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Dissertation Looking for independence reducing the risk of groups of long

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					     Looking for independence: reducing the risk of
groups of long-short strategies during market swings
                    through correlation forecasts



                   MSc Finance and Economics 2003-2004




                AC402, Financial Risk Analysis, Dissertation




                          London School of Economics
                    Department of Accounting and Finance




                              Candidate Number: 67446


                                  Word Count: 5968
      (as specified, although lengthy, appendices are not part of the word count)




                                   June 24, 2004




“The copyright of this dissertation rest with the author and no quotation from it or
information derived from it may be published without prior written consent of the
author.”
Contents


Abstract                                   2


1 Data                                     3
         1.1 Original versus in US         4
         1.2 Trading costs                 5


2 Long-short models                        6
         2.1 ARMA (lags)                   7
         2.2 GARCH                         8
                2.2.1 Zero mean            8
                2.2.2 Lags                 9
         2.3 Results                       10
                2.3.1 Trading theory       11
                2.3.2 Comparison           12


3 Correlation                              15
         3.1 Theoretical models            16
         3.2 Refinements                   17
         3.3 Evaluating predictions        18
         3.4 Results                       21
                3.4.1 Trading theory       21
                3.4.2 Evaluation           22


4 Conclusion                               24
         4.1 Potential problems            26
         4.2 Beyond correlation            27


References                                 28


Appendices                                 31




                                       1
Abstract


In this dissertation I will look at daily trading strategies for trading index futures
(ARMA forecasts scaled by GARCH forecasts to produce a long-short strategy, in
DJIA, Nasdaq Composite, S&P 500, FTSE 100, [DAX 30], [CAC 40], Nikkei 225,
and [Hang Seng] – both locally, and adjusted for exchange rates). In this initial
analysis, issues relating to the cost of trading and the number of lags to use within the
models will be discussed. This will serve as the benchmark and the basis to generate
the covariance forecasts.


In the second, main, part of the dissertation I will look at forecasting correlation
between the five (not in []) indices mentioned above. Subsequently implementing
these forecasts to reduce the risk of the overall ‘portfolio’ of strategies, and evaluating
the result using Value at Risk methodology. Issues relating to the optimum (minimum
variance) portfolio and non-synchronous trading of foreign markets will be discussed.
While the optimum portfolio’s results will be compared against an equally weighted
benchmark.


In conclusion the results will be discussed and possible explanations for the
observations will be presented. As well as a brief discussion of potential problems
with a practical implementation, and a mention of alternative ways of looking at
interdependence between asset returns.




                                            2
1 Data


The data used was acquired from DataStream (Thomson Financial) and spans from
January 1st 1990 to December 31st 2003, consisting of daily data. The following time
series were obtained:


Price Indices:
Dow Jones Industrials
Nasdaq Composite
S&P 500 Composite
FTSE 100
DAX 30 Performance
France CAC 40
Nikkei 225 Stock Average
Hang Seng

Interest rates:
US Interbank
UK Interbank
Germany Interbank
France Interbank
Japan Interbank
Hong Kong Interbank

Foreign exchange rates:
US $ to UK £ (GTIS)
Euro to US $ (ECU History)
US $ to Japanese Yen (GTIS)
US $ to Hong Kong $ (GTIS)

From this a dataset of FTSE, DAX, CAC, NIKK, and HSENG in US Dollars was
                                     Pt
derived and simple returns ( rt         1) calculated for both the original and the in US
                                    Pt1
Dollars indices. This leads to the results of any trading strategies being linear, since
the scale of all trades is constant (to convert to ‘true’ results the y axis can be though
                    
of as being logarithmic). It should be noted that while all other exchange rates are the
GTIS rates, the Euro rate is the ECU History rate, since it was the only rate that
already had the values calculated for the time period prior to the launch of the Euro.
This allowed a direct conversion of the DAX and CAC index prices from Euro to US
Dollars.


                                              3
1.1 Original versus in US


Since the ultimate goal is to look at a portfolio of long-short strategies it is important
to decide how the exchange rates will be handled. There are a number of possibilities;
the two relevant here (since modeling exchange rates is beyond the scope of this
                                                                                                                                                                                                                                                                                                                                       [1]
dissertation) are the use of original returns (assuming the use of quanto                                                                                                                                                                                                                                                                    futures
contracts, therefore eliminating all exchange rate uncertainty). The other is to instead
make the foreign exchange forward an explicit part of the trading cost, this however
would add additional complexity without bringing any clear benefits, and so the
original returns along with the implicit quanto assumption will be used.


To support this decision graphs of the results of a simple long (buying a single future
every day) strategy are presented in Figure 1.1.1 (in original currency) and Figure
1.1.2 (in US Dollars). Both are calculated assuming zero costs. Appendix 1 also
contains comparisons between the individual ARMA GARCH long-short trading
strategies in original currencies as well as in US Dollars.




  100



   80



   60
                                                                                                                                                                                                                                                                                                                                             DJIA
                                                                                                                                                                                                                                                                                                                                             NASD
   40                                                                                                                                                                                                                                                                                                                                        SP500
                                                                                                                                                                                                                                                                                                                                             FTSE
                                                                                                                                                                                                                                                                                                                                             DAX
   20                                                                                                                                                                                                                                                                                                                                        CAC
                                                                                                                                                                                                                                                                                                                                             NIKK
                                                                                                                                                                                                                                                                                                                                             HSENG
    0
        19910102
                   19910614
                              19911126
                                         19920507
                                                    19921019
                                                               19930331
                                                                          19930910
                                                                                     19940222
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                                                                                                           19950116
                                                                                                                      19950628
                                                                                                                                 19951208
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                                                                                                                                                       19961031
                                                                                                                                                                  19970414
                                                                                                                                                                             19970924
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                                                                                                                                                                                                                                               20000602
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                                                                                                                                                                                                                                                                                           20020320
                                                                                                                                                                                                                                                                                                      20020830
                                                                                                                                                                                                                                                                                                                 20030211
                                                                                                                                                                                                                                                                                                                            20030724




  -20



  -40



                                                                           Figure 1.1.1 (long returns, in original currency)




                                                                                                                                                                                        4
  100



   80



   60
                                                                                                                                                                                                                                                                                                                               DJIA
                                                                                                                                                                                                                                                                                                                               NASD
   40                                                                                                                                                                                                                                                                                                                          SP500
                                                                                                                                                                                                                                                                                                                               FTSE_US
                                                                                                                                                                                                                                                                                                                               DAX_US
   20                                                                                                                                                                                                                                                                                                                          CAC_US
                                                                                                                                                                                                                                                                                                                               NIKK_US
                                                                                                                                                                                                                                                                                                                               HSENG_US
       0
           19910102
                      19910621
                                 19911210
                                            19920528
                                                       19921116
                                                                  19930505
                                                                             19931022
                                                                                        19940412
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                                                                                                              19950320
                                                                                                                         19950906
                                                                                                                                    19960223
                                                                                                                                               19960813
                                                                                                                                                          19970130
                                                                                                                                                                     19970721
                                                                                                                                                                                19980107
                                                                                                                                                                                           19980626
                                                                                                                                                                                                      19981215
                                                                                                                                                                                                                 19990603
                                                                                                                                                                                                                            19991122
                                                                                                                                                                                                                                       20000510
                                                                                                                                                                                                                                                  20001027
                                                                                                                                                                                                                                                             20010417
                                                                                                                                                                                                                                                                        20011004
                                                                                                                                                                                                                                                                                   20020325
                                                                                                                                                                                                                                                                                              20020911
                                                                                                                                                                                                                                                                                                         20030228
                                                                                                                                                                                                                                                                                                                    20030819
  -20



  -40



                                                                                            Figure 1.1.2 (long returns, in US Dollars)


1.2 Trading costs


Since a decision has been made to use the original currency returns, and implicitly
assume the use of a quanto futures contract, the cost consists of the two futures
(difference between the closing an the futures price) that make up the quanto, and is
therefore influenced by both the foreign and the domestic interest rates. However
approximating this cost for daily trades, and also considering that there will be short
trades as well (which are affected in the opposite direction), this theoretical cost is
overshadowed by the practical costs of trading. More relevant is an approximation of
trading costs based on the absolute value of the trade size (which accounts for
commission and potential market impact) and domestic interest rates (which account
for the cost of capital which would have to be held, domestically, to cover potential
margin calls). So only the US interest rate will be used in calculating the costs after
all.

                                                                                                                                                                                                                                                                                              [2]
Margin requirements seem to vary widely between exchanges                                                                                                                                                                                                                                                (20% in Chicago to
50% on some others), so the top limit of 50% seems reasonable if it is also considered
to incorporate the potential market impact and other transaction costs. This leads to
the cost equation being Ct  abs(tradet )  0.5  rf ,t . To illustrate the impact of trading
costs the cumulative returns of a NASD long-short ARMA GARCH trading strategy

                                                       
                                                                                                                                                                                           5
(to be discussed later in more detail) are plotted together with the results adjusted for
trading costs in Figure 1.2.1.



