estimating value at risk via markov switching arch models an empirical study on stock index returns
TRANSCRIPT
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%
VARα 0 μ
Absolute VaR
Relative VaR
%1
The figure presents the definition for the VaR. VaR concept focuses on point VaRα, or the left-tailed maximum loss with confidence interval 1-α
The Keys of Estimating VaR
Non-normality Properties: Skewness
Kurtosis,
Tail-fatness
(a)PDF of Dow Jones Index Return Shock: Linear Model
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
-5 -4.7 -4.4 -4.1 -3.8 -3.5 -3.2 -2.9 -2.6 -2.3 -2 -1.7 -1.4 -1.1 -0.8 -0.5 -0.2 0.12 0.42 0.72 1.02 1.32 1.62 1.92 2.22 2.52 2.82 3.12 3.42 3.72 4.02 4.32 4.62 4.92
Table 1. Skewness, Kurtosis, and 1%, 2.5%, 5% Critical Values for Returns
Shocks of Various Indices
Statistics Coefficients Dow Jones FCI FTSE Nikkei
Skewness Coefficients (N=0) -2.26 2.20 -0.53 0.17
Kurtosis Coefficients (N=3) 58.21 157.14 22.92 18.03
1% Left-tailed Critical Value (N= -2.33) -2.43 -2.46 -2.49 -2.78
2.5% Left-tailed Critical Value (N= -1.96) -1.90 -1.69 -1.87 -2.10
5% Left-tailed Critical Value (N= -1.65) -1.45 -1.26 -1.46 -1.55
1% Right-tailed Critical Value (N=2.33) 2.43 2.24 2.32 2.82
2.5% Right-tailed Critical Value (N=1.96) 1.92 1.48 1.75 1.97
5% Right-tailed Critical Value (N=1.65) 1.44 1.15 1.36 1.42
Number of Observations 4838 4758 3801 5045
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 S
etting Historical Simulation Are the data used to simulate the underlying distributi
on representative?
The Shortcomings of Student t Setting Can not picture the Skewness for the Return Distributi
ons
0.00
0.01
0.02
-5 -4.2 -3.5 -2.7 -2 -1.2 -0.5 0.271.021.772.523.274.024.77
0.00
0.01
0.02
-5 -4.2 -3.4 -2.6 -1.8 -1 -0.2 0.62 1.42 2.22 3.02 3.82 4.62
0.00
0.01
0.02
-5 -4.2 -3.5 -2.7 -2 -1.2 -0.5 0.27 1.02 1.77 2.52 3.27 4.024.77
x11,x12,x13,x14,..
x21
x22
x23
…
x21
x22
x23
x11,x12, .……………… x13,x14
-----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
tt euR
32.2VaR
Model Specifications
• ARCH and GARCH Models:
p
i itiit
q
i it
ttt
tttt
baa
e
NDiideeuR
1
22
10
)1,0(~,
ttVaR 32.2
Model Specifications
• SWARCH Models
tst tuR
tst wgt
ttt ehw
2222
2110 ... qtqttt wawawaa
Markov Chain Process
In a special two regimes setting, set st=1 for the regime with low return volatility and st=2 for the one with high return volatility.
The transition probabilities can be presented as:
211221
121111
)2|1(,)2|2(
)1|2(,)1|1(
pssppssp
pssppssp
tttt
tttt
VaR Estimate by SWARCH
ts s
qttt
t qt
sspVaR
32.2),,(...2
1
2
1
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.
000,1
1 iitR
Empirical Analyses
• 4,838 trading days during the sample period • For our tests with 1,000 prior-trading-day estimat
ion 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
(B) The Estimates of g2 Parameter
0
2
4
6
8
10
1983 1985 1987 1989 1991 1993 1995 1997
Year
Empirical Analyses
(D) The Predicting Probabilities of Regime 2
0
0.2
0.4
0.6
0.8
1
1.2
1983 1985 1987 1989 1991 1993 1995 1997
Year
Empirical Analyses
(C) The Predicting VaR for Confidence Interval 99%
-25.00
-20.00
-15.00
-10.00
-5.00
0.00
5.00
10.00
15.00
1983 1985 1987 1989 1991 1993 1995 1997
Dow Junes Index Returns Predicting VaR for 99%
Empirical Analyses
Empirical Analyses
Empirical Analyses