ANTICIPATING
CORRELATIONS
Robert Engle
Stern School of Business
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?
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.
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.”
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?
CORRELATIONS – WHAT ARE
THEY?
• CORRELATIONS MEASURE THE
DEGREE TO WHICH TWO SERIES
MOVE TOGETHER
• THEORETICAL DEFINITION:
Let r1 and r2 be mean zero random variables, then
1,2 
E  r1r2 
E  r12  E  r22 
, and
E  r1r2   1,2 E  r12  E  r22 
4
3
3
2
2
1
1
Y_50
Y_00
4
0
0
-1
-1
-2
-2
-3
-3
-4
-4
-4
-3
-2
-1
0
1
2
3
4
-4
-3
-2
-1
1
2
3
4
1
2
3
4
X
4
4
3
3
2
2
1
1
Y__50
Y_90
X
0
0
0
-1
-1
-2
-2
-3
-3
-4
-3
-2
-1
0
X
1
2
3
4
-4
-3
-2
-1
0
X
.15
AXP
.05
.00
-.05
-.10
-.15
.15
.10
JPM
.05
.00
-.05
-.10
-.15
.20
.15
.10
INTC
10 YEARS OF
LARGE CAP STOCKS
.10
.05
.00
-.05
-.10
-.15
.2
AXP
MSFT
JPM
.1
.0
-.1
INTC
-.2
.12
MSFT
.08
MRK
MRK
.04
.00
-.04
-.08
-.12
-.15
-.10
-.05
.00
AXP
.05
.10
.15 -.15
-.10
-.05
.00
JPM
.05
.10
.15 -.15 -.10 -.05
.00
.05
INTC
.10
.15
.20 -.2
-.1
.0
MSFT
.1
.2 -.15
-.10
-.05
.00
MRK
.05
.10
DAILY CORRELATIONS
AXP
JPM
INTC
MSFT
MRK
AXP
JPM
INTC
MSFT
MRK
1.000000
0.554172
0.285812
0.283375
0.224685
0.554172
1.000000
0.318260
0.310113
0.228688
0.285812
0.318260
1.000000
0.551379
0.130294
0.283375
0.310113
0.551379
1.000000
0.186004
0.224685
0.228688
0.130294
0.186004
1.000000
T3
MONTH
T5
YRRET
T20
YRRET
CAN$
POUND$
AUS$
YEN$
SP500
T3MONTH
1.000
0.329
0.206
0.011
0.076
0.025
0.031
-0.031
T5YRRET
0.329
1.000
0.875
-7E-04
0.136
0.007
0.005
-0.057
T20YRRET
0.206
0.875
1.000
0.007
0.103
-0.002
-0.049
-0.016
CAN$
0.011
-7E-04
0.007
1.000
0.117
0.415
0.145
0.015
POUND$
0.076
0.136
0.103
0.117
1.000
0.253
0.224
-0.018
AUS$
0.025
0.007
-0.002
0.415
0.253
1.000
0.269
0.040
YEN$
0.031
0.005
-0.049
0.145
0.224
0.269
1.000
-0.003
SP500
-0.031
-0.057
-0.016
0.015
-0.018
0.040
-0.003
1.000
WEEKLY EQUITY CORRELATIONS
1987-2002
US
US
ITALY
FRANCE
JAPAN
ITALY
FRANCE
JAPAN
HONG
KONG
0.223
0.465
0.223
0.308
0.537
0.237
0.269
0.340
0.347
0.229
WHY DO WE NEED
CORRELATIONS?
WHY DO WE NEED
CORRELATIONS?
• CALCULATE PORTFOLIO RISK
• FORM OPTIMAL PORTFOLIOS
• PRICE, HEDGE, AND TRADE
DERIVATIVES
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.
PORTFOLIO RISK
• Portfolio risk depends upon the
volatilities and correlations of all the
components.
• For weights w and covariance matrix
Omega
  w ' w
2
P
FINDING THE OPTIMAL
PORTFOLIO
• Minimize portfolio variance subject to a
required return. “The Markowitz Problem”
With covariance matrix  and expected
excess returns above a riskless rate of 
min w ' w
s . t . w '   0
 