  140


  120


  100


   80

                                                                                                                                                                                                                                                                  NASD
   60
                                                                                                                                                                                                                                                                  NASD - cost

   40


   20


    0
        19910102
                   19910813
                              19920323
                                         19921030
                                                    19930610
                                                               19940119
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                                                                                                19951117
                                                                                                           19960627
                                                                                                                      19970205
                                                                                                                                 19970916
                                                                                                                                            19980427
                                                                                                                                                       19981204
                                                                                                                                                                  19990715
                                                                                                                                                                             20000223
                                                                                                                                                                                        20001003
                                                                                                                                                                                                   20010514
                                                                                                                                                                                                              20011221
                                                                                                                                                                                                                         20020801
                                                                                                                                                                                                                                    20030312
                                                                                                                                                                                                                                               20031021
  -20




                     Figure 1.2.1 (NASD long-short ARMA GARCH results with costs)


2 Long-Short Models


The basis of the analysis is the setup of individual daily long-short trading strategies
                                                                                                                                                                                                                                                          [3]
for each market. For this the simplest possible category of models                                                                                                                                                                                              will be used so
as not to overshadow the main goal of looking at reducing risk through correlation
forecasts. Two models will be used for each security, an ARMA model to forecast the
conditional mean, and a GARCH model to forecast the conditional variance. The
models will only rely on past returns, and will be estimated yearly, on the previous
year’s returns, i.e. a one year sliding window. No attempt will be made to optimize
model settings beyond the ‘reasonable’ inputs so as to prevent data snooping. Such a
                                                                                                                                                                                                                                    [4]
setup brings up the obvious question of market efficiency                                                                                                                                                                                 . It should be noted
however that outperforming the market does not necessarily violate market efficiency
since the trading cost assumptions made above are in no way exhaustive. And there is
still a possibility of data snooping, even if care is taken to estimate the models out of
sample, since the data and the models being evaluated are being looked at after the


                                                                                                                                                       6
     fact rather than committing to a setup in 1990 and running it through to the end of
     2003.


     2.1 ARMA (lags)


     Auto       Regressive       Moving         Average        models        (general      form:
     rt  0  1rt1  2rt2  ... 1t1  2t2  ... t where t ~ WN(0) ) will be used to
     forecast the conditional mean of returns. At this point it should be noted that returns
     are not strongly autocorrelated (as can be seen from the correlogram of DJIA in
                                               
     Figure 2.1.1, correlograms for the remaining indices are presented in Appendix 2),
     and so the selection of models based on autocorrelation Q statistics is likely to fail.
     Considering this, and that any such selections based on in-sample autocorrelation Q
     statistics, or model evaluation criteria such as R2, AIC, and BIC are likely to
     significantly contribute to the possibility of data snooping, an intuitive selection
     verified by the consistency (rather than profitability or goodness of fit) of empirical
     results will be used.




                Figure 2.1.1 (correlogram of DJIA returns, produced by EViews)




                                                  7
     Insignificant autocorrelations in returns also suggest that the MA (Moving Average)
     part of the model may not be particularly relevant as the error term will vary quite
     significantly. This is supported by empirical results where the ARMA predictions of
     any model with a non-zero number of MA terms tend to ‘explode’ following the
     model encountering a few unexpected returns (and so generating large error terms).


     This leaves only AR (Auto Regressive) models to consider, but still poses the
     problem that due to insignificant autocorrelations and the possibility of data snooping
     the number of lags can not be chosen using Q statistics. This leaves an intuitive
     choice of the number of lags to use, and so one, two, and three weeks (5, 10, and 15
     trading days) are considered. All of which generate reasonable forecasts and so a final
     selection will be made once this is combined with the GARCH model, since it will
     also potentially (if returns can not be assumed to have a zero mean) rely on the
     ARMA models’ forecasts.


     2.2 GARCH


     Generalized Autoregressive Conditional Heteroskedasticity models (general form:
      t2    1t1   2t2  ... 1 t1  2 t2  ...) will be used to model the conditional
                   2         2              2         2


     volatility of returns, individually for each index.


     2.2.1 Zero mean


     Since the GARCH class of models requires an input of zero mean residuals (otherwise
     we will get erroneous predictability), it is necessary to either verify that returns are in
     fact mean zero, or if they are not, to de-mean them. In order to test whether returns
     are in fact already mean zero a test is carried out on each series of returns with a null
     hypothesis that returns are mean zero, and an alternative that they are not. The results
     are presented in Figure 2.2.1.1. Looking at the low p-values it is easy to reject the null
     hypothesis for all three of the US indices and for HSENG. The evidence for other
     markets is not as straightforward, however it should be noted that the p-values are still
     reasonably low which suggests that this can not be taken as strong evidence in favor
     of zero mean returns. Considering this and the desire to keep the modeling procedure



                                                    8
consistent for all markets it is reasonable to say that returns can not be assumed to be
mean zero, and so will have to be de-meaned using the ARMA models discussed
above.


                               Mean             T-Stat         P-Value
             DJIA              0.041766         2.469618          0.0136
             NASD              0.053031          2.03162          0.0423
             SP500             0.036753         2.144705           0.032
             FTSE              0.022399          1.28082          0.2003
             FTSE_US           0.025974         1.387994          0.1652
             DAX               0.032717         1.337272          0.1812
             DAX_US            0.034372         1.401565          0.1611
             CAC               0.025178         1.108506          0.2677
             CAC_US             0.02653         1.184123          0.2364
             NIKK             -0.024144        -0.970766          0.3317
             NIKK_US           -0.01293        -0.458652          0.6465
             HSENG             0.054244         1.995999           0.046
             HSENG_US          0.054436         1.999734          0.0456


                            Figure 2.2.1.1 (zero mean tests)


2.2.2 Lags


What remains is to test that in fact volatility is significant and to determine the
number of lags to use in the final model. As it has been noted above, since it is not
reasonable to assume that we have zero mean returns, the choice of the ARMA model
will impact on how returns are de-meaned, which will in turn impact on the choice of
the GARCH model. The ‘ideal’ solution would be to iteratively run the models until
the best combination of settings is found, however, since evaluating in-sample data,
this will significantly increase the chance of data snooping. Instead the faulty
assumption that returns are mean zero is made for the moment and the
autocorrelations of squared returns are evaluated (due to returns not actually being
mean zero it should be noted that this will lead to stronger autocorrelations than is in
fact the case).


It is well known that squared returns are strongly correlated, even when de-meaned,
so this is mainly a side note to confirm this for the chosen dataset. And, as can be seen
from Figure 2.2.2.1, the correlogram of squared DJIA returns (correlograms for the



                                           9
remaining indices are presented in Appendix 3), this is definitely confirmed for the
dataset being used. It is also safe to say, considering the significance of the observed
Q statistics, that de-meaning the returns using any reasonable model will still lead to
strong autocorrelation in squared de-meaned returns.




     Figure 2.2.2.1 (correlogram of squared DJIA returns, produced by EViews)


Given these results it can be concluded that a GARCH model using pretty much any
combination of lags will be useful in forecasting volatility. As with the choice of lags
for the ARMA models, consistency rather than performance of models will be
evaluated at this stage to prevent data snooping in the choice of settings.


2.3 Results


From here on in the modeling is done in MatLab using the “Econometrics Toolbox”
                                                                              [5]
by James LeSage and the “UCSD GARCH Toolbox” by Kevin Sheppard                      . While
the code written for the ARMA GARCH models can be found in Appendix 4.
Considering that the code for the multivariate GARCH models, used later on, uses
normal errors the GARCH models were set to use normal errors as well (the


                                           10
multivariate GARCH code could potentially be modified to use student’s t or GED
errors in a ‘production environment’ however doing so here and comparing the results
would add little while once again introducing the possibility of data snooping).


With long-short strategies there is the obvious problem of how to compare the results,
particularly how to make the y-axis scales compatible. The simplest solution is to
scale the results by the sum of the absolute values of the trades, which will be done
here. Since the alternative of scaling by average variance poses the problem that the
trades themselves have already been scaled by the conditional variances, and scaling
by any entirely ex-post statistic (such as VaR, which is in fact used later on to
evaluate the results) would not be particularly valid. Scaling the results by the sum of
the absolute values of the trades gives a consistent and easily comparable scale for
results throughout the dissertation (the long ‘strategies’ presented earlier were also
scaled in this way).


2.3.1 Trading theory


Logic suggests that one would want to take a position proportional to the expected
move (conditional mean prediction from the ARMA models) and inversely
proportional to how volatile this prediction is (conditional variance prediction from
the GRACH models). However this does not give us a definite equation to calculate
the long-short positions, and so, given certain assumptions, a specific market model
needs to be used. Considering that the ARMA and GARCH model predictions can be
treated as price signals a version of the Grossman model of fully revealing prices
(Appendix 5 contains the relevant derivation and references) seems like a reasonable
way to determine the optimum position. From this, considering that the price (future
price less the closing price) is close to zero for daily trading, and letting the risk
aversion coefficient equal to one (since it just scales the trades anyway), we have that
                                        E(rt | It1)
the optimum position to take is t                   i.e. the conditional mean prediction
                                       Var(rt | It1)
over the conditional variance prediction. This is what one would expect based on the
logical requirements outlined above, while the assumptions made, although not
                        
entirely correct, are not likely to be far from the truth, and so should not impact the
accuracy of the suggested trading strategy.


