. Gonçalves (2005) Estimation Risk in Financial Risk Management. Journal of Risk 7, 1--28.

Comte, F. and O. Lieberman(2003) Asymptotic Theory for Multivariate GARCH Processes. Journal of Multivariate Analysis 84, 61--84.

Embrechts, P. and G. Puccetti (2010) Risk Aggregation. In P.Jaworski et al. (eds.), Copula Theory and Its Applications, Lecture Notes in Statistics 198, Springer, Berlin/Heidelberg.

Engle, R.F. (2002) Dynamic conditional correlation: a simple class of multivariate generalized autoregressive conditional heteroskedasticity models. Journal of Business and Economic Statistics 20, 339--350.

Engle, R.F. and K. Kroner (1995) Multivariate simultaneous GARCH. Econometric Theory 11, 122--150.

Escanciano, J.C., and J. Olmo (2010) Backtesting parametric VaR with estimation risk. Journal of Business and Economic Statistics 28, 36--51.

Escanciano, J.C. and J. Olmo (2011) Robust backtesting tests for value-at-risk models. Journal of Financial Econometrics 9, 132--161.

Francq, C., Horváth, L. and J.M. Zakoïan (2015) Variance targeting estimation of multivariate GARCH models. Forthcoming in the Journal of Financial Econometrics.

Francq, C., Jiménez Gamero, M.D. and S. Meintanis (2015) Tests for sphericity in multivariate GARCH models. MPRA Paper No. 67411.

Francq, C., and J.M. Zakoïan (2010) GARCH models: structure, statistical inference and financial applications. Chichester: John Wiley.

Francq, C., and J.M. Zakoïan (2012) QML estimation of a class of multivariate asymmetric GARCH models. Econometric Theory 28, 179--206.

Francq, C. and J.M. Zakoïan (2015a) Risk-parameter estimation in volatility models. Journal of Econometrics 184, 158--173.

Francq, C. and J.M. Zakoïan (2015b) Estimating multivariate GARCH models equation by equation. Forthcoming in the Journal of the Royal Statistical Society: Series B (Statistical Methodology).

Gouriéroux, C. and J.M. Zakoïan (2013) Estimation adjusted VaR. Econometric Theory 29, 735--770.

Koenker, R. and Z. Xiao (2006) Quantile autoregression. Journal of the American Statistical Association 101, 980--990.

Ling, S. (2004) Estimation and testing stationarity for double-autoregressive models. Journal of the Royal Statistical Society B 66, 63--78.

Ling, S. (2005) Self-weighted least absolute deviation estimation for infinite variance autoregressive models. Journal of the Royal Statistical Society B 67, 381--393.

Mancini, L. and F. Trojani (2011) Robust Value-at-Risk prediction. Journal of Financial Econometrics 9, 281--313.

McAleer, M. and B. da Veiga (2008) Single-index and portfolio models for forecasting Value-at-Risk. Journal of Forecasting 27, 217--235.

Pedersen, R.S. and A. Rahbek (2014) Multivariate variance targeting in the BEKK-GARCH model. The Econometrics Journal 17, 24--55.

Silvennoinen, A. and T. Teräsvirta (2009) Multivariate GARCH models. Handbook of Financial Time Series T.G. Andersen, R.A. Davis, J-P. Kreiss and T. Mikosch, eds. New York: Springer.

Santos, A.A.P., Nogales F.J. and E. Ruiz (2013) Comparing univariate and multivariate models to forecast portfolio Value-at-Risk. Journal of Financial Econometrics 11, 400--441.

Spierdijk, L. (2014) Confidence Intervals for ARMA-GARCH Value-at-Risk: The Case of Heavy Tails and Skewness. Computational Statistics and Data Analysis, forthcoming.

Tsay, R.S. (2014) Multivariate time series: with R and financial applications. John Wiley.

Zhu, D. and V. Zinde-Walsh (2009) Properties and estimation of asymmetric exponential power distribution. Journal of Econometrics 148, 86--99.