Engle

Document Sample
Engle Powered By Docstoc
					ANTICIPATING
CORRELATIONS

FMA ANNUAL MEETING
CARIBE ROYALE HOTEL, ORLANDO 2007
Robert Engle
Stern School of Business
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?
                            .15


                            .10




LARGE CAP STOCKS
                            .05




                   AXP
                            .00


                            -.05


                            -.10


                            -.15

                            .15
10 YEARS OF

                            .10


                            .05




                   JPM
                            .00


                            -.05


                            -.10


                            -.15

                            .20

                            .15

                            .10

                            .05
                   INTC




                            .00

                            -.05

                            -.10

                            -.15

                              .2



AXP                           .1
                     MSFT




                              .0

JPM
                             -.1



INTC                         -.2

                            .12



MSFT                        .08


                            .04
                   MRK




                            .00

MRK                         -.04


                            -.08


                            -.12
                                  -.15   -.10   -.05   .00   .05   .10   .15 -.15   -.10   -.05   .00   .05   .10   .15 -.15 -.10 -.05   .00   .05   .10   .15   .20 -.2   -.1    .0    .1   .2 -.15   -.10   -.05   .00   .05   .10
                                                       AXP                                        JPM                                      INTC                                  MSFT                            MRK
DAILY CORRELATIONS
       AXP        JPM        INTC       MSFT       MRK

AXP    1.000000   0.554172   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
MSFT   0.283375   0.310113   0.551379   1.000000   0.186004
MRK    0.224685   0.228688   0.130294   0.186004   1.000000
                       .8


                       .6


                       .4




               RCHA
                       .2


FOUR ASIAN             .0


FINANCIAL              -.2

                      .12
INDICES               .08

                      .04
CHINA A               .00
             RKOR




                      -.04

KOREA                 -.08

                      -.12

SINGAPORE             -.16

                      .10

TAIWAN
                      .05
             RSNG




                      .00



                      -.05



                      -.10

                      .15

                      .10

                      .05
             RTAI




                      .00

                      -.05

                      -.10


                      -.15
                             -.2   .0   .2          .4   .6   .8 -.15   -.10   -.05   .00    .05   .10   .15 -.10   -.05   .00    .05   .10 -.15   -.10   -.05   .00    .05   .10   .15
                                             RCHA                                     RKOR                                 RSNG                                  RTAI
CORRELATIONS

       RCHA RKOR RSNG RTAI


RCHA    1.000 -0.001   0.033   0.020
RKOR   -0.001 1.000    0.309   0.234
RSNG    0.033 0.309    1.000   0.267
RTAI    0.020 0.234    0.267   1.000
WHY DO WE NEED
CORRELATIONS?
• CALCULATE PORTFOLIO RISK

• FORM OPTIMAL PORTFOLIOS

• PRICE, HEDGE, AND TRADE
  DERIVATIVES
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.
ESTIMATION
• HISTORICAL CORRELATIONS
 – Use a rolling window of N observations for
   both covariances and variances.
• EXPONENTIAL SMOOTHING
 – Use an exponential smoother for both
   covariances and variances using the same
   smoothing parameter.
.9
.8
.7

.6
.5
.4

.3
.2
.1
.0
     94   95   96   97   98   99   00   01   02   03   04

                         C100_AXP_GE
GENERAL ELECTRIC PROFITS
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.
.7

.6

.5

.4

.3

.2

.1

.0

-.1
      94   95   96   97   98   99   00   01   02   03   04

                          C100_GM_BA
FORMULATE MODELS
DYNAMIC CONDITIONAL CORRELATION
ONE FACTOR MODEL
MANY FACTOR MODEL
MULTIVARIATE GARCH
FACTOR DCC
DYNAMIC EQUICORRELATION
     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).
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 TO UPDATE CORRELATIONS

• When two assets move in the same direction,
  the correlation is increased slightly. When they
  move in the opposite direction it is decreased.

• This effect may be stronger in down markets.

• The correlations often are assumed to only
  temporarily deviate from a long run mean
CORRELATIONS UPDATE LIKE
GARCH
• Approximately,
1,2t  1,2  1,t 1 2,t 1  1,2,t 1
         1,2
1,2          , or 1,2  1,2 1     
       1  
• And the parameters alpha and beta are assumed the
  same for all pairs. Consequently there are only 2
  parameters to estimate, no matter how many assets
  there are!
.8

.7

.6

.5

.4

.3

.2

.1

.0
     94   95   96   97   98   99   00   01   02   03   04

                         DCC AXP AND GE
               AXP AND GE CORRELATIONS
.9

.8

.7

.6

.5

.4
.3

.2

.1

.0
     94   95   96   97   98    99   00   01   02   03   04

          ASYDCC              DCC        HISTORICAL100
 The DCC Model

Vt 1  rt   Dt Rt Dt , Dt ~ Diagonal , Rt ~ Correlation Matrix
 t  Dt1rt
Rt  diag  Qt            Qt diag  Qt 
                     1/ 2                  1/ 2