                                           11
2.3.2 Comparison


The goal is to find settings, the number of lags, which produce consistent models and
therefore consistent results. In order to do this the simplest possible volatility model,
GARCH(1,1), was combined with the three earlier proposed ARMA models
(ARMA(5,0), ARMA(10,0), and ARMA(15,0)). An ARMA(5,0) model can be seen
in Figure 2.3.2.1, an ARMA(10,0) in Figure 2.3.2.2, and an ARMA(15,0) in Figure
2.3.2.3. From these results, and from looking at the predicted conditional means
directly, the ARMA(10,0) produces the most consistent and least volatile results, with
the ARMA(5,0) seeming to follow trends too easily and the ARMA(15,0) failing to
adapt sufficiently quickly during drops in the market (most notably the NASD
reaction to the 1998 crisis). An interesting feature is that the results for the NASD
long-short trading strategy are by far the best of the lot, and although discussion of
possible causes for this is beyond the scope of this dissertation, it could likely be
attributed to the index consisting of stocks which seem to exhibit strong momentum
effects which are picked up by the ARMA models.



   200



   150



   100                                                                                                                                                                                                                                                                           DJIA
                                                                                                                                                                                                                                                                                 NASD
                                                                                                                                                                                                                                                                                 SP500
                                                                                                                                                                                                                                                                                 FTSE
    50
                                                                                                                                                                                                                                                                                 DAX
                                                                                                                                                                                                                                                                                 CAC
                                                                                                                                                                                                                                                                                 NIKK
     0                                                                                                                                                                                                                                                                           HSENG
         19910102
                    19910726
                               19920218
                                          19920910
                                                     19930405
                                                                19931027
                                                                           19940520
                                                                                      19941213
                                                                                                 19950706
                                                                                                            19960129
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                                                                                                                                             19971007
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                                                                                                                                                                              19990616
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                                                                                                                                                                                                                                                20021101
                                                                                                                                                                                                                                                           20030527
                                                                                                                                                                                                                                                                      20031218




   -50



  -100


                               Figure 2.3.2.1 (results of an ARMA(5,0) with a GARCH(1,1))




                                                                                                                                                   12
  150




  100


                                                                                                                                                                                                                                                                                 DJIA
                                                                                                                                                                                                                                                                                 NASD
   50
                                                                                                                                                                                                                                                                                 SP500
                                                                                                                                                                                                                                                                                 FTSE
                                                                                                                                                                                                                                                                                 DAX
                                                                                                                                                                                                                                                                                 CAC
    0
                                                                                                                                                                                                                                                                                 NIKK
         19910102
                    19910726
                               19920218
                                          19920910
                                                     19930405
                                                                19931027
                                                                           19940520
                                                                                      19941213
                                                                                                 19950706
                                                                                                            19960129
                                                                                                                       19960821
                                                                                                                                  19970314
                                                                                                                                             19971007
                                                                                                                                                        19980430
                                                                                                                                                                   19981123
                                                                                                                                                                              19990616
                                                                                                                                                                                         20000107
                                                                                                                                                                                                    20000801
                                                                                                                                                                                                               20010222
                                                                                                                                                                                                                          20010917
                                                                                                                                                                                                                                     20020410
                                                                                                                                                                                                                                                20021101
                                                                                                                                                                                                                                                           20030527
                                                                                                                                                                                                                                                                      20031218
                                                                                                                                                                                                                                                                                 HSENG


   -50




  -100


                           Figure 2.3.2.2 (results of an ARMA(10,0) with a GARCH(1,1))



  150




  100


                                                                                                                                                                                                                                                                                 DJIA
                                                                                                                                                                                                                                                                                 NASD
   50
                                                                                                                                                                                                                                                                                 SP500
                                                                                                                                                                                                                                                                                 FTSE
                                                                                                                                                                                                                                                                                 DAX
                                                                                                                                                                                                                                                                                 CAC
    0
                                                                                                                                                                                                                                                                                 NIKK
         19910102
                    19910726
                               19920218
                                          19920910
                                                     19930405
                                                                19931027
                                                                           19940520
                                                                                      19941213
                                                                                                 19950706
                                                                                                            19960129
                                                                                                                       19960821
                                                                                                                                  19970314
                                                                                                                                             19971007
                                                                                                                                                        19980430
                                                                                                                                                                   19981123
                                                                                                                                                                              19990616
                                                                                                                                                                                         20000107
                                                                                                                                                                                                    20000801
                                                                                                                                                                                                               20010222
                                                                                                                                                                                                                          20010917
                                                                                                                                                                                                                                     20020410
                                                                                                                                                                                                                                                20021101
                                                                                                                                                                                                                                                           20030527
                                                                                                                                                                                                                                                                      20031218




                                                                                                                                                                                                                                                                                 HSENG


   -50




  -100


                           Figure 2.3.2.3 (results of an ARMA(15,0) with a GARCH(1,1))


After determining that using an ARMA(10,0) gives the most consistent results an
attempt to vary the lags in the GARCH model was made to see if the GARCH(1,1)
could be improved on. At this point it should be noted that using a high (anything


                                                                                                                                                   13
     where p  q  4) number of lags in the GARCH model resulted in errors (singular
     matrices during estimation) while running the GARCH model. Which seems to
     happen due to the fact the additional lags, while significant, do not carry any new

     information that is not already contained in the earlier lags. A GARCH(1,3) and a
     GARCH(2,2) models are ultimately compared (GARCH(3,1) encountered the
     singular matrix problem, and in-between models did not show any interesting results).
     A GARCH(1,1) model can be seen in Figure 2.3.2.2, a GARCH(1,3) in Figure
     2.3.2.4, and a GARCH(2,2) in Figure 2.3.2.5. The GARCH(1,3) gives quite similar
     results to GARCH(1,1), as would be expected since it is reasonable to assume that a
     lot of information is already carried by the first lag, and so the additional lags using
     error data, are unlikely to carry significant additional information. Compared to these
     the GARCH(2,2) clearly gives significantly more volatile results, and also seems to
     perform poorly past the market peak, likely failing to adapt quickly enough in a
     volatile market due to its higher persistence.



        150




        100


                                                                                                                                                                                                                                                                                      DJIA
                                                                                                                                                                                                                                                                                      NASD
         50
                                                                                                                                                                                                                                                                                      SP500
                                                                                                                                                                                                                                                                                      FTSE
                                                                                                                                                                                                                                                                                      DAX
                                                                                                                                                                                                                                                                                      CAC
          0
                                                                                                                                                                                                                                                                                      NIKK
              19910102
                         19910726
                                    19920218
                                               19920910
                                                          19930405
                                                                     19931027
                                                                                19940520
                                                                                           19941213
                                                                                                      19950706
                                                                                                                 19960129
                                                                                                                            19960821
                                                                                                                                       19970314
                                                                                                                                                  19971007
                                                                                                                                                             19980430
                                                                                                                                                                        19981123
                                                                                                                                                                                   19990616
                                                                                                                                                                                              20000107
                                                                                                                                                                                                         20000801
                                                                                                                                                                                                                    20010222
                                                                                                                                                                                                                               20010917
                                                                                                                                                                                                                                          20020410
                                                                                                                                                                                                                                                     20021101
                                                                                                                                                                                                                                                                20030527
                                                                                                                                                                                                                                                                           20031218




                                                                                                                                                                                                                                                                                      HSENG


        -50




       -100


                                Figure 2.3.2.4 (results of an ARMA(10,0) with a GARCH(1,3))




                                                                                                                                                        14
  140

  120

  100

   80
                                                                                                                                                                                                                                                                                DJIA
   60                                                                                                                                                                                                                                                                           NASD
                                                                                                                                                                                                                                                                                SP500
   40
                                                                                                                                                                                                                                                                                FTSE
                                                                                                                                                                                                                                                                                DAX
   20
                                                                                                                                                                                                                                                                                CAC
    0                                                                                                                                                                                                                                                                           NIKK
                                                                                                                                                                                                                                                                                HSENG
        19910102
                   19910724
                              19920212
                                         19920902
                                                    19930324
                                                               19931013
                                                                          19940504
                                                                                     19941123
                                                                                                19950614
                                                                                                           19960103
                                                                                                                      19960724
                                                                                                                                 19970212
                                                                                                                                            19970903
                                                                                                                                                       19980325
                                                                                                                                                                  19981014
                                                                                                                                                                             19990505
                                                                                                                                                                                        19991124
                                                                                                                                                                                                   20000614
                                                                                                                                                                                                              20010103
                                                                                                                                                                                                                         20010725
                                                                                                                                                                                                                                    20020213
                                                                                                                                                                                                                                               20020904
                                                                                                                                                                                                                                                          20030326
                                                                                                                                                                                                                                                                     20031015
  -20

  -40

  -60

  -80


                              Figure 2.3.2.5 (results of an ARMA(10,0) with a GARCH(2,2))


Based on this, the combination of an ARMA(10,0) and a GARCH(1,1) is chosen, the
results produced by which can be seen in Figure 3.2.2.2. These settings will also be
carried over to, and used for consistency, in the correlation models.


