DYNAMIC CONDITIONAL CORRELATION MODELS OF TAIL DEPENDENCE

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DYNAMIC CONDITIONAL CORRELATION MODELS OF TAIL
DEPENDENCE

• Robert Engle
• NYU Stern
• DEPENDENCE MODELING FOR CREDIT PORTFOLIOS
• Venice 2003

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Two Frontiers

• We are celebrating over 20 years of research in
  Volatility Modeling
• The simple GARCH(1,1) has transformed our risk
  measurement
• And there are many many extensions
• Multivariate Volatility
• High Frequency or Real Time Volatility

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MULTIVARIATE

• Multivariate GARCH has never been widely used
• Asset allocation and risk management problems
  require large covariance matrices
• Credit Risk now also requires big correlation
  matrices to accurately model loss or default
  correlations

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WHERE DO WE USE CORRELATIONS?

• TWO CANONICAL PROBLEMS
• Forecasting risk and Forming optimal portfolios
• Pricing derivatives on multiple underlyings
• Credit Risk uses both of these tools

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Joint Density
P2,T
P1,T
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APPLICATIONS

• Portfolio Value at Risk
• joint empirical distribution
• Option payoff
• joint risk neutral distribution
• Payoff of option that both assets are below
  strikes
• Probability of one or more Defaults
• joint empirical distribution of firm value
• Pricing Credit Derivatives
• joint risk neutral distribution of firm value

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P2,T
P1,T
P1,T lt P1,0 -VaR
Probability that the portfolio looses more than K
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P2,T
K1
Put Option on asset 1 Pays
P1,T
K2
Option on asset 2 Pays
Both options Payoff
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OPTIONS

• Value of each option depends only on the marginal
  risk neutral distribution
• Correlation between the payoffs depends on the
  joint distribution.
• Optimal portfolios including options
• Value an option that pays only when both are in
  the money.

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CREDIT RISK

• Credit Risk correlation is like this problem
  where Ks are default points and prices are firm
  values
• Credit Default Swaps (CDS) are like options
  (written puts) in firm value.
• Credit Default Obligations (CDO) in lower
  tranches are minima of sums of firm values.

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Symmetric Tail Dependence
P2,T
P1,T
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Lower Tail Dependence
P2,T
P1,T
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P2,T
K1
Put Option on asset 1 Pays
P1,T
K2
Option on asset 2 Pays
Both options Payoff
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Joint Distribution

• Under joint log normality, these probabilities
  can be calculated
• Under other distributions, simulations are
  required
• Copulas are a new way to formulate such joint
  density functions
• How to parameterize a Copula to match this
  distribution?

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JOINT DISTRIBUTIONS

• Dependence properties are all summarized by a
  joint distribution
• For a vector of kx1 random variables Y with
  cumulative distribution function F
• Assuming for simplicity that it is continuously
  differentiable, then the density function is

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UNIVARIATE PROPERTIES

• For any joint distribution function F, there are
  univariate distributions Fi and densities fi
  defined by
• is a uniform random variable on
  the interval (0,1)
• What is the joint distribution of

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COPULA

• The joint distribution of these uniform random
  variables is called a copula
• it only depends on ranks and
• is invariant to monotonic transformations.
• Equivalently

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COPULA DENSITY

• Again assuming continuous differentiability, the
  copula density is
• From the chain rule or change of variable rule,
  the joint density is the product of the copula
  density and the marginal densities

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Tail Dependence

• Upper and lower tail dependence
• For a joint normal, these are both zero!

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DEFAULT CORRELATIONS

• Let Ii be the event that firm i defaults,
• Then the default correlation is the correlation
  between I1 and I2 which can be computed
  conditional on todays information set.
• If the probability of default for each firm is ?
  , then

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Default Correlations and Tail Dependence

• When defaults are unlikely, these are related to
  the tail dependence measure
• Take the limit as ? becomes small
• Under normality or independence, the limiting
  default correlation is zero
• Under lower tail dependence it is positive.

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Asset Allocation

• Optimal portfolios will be affected by such
  asymmetries.
• The diversification is not as great in a down
  market as it is in an up market. Thus the risk
  is greater than implied by an elliptical
  distribution with the same correlation.
• The optimal portfolio here would probably hold
  more of the riskless asset.

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DYNAMIC CORRELATIONS

• A joint distribution can be defined for any
  horizon. Long horizon distributions can be built
  up from short horizons
• Multivariate GARCH gives many possible models for
  daily correlations. The implied multi-period
  distribution will generally show symmetric tail
  dependence
• Special asymmetric multivariate GARCH models give
  greater lower tail dependence.

