statistical arbitrage models of ftse 100

STATISTICAL ARBITRAGE MODELS OF THE FTSE 100

A. N. BURGESSDepartment of Decision Science

London Business SchoolSussex Place, Regents Park, London, NW1 4SA, UK

E-mail: [email protected]

In this paper we describe a set of statistical arbitrage models which exploit relative value relationshipsamongst the constituents of the FTSE 100. Rather than estimating cointegration vectors of highdimensionality, a stepwise regression approach is used to identify the most appropriate subspace for thestochastic detrending of each individual equity price. A Monte Carlo simulation is used to identify theempirical distribution of the Variance Ratio profile of the regression residuals, under the null hypothesisof random walk behaviour. Both a chi-squared test on the joint distribution of the Variance Ratioprofile, and additional tests based on its eigenvectors, indicate that as a whole the stochasticallydetrended stock prices deviate significantly from random walk behaviour and hence may containpredictable components. A combined cross-sectional and time-series model indicates that the relative“mispricing” of the equities tends to trend in the short-term and revert in the longer term. The out-of-sample performance of the models is consistently profitable using a simple trading rule, with thecombined portfolio suggesting a possible annualised Sharpe Ratio of over 7 for a trader with costs of 50basis points. Furthermore, information derived from the in-sample variance ratio profile is shown to besignificantly correlated with the out-of-sample profitability of the individual models – suggesting thatthe performance may be improved further by modelling the time-series properties conditionally onsuch information.

1 Introduction

In many cases the volatility in asset returns is largely due to movements which are market-wide or evenworld-wide in nature rather than specific characteristics of the particular asset; consequently there is arisk that this “market noise” will overshadow any predictable component of asset returns. A number ofauthors have recently suggested approaches which attempt to reduce this effect by suitablytransforming the financial time-series. Lo and MacKinley (1995) create “maximally predictable” portfoliosof assets, with respect to a particular information set. Bentz et al (1996), use a modelling framework inwhich prices are relative to the market as a whole, and returns are also calculated on this basis; this “de-trending” removes typically 90% of the volatility of asset returns, as is consistent with the Capital AssetPricing Model (CAPM) of finance theory. Burgess and Refenes (1996) use a cointegration framework inwhich FTSE returns are calculated relative to a portfolio of international equity indices, with theweightings of the portfolio given by the coefficients of the cointegrating regression. Steurer and Hann(1996) also adopt a cointegration framework, modelling exchange rates as short-term fluctuations aroundan “equilibrium” level dictated by monetary and financial fundamentals. This type of approach ingeneral is characterised as “statistical arbitrage” in Burgess (1996) where a principle componentsanalysis is used to create a eurodollar portfolio which is insulated from shifts and tilts in the yield curveand optimally exposed to the third, “flex” component; the returns of this portfolio are found to be partlypredictable using neural network methodology but not by linear techniques.

We define statistical arbitrage as a generalisation of traditional “zero-risk” arbitrage. Zero-risk arbitrageconsists of constructing two combinations of assets with identical cash-flows, and exploiting anydiscrepancies in the price of the two equivalent assets. The portfolio Long(combination1) +Short(combination2) can be viewed as a synthetic asset, of which any price-deviation from zerorepresents a “mispricing” and a potential risk-free profit1. In statistical arbitrage we again constructsynthetic assets in which any deviation of the price from zero is still seen as a “mispricing”, but this timein the statistical sense of having a predictable component to the price-dynamics.

