and mt = Xt − (0.632Yt+ 0.703Zt )
the long-term volatility, thus providing evidence for a substantial mean-reverting component in the mispricing dynamics.Let us now evaluate the effectiveness of the cointegration procedure at “reverse engineering” the underlying factor dynamics. In attempting to replicate the “target” timeseries X the cointegrating regression procedure creates the “synthetic asset” 0.632Y +0.703Z which has similar exposures to the common factors f1 and f2 . Thus in the mispricing time series Xt − (0.632Yt + 0.703Zt ) the net exposure to the common factors isclose to zero, allowing the mean-reverting asset-speciﬁc effects ε1 , ε2 , ε3 to dominate themispricing dynamics. This “statistical hedging” of the common risk factors is quantiﬁedin Table 2.2, which reports the proportion of the variance of each observed time serieswhich is associated with each of the underlying factors.This demonstrates that the use of cointegrating regression can immunise against common underlying factors which are not observed directly but instead proxied by theobserved asset prices. Whilst the variance of changes in the original time series X, Yand Z is primarily (70–90%) associated with the common risk factors f1 and f2 , theeffect of these factors on the mispricing m is minimal (0.2%). Conversely, the relativeeffect of the asset-speciﬁc factors is greatly magniﬁed, growing from 10–30% in theoriginal time series to 99.8% in the relative mispricing m.By magnifying the component of the dynamics which is associated with asset-speciﬁceffects, we would expect to magnify the predictable component which (by construction)is present in the asset-speciﬁc effects but not in the common factors. This effect can be
Cointegration to Hedge and Trade International Equities
53
Table 2.2 Sensitivity of price changes of the originaltime series X, Y and Z and the “mispricing” time seriesXt − (0.632Yt + 0.703Zt ). The table entries show the proportion of the variance of each time series which is associated with changes in common risk factors f1 and f2 andasset-speciﬁc effects ε1 , ε2 , ε3X
Y
Z
m
f1f2ε1ε2ε3
46.1%41.5%11.5%0.5%0.4%
64.2%10.1%0.2%25.1%0.4%
16.8%67.8%0.0%0.1%15.3%
0.2%0.0%54.5%23.7%21.6%
Total
100.0%
100.0%
100.0%
100.0%
quantiﬁed by considering the Dickey–Fuller statistics obtained from simple ECMs of thetime series dynamics:DF(st ) = β̂/σβ̂ from regression st = α − β̂st + ηt
where ηt is a noise term(2.11)i.e. we regress changes in the time series (st ) against the level of the series (st ) and testfor a statistically signiﬁcant error-correcting coefﬁcient β. The details of the estimatedECMs for our experiment are presented in Table 2.3.The DF statistic approximately follows a t-distribution so, roughly speaking, DF valuesgreater than two indicate signiﬁcant evidence for a mean-reverting/error-correction effect.For the underlying (but unobserved) factors, the low DF statistics for f1 and f2 conﬁrmthe lack of predictable components in these common factors, whilst the high DF values forε1 , ε2 and ε3 (4.908, 5.644 and 4.454 respectively) conﬁrm the highly signiﬁcant degreeof mean reversion in the asset-speciﬁc effects.In the observed series, X, Y and Z, the mean-reverting effect is “watered down” bythe unpredictable factor effects, with the result that the corresponding DF statistics aresmall (actually slightly negative) and present no evidence of a predictable component.Table 2.3 Details of simple error-correction models estimated to quantify the mean-revertingcomponent in both the unobserved factors and the observed time series. Values in bold correspondto cases where the estimated mean-reversion coefﬁcient β̂ is signiﬁcant at the 0.1% level. Therows in the table are: estimated reversion parameter β̂; standard error of estimate; associated DFstatistic (approximately equivalent to the t-statistic in a standard regression); proportion of varianceexplained by model (R 2 )Factor/asset
f1
f2
ε1
ε2
ε3
X
Y
Z
Estimated β̂0.014 −0.0010.0960.1240.079 −0.001 −0.000 −0.0010.0080.0010.0200.0220.0180.0020.0030.001Std. error σβ̂1.631 −0.5804.9085.6444.454 −0.452 −0.004 −0.591DF(st )0.5%0%4.8%6.2%4.0%0%0%0%R2
m0.0770.0184.3983.9%
54
Applied Quantitative Methods for Trading and Investment
