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Figure 9.7: Simulated Threshold Correlations from the Symmetric t Copula with Various Parameters

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



46



t Copula



47



• 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



48



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



49



• 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



3



4



z1



z2



z2



4



50



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



3



4



51



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



52



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



53



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



54



• 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.

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55



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Figure 9.7: Simulated Threshold Correlations from the Symmetric t Copula with Various Parameters

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