Our methodology for exploiting statistical arbitrage consists of three stages:

1 Subject to transaction costs, bid-ask spreads and price slippage

• constructing “synthetic assets” and testing for predictability in the price-dynamics• modelling the error-correction mechanism between relative prices• implementing a trading system to exploit the predictable component of asset returns

In this paper we adopt an approach to statistical arbitrage which is essentially a generalisation of theeconometric concept of cointegration. We modify the standard cointegration methodology in two mainways: firstly we replace the cointegration tests for stationarity with more powerful variance ratio testsfor “predictability”, and secondly we construct the cointegrating regressions by a stepwise approachrather than the standard regression or principal components methodologies which are found in theliterature. These two innovations are easily motivated: firstly, variance ratio tests are more powerfulagainst a wide range of alternative hypotheses than are standard cointegration tests for stationarity,and hence are more appropriate for identifying statistical arbitrage opportunities; secondly, the highdimensionality of the problem space (approx. 100 constituents of the FTSE 100 index) necessitates theuse of a methodology for reducing the models to a manageable (and tradable!) complexity, but in asystematic and principled manner – for which the “subset” approach of stepwise regression is ideallysuited. The predictive model is simply a linear error-correction model using the cointegration residuals(asset “mispricings”) and lagged relative returns to forecast future relative returns on a one-day aheadbasis. The trading system described in this paper is very simple – simply taking offsetting long andshort positions which are proportional to the forecasted relative return. For a discussion of more-sophisticated trading rules for statistical arbitrage, see Towers and Burgess (1998a, b).

The paper is organised as follows. Section 2 describes the stepwise cointegration methodology and theMonte Carlo experiments to determine the distribution of the variance ratio profile under the nullhypothesis that the variables are all random walks. Section 3 describes the tests for predictability whichare based on the variance ratio analysis, and the results of applying these tests to the statistical“mispricings” obtained from the stepwise regressions. Section 4 describes the time-series model forforecasting changes in the mispricings and section 5 analyses the out-of-sample performance of thismodel. Section 6 explores the relationship between the characteristics of the variance ratio for a givenmispricing and the profitability of the associated statistical arbitrage model. Finally, a discussion andbrief conclusions are presented in section 7.

2 Distribution of the Variance Ratio profile of stepwise regression residuals

Our methodology for creating statistical arbitrage models is based on the econometric concept ofcointegration. Cointegration can be formally defined as follows: if a set of variables y are integrated oforder d (i.e. must be differenced d times before becoming stationary) and the residuals of thecointegrating regression are integrated of order d-b where b > 0 then the time-series are said to becointegrated of order (d,b).

i.e. if each yi is I(d) and εt is I(d - b) b >0 then y ~ CI( d, b)

The most common and useful form of cointegration is CI(1,1) where the original series are random walksand the residuals of the regression are stationary according to a “unit root” test such as the Dickey-Fuller (DF), Augmented Dickey-Fuller (ADF), suggested by Engle and Granger (1987) or thecointegrating regression Durbin-Watson (CRDW) proposed by Sargan and Bhargava (1983). Testsbased on a principal components or canonical correlation approach have been developed by Johansen(1988) and Phillips and Ouliaris (1988) amongst others.

In our case, however, the data consists of 93 constituents 2 of the FTSE 100 together with the indexitself, giving a dimensionality of 94- much higher than normal for cointegration analysis, and largerelative to the sample size of 400 (see section 3 for a description of the data). In order to reduce thedimensionality of the problem we decided to identify relationships between relatively small subsets ofthe data. There remains the problem of identifying the most appropriate subsets to form the basis of the

2 the remaining FTSE constituents were excluded from the analysis due to insufficient historical databeing available (e.g. for newly quoted stocks such as the Halifax building society)

statistical arbitrage models. In order ensure a reasonable span of the entire space, we decided to useeach asset in turn as the dependent variable of a cointegrating regression. To identify the mostappropriate subspace for the cointegrating vector we use a stepwise regression methodology in place ofthe standard “enter all variables” approach. Before moving on to analyse these models further, we willdescribe the basis of the “Variance Ratio” methodology which we use to test for potential predictability.

The variance ratio test follows from the fact that the variance of the innovations in a random walk seriesgrows linearly with the period over which the increments are measured. Thus the variance of theinnovations calculated over a period τ should approximately equal τ times the variance of single periodinnovations. The basic VR(τ) statistic is thus:

( )( )

VR( )ττ

τ τ

=−

−

∑

∑

∆ ∆

∆ ∆

d d

d d

tt

tt

2

2 (1)

The variance ratio is thus a function of the period τ. For a random walk the variance ratio will be close to1 and this property has been used as the basis of statistical tests for deviations from random walkbehaviour by a number of authors since Lo and McKinley (1988) and Cochrane (1988).

