HIGH DIMENSION DYNAMIC CORRELATIONS

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HIGH DIMENSION DYNAMIC CORRELATIONS

• By
• Robert Engle
• Stern School of Business

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FINANCIAL FORECASTING

• Forecasting Risk
• In high dimension systems this means forecasting
  volatilities and correlations
• Predictive Failure?
• Black Swans?

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WHY DO WE NEED CORRELATIONS?

• CALCULATE PORTFOLIO RISK
• FORM OPTIMAL PORTFOLIOS
• PRICE, HEDGE, AND TRADE DERIVATIVES

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ARE CORRELATIONS TIME VARYING?

• Yes Correlations are time varying
• Derivative prices of correlation sensitive
  products imply changes.
• Derivatives on correlation now are traded.
• Time series estimates change. There are many
  varieties.

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GENERAL ELECTRIC PROFITS
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CHANGING EXTERNAL EVENTS

• CONSIDER BOEING AND GENERAL MOTORS
• CORRELATIONS MAY HAVE CHANGED BECAUSE OF CHANGING
  ENERGY PRICES. ENERGY PRICE VOLATILITY WILL MAKE
  THE RETURNS MORE CORRELATED.

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Dynamic Conditional Correlation

• A simple approach to estimating and forecasting
  large covariance matrices
• Engle(2002), (200?)
• Based on standardized returns s

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TWO SPECIFICATIONS

• Where in the integrated model
• And in the mean reverting model

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CORRELATION TARGETING FOR THE INTERCEPT

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ESTIMATION

• Two steps
• GARCH on returns
• MLE on standardized returns

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OOps

• This is not a good solution for very large
  systems
• Inversion of R many many times is slow
• Downward bias in alpha for large systems- see
  Engle and Sheppard
• Possibly need more structure

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THE MACGYVER METHOD
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A SOLUTION

• Estimate all bivariate models. Each should be
  consistent for the same parameters.
• Combine these estimates
• Calculate all the correlations

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MONTE CARLO

• Generate 1000 observations on n dimensional
  systems which are gaussian DCC with unit
  conditional variance.
• Construct with the following parameters
• N3,5,10,20,30,50

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MLE ESTIMATION

• Bivariate Log likelihood function
• Although the parameters should be positive and
  sum to less than one, these restrictions are not
  imposed.

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RESTRICTED MLE ESTIMATION

• Log likelihood function
• To constrain the sum of alpha and beta to be less
  than one a penalty is assigned whenever they
  exceed one.

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ESTIMATORS

• Mean, Median and trimmed Mean where the 5
  largest and smallest values are deleted before
  taking the mean.
• When n3, median and trimmed mean are the same.
• The estimators are computed for both restricted
  and unconstrained Bivariate MLE.

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EXPERIMENTS
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RMS ERRORS ON BETA
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RMS ERRORS ON ALPHA
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BIAS ON BETA
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BIAS ON ALPHA
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BIAS OF MEDIAN ESTIMATOR
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ADVANTAGES OF THIS APPROACH

• Requires only bivariate estimation
• Does not require the same number of observations
  on all assets
• Robust to occasional non-convergence and
  diverging estimates
• Potential to investigate validity of DCC
  assumptions relative to block DCC, GDCC
• Possible to estimate only a subset of pairs and
  apply parameters to all.

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DYNAMIC EQUICORRELATION

• Robert Engle and Bryan Kelly

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Dynamic EquiCOrrelation (DECO)

• Suppose all pairs of assets have the same
  correlation, but that this changes over time. It
  is like the average correlation.
• Can you estimate this directly?

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PLAUSIBILITY?

• Elton and Gruber use this model for Asset
  Allocation. See their textbook.
• Derivatives are priced with a single average
  correlation
• Credit Risk often assumes homogeneity
• In many cases this can be interpreted as an
  average correlation
• A block equicorrelation model allows more
  flexibility.

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ADVANTAGES

• The likelihood is very simple
• It can be estimated about as fast as GARCH no
  matter how many assets there are.
• It is easy to interpret and analyze.

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Estimation

• Matrix Results

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Estimation

• Assuming Normality and examining the second step
  as in DCC

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UPDATING

• Each period, new information becomes available on
  the equicorrelation. We specify this in several
  ways.
• Where U can be various measures. i.e.

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FACTOR DCC

• And its special cases
• FACTOR ARCH
• FACTOR DOUBLE ARCH

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FACTOR ARCH

• Introduced by Engle Ng and Rothschild(1990)(1992)
• Or, assuming normality

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ESTIMATION OF FACTOR ARCH

• MLE is estimation by OLS to get the betas
• GARCH estimation of the factor
• Correlations are calculated from

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FACTOR DOUBLE ARCH

• Idiosyncrasies are also GARCH models
• or

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ESTIMATION OF FACTOR DOUBLE ARCH

• Regress returns on the factor assuming the errors
  are GARCH. This is MLE by weak exogeneity
  argument.
• Model the factor volatility with GARCH
• Calculate correlations

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FACTOR DCC

• Model the residuals of the FACTOR DOUBLE ARCH and
  the residuals of the factor with DCC
• Two step method is not MLE
• If residual correlations are all zero then there
  is no change from FACTOR DOUBLE ARCH

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COVARIANCE MATRIX

• Now four terms in covariance matrix
• and correlations are given by

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FEATURES

• This model has time varying correlations between
  the idiosyncrasies.
• This is what would be observed if there are
  multiple factors with time varying volatilities
• This model has time varying covariances with the
  factor that is time varying betas.

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PERFORMANCE
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DATA

• 18 LARGE CAP STOCKS AND SP500
• aa, axp, ba, cat, dd, dis, ge, gm, ibm, ip, jnj,
  jpm, ko, mcd, mmm, mo, mrk, msft
• 1994-2004 DAILY FOR 2771 Obs

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MacGyver

• Use TARCH or GJR-GARCH for all 18
• Estimate 153 bivariate models without
  restrictions
• Median alpha0.0157
• Median beta. 9755

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Bivariate Alphas
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Bivariate Betas
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RESULTS

• 153 TIME SERIES
• LOOK AT AVERAGE CORRELATION BY DCC AND BY 100 DAY
  HISTORICAL OR MOVING AVERAGE

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FACTOR DCC

• Estimate DCC on residuals from FACTOR DOUBLE ARCH
• Use MacGyver method again
• Median alpha.009
• Median beta .925
• Less persistent and less responsive to shocks
  than for DCC.

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UPDATING

• USING PARAMETER VALUES ESTIMATED FROM 1994-2004,
  CALCULATE CORRELATIONS THROUGH SEPTEMBER 14,2007
• WHAT DO THE CORRELATIONS SHOW THROUGH THE SUMMER
  OF 2007 WITH THE MARKET TURBULENCE?

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WHICH CORRELATION ESTIMATE IS MOST ACCURATE?

• CONSIDER AN ASSET ALLOCATION SOLUTION.
• FOR EACH PAIR OF STOCKS FORM THE MINIMUM VARIANCE
  PORTFOLIO AND SEE WHICH COVARIANCE ESTIMATOR
  ACHIEVES THE SMALLEST VARIANCE.

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WHICH HEDGE GIVES THE BEST LONG - SHORT PORTFOLIO?

• For each pair of assets hold one long and take a
  short position in the other to minimize variance.

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Average standard deviation of the minimum
variance portfolio of all pairs of stocks
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Average standard deviation of the minimum
variance long - short portfolio of all pairs of
stocks
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WINNING FRACTIONS MV
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WINNING FRACTIONS HEDGE
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