ANTICIPATING CORRELATIONS

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ANTICIPATING CORRELATIONS

• Robert Engle
• Stern School of Business

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Correlation

• Correlations for Life
• What is the correlation between thunder and rain?
• What is the correlation between exercise and
  health?
• What is the correlation between happiness and
  good food?

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Correlations for Risk

• Stock returns are correlated
• Stocks in one country are correlated with stocks
  in another
• Bond returns on one firm or country or maturity
  are generally correlated with returns on others
• But stock and bond returns sometimes appear
  uncorrelated
• The risk of a portfolio is greater if all the
  assets are highly correlated. It may go down (or
  up) further, if they all move together.

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QUOTATIONS

• It is not the biggest, the brightest or the best
  that will survive, but those who adapt the
  quickest. Charles Darwin
• The secret of life is to be interested in one
  thing profoundly and a thousand things well.
  Henry Walpole
• Studies of high school graduates rarely find any
  correlation between recognition in high school
  and recognition thereafter.

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ANTICIPATING CORRELATIONS

• Can we anticipate future correlations?
• How and why do correlations change over time?
• How can we get the best estimates of correlations
  for financial decision making?

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CORRELATIONS WHAT ARE THEY?

• CORRELATIONS MEASURE THE DEGREE TO WHICH TWO
  SERIES MOVE TOGETHER
• THEORETICAL DEFINITION

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10 YEARS OF LARGE CAP STOCKS
AXP JPM INTC MSFT MRK
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DAILY CORRELATIONS
AXP JPM INTC MSFT MRK AXP  1.000000  0.5
54172  0.285812  0.283375  0.224685 JPM  0.554172
 1.000000  0.318260  0.310113  0.228688 INTC  0.
285812  0.318260  1.000000  0.551379  0.130294 MS
FT  0.283375  0.310113  0.551379  1.000000  0.1860
04 MRK  0.224685  0.228688  0.130294  0.186004  1
.000000
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WEEKLY EQUITY CORRELATIONS 1987-2002
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WHY DO WE NEED CORRELATIONS?
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WHY DO WE NEED CORRELATIONS?

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

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DIVERSIFICATION

• Diversified portfolios have lower variance and
  risk because some assets go one direction while
  others go the opposite.
• There are many thousands of possible stocks,
  bonds and other assets to invest in. Can we
  reduce the risk to zero?
• Clearly not. Assets are not uncorrelated.

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PORTFOLIO RISK

• Portfolio risk depends upon the volatilities and
  correlations of all the components.
• For weights w and covariance matrix Omega

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FINDING THE OPTIMAL PORTFOLIO

• Minimize portfolio variance subject to a required
  return. The Markowitz Problem

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ARE CORRELATIONS TIME VARYING?

• YES
• WHY?
• Because the business practice of the companies
  changes
• Because shocks to the economy affect all
  businesses
• Because shocks to one part of the economy will
  affect only some businesses

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CONDITIONAL CORRELATIONS

• DEFINE BOTH COVARIANCES AND VARIANCES CONDITIONAL
  ON CURRENT INFORMATION

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ESTIMATION

• HISTORICAL CORRELATIONS
• Use a rolling window of N observations for both
  covariances and variances. We will use 100 days.
• DYNAMIC CONDITIONAL CORRELATION or DCC
• Estimates conditional correlations by first
  adjusting for differing variances and then
  updating correlations as new information is
  received.

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100 day historical correlations between AXP and GE
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GENERAL ELECTRIC PROFITS
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CHANGING EXTERNAL EVENTS

• CONSIDER FORD AND HONDA IN 2000
• CORRELATIONS MAY HAVE CHANGED BECAUSE OF CHANGING
  ENERGY PRICES.

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EXTEND GARCH CONFIDENCE INTERVALS
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IMPLICATIONS

• On Jan 1 2000 the market prices of Ford and Honda
  reflected the best analysis of the financial
  markets
• What would happen to energy prices?
• What would happen to the economy?
• What choices would management make?
• Five years later, Ford stock was down and Honda
  was up.
• The market rewarded the company that was prepared
  for higher energy prices.

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HISTORICAL CORRELATIONS
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USE SOME KIND OF MODEL

• ONE FACTOR MODEL
• MANY FACTOR MODEL
• MULTIVARIATE GARCH
• DYNAMIC CONDITIONAL CORRELATION

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MULTIVARIATE MODELS
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Dynamic Conditional Correlation

• DCC is a new type of multivariate GARCH model
  that is particularly convenient for big systems.
  See Engle(2002) or Engle(2005).

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DYNAMIC CONDITIONAL CORRELATION OR DCC

• Estimate volatilities for each asset and compute
  the standardized residuals or volatility adjusted
  returns.
• Estimate the time varying covariances between
  these using a maximum likelihood criterion and
  one of several models for the correlations.
• Form the correlation matrix and covariance
  matrix. They are guaranteed to be positive
  definite.