This picture changes dramatically when we look at the constructed “mispricing” m whichhas a high DF statistic of 4.398 – almost as high as for the true asset-speciﬁc effects.The actual magnitude (as opposed to statistical signiﬁcance) of the detected meanreversion effect is given by the R 2 values in the table. The results conﬁrm that thepredictable component of the dynamics is almost as strongly present in the mispricingtime series as in the underlying, but unobserved, asset-speciﬁc dynamics themselves. Themagnitude of the deterministic component in the mispricing is 3.9%, which is comparableto the 4.8%, 6.2% and 4.0% in the true asset-speciﬁc dynamics, and a negligible amountin the case of the original time series X, Y and Z.These results from our controlled experiment serve to illustrate the power of the cointegration approach to remove market-wide risk factors and highlight the asset-speciﬁccomponents of price dynamics, which in this case were constructed to contain a meanreverting effect. However the qualitative reasoning presented in Sections 2.3 and 2.4,together with quantitative evidence from other sources, suggests that similar results may beobtained for real asset prices. In the following section we apply essentially the same techniques as those used in this controlled experiment to analyse price relationships betweenreal assets, namely the equities which constitute the European-wide STOXX 50 index.
2.6 APPLICATION TO INTERNATIONAL EQUITIESIn this section we describe an application of the cointegration tools and techniquesdescribed above to data from those international equities which comprised the STOXX50 index as of 4 July 2002. We describe this analysis with reference to the accompanyingExcel workbook named “equity coint.xls” on the CD-Rom.The set of equities which constitute our universe are listed in the ﬁrst sheet of theworkbook (named “Constituents”). The full set of equities included in the analysis arelisted in Table 2.4.The second sheet in the workbook is named “Prices” and contains the raw data forthe analysis. This consists of daily closing prices which have been adjusted to removethe effects of stock splits, dividends and other corporate actions. The time frame for theanalysis is from 14 September 1998 to 3 July 2002, which is the longest period overwhich continuous data is available across the whole set of stocks. This comprises almost4 years of data, giving 993 daily observations.Note that this data does not provide a true “snapshot” of the European equity markets due to the complication that the closing times differ across the different nationalexchanges. For a practical trading system this would induce serious distortions to ourmodels, but for our purposes here the close prices serve adequately to illustrate the useof the tools we have described above.The third sheet (“Pairs”) contains a simple relative value analysis of a pair of assetsat a time. The sheet also serves to illustrate the data itself and the use of variance ratiofunctions to identify the underlying time series dynamics. A screen shot of this worksheetis shown in Figure 2.10.Cells D37 and D38 are used to select two equities whose prices we wish to compare.The equities are selected by entering numbers from 1 to 50 corresponding to the referencenumbers shown in Table 2.4. The example given shows the case of selecting the Britishoil-stock BP (BP.L, number 6 in the set) and the French oil-stock Total-Fina (TOTF.PA,number 27 in the set). The lower chart plots BP against Total and also shows the synthetic
Cointegration to Hedge and Trade International Equities
55
Table 2.4 The list of companies included in the analysis. The 50 stocks correspond to the constituents of the pan-European STOXX 50 index as of 4 July 2002Ref. Name12345678910111213141516171819202122232425
British TelecomGlaxo SmithklineAlcatelUBSDaimler ChryslerBPAstro-ZenecaNokiaNovartisEricssonPhilipsINGABN AmroAegonUnileverRoyal DutchSwiss ReRocheVivendiBSCHNestleCarrefourBNP-ParibasAvivaAXA
SymbolBT.LGSK.LCGEP.PAUBSZn.VXDCXGn.DEBP.LAZN.LNOK1V.HENOVZn.VXERICb.STPHG.ASING.ASAAH.ASAEGN.ASUNc.ASRD.ASRUKZn.VXROCZg.VXEAUG.PASAN.MCNESZn.VXCARR.PABNPP.PAAV.LAXAF.PA
Ref. Name26272829303132333435363738394041424344454647484950
Zurich FinancialTotal-FinaSuezOrealTelecom ItaliaENIEonSiemensDeutsche BankGeneraliDeutsche TelecomBBVAAllianzBayerBarclaysHSBCDiageoLloyds BankPrudentialRoyal Bank of ScotlandShellVodafoneTelefonicaMunich ReCredit Swiss
SymbolZURZn.VXTOTF.PALYOE.PAOREP.PATIT.MIENI.MIEONG.DESIEGn.DEDBKGn.DEGASI.MIDTEGn.DEBBVA.MCALVG.DEBAYG.DEBARC.LHSBA.LDGE.LLLOY.LPRU.LRBOS.LSHEL.LVOD.LTEF.MCMUVGn.DECSGZn.VX
