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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AXP JPM INTC MSFT MRK 10 YEARS OF LARGE CAP STOCKS
WEEKLY EQUITY CORRELATIONS 1987-2002 USITALYFRANCEJAPANHONG
KONG US 0.2230.4650.2230.308 ITALY 0.5370.2370.269 FRANCE
0.3400.347 JAPAN 0.229
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WHY DO WE NEED CORRELATIONS?
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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 1.Estimate volatilities
for each asset and compute the standardized residuals or volatility
adjusted returns. 2.Estimate the time varying covariances between
these using a maximum likelihood criterion and one of several
models for the correlations. 3.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 S&P500. 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 S&P500 plotted on right axis
REGRESSION IN DIFFERENCES Dependent Variable: D(MEANCOR9F)
Method: Least Squares Date: 09/09/06 Time: 11:37 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 truncation=8) VariableCoefficientStd.
Errort-StatisticProb. C-2.57E-069.18E-05-0.0280540.9776
D(V9F_SPRET)7.7554170.61275712.656600.0000
AR(1)0.0701290.0238812.9366530.0033
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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.