DYNAMIC CONDITIONAL CORRELATION MODELS OF TAIL DEPENDENCE Robert Engle NYU Stern DEPENDENCE MODELING FOR CREDIT PORTFOLIOS Venice 2003

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

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

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

P 1,T P 2,T Joint Density

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

P 1,T P 2,T Probability that the portfolio looses more than K P 1,T < P 1,0 -VaR

P 1,T P 2,T K1K1 K2K2 Put Option on asset 1 Pays Option on asset 2 Pays Both options Payoff

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.

CREDIT RISK Credit Risk correlation is like this problem where K’s 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.

P 1,T P 2,T Symmetric Tail Dependence

P 1,T P 2,T Lower Tail Dependence

P 1,T P 2,T K1K1 K2K2 Put Option on asset 1 Pays Option on asset 2 Pays Both options Payoff

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?

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:

UNIVARIATE PROPERTIES For any joint distribution function F, there are univariate distributions F i and densities f i defined by: is a uniform random variable on the interval (0,1) What is the joint distribution of

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

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

Tail Dependence Upper and lower tail dependence: For a joint normal, these are both zero!

DEFAULT CORRELATIONS Let I i be the event that firm i defaults, Then the default correlation is the correlation between I 1 and I 2 which can be computed conditional on today’s information set. If the probability of default for each firm is , then:

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.

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.

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.

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 High variance Low variance

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

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:

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

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

GENERAL SPECIFICATION

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

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.

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.

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.

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 S&P 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

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

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?

Two period Joint Returns

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.

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

Europe  BELGIUM*  DENMARK*  FRANCE*  GERMANY*  IRELAND*  AUSTRIA*  ITALY  THE NETHERLANDS*  SPAIN  SWEDEN*  SWITZERLAND*  NORWAY  UNITED KINGDOM* Australasia  AUSTRALIA  HONG KONG  JAPAN*  NEW ZEALAND  SINGAPORE Americas  CANADA*  MEXICO  UNITED STATES*

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

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

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