Qt    a t 1 t 1 ' bQt 1
  R 1  a  b 
AN INTERESTING CORRELATION




• Examine the relation between the
  Shanghai A shares index and the MSCI
  China index.
 MSCI CHINA INDEX
                                               .12
                                               .08
                                               .04
                                               .00
60
                                               -.04
50
                                               -.08
40                                             -.12

30

20

10
     1999 2000 2001 2002 2003 2004 2005 2006

                MSCI_CHINA      RMSCI
 SHANGHAI A SHARES
                                                        .10

                                                        .05

                                                        .00
5000

                                                        -.05
4000

                                                        -.10
3000

2000

1000
       2000   2001   2002   2003   2004   2005   2006

                      CHSASHR         RCHA
WHAT IS THE DIFFERENCE?
• MSCI is an index of China stocks that
  foreign investors can buy.
• It includes B shares, H shares (traded in
  Hong Kong), N shares (traded on Nasdaq)
  and Red Chips
  VOLATILITY OF MSCI AND
  SHANGHAI A SHARES
100
90

80

70

60

50

40

30

20
10
      2000   2001   2002   2003   2004   2005   2006

                       VCHA        VMSCI
 DYNAMIC CORRELATIONS
.4



.3



.2



.1



.0
     2000   2001   2002   2003   2004   2005   2006

                      C0_RMSCI_RCHA
FINDINGS
• The volatility of the A shares is greater
  than the internationally traded China
  shares
• The correlation between these is small but
  increasing
MULTIVARIATE MODELS
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  m ,t .
                                       2


               ri ,t   i rm ,t  e i ,t
                i ,i ,t   i2 m ,t  i
                                 2



• If the market has asymmetric volatility, then
  individual stocks will too.
CORRELATIONS

• Between stock i and stock j assuming
  idiosyncrasies are uncorrelated.
          i , j ,t   i  j  m ,t
                                2


                                   i  j  m ,t
                                            2

         i , j ,t 
                       i   i2 m ,t  j   j2 m ,t 
                                   2                  2



• Assuming betas are both positive, correlations
  range from zero to one and increase with
  market volatility.
• Returns will have lower tail dependence if the
  market is negatively skewed.
HOW TO ESTIMATE A ONE
FACTOR MODEL
• FIT THE VOLATILITY OF THE MARKET
• ESTIMATE THE BETAS OF THE
  STOCKS AND THE VARIANCE OF THE
  IDIOSYNCRACIES
• CALCULATE THE TIME VARYING
  CORRELATIONS
• CALCULATE THE VaR
AXP AND GE AGAIN
.9

.8

.7

.6

.5

.4

.3

.2

.1
     94   95   96   97   98   99   00   01   02   03   04

                          C4_AXP_GE
MARKET VOLATILITY
 .030

 .025


 .020


 .015


 .010


 .005

 .000
        94   95   96    97   98   99   00   01   02     03   04

                       Conditional Standard Deviation
ANTICIPATING CORRELATIONS
• FORECASTING FACTOR VOLATILITIES IS
  PART OF THE ANSWER. USE SPLINE
  GARCH RESULTS TO LINK WITH THE
  MACRO ECONOMY
• HOW CAN WE MAKE THIS WORK BETTER,
  PARTICULARLY FOR CHANGING ECONOMIC
  ENVIRONMENT?
• ENGLE AND RANGEL forthcoming RFS
MULTIPLE REGRESSIONS

              All Countries
 emerging          0.0376
              ( 0.0131 )**           Time Effects
 transition       -0.0178
              ( 0.0171 )
 log(mc)          -0.0092     0.25
              ( 0.0055 )*
 log(gdpus)        0.0273
              ( 0.0068 )**     0.2
 nlc            -1.8E-05
              ( 5.4E-06 )**
 grgdp            -0.1603     0.15
              ( 0.1930 )
 gcpi              0.3976
              ( 0.1865 )**     0.1
 vol_irate         0.0020
              ( 0.0008 )**
 vol_gforex        0.0222
                              0.05
              ( 0.0844 )
 vol_grgdp         0.8635       0
              ( 0.1399 )**
 vol_gcpi          0.9981       1990 1994 1998 2002
              ( 0.3356 )**
FACTOR DCC
WHAT IS WRONG WITH
ONE FACTOR ARCH?
• Idiosyncratic volatilities do change