3 Correlation


First of all, since any correlation (multivariate GARCH) model will have more
parameters to be estimated than the univariate version, one would expect that a longer
window for the estimation of models would be required. This is in fact confirmed
empirically, with a one-year estimation window leading to inconsistent results and
estimation problems. In light of this the estimation window is expanded to two years,
but the model is still re-estimated every year (on the previous two years’ data, except
for 1991, since only one year is available then). The expanded estimation window
changes the setup somewhat from the ARMA GARCH models described previously,
however the fact that models are estimated out of sample and yearly re-estimation
remains, making comparisons between results just as valid. And any direct
comparisons in this section are made against models with the expanded two-year
estimation window.


                                                                                                                                                       15
     3.1 Theoretical models


     There are two main desirable properties, in addition to fitting the data well, of
     correlation models: positive definiteness and parsimony. Positive definiteness
     basically means that the generated conditional covariance matrix is valid, and does
     not imply an inconsistency in the relationships between the variances and
     covariances. Parsimony is the growth in the number of parameters being estimated as
     the number of assets increases, it is important for a number of reasons, mainly that
     estimation times increase significantly as the number of parameters to be estimated
     grows. The other reason being that as the number of parameters is increased, while
     keeping the sample size constant, this generally leads to worse estimates of the
     parameters, and so the fixed sample size performance of the model will deteriorate
     (since the number of parameters grows faster than linearly in the number of assets in
     most models).

                                                                           [6]
     The first multivariate volatility model to be proposed was the vech         model, and so
     potentially should be used as the benchmark. However the vech model has significant
     drawbacks in that it does not guarantee positive definiteness and the number of
     parameters quickly becomes unmanageable as the number of assets increases. Due to
     this, and the fact that an economic benchmark is a portfolio where covariance is not
     managed (an equally weighted portfolio), the vech model will not be considered
     empirically. In addition to the models that will be considered here, due to their
     benefits that are discussed below, there are a number of other multivariate volatility
     models [7] that could potentially be useful in more specific cases.


     A model that satisfies the desired requirements most closely is the DCC (Dynamic
     Conditional Correlation multivariate GARCH) [8]. This model is formally represented
     by Ht  Dt Rt Dt where H t is the conditional covariance matrix. The advantage of this
     over the CCC model is that the correlation is allowed to vary through time, and is
     represented by Rt  Qt*1Qt Qt*1 where Qt* is the square root of the diagonal of Qt ,
                
     which follows a form of a GARCH process ( Qt  W  Qt1  t1't1 ). In this setup
     the model can be estimated in stages, first the individual GARCH models and then the
                                                                       
                                          
                                                16
     correlation matrix, as well as having a very reasonable number of parameters
          k 2  5k  4
     (P               where k is the number of securities), making it very desirable in
                2
     terms of computational complexity, while also ensuring positive definiteness.


     Another model that also comes close to satisfying the desired requirements is the
     BEKK [9]. However the number of parameters grows faster than for the DCC, making
     it less suited to a scenario with more than a few securities being considered. On the
     other hand the extra parameters enable the model to be more flexible than the DCC.
     The model is formally represented by Ht  C'C  B' Ht1B  A'et1e't1A , where C is
     restricted to being lower triangular while A and B are unrestricted. The BEKK also
     guarantees positive definiteness, so the only remaining issue is the number of
                                   
                      5k 2  k
     parameters ( P           ), which grows significantly faster in the number of
                         2
     securities, than with the DCC.

        
     3.2 Refinements


     Considering that the BEKK model is the most flexible multivariate GARCH model
     that can be practically applied while the DCC makes some flexibility tradeoffs in
     order to reduce the number of parameters and therefore computational time, it is
     interesting to consider whether the added flexibility is a benefit when the models are
     applied practically. Noting that the added flexibility, due to a larger number of
     parameters, inevitably leads to more noisy correlation forecasts, as can be seen from
     Section 3.3. Two ways of looking at this would be to either consider a third model,
     with fewer parameters than the DCC, or to smooth the predictions already available.
     Considering a third model seem problematic since it would be the CCC, which does
     not allow varying correlations and so it is of little use, while other models are not
     similar enough for this to be a relevant comparison. This leaves the option of
     smoothing the predictions of the two models being considered. Smoothing predictions
     of the BEKK poses the problem that it is not possible to rank the flexibility of the
     smoothed BEKK above or below that of the DCC, which leaves only the smoothed
     DCC to be considered. There are many various running average formulae and settings
     that can be used to smooth the predictions, and once again choosing the one that leads


                                              17
     to the best results is likely to lead to data snooping. So it is proposed to use a five-day
     (including the actual prediction) window and smooth the covariance matrix elements
     individually    according    to    an   exponential     formula     (using    weights    of
                 1                                    1
     exp(                ) i.e. exp(1) for today, exp( ) for yesterday, etc., which are
            days_ ago  1                             2
     subsequently normalized for the five day window).
                     
                                  
     Another issue that needs to be addressed is the non-synchronous nature of the
     markets, which causes a number of problems, the main one being that when
     forecasting correlation between securities traded on exchanges around the world all of
     ‘yesterdays’ returns will not be available before the start of the day, due to time
     differences. And so the model becomes unimplementable unless this is addressed. A
     potentially interesting approach is to borrow an idea applied to volatility forecasts
     where intraday data is used to calculate the true daily sample volatility, and use the
     intraday data during the hours when the opening of the exchanges overlaps to
     calculate actual daily sample correlations and base the correlation model off that. An
     easier way to handle this, and the one that will be used here, is to offset the covariance
     predictions by a day when dealing with non-synchronous markets by using the
                                                                                   ˆ
     forecast for yesterday instead of the forecast for today (essentially setting H t  H t1 ,
     which, it should be noted, leads to the first day of results being a day later than
     initially).
                                                                           

     3.3 Evaluating predictions


     The predictions are ultimately evaluated by assessing the results of the trading
     strategies based on them, which is similar to using an economic loss function. To
     evaluate the predictions directly, numerically, a proxy of ri,t rj,t could be used for

     Cov(ri,t ,rj,t ) (where ri,t and r j,t are de-meaned returns), however since the goal is not

     to chose the best combination of a model and it’s settings, but to evaluate their
                                                        
     usefulness when applied to trading strategies, these tests will not be conducted here.
                  
     Instead a graphical representation of predicted correlation versus actual sample
     correlations (quarterly samples, since anything shorter gives results which are too
     noisy and half yearly does not pick up on the changes enough) is presented.


                                                 18
Due to the excessive estimation time of the BEKK with five securities, the
evaluations were done with the subset of the three US indices, to preserve the
comparability of the results. Out of these, the combination of the DJIA and the NASD
provided the best visible pattern, and so will be used here. Figure 3.3.1 shows the
results for the BEKK model, with the sample correlations represented by the dashed
line, Figure 3.3.2 shows the DCC, and Figure 3.3.3 the smoothed DCC, as discussed
above. By visually evaluating these figures it is clear that predictions from both base
models are far too noisy to represent true correlations, with oscillations that often
exceed, especially for the BEKK, long run changes in sample correlations. The
smoothed DCC seems to remove some of the daily volatility, however spikes can still
be observed.




                                     Figure 3.3.1




                                          19
Figure 3.3.2




Figure 3.3.3


    20
          3.4 Results


          For the reasons mentioned previously (namely data snooping), and following the logic
          used in the selection of the ARMA and GARCH model settings, the same settings will
          be used for the correlation models as have been used in the individual volatility
          models. This leads to a DCC(1,1,1,1) and a BEKK(1,1) being considered. The only
          difference being that, due to excessive estimation time, the BEKK model will only
          have a one-year estimation window (except for forecasts for 1999 through to the end
          of 2002, where the two year window is used, due to excess volatility causing
          estimation to fail otherwise). The BEKK will also only be estimated for the group of
          the three US indices, hence part of the comparisons being done with this subgroup.
          And, as noted previously, a smoothed DCC and an equally weighted benchmark are
          also evaluated.