Rather than testing individual VR statistics, we prefer to test the variance ratio profile as a whole, firstlybecause there is no a priori “best” period for the comparison and secondly because it can summarise thedynamic properties of the time series: a positive gradient to the variance ratio function (VRF) indicatespositive autocorrelation and hence trending behaviour; conversely a negative gradient to the VRFindicates negative autocorrelations and mean-reverting or cyclical behaviour. Figure 1, below, showsthe VRFs for the Dax and Cac indices together with the VRF for the relative value of the two indices.

0.5

0.6

0.7

0.8

0.9

1

1.1

1 11 21 31 41Period

Var

ian

ce R

atio

Dax Cac

Relative RW

Figure 1: the Variance Ratio profile of the Dax and Cac indices individually and in relative terms. The xaxis is the period over which asset returns are calculated (in days), the y axis is the normalised variance ofthe returns. In this case, the fact that the relative price deviates further from random-walk behavioursuggests that it may be easier to forecast than the individual series

The usefulness of the variance ratio profile can be seen from the fact that it indicates the degree towhich the time-series departs from random walk behaviour – which may be taken as a measure of thepotential predictability of the time-series. This is unlike standard tests for cointegration which areconcerned with the related but different issue of testing for stationarity – a series may be nonstationarybut still contain a significant predictable component and thus the variance ratio will identify a widerrange of opportunities than the more restrictive approach of testing for stationarity. For both the Daxand the Cac the VRFs fall below 1, suggesting a certain degree of predictability - even though bothseries are nonstationary. Note also that the VRF for the relative price series is consistently below thoseof the individual series, indicating that the relative price exhibits a greater degree of potentialpredictability than either of the individual assets.

A problem with using the Variance Ratio test in conjunction with a cointegration methodology is thatthe residuals of a cointegrating regression (even when the variables are random walks) will not behaveentirely as a random walk – for instance, they are forced, by construction, to be zero mean. Moreimportantly, the regression induces a certain amount of spurious “mean-reversion” in the residuals andthe impact of this on the distribution of the VR function must be taken into account. In our case, there isone further complication in that we are using stepwise regression and hence the selection bias inherentin choosing m out of n > m regressors must also be accounted for. This is akin to the “data snooping”issue highlighted by Lo and McKinley (1990)

We thus performed a Monte Carlo simulation to identify the joint distribution of the variance ratioprofile under the null hypothesis of regressing random walk variables on other random walks (i.e. nopredictable component), accounting in particular for the impact of (a) the mean-reversion induced by theregression itself, and (b) the selection bias introduced by the use of the stepwise procedure. Thedistribution was calculated from 1000 simulations, in each case the parameters of the simulation matchthose of the subsequent statistical arbitrage modelling: namely a 400 period realisation of a random walkis regressed upon 5 similarly generated series from a set of 93 using a forward stepwise selectionprocedure, and the variance ratio profile calculated from the residuals of the regression3. The varianceratio is calculated for returns varying from one-period up to fifty periods. Note however, that byconstruction the value of VR(1) can only take the value 1.

From these 1000 simulations, both the average variance ratio profile and the covariance matrix ofdeviations from this profile were calculated. As we are interested in the “shape” of the VR profile wealso conducted a principle component analysis to characterise the structure of the deviations from theaverage profile. The scree plot of the normalised eigenvalues is shown below:

Figure 2: the scree plot of normalised eigenvalues for the covariance matrix of the variance ratio profile. Thefact that almost the entire variability can be represented by the first few factors (out of a total of 49) showsthat deviations from the average profile tend to be highly structured and can be characterised by only a smallnumber of parameters.

