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Figure 9.4: Simulated Threshold Correlations from the Symmetric t Distribution with Various Parameters

# Figure 9.4: Simulated Threshold Correlations from the Symmetric t Distribution with Various Parameters

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Multivariate Standardized t

Distribution

17

• In the case of n assets we have the multivariate t distribution

• Where

• Using the density definition we can construct the

likelihood function

• which can be maximized to estimate d

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

Multivariate Standardized t

Distribution

18

• The correlation matrix can be preestimated using

• The correlation matrix  can also be made dynamic, and

can be estimated using the DCC approach

• An easier estimate of d can be obtained by computing the

kurtosis, 2, of each of the n variables

• The relationship between excess kurtosis and d is

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

Multivariate Standardized t

Distribution

19

• Using all the information in the n variables we can estimate d using

• where 2,i is sample excess kurtosis of ith variable

• The standardized symmetric n dimensional t variable can

be simulated as follows

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

Multivariate Standardized t

Distribution

20

• where W is a univariate inverse gamma random variable,

• where U is a vector of multivariate standard normal variables,

• where U and W are independent

• The simulated z will have a mean of zero, a standard deviation

of one, and a correlation matrix 

• Once we have simulated MC realizations of vector z we can

simulate MC realizations of the vector of asset returns and

from this the portfolio VaR and ES can be computed by

simulation as well

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

Multivariate Asymmetric t

Distribution

Let be an nì1 vector of asymmetry parameters

The asymmetric t distribution is then defined by

where

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

21

Multivariate Asymmetric t

Distribution

22

• where

is the so-called modified Bessel function of

the third kind, which can be evaluated in Excel using the

formula besselk(x, (d+n)/2)

• Note that the vector and matrix are constructed so that

the vector of random shocks z will have a mean of zero, a

standard deviation of one, and the correlation matrix 

• Note also that if  = 0 then

• Note that the asymmetric t distribution will converge to the

symmetric t distribution as the asymmetry parameter

vector  goes to a vector of zeros

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

Figure 9.5: Simulated Threshold Correlations from the

Asymmetric t Distribution with Various Linear

Correlations

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

23

24

Multivariate Asymmetric t Distribution

• From the density

function

we can construct the likelihood

• which can be maximized to estimate the scalar d and

vector 

• The correlation matrix can be preestimated using

• The correlation matrix  can be made dynamic, t, and

can be estimated using the DCC approach

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

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Figure 9.4: Simulated Threshold Correlations from the Symmetric t Distribution with Various Parameters

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