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Figure 9.3: Simulated Threshold Correlations from Bivariate Normal Distributions with Various Linear Correlations

Figure 9.3: Simulated Threshold Correlations from Bivariate Normal Distributions with Various Linear Correlations

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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 â 2012 by Peter Christoffersen



Multivariate Standard Normal

Distribution



13



The 1-day ES is also easily computed using



• In multivariate normal distribution, linear combination of

multivariate normal variables is normally distributed

• The multivariate normal distribution does not adequately

capture the (multivariate) risk of returns

• This means that convenience of normal distribution comes

at a too-high price for risk management purposes

• We therefore consider the multivariate t distribution

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



Multivariate Standardized t

Distribution

• In Chapter 6 we considered the univariate standardized t

distribution that had the density



• where the normalizing constant is



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



14



Multivariate Standardized t

Distribution



15



The bivariate standardized t distribution with correlation

takes the following form:



• where



• Note that d is a scalar here and so the two variables have

the same degree of tail fatness

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



Figure 9.4: Simulated Threshold Correlations from the

Symmetric t Distribution with Various Parameters



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



16



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



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Figure 9.3: Simulated Threshold Correlations from Bivariate Normal Distributions with Various Linear Correlations

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