• Betas may change

• New factors may become important

• Recall the economic examples
THREE ONE FACTOR MODELS
1. One Factor with constant idiosyncratic
   volatility – FACTOR ARCH of Engle Ng
   and Rothschild
2. One Factor with Garch idiosyncratic
   volatility FACTOR DOUBLE ARCH
3. One Factor with Garch idiosyncratic
   volatility and DCC estimated correlations
   between all residuals. FACTOR DCC
DOUBLE ARCH
• Each asset return is regressed on the
  market return with GARCH error terms
• Thus each return has two ARCH models,
  one from the factor and one from the
  idiosyncrasy.
                                  i  j  m ,t
                                           2

      i , j ,t 
                    i ,t   i2 m ,t  j ,t   j2 m ,t 
                                   2                     2
FACTOR DOUBLE ARCH and
FACTOR DCC

  rt  h 
   m
                   t
                    m
                        t
                         m
                               ~ TARCH
  ri ,t   rt  hi ,t  i ,t
                   m
                                     ~ TARCH

     t
        m
            , 1,t ,...,  n ,t    ~ DCC
     THE CORRELATIONS

        i , j ,t 
                                   
                             Et 1 i rm  hi ,t  i ,t
                                       t
                                                                       j rm  h j ,t  j ,t
                                                                             t
                                                                                                 
                                                                                                  
                                                               2                                         2
                         Et 1  r  hi ,t  i ,t
                                     t
                                   i m                             Et 1  r  h j ,t  j ,t
                                                                                t
                                                                              j m




               i  j htm  idcc,t hi ,t h j ,t   j idcc,t hi ,t htm  i  dcc ,t h j ,t htm
 i , j ,t 
                              ,j                         ,m                     j ,m


                  i2 htm  hi ,t  2 idcc,t hi ,t htm
                                         ,m                       j2 htm  h j ,t  2  dcc ,t h j ,t htm
                                                                                           j ,m                
FLEXIBILTY
• Idiosyncracies can be temporarily
  correlated with each other such as when a
  latent factor becomes important
• Idiosyncracies can be temporarily
  correlated with the market factor implying
  time varying betas.
1.0
0.9
0.8

0.7
0.6

0.5

0.4
0.3
0.2
0.1
      94   95   96   97   98   99   00   01   02   03   04

      CF1_GE_AXP           CF2_GE_AXP              CF3_GE_AXP
.8

.7

.6

.5

.4

.3

.2

.1

.0
     94   95   96   97   98   99   00   01   02   03   04

     CF1_GM_DD            CF2_GM_DD               CF3_GM_DD
.9
.8
.7
.6
.5
.4
.3
.2
.1
.0
     94   95   96   97   98    99   00   01   02   03   04

      CF1_DD_AA               CF2_DD_AA            CF3_DD_AA
.8

.7

.6

.5

.4

.3

.2

.1

.0
      94   95   96   97   98   99   00   01   02   03   04

     CF1_MRK_JNJ           CF2_MRK_JNJ             CF3_MRK_JNJ
AVERAGE CORRELATIONS OVER
153 PAIRS
.8

.7

.6

.5

.4

.3

.2

.1

.0
     94    95   96   97   98    99   00   01   02   03   04

          MEAN_CF1             MEAN_CF2             MEAN_CF3
.24


.20


.16


.12


.08


.04


.00
      94   95   96    97   98   99   00   01   02   03   04

                     STD CORR FACTOR ARCH
                     STD CORR FACTOR DOUBLE ARCH
                     STD CORR DCC
.20



.16



.12



.08



.04



.00
      94   95   96    97   98   99   00   01   02   03   04

                     STD RESID CORR FACTOR DCC
                     MEAN RESID CORR FACTOR DCC
.28


.24


.20


.16


.12


.08


.04


.00
  94

         95

                96

                       97

                              98

                                     99

                                            00

                                                   01

                                                          02

                                                                 03

                                                                        04
3/

       2/

              1/

                     1/

                            1/

                                   1/

                                          3/

                                                 1/

                                                        1/

                                                               1/

                                                                      1/
1/

       1/

              1/

                     1/

                            1/

                                   1/

                                          1/

                                                 1/

                                                        1/

                                                               1/

                                                                      1/
                                 STDCOR100
                                 STDCOR DOUBLE ARCH
                                 STDCOR FACTOR DCC
DYNAMIC EQUICORRELATION
Robert Engle and Bryan Kelly
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?
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.
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.
Estimation
   Vt 1  rt   Dt Rt Dt , Dt  diag ( hi ,t ), rt  Dt rt
                                                         1


    Rt  1  t  I nxn  t J nxn , J nxn   '

• Matrix Results
     1     1          t          1
   Rt          In                         J nxn
         1  t      1  t 1   n  1 t
     det( Rt )  1  t           1   n  1 t 
                             n 1
                                                    
Estimation
• Assuming Normality and examining the
  second step as in DCC:
                1 T
  L rt      log Rt  rt ' Rt1rt 
                                        
