constructed with λ = 0.01. Let us nowcompare this model to the model we obtain by leaving all other parameters the same butreplacing the “RidgeFac” value in cell F4 with 0.1. The composition of the new syntheticasset is shown in Figure 2.19.With this higher degree of regularisation, the weights become more uniform. Very feware now negative and the highest weight (for Shell, SHEL.L) is reduced from approximately 18% to only 6%. The “sum” of the weights falls to 98% and the “sumabs” to103%. In this case, however, the new synthetic asset is a less effective hedge for pricemovements in BP. The residual variance of the hedged portfolio rises to 702.56 (fromthe 423 shown in Figure 2.17), and the reduction in variance due to the hedge is nowonly 46% (from 67% previously). Thus, in this particular case, increasing the degree of7%SHEL.L
RD.AS
6%
CARR.PA
5%TOTF.PA
4%
BT.LENI.MI
AAH.AS
3% constant DCXGn.DE2%1%0%
BNPP.PANESZn.VXAV.LSAN.MCUNc.AS
UBSZn.VXAZN.LROCZg.VXPHG.ASGSK.LRUKZn.VXNOVZn.VXINGAS.AEGN.ASEAUG.PACGEP.PANOK1V.HE
EONG.DE
BARC.LCSGZn.VXLLOY.LRBOS.LHSBA.L
VOD.LDGE.LDBKGn.DE BAYG.DEPRU.LOREP.PAAXAF.PABBVA.MCTEF.MCDTEGn.DELYOE.PASIEGn.DEZURZn.VXGASI.MITIT.MI
ERICb.ST
−1%
ALVG.DEMUVGn.DE
−2%BP.L
Figure 2.19
Effective weights for the synthetic asset for British Petroleum (BP.L) λ = 0.1
64
Applied Quantitative Methods for Trading and Investment
regularisation has decreased the effectiveness of the synthetic asset as a hedging portfolio.This should not be surprising, as in the limit we would expect a heavily regularised synthetic asset to closely match the equally-weighted portfolio, which we know is not a verygood hedge in this case.It is easy to conﬁrm that moving to the opposite extreme also leads to a performancedegradation: with the ridge factor set to zero, not only does the sum of absolute weightsrise to the unattractive 404% mentioned above, but the residual variance of 611.01 (53%reduction) is also worse than the 423 (67%) for the intermediate case of λ = 0.01. Theseresults indicate a pattern which is typical of much statistical modelling: a certain degreeof regularisation tends to be beneﬁcial, but beyond a certain point the smoothing becomesexcessive and begins to degrade the model performance.Whilst the case of this one model, for BP, is both interesting and illustrative, it isimportant to know whether these results are merely a lucky “one off ” or whether theyrepresent an approach which can be applied more generally. For this reason the ﬁnalworksheet in the analysis, called “Results Summary”, contains a table of results generatedby taking each of the 50 assets in turn as the “target” asset. A particular sample of theseresults is shown in Table 2.5.Table 2.5 Performance of cointegration model across the universe of equitiesStockBP.LBT.LGSK.LCGEP.PAUBSZn.VXDCXGn.DEBP.LAZN.LNOK1V.HENOVZn.VXERICb.STPHG.ASING.ASAAH.ASAEGN.ASUNc.ASRD.ASRUKZn.VXROCZg.VXEAUG.PASAN.MCNESZn.VXCARR.PABNPP.PAAV.LAXAF.PAZURZn.VX
VarRed MktRed RelImp HdgeFac In-sample Out-sample SumWts AbsSumWts67%47%56%82%46%−1%67%54%62%−37%67%31%39%39%86%21%68%69%−60%49%86%32%39%3%26%60%75%
2%
65%
0.01
700
293
100%
166%
31%16%38%18%58%24%49%−3%46% −47%2%65%37%17%36%27%−50%13%42%25%31%0%60% −21%40%−1%64%22%−68%88%67%1%62%8%46% −105%41%8%79%7%−43%75%50% −11%−42%45%40% −15%56%4%64%11%
0.010.010.010.010.010.010.010.010.010.010.010.010.010.010.010.010.010.010.010.010.010.010.010.010.010.01
700700700700700700700700700700700700700700700700700700700700700700700700700700