The average profile and selected eigenvectors are shown in figure 3, below. The average profile shows asignificant negative slope which would imply a high degree of mean reversion if this were a standard

3 Clearly it would be straightforward to repeat the procedure for other experimental parameters, samplesize, number of variables etc, but the huge number of possible combinations leads towards recalibratingonly for particular experiments rather than attempting to tabulate all possible conditional distributions.

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

1 2 3 4 5 6 7 8 9 10 11

NormalisedEigenvalue

Cumulative

variance ratio test. In our case it merely represents an artefact of the regression methodology which canbe taken as a “baseline” for comparing the variance ratio profiles of actual statistical

“mispricings”. Note also the highly structured nature of the eigenvectors – indicating that deviationsfrom the average profile have a tendency to be correlated across wide regions of lag-space rather thanshowing up as “spike” in the VR profile. The first eigenvector represents a low frequency deviation inwhich the variance is consistently higher than the average profile – patterns with a positive projectionon this eigenvector will tend to be trending whilst a negative projection will tend to indicate mean-reversion. The second eigenvector has a higher “frequency” and characterises profiles which mean-revert in the short term and trend in the longer term (or vice versa). Similarly the third eigenvectorrepresents a pattern of trend-revert-trend. The higher-order eigenvectors (not shown in the figure) tendto follow this move towards higher frequency deviations. The fact that the associated eigenvalues arelarge only for the first few components tells us that the residuals derived from random walk time-seriestend to deviate from the average profile only in very simple ways, as represented by the low-ordereigenvectors shown in the diagram.

0

0.2

0.4

0.6

0.8

1

1.2

Average

EigenVec1

EigenVec2

EigenVec3

Figure 3: Variance Ratio profiles for: average residual of regression from simulated random-walk data;characteristic deviations from the average profile as represented by selected eigenvectors

3 Analysis of Variance Ratio profiles of statistical “mispricings” of FTSE 100 stocks

Given the average profile and covariance matrix of the profile under the null hypothesis of random walkbehaviour, we can test the residuals of actual statistical arbitrage models for significant deviations fromthese profiles. The data used consist daily closing prices of the FTSE 100 and 93 of its constituentstocks. The prices were obtained from the Reuters TS1 database and it total consist of 500 observationsfrom 13 June 1996 to 13 May 1998. Of these 400 observations were used to estimate the cointegratingregressions and the final 100 observations were reserved for the purposes of out-of-sample evaluation.

Each asset in turn was used as the dependent variable in a stepwise regression, with constant term andfive regressors selected from the possible 93, and the VR profile of the resulting statistical mispricingtested for potential predictability in the form of deviation from random walk behaviour.

Two types of test were used, the first treating the distribution of the VR profile as multivariate normaland measuring the Mahalanobis distance of the observed profile from the average profile under the nullhypothesis. This approach to joint testing of VR statistics has previously been used by Eckbo and Liu(1996) and it is easy to show that the test statistic should follow a chi-squared distribution with degrees

of freedom equal to the dimensionality of the test. The second set of tests are designed to identifydifferent types of deviation from the average profile and are based on the projection of the deviationonto the different eigenvectors – under the null hypothesis these statistics should follow a standardnormal distribution. Figure 4, below, shows Variance Ratio profiles of the mispricings for selectedstatistical arbitrage models:

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Model 1 (FTSE+ )

Model 2 (ABF+ )

Model 76 (SDRt+ )

Model 87 (ULVR+ )

Figure 4: Selected variance ratio profiles for statistical mispricings obtained through stepwise regression of asseton remaining assets in FTSE 100 universe

The test results are shown in the table below; in order to account for deviations from (multivariate)normality we report the nominal size but also the empirical size of the tests – calculated from thecalibration data from the original simulation and also a test set from a second similar but independentsimulation. Eigenvectors derived from both the correlation and the covariance matrix are used in theanalysis.

