Estimating Value at Risk via Markov Switching ARCH models An Empirical Study on Stock Index Returns

Value at Risk (hereafter, VaR) is at the center of the recent interest in the risk management field.

 Bank for International Settlements (BIS)  The measure of the banks’ capital adequacy ratios.  The measure of default risk, credit risk, operation risks and liquidity risk

The Definition of VaR VaR for a Confidence Interval of 99% VAR α 0 μ Absolute VaR Relative VaR The figure presents the definition for the VaR. VaR concept focuses on point VaR α, or the left-tailed maximum loss with confidence interval 1-α

The Keys of Estimating VaR Non-normality Properties:  Skewness  Kurtosis,  Tail-fatness

The Solutions for Non-Normality  Non-Parametric Setting  Historical Simulation  Student t Setting  Stochastic Volatility Setting

Why We Propose Stochastic Volatility?  The Shortcomings of Non-Parametric Setting Historical Simulation  Are the data used to simulate the underlying distribution representative? The Shortcomings of Student t Setting  Can not picture the Skewness for the Return Distributions

x 11,x 12,x 13,x 14,.. x 21 x 22 x 23 … x 21 x 22 x 23 x 11,x 12,.……………… x 13,x Distribution 1: A high Volatility Distribution _____ Distribution 2: A Low Volatility Distribution ---- Distribution 1 ___ Distribution 2

Normal +Normal=Normal? Normal +Normal=Normal But, state-varying framework is –some observations from Dist. 1 –other observations from Dist. 2 How to decide the sample from distribution 1 or distribution 2? Two mechanisms: threshold systems and Markov-switching models

The Most Popular Stochastic Volatility Setting  The ARCH and GARCH models

Why We Propose Hamilton and Susmel (1994)’s SWARCH Model?  The Structure Change During the Estimating Periods  The SWARCH Models Incorporate Markov Switching (MS) and ARCH models  Use the MS to Control the Structural Changes and Thus Mitigate the Returns Volatility High Persistence Problems in ARCH models.

Model Specifications Linear Models

Model Specifications ARCH and GARCH Models:

Model Specifications SWARCH Models

Markov Chain Process  In a special two regimes setting, set s t =1 for the regime with low return volatility and s t =2 for the one with high return volatility.  The transition probabilities can be presented as:

VaR Estimate by SWARCH

Empirical Analyses Data: –Dow Jones, Nikkei, FCI and FTSE index returns. Sample period is between January 7, 1980 and February 26, 1999 Models: –ARCH, GARCH and SWARCH to control non- normality properties

Empirical Analyses 1,000-day windows in the rolling estimation process. The research design begins with our collecting the 1,000 pre-VaR daily returns,, for each date t.

Empirical Analyses 4,838 trading days during the sample period For our tests with 1,000 prior-trading-day estimation window and one-day as the order of the lagged term, we have 3,837 out-sample observations of violation rates. If the VaR estimate is accurate, the violation rate should be 1%, or the violation number should be approximately equal to 38

Empirical Analyses