Figure 9.7: Simulated Threshold Correlations from the Symmetric t Copula with Various Parameters
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t Copula
• The t copula can generate large threshold correlations
for extreme moves in the assets
• Furthermore it allows for individual modeling of the
marginal distributions, which allows for much
flexibility in the resulting multivariate distribution
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
45
t Copula
• In the general case of n assets we have t copula CDF
• and the t copula PDF
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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t Copula
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• Notice that d is a scalar, which makes the t copula somewhat
restrictive but also makes it implementable for many assets
• Maximum likelihood estimation can again be used to
estimate the parameters d and * in the t copula.
• We need to maximize
• defining again the copula shocks for asset i on day t as
follows:
Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen
t Copula
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In large dimensions we need to target the copula correlation
matrix, which can be done as before using
• With this matrix preestimated we will only be searching for
the parameter d in the maximization of lnLg earlier
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Other Copula Models
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• An asymmetric t copula can be developed from the
asymmetric multivariate t distribution
• Only a few copula functions are applicable when the
number of assets, n, is large
• So far we have assumed that the copula correlation matrix,
*, is constant across time
• However, we can let the copula correlations be dynamic
using the DCC approach
• We would now use the copula shocks z*i,t as data input into
the estimation of the dynamic copula correlations instead of
the zi,t
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Figure 9.7: Simulated Threshold Correlations from the
Symmetric t Copula with Various Parameters
Norma l Copul a , * = 0.5
3
3
2
2
1
1
0
0
-1
-1
-2
-2
-3
-3
-4
-4
-3
-2
-1
0
1
2
t Copul a , * = 0.5, d = 10
4
z2
z2
4
3
-4
-4
4
-3
-2
-1
z1
Skewed t Copul a , * = 0.5, d = 10, = -0.5
4
3
3
2
2
1
1
0
0
-1
-1
-2
-2
-3
-3
-4
-4
-3
-2
-1
0
z1
0
1
2
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z1
z2
z2
4
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1
2
3
4
-4
-4
Skewed t Copul a , * = 0.5, d = 10, = +0.5
-3
-2
-1
0
1
2
z1
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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4
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Risk Management Using Copula Models
• To compute portfolio VaR and ES from copula models, we
need to rely on Monte Carlo simulation
• Monte Carlo simulation essentially reverses the steps taken
in model building
• Recall that we have built the copula model from returns as
follows:
• First, estimate a dynamic volatility model, i,t, on each
asset to get from observed return Ri,t to shock zi,t = ri,t / i,t
• Second, estimate a density model for each asset to get the
probabilities ui,t = Fi(zi,t) for each asset
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Risk Management Using Copula Models`
• Third, estimate the parameters in the copula model using
lnLg = Tt=1ln g(u1,t,…,un,t)
• When we simulate data from copula model we need to
reverse steps taken in the estimation of the model
• We get the algorithm:
• First, simulate the probabilities (u1,t,…,un,t) from the copula
model
• Second, create shocks from the copula probabilities using
the marginal inverse CDFs zi,t = F-1i(ui,t) on each asset
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Risk Management Using Copula Models
• Third, create returns from shocks using the dynamic
volatility models, ri,t = i,tzi,t on each asset
• Once we have simulated MC vectors of returns from the
model we can easily compute the simulated portfolio
returns using a given portfolio allocation
• The portfolio VaR, ES, and other measures can then be
computed on the simulated portfolio returns
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Integrated Risk Management
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• Integrated risk management is concerned with the
aggregation of risks across different business units within
an organization.
• Senior management needs a method for combining
marginal distributions of returns in each business unit
• In the simplest case, we can assume that the multivariate
normal model gives a good description of the overall risk of
the firm.
• If the correlations between all the units are one then we get
a very simple result.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Integrated Risk Management
• Consider first the bivariate case
• where we have assumed the weights are positive
• The total VaR is simply the (weighted) sum of the two
individual business unit VaRs under these specific
assumptions.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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