          3.4.1 Trading theory


          Given a variance forecast and a goal of reducing risk, the trading strategy is to
                                                               [10]
          construct a minimum variance portfolio                      . Derivation of this is covered in the
                                                                                                                    Bt
          referenced sources, so only the result is presented here, which is that E t1(rt* ) 
                                                                                                                    At
                           1
          and E t1 ( t2 )  (where At  1' E t1 (V t)11 and Bt 1' E t1 (Vt )1 E t1(rt ) ). This allows
                          At
                                                                                 
          us to use the same trading strategy as with the individual ARMA and GARCH models
                           
          (Section 2.3.1) to determine the size  trade to make. In order to calculate the
                                                 of
     
          outcome of the trade and obtain the results of the trading strategy, the weighting of
          the global minimum variance portfolio is required, derivation of which is also covered
          in the referenced sources. The resulting expression for the weights of the global
          minimum variance portfolio is                  w*  t E t1 (Vt )11  t E t1(Vt )1 E t1 (rt )
                                                          t                                                     (where
               Ct  Bt E t1 (rt* )                       At E t1 (rt* )  Bt
          t                       ,                t                       ,                      t  At Ct  Bt2 ,
                     t                                           t
                                                 
          Ct  E t1(rt )' E t1(Vt )1 E t1(rt ) , and At , Bt , and E t1(rt* ) are as defined above).
                                                                                       
                                      
                                                 
                                                             21
3.4.2 Evaluation


The MatLab code for the three different setups (DCC, smoothed DCC, and BEKK) is
quite similar, and so only the DCC code is included, and can be found in Appendix 6,
while the code for the remaining models and for the correlation comparison graphs
can be found attached along with the datasets.


The models are evaluated by looking the at VaR (Value at Risk) values, and
                                                   [11]
subsequently at the full distribution of returns          . The VaR evaluation, although not
as general, allows us to look at only the far out negative tail of the returns, and
therefore get more of a handle on the extreme risk properties of the models’ returns.
Various VaR values are presented in Figure 3.4.2.1, from where it can be seen that
diversifying by investing in all five indices consistently reduces the risk of the
different models. Interestingly enough the BEKK performs worse than the DCC,
while the smoothed DCC performs marginally worse than the DCC, suggesting that
the optimum amount of generality in a model is around what the DCC provides (in
this case). Lagged models, which are the ones to look at, obviously give slightly
worse results since the non-lagged models are not replicable in real life due to
asynchronous trading.


Observations                       3391

VaR %                               99.9           99.5             99       97.5         95
Observation                            3             16             33         84        169

DCC 5                             -0.479        -0.373          -0.319     -0.243     -0.194
DCC 5 lagged                      -0.623        -0.407          -0.317     -0.244     -0.192
smoothed DCC 5                    -0.514        -0.393          -0.312     -0.242     -0.192
smoothed DCC 5
lagged                            -0.658        -0.396          -0.321     -0.240     -0.194
Equally weighted 5                -1.750        -1.123          -0.940     -0.663     -0.470

BEKK 3                            -1.175        -0.699          -0.571     -0.427     -0.309
DCC 3                             -0.737        -0.540          -0.452     -0.361     -0.288
smoothed DCC 3                    -0.780        -0.549          -0.446     -0.350     -0.287
Equally weighted 3                -1.893        -1.153          -0.927     -0.679     -0.468


                 Figure 3.4.2.1 (VaR values for the various strategies)




                                           22
The next step is to look at the full distribution of returns, in order to gain a more
general idea of the relative behavior of the models. In this case it is no longer
particularly valid to compare the distributions of returns in the five index case and
those of only the US subset, since return properties are clearly different due to the US
technology boom. So first a comparison of the models trading the three US indices is
presented in Figure 3.4.2.2, where it can be clearly seen that there is not much
difference between the DCC and the smoothed DCC. While the BEKK exhibits a
taller peak which supports the fatter tails that were observed in the VaR figures above,
with the equally weighted portfolio doing so to an even greater extent.




                             500



                             400



                                                                       BEKK 3
                             300
                                                                       DCC 3
                                                                       smoothed DCC 3
                                                                       Equally weighted
                             200



                             100



                               0
   -1            -0.5              0           0.5            1


         Figure 3.4.2.2 (distribution of returns for models trading US indices)


A more distinctive comparison can be observed in the case where all five indices are
traded, which can be seen in Figure 3.4.3.3, between the DCC and the equally
weighted benchmark. Which clearly shows the thinner tail properties of the returns
from the risk managed DCC strategy versus the equally weighted benchmark.




                                          23
                              450


                              400


                              350


                              300


                              250
                                                                          DCC 5 lagged
                                                                          Equally weighted
                              200


                              150


                              100


                               50


                                0
   -1             -0.5              0            0.5             1


        Figure 3.4.2.3 (distribution of returns for models trading all five indices)


4 Conclusion


While the return distribution properties relating to risk, outlined above, show that
correlation models can in fact be used to reduce the risk of trading strategies it is also
of interest to look at the actual results of the optimized trading strategies, since
profitability is a large deciding factor at the end of the day. A comparison of the
results of all the models trading only the US indices can be seen in Figure 4.1, where
it can clearly be observed that as well as being less risky the optimized trading
strategies also outperform the equally weighted one. Even more significantly, they
outperform the best of the simple long strategies (Figure 1.1.1) as well as the best
individual ARMA GARCH strategy (Figure 2.3.2.2) (NASD for both), it should be
noted though that at this point the comparison is restricted to the US indices. Another
feature is that the BEKK, as well as being more risky than the DCC as discussed
above, also underperforms both the DCC and the smoothed DCC (which are very
similar, as could reasonably be expected based on their nearly identical return
distributions observed above). The BEKK based strategy also performs significantly
worse around the 1994 to 1996 period, while the DCC breaks even at the same time,




                                            24
suggesting that the DCC, although potentially poorly estimated on 1992 to 1994 data
does not suffer from this as much as the BEKK.



  250



  200



  150

                                                                                                                                                                                                                                               BEKK 3
                                                                                                                                                                                                                                               DCC 3
  100
                                                                                                                                                                                                                                               smoothed DCC 3
                                                                                                                                                                                                                                               Equally weighted

   50



    0
        19910102
                   19910820
                              19920406
                                         19921120
                                                    19930708
                                                               19940223
                                                                          19941011
                                                                                     19950529
                                                                                                19960112
                                                                                                           19960829
                                                                                                                      19970416
                                                                                                                                 19971202
                                                                                                                                            19980720
                                                                                                                                                       19990305
                                                                                                                                                                  19991021
                                                                                                                                                                             20000607
                                                                                                                                                                                        20010123
                                                                                                                                                                                                   20010910
                                                                                                                                                                                                              20020426
                                                                                                                                                                                                                         20021212
                                                                                                                                                                                                                                    20030730
  -50



                                                    Figure 4.1 (returns of the models trading US indices)


The real test is to evaluate the performance, of the DCC, with all five indices being
traded, so avoiding the potential selection bias in only looking at US indices (since it
would not have been possible to say in 1990 with certainty that the US markets would
be the ones to experience the most significant boom). This is presented in Figure 4.2,
from which we can see that it is still the case, although to a lesser extent, that the
optimized strategy outperforms the equally weighted one, the best of the simple long
ones, and is at least as good as the best of the individual ARMA GARCH strategies.
All without having a country or an index selection bias, beyond potentially a heavier
weighting on the US indices. Looking at the results of the optimized strategy on their
own, there are two points to note: the poor performance at the start and at the end.
Potential reasons for the poor performance in the first year, although it is outside the
scope of this dissertation to fully evaluate this, are either the fact that the DCC model
is estimated on only one year of data, but also that the market, following a late 1980’s
crash is not well suited to being accurately modeled. As for the losses in the final year
the post crash explanation could also apply, in that the volatility of the crash affects



                                                                                                                                                       25
the estimation of the model while the post crash behavior of the market is ill suited to
modeling due to it’s often “directionless” behavior. Although here the market
efficiency definition that considers the availability of forecasting models may also be
applicable.



  160

  140

  120


  100

   80
                                                                                                                                                                                                                                                   DCC 5 lagged
                                                                                                                                                                                                                                                   Equally weighted
   60

   40


   20

    0
        19910102
                   19910822
                              19920410
                                         19921130
                                                    19930720
                                                               19940309
                                                                          19941027
                                                                                     19950616
                                                                                                19960205
                                                                                                           19960924
                                                                                                                      19970514
                                                                                                                                 19980101
                                                                                                                                            19980821
                                                                                                                                                       19990412
                                                                                                                                                                  19991130
                                                                                                                                                                             20000719
                                                                                                                                                                                        20010308
                                                                                                                                                                                                   20011026
                                                                                                                                                                                                              20020617
                                                                                                                                                                                                                         20030204
                                                                                                                                                                                                                                    20030924

  -20




                                           Figure 4.2 (returns of the models trading all five indices)




4.1 Potential problems


Even considering the encouraging results of using correlation forecasts to reduce the
risk of a portfolio of index trading strategies there are potential problems with this
approach. While using conditional correlation forecasts does reduce risk, it does not,
even with a strategy that is allowed to go long as well as short, produce results that
are entirely independent from the underlying markets. Using more robust models may
allow us to approach this goal, but it will never be possible to reach it by relying on
correlation forecasts alone, since dependence goes beyond what can be evaluated with
just an analysis of correlation. This is an important matter to consider since achieving
                                                                                                                                                                                                                                         [12]
market neutrality is important for investment purposes                                                                                                                                                                                          . Considering that
correlation does not fully measure dependence, for example returns tend to exhibit



                                                                                                                                                       26
stronger correlation when the markets are falling (i.e. specifically when managing risk
is of utmost importance), other ways of looking at the dependence of returns are
required.