asset which represents the relative return on the two stocks. All three series have beennormalised to represent log price changes since the beginning of the analysis period. Aclose-up of the chart is shown in Figure 2.11.In this case we see that there appears to be a semi-stable equilibrium which existsbetween the two asset prices. For long periods of time the relative price tends to ﬂuctuatearound an equilibrium or “fair price” level, however signiﬁcant shifts in the relationshipalso occur, such as the 30% shift in the relative value which occurred during the early partof 2000. The apparent existence of a relationship between the two price series, togetherwith the instability in this relationship, serve to respectively illustrate the opportunitiesand the risks which arise from a relative value approach to trading.The top half of the sheet contains a variance ratio analysis of the price dynamics ofthe selected equities and the synthetic asset corresponding to their relative prices. Thecells in the range C5:E34 contain array formulae to calculate the n-period variancesfor each of the three time series, with n varying between 1 and 30. To the right ofthese variances, cells H5:J34 contain the variance ratios, with each n-period variancenormalised by n times the one-period variance. These three functions are plotted in thechart to the right of the numbers, with the example for BP and Total-Fina shown inFigure 2.12.
56
Applied Quantitative Methods for Trading and Investment
Figure 2.10 The “Pairs” worksheet containing a pairwise relative value analysis, the selectedstocks are BP (number 6) and Total-Fina (number 27)0.700.600.500.400.300.200.100.00−0.10−0.20−0.309/14/98
Figure 2.11
BP.LTOTF.PABP.L/TOTF.PA
3/14/99
9/14/99
3/14/00
9/14/00
3/14/01
9/14/01
3/14/02
Relative prices for BP, Total, and the synthetic asset which is the ratio of the two
In this case we see that the variance ratio functions for the two securities show decliningproﬁles, indicating the presence of reverting components in their time series dynamics. Themean-reversion tendency is signiﬁcantly more prominent in the synthetic asset (BP/Total)than for either of the individual assets, providing further evidence to support the presenceof a potentially predictable component in the relative price dynamics.Given such evidence of mean-reverting dynamics we could move on to implementa statistical arbitrage strategy based on the types of trading rules described by Burgess
Cointegration to Hedge and Trade International Equities
57
1.2
1.0
Variance ratio
0.8VR(BP.L)VR(TOTF.PA)VR(BP.L/TOTF.PA)
0.6
0.4
0.2
29
27
25
23
21
19
17
15
13
11
9
7
5
3
1
0.0Window length (days)
Figure 2.12 Variance ratio functions for BP, Total, and the synthetic asset which is the ratio ofthe two
(1999) and Towers (2000). Note however that in this particular case at least some ofthis effect will be due to the non-synchronous sampling of the close price in the Frenchand UK markets. Because of this non-synchronicity a more sophisticated analysis (andprobably additional data) would be needed to evaluate the true magnitude of the meanreverting effect in the relative price of these equities and its viability as the basis for aproﬁtable trading strategy.Whilst pairs analysis works well for some equities, it is highly sensitive to the propertiesof each asset price and works better for some stocks than for others. Essentially it requiresthat for a given equity, there is one (and only one) equity which has similar exposures toeach and every underlying factor. For a given equity there may be zero, one or more thanone closely matching pairs and only in the case of a single matching pair is the simpleapproach likely to be close to optimal. These complications mean that pairs analysis isessentially opportunistic in nature rather than representing a general strategy which canbe applied across a broad asset universe.Cointegration modelling is essentially an extension of pairs analysis which is designedto overcome these limitations. Rather than requiring the existence of a single perfectmatch we instead create an optimally matching “synthetic asset” in the form of a weightedcombination of one or more assets. The remaining sheets in the workbook demonstratethe workings and results of this more sophisticated form of relative value modelling.Firstly, the sheet “CointAnalysis” illustrates the construction of a synthetic asset tomatch a chosen “target” asset. A screen shot of this worksheet is shown in Figure 2.13. Thetop-left of the worksheet contains various control parameters and diagnostic information.The chart in the top-centre of the worksheet presents a variance ratio analysis of thestatistical mispricing. The bottom-left chart is a visualisation of the synthetic asset andthe chart in the bottom-right shows the evolution of the various price series over time.