Chi-sq EigCov1 EigCov2 EigCov3 EigCov4 EigCov5 EigCor1 EigCor2 EigCor3 EigCor4 EigCor5Cal 1.8% 1.6% 1.4% 1.4% 0.9% 1.4% 1.7% 1.1% 1.7% 1.5% 1.2%Test 4.3% 1.2% 0.9% 1.8% 1.3% 1.2% 1.3% 0.9% 1.3% 1.3% 1.6%Model 36.2% 8.5% 1.1% 2.1% 3.2% 3.2% 8.5% 4.3% 3.2% 4.3% 8.5%

Table 1: Comparison of VR tests for random-walk simulations and actual mispricings, nominal size of test = 1%


Table 2: Comparison of VR tests for random-walk simulations and actual mispricings, nominal size of test = 5%


Table 3: Comparison of VR tests for random-walk simulations and actual mispricings, nominal size of test =10%

The tests indicate that the mispricings of the statistical arbitrage models deviate significantly from thebehaviour of the random data – suggesting the presence of potentially predictable deviations from

( )MISs,t = − +=∑P w P cs t s ii c i s t, , ( , ),1

5

randomness. The table below shows ‘z’ tests of the average scores of the true mispricings whencompared to the simulated test data:

Chi-sq EigCov1 EigCov2 EigCov3 EigCov4 EigCov5 EigCor1 EigCor2 EigCor3 EigCor4 EigCor5AveTest 50.99 -0.01 -0.01 0.00 0.01 0.00 -0.15 0.03 0.10 0.08 -0.07VarTest 135.19 0.22 0.04 0.01 0.01 0.00 33.57 5.56 2.72 1.57 0.83AveModel 70.79 -0.23 0.01 0.07 -0.03 -0.03 -2.34 0.32 1.02 -0.65 0.61VarModel 676.34 0.41 0.04 0.02 0.01 0.00 62.67 7.45 3.83 1.86 1.06z' stat 7.3 -3.2 0.5 4.0 -4.1 -5.0 -2.6 1.0 4.4 -5.0 6.3p-value 0.00000 0.00129 0.61169 0.00006 0.00004 0.00000 0.00890 0.32816 0.00001 0.00000 0.00000

Table 4: Comparison of average values of the various VR tests for random-walk simulations and actualmispricings

This result reinforces the findings that the actual mispricings deviate from random behaviour. In the nextsection we describe a forecasting model based on these mispricings.

4 Modelling the dynamics of the statistical mispricings

In this section we describe the error-correction model which forecasts one-day-ahead changes in thestatistical mispricings of the FTSE 100 stocks.

A single “pooled” model was estimated across the cross-section of 94 mispricing models and sampleperiod of 400 observations. In order to capture any “mean reversion” effects, the one day aheadchanges in the mispricings were regressed on the current level of the mispricing:

(2)

where Pc(i,s) is the price of the i’th constituent asset for the model of stock ‘s’ and ws,i is the associated

regression coefficient (portfolio weighting).

The remaining independent variables were selected in order to capture properties of different segmentsof the lag-space of mispricing dynamics and are of the form:

( )L n m s t, , = −MIS MISs,t-n s,t-m (3)

with the resulting regression of the form:

MIS MIS MISs,t +1 s,t s,t− = + + +

+ + + + +

α β β β

β β β ε0 1 2

3 4 5 1

0 1 1 2

2 5 10 10 20

L L

L L Ls t s t

s t s t s t s t

( , ) ( , )

( , ) (5, ) ( , ), ,

, , , ,

(4)

In total, 94*400 = 37600 observations were used to estimate the model, leaving 94*100=9400 for out-of-sample evaluation. The regression output is shown below:

SUMMARY OUTPUT

Regression StatisticsMultiple R 28.6%R Square 8.2%Adjusted R Square 8.2%Standard Error 0.016Observations 37600

ANOVAdf SS MS F Significance F

Regression 6 0.83 0.14 559.02 0Residual 37593 9.31 0.0002Total 37599 10.14

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 0.000 0.0001 -2.05 0.0401 0.000 0.000MIS -0.188 0.0043 -43.48 0.0000 -0.197 -0.180L1020 0.021 0.0027 7.99 0.0000 0.016 0.027L510 0.030 0.0036 8.24 0.0000 0.023 0.037L25 0.037 0.0043 8.76 0.0000 0.029 0.046L12 0.018 0.0060 2.96 0.0031 0.006 0.029L01 0.107 0.0060 17.97 0.0000 0.096 0.119

