VaR

Forecasting ValueatRisk via GARCH Models
The ValueatRisk (VaR) framework [Jorion, 1996], [Dowd, 2005], [Christoffersen, 2011], [Danielsson, 2011]
has growing importance for banks and regulators as it helps to quantify possible losses due to trading
assets and portfolios. This framework helps us to measure the risk that future returns may exceed a certain
VaR value with a certain probability over a period of time. Statistically the VaR is associated with the
quantiles, that is the probability mass in the tails of the return distribution. The violations of the left
tail indicate the risk for long trading positions, and the violations of the right tail indicates the
risk for short positions.
Our research studies conditional VaR estimation via forecasting the mean and variance of the return distribution
using GARCH models. This includes formulation of innovative GARCH models and implementation of corresponding
calibration algorithms. Recently, we developed: normal mixture GARCH models and heavytail mixture
GARCH models [Nikolaev, Boshnakov and Zimmer, 2013]. Normal mixture models help to achieve better fit to
timevarying data compared to Gaussian models [Haas, Mittnik, and Paolella, 2004]. Heavytail mixture models
capture more accurately data coming from more peaked distributions with longer tails (the excess kurtosis of
the data distribution). These mixture models help us to describe more adequately aberrant changes in time series
arising from occasional events. Especially for mixture GARCH we derive dynamic (temporal) training algorithms
that are used for analytical optimization through generalised expectation maximisation.
The empirical investigations show that the developed mixture GARCH deliver improved forecasts compared to
linear as well nonlinear Gaussian and Studentt noise models with respect to the loglikelihood and several
statistical characteristics. The analytical training leads to mixture GARCH models with quite similar
statistical features to these achieved by Monte Carlo simulation methods [Ausin and Galeano, 2007].

ASV

Asymmetric Stochastic Volatility and ValueatRisk
The stochastic volatility models are also attractive for ValueatRisk estimation, because
they also allow efficient generation of the predictive distribution of returns and its quantiles.
Currently we develop nonlinear filters for calibration of heavytailed Asymmetric Stochastic
Volatility (ASV) models incorporating leverage effects. These filters approximate the moments
of the volatility posterior via recursive numerical convolution, and thus they obtain the
likelihood more efficiently than algorithms that use stochastic sampling.
VaR testing of statespace ASV models is carried out following a bootstrapping methodology
[Rodriguez and Ruiz, 2009]. Bootstrap replicates of the return series are made by drawing samples
from the standardised residuals. Next, the model is reestimated over the replicates, and samples
from the predictive return distribution are obtained. The experiments are performed according to
the methodology of Christoffersen [2011]. It involves precomputing the failure rates as the proportion
of exceptions for the tails that exceed the corresponding valueatrisk, and computation of
likelihood ratio statistics. The obtained results show that the ASV models generate reliable
1% and 5% VaR predictions for long as well as short trading positions.
The research into heavytail ASV and heavytail mixture GARCH models found that they produce
accurate forecasts, which leads to adequate results in conditional VaR evaluation. Out calibration
algorithms for such heavytail volatility models construct efficiently the predictive distribution,
and make them useful tools for assesment of the risk exposure and for mitigating financial risks.

References
Nikolaev,N., Boshnakov,G. and Zimmer,R. (2013). Heavytail Mixture GARCH Volatility Modeling and ValueatRisk
Estimation, Expert Systems with Applications, vol.40, N:6, pp.22332243.
Nikolaev,N., de Menezes,L. and Smirnov,E. (2013). Nonlinear Filtering of Asymmetric Stochastic Volatility Models
and ValueatRisk Estimation, Technical Report 430713, Goldsmiths College, University of London.
Zangari,P. (1996). An Improved Methodology for Measuring VAR, RiskMetrics Monitor, Reuters/JP Morgan, pp.725.
Jorion,P. (1996). Value at Risk: The New Benchmark for Managing Financial Risk, The McGrawHill Co., New York.
Haas,M., Mittnik,S. and Paolella,M. (2004). Mixed Normal Conditional Heteroscedasticity,
Journal of Financial Econometrics, vol.2, pp.211250.
Dowd,K. (2005). Measuring Market Risk, 2nd ed., John Wiley and Sons, England.
Hartz,C., Mittnik,S. and Paolella,M. (2006). Accurate ValueatRisk Forecasting Based on the
NormalGARCH Model, Computational Statistics and Data Analysis, vol.51, N:4, pp.22952312.
Pascual,L., Romo,J. and Ruiz,E. (2006). Bootstrap Prediction for Returns and Volatilities in GARCH Models,
Computational Statistics and Data Analysis, vol.50, pp.22932312.
Ausin,M.C. and Galeano,P. (2007). Bayesian Estimation of the Gaussian Mixture GARCH Model,
Computational Statistics and Data Analysis, vol.51, N:5, pp.26362652.
Rodriguez,A. and Ruiz,E. (2009). Bootstrap Prediction Intervals in StateSpace Models. Journal of
Time Series Analysis, vol.30, N:2, pp.167178.
Danielsson,J. (2011). Financial Risk Forecasting: The Theory and Practice of Forecasting
Market Risk with Implementation in R and Matlab, John Wiley and Sons, England.
Christoffersen,P. (2011). Elements of Financial Risk Management, 2nd ed., Academic Press, San Diego, CA.
Relevant Risk Management Sites
n.nikolaev@gold.ac.uk