293293293293293293293293293293293293293293293293293293293293293293293293293293
103%100%114%100%101%100%98%101%101%93%96%104%100%101%98%102%98%99%105%101%98%100%93%103%103%107%
373%188%506%143%289%166%172%377%170%396%193%165%141%230%179%160%154%158%231%117%148%277%151%163%154%289%
Cointegration to Hedge and Trade International EquitiesTable 2.5Stock
(continued )
VarRed MktRed RelImp HdgeFac In-sample Out-sample SumWts AbsSumWts
TOTF.PA57%−6%LYOE.PA53%29%OREP.PA−140% −161%TIT.MI76%83%ENI.MI44% −48%EONG.DE−28% −104%SIEGn.DE63%37%DBKGn.DE70%64%GASI.MI37%56%DTEGn.DE46%60%BBVA.MC81%86%ALVG.DE88%73%BAYG.DE63%69%BARC.L71% −26%HSBA.L71%44%DGE.L23% −48%LLOY.L−61% −97%PRU.L67%62%RBOS.L66% −48%SHEL.L33%53%VOD.L33%23%TEF.MC77%57%MUVGn.DE70%61%CSGZn.VX47%69%MeanMedian
65
42%53%
24%43%
64%24%21%−6%91%75%26%6%−19%−13%−5%15%−6%97%27%71%36%5%115%−20%10%20%9%−22%
0.010.010.010.010.010.010.010.010.010.010.010.010.010.010.010.010.010.010.010.010.010.010.010.01
700700700700700700700700700700700700700700700700700700700700700700700700
293293293293293293293293293293293293293293293293293293293293293293293293
98%98%97%100%99%98%99%101%103%98%102%100%106%100%99%97%99%98%96%102%104%101%99%102%
161%152%130%224%152%157%211%196%169%379%154%174%185%190%211%155%196%202%219%157%207%237%228%147%
18%12%
0.010.01
700700
293293
100%100%
206%176%
The metrics in the table are precisely the same as those which are presented for individual models on the “CointAnalysis” worksheet (and in fact are directly derived fromthose values). Whilst the performance varies substantially from one equity to another,the ﬁgures for both mean and median performance conﬁrm the general applicability ofthe approach. During what has been a very turbulent time for the equity markets, thesynthetic hedge portfolios manage to reduce the out-of-sample volatility by a factor of42% (mean) or 53% (median) – note that the mean performance is more heavily affectedby one or two particularly unsuccessful “outliers”. This performance represents a signiﬁcant improvement over that obtained by hedging using an equally weighted combinationof assets representing the market portfolio. These results suggest that the cointegrationapproach to hedging asset-speciﬁc risk deserves to be taken seriously.Our study also provides some evidence to support the possibility of creating statisticalarbitrage strategies based on cointegration effects within this set of equities. Evidence forthe presence of mean-reverting components in the asset price dynamics is reﬂected in thedownward-sloping variance ratio proﬁles shown in Figure 2.20.Both the mean and median proﬁles show a signiﬁcant decline in normalised variance, with the 30-day variance just under 70% of that which would be expected from
66
Applied Quantitative Methods for Trading and Investment
1.2
1
0.8
0.6Mean
Median
6
11
0.4
0.2
01
16
21
26
Figure 2.20 Average (mean and median) variance ratio proﬁles for the residual dynamics,calculated across the full set of 50 equities
extrapolating the 1-day variance. This evidence in itself is clearly not conclusive, especiallyconsidering the non-synchronous nature of our data sampling due to the different closingtimes in the various markets. However, this result, together with previous studies (e.g.Burgess, 1999) seems to strongly suggest that reverting components exist in the assetspeciﬁc components of price volatility which may in principle be open to exploitationusing suitable statistical arbitrage strategies.