58
Applied Quantitative Methods for Trading and Investment
Figure 2.13 The “CointAnalysis” worksheet showing the construction of a synthetic asset tomatch asset number 6: British Petroleum (BP.L)
CONTROLS
Figure 2.14
Manual Select X
6
using:
RidgeFac
0.01
total
993
Insampleoutsample
700293
6
The controls for the cointegration analysis
The controls for the cointegration analysis are contained in the top-left of the worksheet.As elsewhere in the workbook, the convention is that user-speciﬁed controls are containedin cells with a black border and yellow background. In this case there are four such cells,as shown in Figure 2.14.Firstly, the target series is speciﬁed in cell F2, using the reference numbers listed inTable 2.4. In this case we remain with the same example as before: asset number 6, BritishPetroleum, or BP.L for short. Note that in order to allow the generation of automatic tables,the actual control cell is H2, and cell F2 acts as a kind of manual override.
Cointegration to Hedge and Trade International Equities
59
The second control cell is F4, labelled “RidgeFac”. This represents an importantmodiﬁcation of the basic methodology, which is necessary to avoid the problems causedby regressing on large numbers of variables. Rather than using a standard regression,this more practical methodology uses a “ridge regression” in which the resulting parameters are in some sense “smoothed” or “regularised” and this cell controls the amount ofsmoothing (Hoerl and Kennard, 1970a,b).The ﬁnal control parameters consist of the number of observations which should be usedto construct the model (the “in-sample” set) and the subsequent number of observationswhich should be used to evaluate the model performance (the “out-of-sample” set). CellF6 indicates the number of observations available in total, which for this analysis is 993.Cell F8 is used to specify the number of “in-sample” observations. In this case we use700 observations, representing approximately two-thirds of the available data. By default,all of the remaining observations are used to perform the “out-sample” evaluation. Thisnumber can be overridden using cell G9, but in this case is left as the default, giving 293observations for the out-of-sample results analysis.The data for the regression is collected on the “CointModel” worksheet. The targetasset is stored in column H of this worksheet; a constant column of ones is placed incolumn J; and the 49 cointegrating assets are remapped to the adjacent columns K throughto BG. It is useful to have the 50 independent variables in contiguous columns in order tosimplify the matrix algebra used to compute the solution to the cointegrating regression.The calculations for the cointegrating regression are performed on the “Workings” sheet.The worksheet performs a “ridge regression” (Hoerl and Kennard, 1970a,b) in which thesolution is given by β = (CT C + λσ I)−1 Ct. The target vector t and data matrix C (the49 other asset price series supplemented by a column of ones) are referenced from the“CointModel” worksheet. The regularisation parameter lambda (λ) is referenced from cellF4 of the “CointAnalysis” worksheet. The covariance matrix CT C is calculated in cellsG4:BD53. The vector Ct is calculated in BI4:BI53. The enhanced covariance matrix,CT C + λσ I, is constructed in cells BM4:DJ53 by re-scaling the diagonal elements ofCT C. The inverse of this enhanced matrix is calculated in cells G56:BD105. Finally thebeta parameters are calculated in cells BI56:BI105 by multiplying this inverse by thevector Ct.With the regularisation parameter set to λ = 0, the solution reduces to the standardOLS regression: β = (CT C)−1 Ct. Lambda acts as a scaling coefﬁcient for the diagonalcomponent of the covariance matrix CT C, proportionally downweighting the off-diagonalcovariance terms and reducing the apparent correlation between the different series. Aswe will see below this has an important effect in stabilising the regression and enablingus to use 50 regressor variables, more than would normally be practically feasible.The resulting beta vector is copied across to cells J25:BG25 of the “CointModel” worksheet and used to construct the synthetic asset. This is calculated as the beta-weightedaverage of the 50 constituent assets (including constant term) and is stored in cellsG40:G1032. Note that once the betas have been estimated from the ﬁrst 700 observations (in this case), the same weights can be applied to subsequent data to calculate thevalues of the synthetic asset during the out-of-sample period. For purposes of visualisingthe composition of the synthetic asset, we take the beta vector and multiply through bythe scale of the individual time series. The resulting “effective weights” are illustrated inthe lower left-hand chart on the “CointAnalysis” worksheet which is also reproduced inFigure 2.15.