The model shows significant predictability in future changes of the statistical mispricings. Thispredictability derives from two sources - firstly a short term trend as represented by the positivecoefficients for the lagged difference terms L(n,m), and secondly a long term error-correction asrepresented by the negative coefficient for the mispricing MIS. Given the size of the dataset from whichthe model was estimated, the results are all highly significant and the adjusted R2 suggests that thepredictable component accounts for 8.2% of total variability in the mispricings. In spite of this, it isunclear how much of this effect is spuriously induced by the cointegrating regression methodologywhich was used to generate the mispricings - the true test of the model is on the out-of-sampleperformance, an evaluation of which is presented in the following section.

5 Performance Analysis

Firstly let us consider the aggregate performance which is achieved by averaging the cross-sectionperformance of the models - this is equivalent to trading a portfolio with an equal weight in each of theindividual statistical arbitrage models.

The out-of-sample aggregate equity curve is shown in figure 5, below:

0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

0 10 20 30 40 50 60 70 80 90

Time (Days)

Cu

mu

lati

ve P

rofi

t

Combined Models

With costs = 50bp

Figure 5: Aggregate equity curve, averaged across the performance of the 94 statistical arbitrage models

A set of performance metrics for the aggregate performance are reported in table 5 below:

Profitable Ave Ret SD ret Ret (Annual) SD (Annual) SharpeNo costs 85% 0.16% 0.14% 31.75% 2.03% 15.7Costs = 50bp 67% 0.08% 0.14% 15.73% 2.02% 7.8

Table 5: Aggregate cross-section performance of the statistical arbitrage models: the first row showsperformance excluding trading costs, the second row shows performance with trading costs assumed equal to 50basis points (0.5%) The metrics are directional ability (percentage of periods in which profits are positive),daily and annualised return and risk (measured as standard deviation of return), and Sharpe Ratio of annualisedreturn to annualised risk.

The trading performance suggests that the model is highly successful - the diversification acrossmodels means that on this aggregate level the strategy is profitable in 85% of the out-of-sample periods(falling to 67% when costs are included). After costs the annualised return is just over 15% which isvery satisfactory given that the trading is market neutral and could be overlaid on an underlying longposition in the market. Alternatively the Sharpe Ratio suggests that the returns are large when comparedto the capital requirements of covering the associated risks and that in this risk-adjusted sense thesystem is highly attractive. Note that the performance is highly sensitive to the assumed level of tradingcosts - one-way costs of 50bp reduce the return by half, with the break-even point lying close totransaction costs of 1%. From this perspective the usefulness of such a system is conditional on thecircumstances of the user - whilst a bank may have costs as low as 10-20 basis points, the equivalentcost for an individual is likely to be over 1%, hence negating the information advantage provided by themodel.

The table below summarises the performance metrics of the individual models; the detailed results arepresented in Appendix C.

Model Correlation Direction Return Risk Sharpe Direction(Adj) Return(adj) Risk(adj) Sharpe (adj)Min 0.006 46% -7.0% 5.8% -0.3 26% -38.0% 5.8% -6.5Max 0.386 66% 184.4% 67.6% 5.4 59% 160.3% 67.2% 4.0Ave 0.224 56% 58.2% 21.6% 2.8 44% 25.3% 21.4% 1.0

Table 6: Summary of the performance metrics evaluated for individual models; the table reports the min, maxand average values of: predictive correlation (between actual and forecasted returns), Directional forecastingability, annualised return risk and Sharpe Ratio, and equivalent figures adjusted for transaction costs at a level of50 basis points (0.5%). Note that the figures in a given row may be derived from different models.