2.7 DISCUSSION AND CONCLUSIONSThe major objective of our methodology is to construct combinations of time series whichare either decorrelated with major sources of economic risk and/or contain a deterministic(and hence potentially predictable) component in the dynamics.Given a particular asset there are two main methods for constructing combinations ofone or more assets with similar net exposures to market-wide risk factors. The cointegration tools described in this chapter represent an “implicit” approach to factor modelling in which neither the factors nor the sensitivities are estimated explicitly, but areinstead addressed from the perspective of the impact which they cause on sets of relatedasset prices.The standard approach to estimating explicit factor models is to postulate a set ofﬁnancial and economic variables as the risk factors and to use regression-based techniquesto estimate the sensitivity coefﬁcients βi,j (e.g. the sensitivity of a particular stock to, say,long-term interest rates). Evidence for mean-reverting dynamics in the residuals of factormodels was reported by Jacobs and Levy (1988). However, the task of constructing such
Cointegration to Hedge and Trade International Equities
67
models is a difﬁcult problem in itself, and the interested reader is directed to a recentextensive study by Bentz (1999).A second approach which is sometimes referred to as “statistical” factor modellingis to perform multivariate analysis of asset price returns and reconstruct the unobservedrisk factors as linear combinations of observed asset returns. The technique of “principalcomponents analysis” (PCA) is a natural tool in this case as it generates sequences oflinear combinations (factors) which account for the greatest possible amount of the totalvariance, subject to the constraint of being orthogonal to the previous factors in thesequence. Some examples of applications of PCA to statistical arbitrage modelling aredescribed in Burgess (1996), Schreiner (1998), Towers (1998), Tjangdjaja et al. (1998)and Towers and Burgess (1998).Still within the area of statistical factor modelling, a generalisation of PCA whichhas recently attracted much attention in engineering disciplines is so-called “independentcomponents analysis” (ICA), which is based upon algorithms developed for the blindseparation of signals (Bell and Sejnowski, 1995; Amari et al., 1996). In ICA the orthogonality condition of PCA is strengthened to one of complete statistical independence,thus taking into account higher moments. Whilst ICA techniques offer exciting potential in computational ﬁnance, particularly in that they account not only for expectations(“returns”) but also for variances (“risk”), applications in this domain are still rare (seeMoody and Wu (1997), Back and Weigend (1998)).The explicit and statistical factor modelling approaches are those which are currentlymost used in practice, with explicit factor models typically used for hedging and riskcontrol and statistical factor models typically used from a statistical arbitrage perspective. However we believe that the implicit factor modelling offered by the cointegrationapproach presents a viable alternative to these more established techniques, both for hedging and for trading. By dealing with asset prices directly, cointegration modelling presentsboth technical and conceptual advantages over the other methods. The technical motivation for the cointegration approach is that it avoids the need for explicit estimation of riskfactors and factor sensitivities, thus eliminating a potential source of estimation error fromthe modelling process as a whole. The conceptual beneﬁt of the cointegration approachis that it can be viewed as a sophisticated version of “relative value” analysis. The synthetic assets constructed by the cointegration procedure can be viewed from a hedgingperspective as “tracking baskets” and from a statistical arbitrage perspective as “syntheticpairs” and thus are consistent with the way in which many traders naturally considerrelationships between sets of assets. The cointegration approach also has its weaknesses,for instance there are many different methods for constructing cointegration relationshipsand it is not clear which works best in practice. Furthermore, in some cases it is a positivebeneﬁt to explicitly model market-wide risk factors, particularly if they are felt to containpredictable components. Thus, whilst there are both pros and cons to the cointegrationapproach, we feel that in a practical discipline such as investment ﬁnance it is importantto take a “broad church” view and apply whichever set of tools works best in a givensituation. The aim of this chapter has been to provide a demonstration of and a motivationfor the relatively unused cointegration approach, in order to add another set of tools tothe ﬁnancial toolbox.In conclusion, this chapter has examined the application of the econometric conceptof cointegration as a tool for hedging and trading international equities. Section 2.2introduced cointegration within the perspective of different frameworks for analysing
68
Applied Quantitative Methods for Trading and Investment
and modelling ﬁnancial time series. Sections 2.3 and 2.4 described two complementaryperspectives in which cointegration is viewed as a basis for implicitly hedging unknownrisk factors and also as a basis for suggesting possible opportunities for statistical arbitrage. Section 2.5 provided a controlled simulation in which the cointegration approachwas demonstrated upon artiﬁcial time series with known properties. Finally, the spreadsheet analysis in Section 2.6 demonstrates an application of these tools to a real-worldproblem. The data used for this demonstration consisted of the daily closing prices ofthe 50 equities which constituted the STOXX 50 index as of 4 July 2002, analysed overa period from 14 September 1998 to 3 July 2002. We have noted that the use of dailyclosing prices will introduce some spurious effects due to the non-synchronous closingtimes of the markets on which these equities trade, so the speciﬁc results themselves canonly be taken as indicative of a more realistic study. This caveat admitted, the resultsnevertheless serve both to illustrate the use of the tools and also to suggest the potentialbeneﬁts which may be gained from their intelligent application to the tasks of hedgingand trading international equities.