There is also the issue of costs that go beyond those of holding cash to meet margin
calls and commissions (considered in Section 1.2). First of all there are the setup and
operational costs that would be significant for trading internationally, and then there
are the potential operational risks, which are also not quantified here.


4.1 Beyond correlation


While looking at operational costs and risks is beyond the area of quantitative analysis
discussed here, there are ways to address the fact that correlation is not the same as
dependence. Correlation forecasts imply linear dependence, however there are ways
of accounting for the non-linear dependence structures that are observed in returns.
The simplest of these is to look at correlations based on the magnitude and the sign of
returns, and try to account for higher correlations associated with negative returns in
this way. This approach is well suited to analysis of deviation from linear dependence
but is not of much use at forecasting it. The next step is looking at modeling this non-
                                                                 [13]
linear dependence; this is approached with the use of copulas           where an attempt is
made to describe the distribution of returns of one asset with respect to the returns of
another.




                                           27
References


[1] quanto
A cross-currency derivative, which is denominated in a foreign currency, on a foreign
asset, but settles in domestic currency using a fixed exchange rate.
(http://www.riskglossary.com/articles/quanto.htm)


[2] futures margin requirements
http://www.onechicago.com/010000_learningctr/oc_010500.html


[3] time series models
Basics of ARMA models covered in 2.6 of “Analysis of Financial Time Series” by
Ruey Tsay (2002, John Wiley & Sons).
GARCH model covered in “Generalized autoregressive conditional
heteroskedasticity” by Tim Bollerslev (1986, Journal of Econometrics).
More advanced ARMAX (incorporating other explanatory variables) and various
refinements of the GARCH model could potentially be used to improve the forecasts.


[4] market efficiency
“A market is efficient with respect to information set t if it is impossible to make
economic profits by trading on the basis of information set t ” “Weak-form
efficiency: the information set contains only historical returns” from “Statistical
                                            
versus Clinical Prediction of the Stock Market” by Harry Roberts (1967,
                                                   
Unpublished).
“A market is efficient with respect to the information set t and forecast models Mt
if it is impossible to make economic profits by trading on the basis of forecasts from a
model in Mt using predictor variables in t ” from “Efficient Market Hypothesis and
                                                                    
Forecasting” by Clive Granger and Allan Timmermann (2002, Working paper), which
is more relevant to the use of the DCC model later on, since it was only published in
                               
2002, with earliest mentions in 2001.


[5] MatLab toolboxes
“Econometrics Toolbox” by James LeSage



                                           28
http://www.spatial-econometrics.com/
“UCSD GARCH Toolbox” by Kevin Sheppard
http://www.kevinsheppard.com/research/ucsd_garch/ucsd_garch.htm
Although Kevin Sheppard wrote all of the functions used directly, some of them do
seem to rely on functions in the “Econometrics Toolbox” which were written by
others.


[6] vech model
“A Capital Asset Pricing Model with time-varying covariances” by Tim Bollerslev,
Robert Engle, and Jeffrey Wooldridge (1988, Journal of Political Economy)


[7] survey of multivariate volatility models
“Multivariate GARCH models: a survey” by Luc Bowens, Sebastien Laurent, and
Jeroen Rombouts (2003, Discussion paper)
A theoretic survey of multivariate volatility models, primarily focusing on
multivariate GARCH models, with a mention of some alternative models.


[8] DCC model
“Theoretical and empirical properties of Dynamic Conditional Correlation
multivariate GARCH” by Robert Engle and Kevin Sheppard (2002, Working paper)
and “Dynamic Conditional Correlation – a simple class of multivariate GARCH
models” by Robert Engle (2002, Journal of Business and Economic Statistics)
Which is a dynamic extension of the CCC (Constant Conditional Correlation) model
by Tim Bollerslev (1990, no exact reference beyond being mentioned in textbooks).


[9] BEKK model
“Multivariate simultaneous generalized ARCH” by Robert Engle and Kenneth Kroner
(1995, Econometric Theory)


[10] (global) minimum variance portfolio
General minimum variance portfolio case covered in 3.8 of “Foundations for
Financial Economics” by Chi-fu Huang and Robert Litzenberger (1988, Prentice
Hall), the particular global minimum variance case, and notation, taken from “AC436



                                           29
Financial Economics Michaelmas Term” by Rohit Rahi (2003, LSE AC436 lecture
notes).


[11] risk as the distribution of returns
Some ideas taken from 4.3 of “Value at Risk” by Philippe Jorion (2001, McGraw-
Hill)


[12] market neutrality
“Are ‘Market Neutral’ hedge funds really market neutral?” by Andrew Patton (2004,
Working paper)


[13] copulas
“Correlation and dependence in risk management: properties and pitfalls” by Paul
Embrechts, Alexander McNeil, and Daniel Straumann (2002, in “Risk Management:
Value at Risk and Beyond” Cambridge University Press)


[other] sources consulted
“Mastering MatLab 6 – A Comprehensive Tutorial and Reference” by Duane
Hanselman and Bruce Littlefield (2001, Prentice Hall) was used as a reference while
writing the MatLab code.
“Trade Like a Hedge Fund” by James Altucher (2004, John Wiley & Sons) was
initially consulted for potential ideas on how to fine tune the trading strategy, but
proved to be far too specific to be of any real use.
“The dangers of using correlation to measure dependence” by Harry Kat (2002,
Working paper) was referred to for support of the ideas presented in the conclusion
regarding the importance of looking at ways of handling risk beyond correlation
forecasts.




                                            30
Appendices


1 Original versus in US


This is something of an after the fact comparison, since it relies on the chosen setting
for the ARMA and GARCH models, however, since this is the ultimate goal it can be
argued that these results are in fact the most relevant. An ARMA(10,0) and a
GARCH(1,1) models are used on both datasets (the three US indices appear in both,
but are if fact the same).


An interesting side note is the significant crash in HSENG’s strategy during the 1997
crisis, which is to a large extent a one day hit, which suggests that if applied in reality
care should be taken on looking at whether exchanges have maximum allowed one
day drops, and if not, at how such extreme results can be controlled in other ways
(value cutoffs in the trading strategy may be of some use in preventing ‘next day’
over-betting). Whether or not DAX’s performance seems to potentially forecast this is
beyond the scope of this dissertation.


  150




  100


                                                                                                                                                                                                                                                                                                                                        DJIA
                                                                                                                                                                                                                                                                                                                                        NASD
   50
                                                                                                                                                                                                                                                                                                                                        SP500
                                                                                                                                                                                                                                                                                                                                        FTSE
                                                                                                                                                                                                                                                                                                                                        DAX
                                                                                                                                                                                                                                                                                                                                        CAC
    0
                                                                                                                                                                                                                                                                                                                                        NIKK
         19910102
                    19910617
                               19911128
                                          19920512
                                                     19921023
                                                                19930407
                                                                           19930920
                                                                                      19940303
                                                                                                 19940816
                                                                                                            19950127
                                                                                                                       19950712
                                                                                                                                  19951225
                                                                                                                                             19960606
                                                                                                                                                        19961119
                                                                                                                                                                   19970502
                                                                                                                                                                              19971015
                                                                                                                                                                                         19980330
                                                                                                                                                                                                    19980910
                                                                                                                                                                                                               19990223
                                                                                                                                                                                                                          19990806
                                                                                                                                                                                                                                     20000119
                                                                                                                                                                                                                                                20000703
                                                                                                                                                                                                                                                           20001214
                                                                                                                                                                                                                                                                      20010529
                                                                                                                                                                                                                                                                                 20011109
                                                                                                                                                                                                                                                                                            20020424
                                                                                                                                                                                                                                                                                                       20021007
                                                                                                                                                                                                                                                                                                                  20030320
                                                                                                                                                                                                                                                                                                                             20030902




                                                                                                                                                                                                                                                                                                                                        HSENG


   -50




  -100



           Figure A1.1 (long-short results from the dataset in the original currencies)




                                                                                                                                                                                 31
  150




  100


                                                                                                                                                                                                                                                                                                                             DJIA
                                                                                                                                                                                                                                                                                                                             NASD
   50
                                                                                                                                                                                                                                                                                                                             SP500
                                                                                                                                                                                                                                                                                                                             FTSE_US
                                                                                                                                                                                                                                                                                                                             DAX_US
                                                                                                                                                                                                                                                                                                                             CAC_US
    0
                                                                                                                                                                                                                                                                                                                             NIKK_US
         19910102
                    19910624
                               19911212
                                          19920602
                                                     19921120
                                                                19930512
                                                                           19931101
                                                                                      19940421
                                                                                                 19941011
                                                                                                            19950331
                                                                                                                       19950920
                                                                                                                                  19960311
                                                                                                                                             19960829
                                                                                                                                                        19970218
                                                                                                                                                                   19970808
                                                                                                                                                                              19980128
                                                                                                                                                                                         19980720
                                                                                                                                                                                                    19990107
                                                                                                                                                                                                               19990629
                                                                                                                                                                                                                          19991217
                                                                                                                                                                                                                                     20000607
                                                                                                                                                                                                                                                20001127
                                                                                                                                                                                                                                                           20010517
                                                                                                                                                                                                                                                                      20011106
                                                                                                                                                                                                                                                                                 20020426
                                                                                                                                                                                                                                                                                            20021016
                                                                                                                                                                                                                                                                                                       20030407
                                                                                                                                                                                                                                                                                                                  20030925
                                                                                                                                                                                                                                                                                                                             HSENG_US


   -50




  -100



                                   Figure A1.2 (long-short results from the datasets in US Dollars)


2 Return correlograms (from EViews)


Presented here are the remaining four (out of the five ultimately used in correlation
forecasts) correlograms for returns in original currency. The entire thirteen
correlograms can be found along with the attached datasets.