The key feature of the results in table 6 is the wide range of performance across the individual models.Note that, after adjusting for transactions costs, the models are only profitable in 44% of the out-of-sample periods and yet still return positive profits - suggesting that the models are better at forecastingthe larger moves. The average Sharpe Ratio of the models is only 1.0 but notice that by aggregatingacross the models the average return is unaffected whilst the average risk is significantly reduced. Fromthis perspective the improvement from a Sharpe Ratio of 1.0 on an individual basis, to 7.8 on anaggregate basis (see table 5) would be expected only from models which are almost uncorrelated andhence can significantly reduce risk by means of diversification.

6 Investigation of the relationship between Variance Ratio profile and profitability

In the final phase of the analysis, we investigated the relationship between the insample properties ofthe variance ratio profiles of the different models, and the variability in their profitability during the out-of-sample period. This analysis consisted of regressing the out-of-sample Sharpe Ratios of theindividual models on their VR statistics (Mahalanobis distance and eigenvector projections). Astepwise regression procedure resulted in the model shown below:

Multiple Regression Analysis-----------------------------------------------------------------------------Dependent variable: adjSharpe----------------------------------------------------------------------------- Standard TParameter Estimate Error Statistic P-Value-----------------------------------------------------------------------------CONSTANT 0.330649 0.166633 1.9843 0.0503EigCor3 0.281245 0.075705 3.71502 0.0004EigCor5 1.35602 0.316422 4.28548 0.0000EigCov5 14.3265 5.00645 2.86161 0.0052-----------------------------------------------------------------------------

Analysis of Variance-----------------------------------------------------------------------------Source Sum of Squares Df Mean Square F-Ratio P-Value-----------------------------------------------------------------------------Model 61.4094 3 20.4698 12.67 0.0000Residual 145.455 90 1.61617-----------------------------------------------------------------------------Total (Corr.) 206.864 93

R-squared = 29.6858 percentR-squared (adjusted for d.f.) = 27.342 percentStandard Error of Est. = 1.27129Mean absolute error = 0.915465Durbin-Watson statistic = 1.85346

The regression diagnostics indicate a significant relationship between certain aspects of the varianceratio profile during the insample period, and risk-adjusted return during the out-of-sample. In particularthe projections of the deviations from the average profile onto Eigenvectors 3 and 5 were found to besignificantly related to profitability. In principle this information could be used in two ways: firstly toidentify the models which are more likely to be profitable and weight them appropriately; secondly asadditional conditioning information in an appropriate nonlinear error-correction model equivalent to (4)but allowing for interaction effects between the time-series dynamics and the information derived fromthe variance ratio profile. This aspect is the subject of current research.

7 Conclusion

The concept of cointegration provides a suitable basis for statistical arbitrage models but, beingmotivated by the search for theoretical understanding, is rather too restrictive if applied in the standardmanner. This paper introduces two general directions in which cointegration analysis can be generalisedto statistical arbitrage: the first is the method which is used to generate the “mispricing relationship” - inthis case stepwise regression rather than standard regression - and the second is the nature of the testsemployed - in our case tests for predictability which are based on the variance ratio profile of themispricing time-series. In this case Monte-Carlo analysis is required in order to estimate the jointdistribution of the individual variance ratio statistics and the test results indicate that the assumption ofmultivariate normality is almost - if not quite - accurate. In this paper, we have hardly touched upon themany options which are available for building the error-correction models, and for implementing tradingsystems which best exploit the information in the forecasts which they generate. In spite of the fact that the focus of the work is placed elsewhere, the trading performance of the systemappears to be impressive - returning an annualised Sharpe Ratio of 7.8 at a realistic transaction cost levelof 50 basis points. From this perspective, the key feature of the system is the benefit of combiningdiversified models, in terms of reducing the aggregate risk - the maximum Sharpe Ratio of the individualmodels is 4.0, and the average only 1.0, much less impressive than the overall figure! Finally, the underlying approach is equally applicable to a wide range of asset classes where assetsshare common stochastic trends and hence can be rendered more predictable by modelling in terms ofrelative prices rather than in raw form. Current research projects are concerned with a range of equity,fixed-income and derivatives markets, with sampling frequencies ranging between 10 minutes and daily.

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