60
Applied Quantitative Methods for Trading and Investment
20%SHEL.L
15%
RD.AS
CARR.PA
10%
TOTF.PA
BT.L
5%
0%
ENI.MIBNPP.PACSGZn.VXLLOY.LAV.LBARC.LNESZn.VXEONG.DEAZN.L PHG.ASHSBA.L PRU.L VOD.LGSK.L NOK1V.HEOREP.PA DBKGn.DEUNc.AS SAN.MCDGE.LUBSZn.VXAXAF.PABAYG.DETEF.MCNOVZn.VX
constantDCXGn.DE
AAH.AS
RUKZn.VXSIEGn.DE BBVA.MCROCZg.VXLYOE.PADTEGn.DEEAUG.PAING.AS AEGN.ASZURZn.VX TIT.MI GASI.MIALVG.DE
CGEP.PA ERICb.ST
−5%
RBOS.L
MUVGn.DE
−10%BP.L
Figure 2.15 Effective weights for the synthetic asset for British Petroleum (BP.L) λ = 0.01
In the case of BP, the synthetic asset weights are dominated by other oil stocks, particularly Royal Dutch/Shell (RD.AS and SHEL.L) and Total-Fina (TOTF.PA), howevermost of the other stocks also have non-zero, though small, weightings indicating that thebest historical ﬁt to BP price movements is obtained by taking into account a wide rangeof other stocks.Given the target asset and the constructed synthetic asset we can calculate the differencein price which is equivalent to the residual of the regression. The evolution of this timeseries represents the performance of a hedged portfolio with a long position in the targetasset and an offsetting short position in the synthetic asset. If the synthetic asset is a goodhedge for the target, this residual price should have low volatility and remain close tozero. In order to evaluate the effectiveness of the cointegration procedure we compare thisprice residual to that obtained by a simpler procedure, namely hedging with an equallyweighted “market” portfolio. These time series are visualised in the bottom right-handchart of the “CointAnalysis” worksheet, which is reproduced in Figure 2.16.The vertical line divides the time axis into the in-sample and out-of-sample periods.During the in-sample period we expect the synthetic asset to closely match the target(BP.L) simply by construction; similarly the corresponding “residual” is stable aroundthe zero level. Note that the synthetic asset is an average across a number of stocks andin this case, as would be typical, has a smoother price trajectory than the target assetitself but on the whole does tend to track the longer term price movements observed inthe target series. The synthetic market price obtained as an unweighted average acrossthe set of stocks appears to be less successful in following the price of the target assetand this is also observed in the higher volatility of the corresponding (market) residual.During the out-of-sample period, the synthetic asset price will only track the target assetto the extent to which it has a similar exposure to the underlying risk factors which drive
Cointegration to Hedge and Trade International Equities
61
800700600500BP.Lsyntheticresids-mktm-resid
4003002001000−100
Figure 2.16
5/14/2002
3/14/2002
1/14/2002
9/14/2001
11/14/2001
7/14/2001
5/14/2001
3/14/2001
1/14/2001
9/14/2000
11/14/2000
7/14/2000
5/14/2000
3/14/2000
1/14/2000
9/14/1999
11/14/1999
7/14/1999
5/14/1999
3/14/1999
1/14/1999
9/14/1998
−300
11/14/1998
−200
Hedged and unhedged time series for the cointegration model for BP
asset prices. In this case the model for BP appears quite successful, the residual remainsin a similar price range as during the in-sample period and seems also to be relativelystable around the zero level.The “CointAnalysis” worksheet also displays some basic measures which quantifysome properties of the synthetic asset and the out-of-sample performance. The valuescorresponding to this example are shown in Figure 2.17.The ﬁrst two values characterise the makeup of the synthetic asset. The “sum” ﬁgurecorresponds to the normalised sum of the asset weights, typically we would expect thisto be close to 100%. The “sumabs” ﬁgure corresponds to the normalised sum of theabsolute asset weights; if there are some negative weights, these will typically be offset
RESULTS
sumsumabs
100%166%
RawVarResVarResMkt
1294.93423.151262.60
ReductionMktRedImprove
67%2%65%
Figure 2.17 Characteristics of the cointegration model for BP.L
62
Applied Quantitative Methods for Trading and Investment