                                                                                                                                                                                 32
3 Squared return correlograms (from EViews)


Presented here are the remaining four (out of the five ultimately used in correlation
forecasts) correlograms for squared returns in original currency. The entire thirteen
correlograms can be found along with the attached datasets.




                                         33
4 MatLab code for individual ARMA GARCH models


It should be noted that rFarch_garch.m, presented here, makes use of
mod_armaxfilter_likelihood.m (not presented here, but can be found along with the
attached datasets), which is a modification of armaxfilter_likelihood.m, which inputs
data (handling lags internally) and parameters ultimately returning the ARMA
model’s predictions.


---


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% ARCH GARCG predictions, out of sample, sliding (jumping) window (F.L.) %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Use: run /Applications/MatLabR13/toolbox/ac402/rFarch_dcc.m

% LOADING DATA SET
clear; % all
load '/Applications/MatLabR13/toolbox/ac402/orig_ret.txt' -ascii

date = orig_ret(:,1);
data = orig_ret(:,2:9);
clear orig_ret;
[T,k] = size(data);


% RESTRICTIONS

ar = 10; % 5 10* 15
ma = 0; % 0*

p = 1; % 1* 1 2 3x 5x
q = 1; % 1* 3 2 3x 5x

offset = 50;
days = 261;


F_pred_ARMA = ones(T,k) * 0;
F_pred_GARCH = ones(T,k) * 0;


year = 0;

while ((year + 1)*days < T)

       col = 1;

       while (col <= k)

                 % TRAIN

                 Tstart = year * days + 1;
                 Tend = (year + 1) * days;

                           % ARMA

                       [ARMA_parameters, ARMA_errors, ARMA_LLF , ARMA_SEregression,
ARMA_stderrors, ARMA_robustSE, ARMA_scores,
ARMA_likelihoods]=armaxfilter(data(Tstart:Tend,col:col),1,ar,ma);
                       clear ARMA_errors;
                       clear ARMA_LLF;
                       clear ARMA_SEregression;




                                             34
                         clear   ARMA_stderrors;
                         clear   ARMA_robustSE;
                         clear   ARMA_scores;
                         clear   ARMA_likelihoods;

                       [pred_ARMA_LLF, pred_ARMA, pred_ARMA_likelihoods] =
mod_armaxfilter_likelihood(ARMA_parameters , data(Tstart:Tend,col:col) , ar , ma);
                       clear pred_ARMA_LLF;
                       clear pred_ARMA_likelihoods;

                         % not really the most efficient way, should only use a subset
                         data2 = data;
                         data2(Tstart:Tend,col:col) = data(Tstart:Tend,col:col) -
pred_ARMA;
                         clear pred_ARMA;

                         % GARCH

                       [GARCH_parameters, GARCH_likelihood, GARCH_stderrors,
GARCH_robustSE, GARCH_ht, GARCH_scores] = fattailed_garch(data2(Tstart:Tend,col:col) ,
p , q , 'NORMAL');
                       clear GARCH_likelihood;
                       clear GARCH_stderrors;
                       clear GARCH_robustSE;
                       clear GARCH_ht;
                       clear GARCH_scores;

                         clear data2;

               clear Tstart;
               clear Tend;

               % PREDICT

               year = year + 1;

               Tstart = year * days + 1 - offset;
               Tend = min(T, (year + 1) * days);

                       [pred_ARMA_LLF, pred_ARMA, pred_ARMA_likelihoods] =
mod_armaxfilter_likelihood(ARMA_parameters , data(Tstart:Tend,col:col) , ar , ma);
                       clear pred_ARMA_LLF;
                       clear pred_ARMA_likelihoods;

                         % not really the most efficient way, should only use a subset
                         data2 = data;
                         data2(Tstart:Tend,col:col) = data(Tstart:Tend,col:col) -
pred_ARMA;

                       [pred_GARCH_LLF, pred_GARCH, pred_GARCH_likelihoods] =
fattailed_garchlikelihood(GARCH_parameters , data2(Tstart:Tend,col:col) , p , q, 1); %
1 = 'NORMAL'
                       clear pred_GARCH_LLF;
                       clear pred_GARCH_likelihoods;

                         clear data2;

                         clear ARMA_parameters;
                         clear GARCH_parameters;

               % STORE

                       F_pred_ARMA(Tstart + offset:Tend,col:col) = pred_ARMA(offset +
1:Tend - Tstart + 1,:);
                       F_pred_GARCH(Tstart + offset:Tend,col:col) = pred_GARCH(offset
+ 1:Tend - Tstart + 1,:);

                         clear pred_ARMA;
                         clear pred_GARCH;

               clear Tstart;
               clear Tend;

        col = col + 1
        year = year - 1;

       end



                                              35
      year = year + 1

end



%save mF_arma_predicted.txt F_pred_ARMA -ascii -tabs;
%save mF_garch_predicted.txt F_pred_GARCH -ascii -tabs;

bF_pred_GARCH = F_pred_ARMA ./ F_pred_GARCH;

%save mF_bF_pred_GARCH.txt bF_pred_GARCH -ascii -tabs;

rARMA = data .* F_pred_ARMA;
rGARCH = data .* bF_pred_GARCH;

%save mF_rARMA.txt rARMA -ascii -tabs;
%save mF_rGARCH.txt rGARCH -ascii -tabs;


% Scale by abs(bet size)

absBET = ones(1,k) * 0;
for i=(days+1):T
       absBET(:,:) = absBET(:,:) + abs(bF_pred_GARCH(i,:));
end

scaleRUN = ones(T,k) * 0;
for i=(days+1):T
       scaleRUN(i,:) = scaleRUN(i-1,:) + ((rGARCH(i,:) ./ absBET) * 1000);
end

save scaleRUN.txt scaleRUN -ascii -tabs;



---


5 Grossman model of fully revealing prices


“On the efficiency of competitive stock markets where traders have diverse
information” by Sanford Grossman (1976, Journal of Finance)


Assuming CARA (Constant Absolute Risk Aversion, valid since modeling as though
always trading from the same base) the utility function is U(Wt )  erWt , making the
assumptions of normally distributed returns it can be shown that maximizing utility is
                                  1
                                                given that Wt  pt xt  t (rt  pt ) the
equivalent to maximizing E(W t )  rVar(W t ) , 
                                  2
                                   E(rt )  pt
first order conditions give t                .
                                   rVar(rt )
                                                    
                   

6 MatLab code for the DCC model
                  




                                               36
As mentioned, mainly due to similarity and the length of the code, the code for the
smoothed DCC, the BEKK, and the variations for the correlation comparison graphs,
is not included here, but is instead attached along with the datasets. The modified
function mod_armaxfilter_likelihood.m is once again used as mentioned in Appendix
4.