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HOW IT WORKS

• When two assets move in the same direction, the
  correlation is increased slightly.
• This effect may be stronger in down markets
  (asymmetry in correlations).
• When they move in the opposite direction it is
  decreased.
• The correlations often are assumed to only
  temporarily deviate from a long run mean
• UPDATING IS THE CENTRAL FEATURE

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CORRELATIONS UPDATE LIKE GARCH

• Approximately,

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DCC Correlations AXP and GE
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FACTOR MODELS

• One or more factors influence all assets
• Some assets are more affected by a particular
  factor than others
• Sometimes the factors have little volatility and
  therefore have little influence

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ONE FACTOR ARCH

• One factor model such as CAPM
• There is one market factor with fixed betas and
  constant variance idiosyncratic errors
  independent of the factor. The market has some
  type of ARCH with variance .
• If the market has asymmetric volatility, then
  individual stocks will too.

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MARKET VOLATILITY
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CALCULATE DYNAMIC CORRELATIONS

• When market volatility is high then correlations
  are high. The market/economy in general
  influences both stocks positively.

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AXP AND GE AGAIN
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CORRELATION OF EXTREMES

• How correlated are extreme returns?
• Bankruptcy is an extreme event and corresponds to
  an extremely large negative stock return over a
  period of time.
• Are bankruptcies correlated?

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CREDIT RISK APPLICATION

• This one factor model is the basis of a new
  credit risk model that I have been developing
  with a graduate student and hedge fund quant.
• How correlated are loan defaults?
• When the aggregate market is very low, the
  probability of default is greater for all
  companies. When it is high, the probability of
  default is low for all companies. Hence defaults
  are correlated and the distribution of market
  returns tells how much.

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ASYMMETRY IN MARKET RETURNS

• Aggregate market returns have negative skewness,
  particularly for long horizon returns. Elsewhere
  I have shown that this is due to asymmetric
  volatility.
• Negative skewness in market returns means that
  large declines can happen with the associated
  credit events.

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EXAMINING THE ONE FACTOR MODEL OF CORRELATIONS
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HOW WELL DOES THIS WORK?

• Examine 18 large cap stocks in the US.
• Calculate correlations either historically or
  with Dynamic Conditional Correlation (DCC)
• Relate these correlations to the volatility of
  SP500.
• Does High market volatility mean high correlation?

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RESULTS
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PLOT

• About 30 Correlations of these large cap stocks
  on left axis
• Estimated with DCC not using market data
• Compare with a GARCH of the SP500 plotted on
  right axis

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SP volatility
Correlations
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MEAN CORRELATION AND MARKET VOLATILITY
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REGRESSION

• Dependent Variable MEANCOR9F
• Method Least Squares
• Date 09/10/06 Time 2000
• Sample 1/04/1994 12/31/2004
• Included observations 2770
• Variable Coefficient Std. Error t-Statistic
• C 0.176566 0.003343 52.81508
• V9_SPRET 9.600815 0.296987 32.32740

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REGRESSION IN DIFFERENCES

• Dependent Variable D(MEANCOR9F)
• Method Least Squares
• Date 09/09/06 Time 1137
• Sample (adjusted) 1/06/1994 12/31/2004
• Included observations 2768 after
  adjustments
• Convergence achieved after 4 iterations
• Newey-West HAC Standard Errors Covariance (lag
  truncation8)
• Variable Coefficient Std. Error t-Statistic Prob. 
   
• C -2.57E-06 9.18E-05 -0.028054 0.9776
• D(V9F_SPRET) 7.755417 0.612757 12.65660 0.0000
• AR(1) 0.070129 0.023881 2.936653 0.0033

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FINDINGS

• MARKET VOLATILITY IS PART OF THE STORY
• THE CURRENT DECLINE IN MARKET VOLATILITY HAS NOT
  LEAD TO THE EXPECTED DROP IN CORRELATIONS.

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ANTICIPATING CORRELATIONS

• FORECASTING FACTOR VOLATILITIES IS PART OF THE
  ANSWER
• HOW CAN WE MAKE THIS WORK BETTER?
• Research Agenda!
• Build DCC models on the residuals
• Build Factor DCC models

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HOW DO WE FORECAST FACTOR VOLATILITIES?

• USE GARCH MODELS OR SIMILAR MODELS FOR SHORT RUN
  FORECASTS.
• USE NEW MULTI-COUNTRY RESULTS USING THE SPLINE
  GARCH FOR LONG RUN MACRO BASED FORECASTS.

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SPLINE GARCH FOR LOW FREQUENCY VOLATILITY AND ITS
MACROECONOMIC CAUSES

• Engle and Rangel
• Model the daily volatility of many country equity
  returns
• Extract a low frequency component using the
  spline
• Model how this component depends on the
  macroeconomy

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MULTIPLE REGRESSIONS
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ANTICIPATING CORRELATIONS

• To forecast correlations, we must forecast the
  volatility of the factors that influence the
  companies.
• When volatility is forecast to be high, then
  correlations will be high.
• Inflation, slow growth, macroeconomic instability
  forecast high market volatility.
• This does not work well when companies are
  changing their business. May need to update
  residual correlations using factor DCC.

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