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Figure 9.2: Threshold Correlation for S&P versus 10-Year Treasury Bond GARCH Shocks

Figure 9.2: Threshold Correlation for S&P versus 10-Year Treasury Bond GARCH Shocks

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Multivariate Distributions



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• In this section we consider multivariate distributions that

can be combined with GARCH (or RV) and DCC models

to provide accurate risk models for large systems of assets

• We will first review the multivariate standard normal

distribution, then the multivariate standardized symmetric t

distribution, and finally an asymmetric version of the

multivariate standardized t distribution



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



Multivariate Standard Normal

Distribution

• In the bivariate case we have the standard normal density

with correlation defined by



• where 1-2 is the determinant of the bivariate correlation

matrix



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



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Figure 9.3: Simulated Threshold Correlations from

Bivariate Normal Distributions with Various Linear

Correlations



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



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Multivariate Standard Normal

Distribution



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• In the multivariate case with n assets we have the density

with correlation matrix 



• Note that each pair of assets in the vector zt will have

threshold correlations that tend to zero for large thresholds

• The 1-day VaR is easily computed via



• where we have portfolio weights wt and the diagonal

matrix of standard deviations Dt+1

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



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Figure 9.2: Threshold Correlation for S&P versus 10-Year Treasury Bond GARCH Shocks

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