by positive weights over and above 100% and the sum of the absolute weights willreﬂect this. The ﬁgure indicates that the sum of the absolute weights is 166% in thiscase, reﬂecting negative weights totalling about 33% and offset by approximately 133%of positive weights, to give a total of 166%. Cross-checking against the visualisation ofthe weights in Figure 2.15, these numbers seem to be reasonable.The sum of the absolute weights can be quite an important issue, as it provides anestimate of the quantity of assets we need to buy and sell in order to use the syntheticasset as a hedge. This measure also highlights the importance of using regularisation. Forinstance, with regularisation set to zero, the sum of the absolute weights for the BP.Lsynthetic asset becomes 404%, indicating that each unit of BP needs to be hedged againsta long–short combination of equities totalling four times the value invested in BP!The remaining ﬁgures serve to quantify the effectiveness of the synthetic asset at hedging the volatility in the target asset. These measures are calculated during the out-of-sampleperiod in order to produce unbiased results. The “RawVar” ﬁgure corresponds to thevolatility of the asset, measured in terms of price variance, the “ResVar” is the residualvariance when hedged by the synthetic asset, and “ResMkt” is the residual variance whenhedged against an equally-weighted “market” portfolio. The ﬁnal three ﬁgures representthe proportional effectiveness of the hedging procedure. Thus in this case, the “Reduction” of 67% indicates that the synthetic asset hedge removes 67% of the out-of-samplevolatility. The “market” portfolio is not a good hedge in this case, only removing 2% ofthe volatility in BP, and thus the cointegration approach improves on the market hedge by65% of the original volatility. This particular example serves to highlight the potentiallylarge improvement which can be obtained by replacing market-based hedging with thecointegration approach, but it is only fair to note that in other cases the market hedgeperforms equally well or even better than the cointegration approach. A fairer comparison,across the whole set of 50 stocks, will be presented towards the end of this section.The ﬁnal part of the “CointAnalysis” worksheet presents a variance ratio analysis ofthe time series dynamics of the hedged portfolio, with the result being shown in the chartat the top of the worksheet. The calculations underlying this chart are contained in thetop-left corner of the “CointModel” worksheet. The results for the BP.L model are shownin Figure 2.18.
1.4
900
1.3
Variance ratio (LH axis)
800
1.2
Variance (RH axis)
700
1.1
600
1
500
0.9
400
0.8
300
0.70.6
200
0.5
1000
0.41 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Length of time period
Figure 2.18 Variance ratio analysis of the cointegration model for BP.L
Cointegration to Hedge and Trade International Equities
63
The variance ratio chart clearly indicates the decaying pattern which corresponds tomean-reverting dynamics. The 30-period variance ratio is just below 0.5, indicating thatthe variance computed over 30-day intervals is less than half as high as would be expected,given the observed 1-day variance. This suggests that, from a relative value perspective,over 50% of the short-term volatility in BP is essentially spurious price ﬂuctuation whichhas a strong tendency to cancel itself out over a longer time-scale. This pattern can alsobe observed, though less clearly, from the concave shape of the variance curve itself.As these ﬁgures are out-of-sample results they would suggest the possibility of ﬁndinga suitable statistical arbitrage strategy to exploit this mean-reverting component in therelative price dynamics of BP against the synthetic asset portfolio. In this case, however,the same caveat as before applies in that the non-synchronous nature of our close-pricedata may overstate the size of the true reversion effect.Before moving on to consider the performance across our broader universe, let us ﬁrstconsider the importance of the regularisation parameter. Remember that the results wehave been describing above correspond to a model