                2 t 1
          1 T
          (n  1) log 1  t   log 1   n  1 t  
          2 t 1                                            

                       1       n 2               t           n
                                                                        
                                                                        2

                               ri ,t                       ri ,t  
                  2 1  t   i 1
                                         1   n  1 t   i1      
UPDATING
• Each period, new information becomes
  available on the equicorrelation. We
  specify this in several ways.
      t     ut 1  t 1
• Where U can be various measures.
.6


.5


.4


.3


.2


.1

.0
     94   95   96   97   98   99   00   01   02   03   04

                          DECO_SS2
.8

.7

.6

.5

.4

.3

.2

.1

.0
     94   95   96    97   98   99   00   01   02   03   04

                    MEAN_CF1             MEANDCC
                    MEAN_CF3             DECO_SS2
OBSERVATIONS
• DECO looks much like average DCC

• F3 is more volatile

• F1 falls at the end of sample
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?
                     1/




                                  .0
                                       .1
                                            .2
                                                 .3
                                                      .4
                                                           .5
                                                                .6
                                                                     .7
                                                                          .8
                       3/
                          9   4
                     1/
                       2/
                          9   5
                     1/
                       1/
                          9   6
                     1/
                       1/
                          9   7
                     1/
                       1/
                          9   8
                     1/
                       1/
                          9   9
                     1/
                       3/
                          0   0
                     1/
                       1/
                          0   1
                     1/
                       1/




MEANCOR FACTOR DCC
                          0   2
                     1/
                       1/
                          0   3
                     1/
                       1/
                          0   4
                     1/
                       3/
                          0   5
                     1/
                       2/
                          0   6
MEANCOR100




                     1/
                       1/
                          0   7
                      1/




                                  .1
                                       .2
                                            .3
                                                 .4
                                                      .5
                                                           .6
                                                                .7
                        3/
                           05
                      4/
                        1/
                           05
                      7/
                        1/
                           05
                     10
                       /3
                          /  05
                      1/
                        2/
                           06
                      4/
                        3/
                           06
                      7/
                        3/
                           06




MEANCOR FACTOR DCC
                     10
                       /2
                          /  06
                      1/
                        1/
                           07
                      4/
                        2/
                           07
                      7/
                        2/
MEANCOR100




                           07
PERFORMANCE
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.
      rport ,t  wt ri ,t  1  wt  rj ,t
                  h j ,t  hi , j ,t
      wt 
             hi ,t  h j ,t  2hi , j ,t
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.
 rport ,t  ri ,t  i , j ,t rj ,t , i , j ,t  hi , j ,t / hj ,t
Average standard deviation of the
minimum variance portfolio of all
pairs of stocks
   0.0159
   0.0158
                                      var100
   0.0157                             DCC
   0.0156                             DECO
   0.0155                             F1

   0.0154                             F2
                                      F3
                          1

                              2

                                  3
                      O
          0

              C




                          F

                              F

                                  F
          0

              C

                  C
      r1




                  E
              D

                  D
     va
Average standard deviation of the
minimum variance long - short
portfolio of all pairs of stocks
   0.0191
    0.019
                                      var100
   0.0189                             DCC
   0.0188                             DECO
   0.0187                             F1

   0.0186                             F2
                                      F3
                          1

                              2

                                  3
                      O
          0

              C




                          F

                              F

                                  F
          0

              C

                  C
      r1




                  E
              D

                  D
     va
WINNING FRACTIONS MV
1.2




 1




0.8




0.6




0.4




0.2




 0

      mv100   mvdcc       mvdeco        mvcf1           mvcf2     mvcf3

               mv100   mvdcc   mvdeco   mvcf1   mvcf2     mvcf3
WINNING FRACTIONS HEDGE
1.2




 1




0.8




0.6




0.4




0.2




 0
      beta100      betaadcc     betadecoasy        betacf1             betacf2        betacf3

        beta100   betaadcc    betadecoasy     betacf1        betacf2        betacf3
FINDINGS
• The factor models appear to be the best.

• The flexible model performs the best.

• The differences are not very big but are
  systematic. E.g. for 82% of the pairs, FACTOR
  DCC has smaller volatility than DOUBLE ARCH.
ANTICIPATING CORRELATIONS
• MARKET VOLATILITY IS A BIG COMPONENT OF
  CORRELATIONS
• THE CURRENT DECLINE IN MARKET VOLATILITY
  HAS NOT LEAD TO THE EXPECTED DROP IN
  CORRELATIONS SO FACTOR MODELS SHOULD BE
  AUGMENTED.
• DYNAMIC EQUICORRELATION IS AN ATTRACTIVE
  SOLUTION FOR BIG SYSTEMS.
• TWO HEDGING PERFORMANCE MEASURES
  REVEAL THE BENEFITS OF MORE FACTOR DCC.

				
DOCUMENT INFO