---


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% ARCH DCC predictions, out of sample, sliding (jumping) window (F.L.) %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Use: run /Applications/MatLabR13/toolbox/ac402/rFarch_dcc.m

% LOADING DATA SET
clear; % all
load '/Applications/MatLabR13/toolbox/ac402/orig_ret.txt' -ascii

date = orig_ret(:,1);
data = orig_ret(:,[2 3 4 5 8]); %[2:djia 3:nasd 4:sp500 5:ftse 8:nikk]);
clear orig_ret;
[T,k] = size(data);

%lag = 0; % TRUE/FALE setting for whether to lag the covariance matrix by a day
lag = 1;

% RESTRICTIONS

ar = 10; % 5 10* 15
ma = 0; % 0*

p = 1; % 1* 1 2 3x 5x
q = 1; % 1* 3 2 3x 5x


dccP = 1;
dccQ = 1;
archP = 1;
garchQ = 1;   % settings carried over from ARMA GARCH

DCC_garchQ=ones(1,k)*garchQ; % k = data series
DCC_archP=ones(1,k)*archP; % k = data series

offset = 50;
days = 261;
grow = 1; % to how many extra years back the training window grows


F_pred_ARMA = ones(T,k) * 0;
F_pred_GARCH = ones(T,k) * 0;

F_pred_CORR = ones(k,k,T) * 0;


year = 0;

while ((year + 1)*days < T)

       col = 1;
       data2 = data;
       data3 = data;

       while (col <= k)

                 % TRAIN




                                          37
                Tstart = max(1, (year - grow) * days + 1);
                Tend = (year + 1) * days;

                          % ARMA

                       [ARMA_parameters, ARMA_errors, ARMA_LLF , ARMA_SEregression,
ARMA_stderrors, ARMA_robustSE, ARMA_scores,
ARMA_likelihoods]=armaxfilter(data(Tstart:Tend,col:col),1,ar,ma);
                       clear ARMA_errors;
                       clear ARMA_LLF;
                       clear ARMA_SEregression;
                       clear ARMA_stderrors;
                       clear ARMA_robustSE;
                       clear ARMA_scores;
                       clear ARMA_likelihoods;

                       [pred_ARMA_LLF, pred_ARMA, pred_ARMA_likelihoods] =
mod_armaxfilter_likelihood(ARMA_parameters , data(Tstart:Tend,col:col) , ar , ma);
                       clear pred_ARMA_LLF;
                       clear pred_ARMA_likelihoods;

                          % not really the most efficient way, should only use a subset
                          data2(Tstart:Tend,col:col) = data(Tstart:Tend,col:col) -
pred_ARMA;
                          clear pred_ARMA;

             % GARCH

                       [GARCH_parameters, GARCH_likelihood, GARCH_stderrors,
GARCH_robustSE, GARCH_ht, GARCH_scores] = fattailed_garch(data2(Tstart:Tend,col:col) ,
p , q , 'NORMAL');
                       clear GARCH_likelihood;
                       clear GARCH_stderrors;
                       clear GARCH_robustSE;
                       clear GARCH_ht;
                       clear GARCH_scores;

                clear Tstart;
                clear Tend;

                % PREDICT

                year = year + 1;

                Tstart = year * days + 1 - offset;
                Tend = min(T, (year + 1) * days);

                       [pred_ARMA_LLF, pred_ARMA, pred_ARMA_likelihoods] =
mod_armaxfilter_likelihood(ARMA_parameters , data(Tstart:Tend,col:col) , ar , ma);
                       clear pred_ARMA_LLF;
                       clear pred_ARMA_likelihoods;

                          % not really the most efficient way, should only use a subset
                          data3(Tstart:Tend,col:col) = data(Tstart:Tend,col:col) -
pred_ARMA;

            [pred_GARCH_LLF, pred_GARCH, pred_GARCH_likelihoods] =
fattailed_garchlikelihood(GARCH_parameters , data2(Tstart:Tend,col:col) , p , q, 1); %
1 = 'NORMAL'
                       clear pred_GARCH_LLF;
                       clear pred_GARCH_likelihoods;

                        clear ARMA_parameters;
             clear GARCH_parameters;

                % STORE

                       F_pred_ARMA(Tstart + offset:Tend,col:col) = pred_ARMA(offset +
1:Tend - Tstart + 1,:);
                       F_pred_GARCH(Tstart + offset:Tend,col:col) = pred_GARCH(offset
+ 1:Tend - Tstart + 1,:);

                          clear pred_ARMA;
                          clear pred_GARCH;

                clear Tstart;
                clear Tend;



                                              38
       col = col + 1
       year = year - 1;

       end

       % DCC TRAIN (using overfit data2)

                 Tstart = max(1, (year - grow) * days + 1);
                 Tend = (year + 1) * days;

               [DCC_parameters, DCC_loglikelihood,
DCC_Ht]=dcc_mvgarch(data2(Tstart:Tend,:),dccP,dccQ,archP,garchQ);
               clear DCC_loglikelihood;
               clear DCC_Ht;

                 clear data2;

                 clear Tstart;
                 clear Tend;

       % DCC PREDICT (using true data3 - remember to trim)

                 year = year + 1

                 Tstart = year * days + 1 - offset;
                 Tend = min(T, (year + 1) * days);

               [pred_DCC_logL, pred_DCC_Rt,
pred_DCC_likelihoods]=dcc_mvgarch_full_likelihood(DCC_parameters,
data3(Tstart:Tend,:), DCC_archP,DCC_garchQ,dccP,dccQ);
        clear pred_DCC_logL;
               clear pred_DCC_likelihoods;

                 F_pred_CORR(:,:,Tstart + offset:Tend) = pred_DCC_Rt(:,:,offset + 1:Tend
- Tstart + 1);

                 clear pred_DCC_Rt;
                 clear data3;

                 clear Tstart;
                 clear Tend;

end

% DCC Returned a CORRELATION matrix, now to mix with the GARCH and get a COVARIANCE
matrix
F_pred_COV = ones(k,k,T) * 0;
F_pred_GARCH_STDEV=F_pred_GARCH.^(0.5);
for i=1:T

F_pred_COV(:,:,i)=diag(F_pred_GARCH_STDEV(i,:))*F_pred_CORR(:,:,i)*diag(F_pred_GARCH_S
TDEV(i,:));
end
% The non synchronous shift
if (lag == 1)
    F_pred_COV(:,:,2:T) = F_pred_COV(:,:,1:T-1);
    days = days + 1;
end

% OPTIMUM PORTFOLIO/BETS AND RESULTS
F_pred_A = ones(T,1) * 0;
F_pred_B = ones(T,1) * 0;
F_pred_C = ones(T,1) * 0;

for i=(days+1):T
    F_pred_A(i,:) = ones(k,1)' * F_pred_COV(:,:,i)^(-1) * ones(k,1);
    F_pred_B(i,:) = ones(k,1)' * F_pred_COV(:,:,i)^(-1) * F_pred_ARMA(i,:)';
    F_pred_C(i,:) = F_pred_ARMA(i,:) * F_pred_COV(:,:,i)^(-1) * F_pred_ARMA(i,:)';
end

F_pred_VAR = ones(T,1) ./ F_pred_A; % of the optimum portfolio
F_pred_RET = F_pred_B ./ F_pred_A; % of the optimum portfolio
bF_pred_DCC = F_pred_RET ./ F_pred_VAR; % of the optimum portfolio

F_pred_W = ones(T,k) * 0;
for i=(days+1):T



                                           39
    DELTA = F_pred_A(i,:) * F_pred_C(i,:) - F_pred_B(i,:)^(2);
    LAMDAH = (F_pred_C(i,:) - F_pred_B(i,:) * F_pred_RET(i,:)) / DELTA;
    SINGY = (F_pred_A(i,:) * F_pred_RET(i,:) - F_pred_B(i,:)) / DELTA;
    F_pred_W(i,:) = (LAMDAH * F_pred_COV(:,:,i)^(-1) * ones(k,1) + SINGY *
F_pred_COV(:,:,i)^(-1) * F_pred_ARMA(i,:)')';
end

% Results
dataW = ones(T,1) * 0;
for i=(days+1):T
    dataW = data(i,:) * F_pred_W(i,:)';
end

rDCC = dataW .* bF_pred_DCC;
rDCC_VAR = F_pred_VAR .* (bF_pred_DCC.^(2)); % actually ends up ret^2/var (if sub in
for)

% EQUALLY WEIGHTED
bF_pred_GARCH = F_pred_ARMA ./ F_pred_GARCH;
rGARCH = data .* bF_pred_GARCH;

rGARCH_EW = (rGARCH * ones(k,1)) ./ k;
rGARCH_EW_VAR = ones(T,1) * 0;
bF_pred_EW = ones(T,1) * 0;
for i=(days+1):T
    rGARCH_EW_VAR(i,:) = (ones(k,1)' / k) * F_pred_COV(:,:,i) * (ones(k,1) / k);
    bF_pred_EW(i,:) = bF_pred_GARCH(i,:) * (ones(k,1) / k);
end

% Running, comparable on bet size

DCCsum = 0;
EWsum = 0;

for i=(days+1):T
    DCCsum = DCCsum + abs(bF_pred_DCC(i));
    EWsum = EWsum + abs(bF_pred_EW(i));
end

rDCC = rDCC / DCCsum * 1000;
rGARCH_EW = rGARCH_EW / EWsum * 1000;

save rDCC.csv rDCC -ascii;
save rGARCH_EW.csv rGARCH_EW -ascii;

rDCCsum = ones(T,1) * 0;
rEWsum = ones(T,1) * 0;

for i=(days+1):T
    rDCCsum(i,:) = rDCCsum(i-1,:) + rDCC(i,:);
    rEWsum(i,:) = rEWsum(i-1,:) + rGARCH_EW(i,:);
end

save rDCCsum.csv rDCCsum -ascii;
save rEWsum.csv rEWsum -ascii;



---




                                          40

				
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