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Figure 6.5: Skewness and Kurtosis in the Asymmetric t Distribution

# Figure 6.5: Skewness and Kurtosis in the Asymmetric t Distribution

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The Asymmetric t distribution

• Notice that the symmetric t distribution is nested in the

asymmetric t distribution and can be derived by setting

d1 = d, d2 = 0 which implies A = 0 and B = 1

• Therefore,

• which yields

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

45

Estimation of d1 and d2

• MLE can be used to estimate the parameters of the

asymmetric distribution, d1 and d2

• The only complication is that the shape of the

distribution on each day depends on zt

• As before the likelihood function for the shock can

be defined as,

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

46

Estimation of d1 and d2

• Where

• This assumes the estimation of

is done without estimation error.

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

47

Estimation of d1 and d2

• Alternatively the joint estimation of volatility and

distribution parameters can be done via,

• We can also estimate d1 and d2 using the sample

moments.

• Equations below can be solved numerically

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

48

Calculating VaR and ES

• Having estimated d1 and d2 we can calculate the

VaR of the portfolio return

• as

• Where

is the pth percentile of the

asymmetric t distribution.

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

49

Calculating VaR and ES

is given by

• The ES is given by

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

50

QQ Plots

51

• Knowing the CDF we can construct the QQ plot as,

• where zi denotes the ith sorted standardized return

• The asymmetric t distribution is cumbersome to estimate

and implement but it is capable of fitting GARCH shocks

from daily asset returns quite well

• The t distributions attempt to fit the entire range of

outcomes using all the data available

• Consequently, the estimated parameters in the distribution

(for example d1 and d2 ) may be influenced excessively by

data values close to zero

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Figure 6.6: QQ Plot of S&P 500 GARCH Shocks

against the Asymmetric t Distribution

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

52

Extreme Value Theory (EVT)

53

• Typically, the biggest risks to a portfolio is the sudden

occurrence of a single large negative return.

• Having an as precise as possible knowledge of the

probabilities of such extremes is therefore at the essence

of financial risk management.

• Consequently, risk managers should focus attention

explicitly on modeling the tails of the returns distribution.

• Fortunately, a branch of statistics is devoted exactly to the

modeling of these extreme values.

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

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Figure 6.5: Skewness and Kurtosis in the Asymmetric t Distribution

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