Estimating Value at Risk via Markov Switching ARCH models

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