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TWO PERIOD RETURNS

• Two period return is the sum of two one period
  continuously compounded returns
• Look at binomial tree version
• Asymmetry gives negative skewness

Low variance
High variance
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Two period Joint Returns

• If returns are both negative in the first period,
  then correlations are higher.
• This leads to lower tail dependence

Up Market
Down Market
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Dynamic Conditional Correlation

• DCC model is a new type of multivariate GARCH
  model that is particularly convenient for big
  systems. See Engle(2002) or Engle(2004)
• Motivation the conditional correlation of two
  returns with mean zero is

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DCC

• Then defining the conditional variance and
  standardized residual as
• All the volatilities cancel, giving

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DCC

• The DCC method first estimates volatilities for
  each asset and computes the standardized
  residuals.
• It then estimates the covariances between these
  using a maximum likelihood criterion and one of
  several models for the correlations
• The correlation matrix is guaranteed to be
  positive definite

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GENERAL SPECIFICATION

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DCC Example

• Let epsilon be an nx1 vector of standardized
  residuals
• Then let
• And
• The criterion to be maximized is

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DCC Details

• This is a two step estimator because the
  volatilities are estimated rather than known
• It uses correlation targeting to estimate the
  intercept.
• There are only two correlation parameters to
  estimate by MLE no matter how big the system.
• On average the correlations will be the same as
  in the data.

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Intuition and Asymmetry

• More specifically
• So that correlations rise when returns move
  together and fall when they move opposite.
• By adding another term we can allow them to rise
  more when both returns are falling than when they
  are both rising.

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DCC and the Copula

• A symmetric DCC model gives higher tail
  dependence for both upper and lower tails of the
  multi-period joint density.
• An asymmetric DCC gives higher tail dependence
  in the lower tail of the multi-period density.

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REFERENCES

• Engle, 2002, Dynamic Conditional Correlation-A
  Simple Class of Multivariate GARCH Models,
  Journal of Business and Economic Statistics
• Bivariate examples and Monte Carlo
• Engle and Sheppard, 2002 Theoretical and
  Empirical Properties of Dynamic Conditional
  Correlation Multivariate GARCH, NBER Discussion
  Paper, and UCSD DP.
• Models of 30 Dow Stocks and 100 SP Sectors
• Cappiello, Engle and Sheppard, 2002, Asymmetric
  Dynamics in the Correlations of International
  Equity and Bond Returns, UCSD Discussion Paper
• Correlations between 34 International equity and
  bond indices

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TERM STRUCTURE OF DEFAULT CORRELATIONS

• SIMULATE FIRM VALUES, AND CALCULATE DEFAULTS
• VARIOUS ASSUMPTIONS CAN BE MADE.
• FOR EXAMPLE, ONCE DEFAULT OCCURS, FIRM REMAINS IN
  DEFAULT
• OR, FIRMS CAN EMERGE FROM DEFAULT FOLLOWING THE
  SAME PROCESS
• OR, CALCULATE THE HAZARD RATE PROBABLILITY OF
  DEFAULT GIVEN NO DEFAULT YET.
• LOSS COULD BE STATE DEPENDENT OR DEPEND ON HOW
  FAR BELOW THRESHOLD THE VALUE GOES

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UPDATING

• As each day passes, the remaining time before
  maturity of a credit derivative will be shorter
  and the joint distribution of the outcome will
  have changed.
• How do you update this distribution?
• How do you hedge your position?

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Two period Joint Returns
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CONCLUSIONS

• Dynamic Correlation models give a flexible
  strategy for modeling non-normal joint density
  functions or copulas.
• Updating can be used to re-price or re-hedge
  positions
• The model can be of equity values or firm values
  and risk neutral or empirical measures, depending
  on the application.

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Data

• Weekly returns Jan 1987 to Feb 2002 (785
  observations)
• 21 Country Equity Series from FTSE All-World
  Index
• 13 Datastream Benchmark Bond Indices with 5 years
  average maturity

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Europe BELGIUM DENMARK FRANCE GERMANY IRELAND AUSTRIA ITALY THE NETHERLANDS SPAIN SWEDEN SWITZERLAND NORWAY UNITED KINGDOM Australasia AUSTRALIA HONG KONG JAPAN NEW ZEALAND SINGAPORE
Europe BELGIUM DENMARK FRANCE GERMANY IRELAND AUSTRIA ITALY THE NETHERLANDS SPAIN SWEDEN SWITZERLAND NORWAY UNITED KINGDOM Americas CANADA MEXICO UNITED STATES
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GARCH Models(asymmetric in orange)

• GARCH
• AVGARCH
• NGARCH
• EGARCH
• ZGARCH
• GJR-GARCH
• APARCH
• AGARCH
• NAGARCH

• 3EQ,8BOND
• 0
• 1BOND
• 6EQ,1BOND
• 8EQ,1BOND
• 3EQ,1BOND
• 0
• 1EQ,1BOND
• 0

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Parameters of DCC Asymmetry in red (gamma) and
Symmetry in blue (alpha)
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RESULTS

• Asymmetric Correlations correlations rise in
  down markets
• Shift in level of correlations with formation of
  Euro
• Equity Correlations are rising not just within
  EMU-Globalization?
• EMU Bond correlations are especially high-others
  are also rising