w
0
1
 ' 
1
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
CONDITIONAL CORRELATIONS
• DEFINE BOTH COVARIANCES AND
VARIANCES CONDITIONAL ON
CURRENT INFORMATION
1,2,t 
Et 1  r1,t r2,t 
Et 1  r1,2t  Et 1  r22,t 
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.
100 day historical correlations
between AXP and GE
.9
.8
.7
.6
.5
.4
.3
.2
.1
.0
94
95
96
97
98
99
00
01
C100_AXP_GE
02
03
04
GENERAL ELECTRIC PROFITS
CHANGING EXTERNAL EVENTS
• CONSIDER FORD AND HONDA IN 2000
• CORRELATIONS MAY HAVE CHANGED
BECAUSE OF CHANGING ENERGY
PRICES.
10.0
5.0
4.0
3.0
2.5
2.0
1.5
1.0
0.5
1990 1992 1994 1996 1998 2000 2002 2004
FNBEFORE
HNBEFORE
EXTEND GARCH CONFIDENCE
INTERVALS
50.0
30.0
20.0
15.0
10.0
5.0
3.0
2.0
1.5
1.0
0.5
1990
1992
1994
1996
1998
FNBEFORE
HNBEFORE
FNUP
2000
2002
FNDOWN
HNUP
HNDOWN
2004
50.0
30.0
20.0
15.0
10.0
5.0
3.0
2.0
1.5
1.0
0.5
1990
1992
1994
FNBEFORE
HNBEFORE
FNUP
1996
1998
2000
FNDOWN
HNUP
HNDOWN
2002
2004
FNAFTER
HNAFTER
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.
HISTORICAL CORRELATIONS
.6
.5
.4
.3
.2
.1
.0
-.1
-.2
-.3
1990 1992 1994 1996 1998 2000 2002 2004 2006
C100_FORDR_HONDAR
USE SOME KIND OF MODEL
• ONE FACTOR MODEL
• MANY FACTOR MODEL
• MULTIVARIATE GARCH
• DYNAMIC CONDITIONAL
CORRELATION
MULTIVARIATE
MODELS
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).
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.
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
CORRELATIONS UPDATE LIKE
GARCH
• Approximately,
t    1,t 1 2,t 1  t 1


1  
DCC Correlations AXP and GE
.9
.8
.7
.6
.5
.4
.3
.2
.1
.0
95
96
97
98
99
00
01
C9_AXP_GE
02
03
04
.9
.8
.7
.6
.5
.4
.3
.2
.1
.0
95
96
97
C100_AXP_GE
98
99
00
01
C4_AXP_GE
02
03
04
C9_AXP_GE
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
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
2

some type of ARCH with variance m ,t.
ri ,t   i rm ,t  e i ,t
 i ,i ,t   i2 m2 ,t  i
• If the market has asymmetric volatility, then
individual stocks will too.
MARKET VOLATILITY
.030
.025
.020
.015
.010
.005
.000
94
95
96
97
98
99
00
01
02
Conditional Standard Deviation
03
04
CALCULATE DYNAMIC
CORRELATIONS
t 
 
2
1
1 2 m2 ,t
2
m ,t

2
1
  
2
2
2
m ,t

2
2

• When market volatility is high then correlations
are high. The market/economy in general
influences both stocks positively.
AXP AND GE AGAIN
.9
.8
.7
.6
.5
.4
.3
.2
.1
94
95
96
97
98
99
00
01
C4_AXP_GE
02
03
04
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?
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.
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.
EXAMINING THE ONE
FACTOR MODEL OF
CORRELATIONS
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?
RESULTS
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
.024
.020
S&P volatility
.016
1.2
.012
0.8
.008
.004
0.4
0.0
Correlations
-0.4
94
95
96
97
98
99
00
01
02
03
04
MEAN CORRELATION AND MARKET
VOLATILITY
.032
.50
.028
.45
.024
.40
.020
.35
.016
.30
.012
.25
.008
.20
.004
.15
.000
.10
94
95
96
97
98
99
MEANCOR9F
00
01
02
03
V9F_SPRET
04
REGRESSION
•
•
•
•
•
•
•
•
•
•
•
•
•
Dependent Variable: MEANCOR9F
Method: Least Squares
Date: 09/10/06 Time: 20:00
Sample: 1/04/1994 12/31/2004
Included observations: 2770
Variable
Coefficient
Std. Error
t-Statistic
C
V9_SPRET
0.176566
9.600815
0.003343
0.296987
52.81508
32.32740
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)
•
•
•
•
•
•
•
•
•
Variable
Coefficient
Std. Error t-Statistic Prob.
C
D(V9F_SPRET)
AR(1)
-2.57E-06
7.755417
0.070129
9.18E-05 -0.028054 0.9776
0.612757 12.65660 0.0000
0.023881 2.936653 0.0033
FINDINGS
• MARKET VOLATILITY IS PART OF THE
STORY
• THE CURRENT DECLINE IN MARKET
VOLATILITY HAS NOT LEAD TO THE
EXPECTED DROP IN CORRELATIONS.
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
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.
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
S&P500
1.2
1.0
0.8
0.6
0.4
0.2
0.0
60
65
70
75
CVOL
80
85
90
UVOL
95
00
MULTIPLE REGRESSIONS
emerging
transition
log(mc)
log(gdpus)
nlc
grgdp
gcpi
vol_irate
vol_gforex
vol_grgdp
vol_gcpi
All Countries
0.0376
( 0.0131 )**
-0.0178
( 0.0171 )
-0.0092
( 0.0055 )*
0.0273
( 0.0068 )**
-1.8E-05
( 5.4E-06 )**
-0.1603
( 0.1930 )
0.3976
( 0.1865 )**
0.0020
( 0.0008 )**
0.0222
( 0.0844 )
0.8635
( 0.1399 )**
0.9981
( 0.3356 )**
Time Effects
0.25
0.2
0.15
0.1
0.05
0
1990 1994 1998 2002
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.