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
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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
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Multivariate Standardized t
Distribution
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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
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Multivariate Standardized t
Distribution
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• 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