REFERENCESAmari, S., A. Cichocki and H. H. Yang (1996), “A New Learning Algorithm for Blind SignalSeparation”, in D. S. Touretzky et al. (eds), Advances in Neural Information Processing Systems8, MIT Press, Cambridge, MA, pp. 757–763.Back, A. D. and A. S. Weigend (1998), “Discovering Structure in Finance using Independent Component Analysis”, in A-P. N. Refenes et al. (eds), Decision Technologies for ComputationalFinance, Kluwer Academic Publishers, Dordrecht, pp. 309–322.Bell, A. and T. Sejnowski (1995), “An Information-maximisation Approach to Blind Separationand Blind Deconvolution”, Neural Computation, 7(6), 1129–1159.Bentz, Y. (1999), Identifying and Modelling Conditional Factor Sensitivities: An Application toEquity Investment Management, unpublished PhD thesis, London Business School.Burgess, A. N. (1996), “Statistical Yield Curve Arbitrage in Eurodollar Futures using Neural Networks”, in A-P. N. Refenes et al. (eds), Neural Networks in Financial Engineering, World Scientiﬁc, Singapore, pp. 98–110.Burgess, A. N. (1999), A Computational Methodology for Modelling the Dynamics of StatisticalArbitrage, unpublished PhD thesis, London Business School. (http://cocreativity.net/papers.html).Cochrane, J. H. (1988), “How Big is the Random Walk in GNP?”, Journal of Political Economy,96, 5, 893–920.Engle, R. F. and C. W. J. Granger (1987), “Cointegration and Error-correction: Representation,Estimation and Testing”, Econometrica, 55, 251–276.Granger, C. W. J. (1983), “Cointegrated Variables and Error-correcting Models”, UCSD DiscussionPaper.Hoerl, A. E. and R. W. Kennard (1970a), “Ridge Regression: Biased Estimation for NonorthogonalProblems”, Technometrics, 12, 55–67.Hoerl, A. E. and R. W. Kennard (1970b), “Ridge Regression: Application to Nonorthogonal Problems”, Technometrics, 12, 69–82.Hull, J. C. (1993), Options, Futures and Other Derivative Securities, Prentice Hall, EnglewoodCliffs, NJ.Jacobs, B. I. and K. N. Levy (1988), “Disentangling Equity Return Regularities: New Insights andInvestment Opportunities”, Financial Analysts Journal, May–June 1988, 18–43.Lo, A. W. and A. C. MacKinlay (1988), “Stock Market Prices Do Not Follow Random Walks:Evidence from a Simple Speciﬁcation Test”, The Review of Financial Studies, 1988, 1, 1, 41–66.Moody, J. E. and L. Wu (1997), “What is the ‘True Price’? – State Space Models for Highfrequency FX Data”, in A. S. Weigend et al. (eds), Decision Technologies for Financial Engineering, World Scientiﬁc, Singapore, pp. 346–358.
Cointegration to Hedge and Trade International Equities
69
Ross, S. A. (1976), “The Arbitrage Pricing Theory of Capital Asset Pricing”, Journal of EconomicTheory, 13, 341–360.Schreiner, P. (1998), Statistical Arbitrage in Euromark Futures using Intraday Data, unpublishedMSc thesis, Department of Mathematics, King’s College, London.Sharpe, W. F. (1964), “Capital Asset Prices: A Theory of Market Equilibrium”, Journal of Finance,19, 425–442.Tjangdjaja, J., P. Lajbcygier and N. Burgess (1998), “Statistical Arbitrage Using Principal Component Analysis For Term Structure of Interest Rates”, in L. Xu et al. (eds), Intelligent DataEngineering and Learning, Springer-Verlag, Singapore, pp. 43–53.Towers, N. (1998), Statistical Fixed Income Arbitrage, Deliverable Report D4.6, ESPRIT project“High performance Arbitrage detection and Trading” (HAT), Decision Technology Centre,London Business School.Towers, N. (2000), Decision Technologies for Trading Predictability in Financial Markets, unpublished PhD thesis, London Business School.Towers, N. and A. N. Burgess (1998), “Optimisation of Trading Strategies using ParametrisedDecision Rules”, in L. Xu et al. (eds), Intelligent Data Engineering and Learning, SpringerVerlag, Singapore, pp. 163–170.
3
Modelling the Term Structure of InterestRates: An Application of Gaussian AfﬁneModels to the German Yield Curve∗NUNO CASSOLA AND JORGE BARROS LUÍS
ABSTRACTThis chapter shows that a two-factor constant volatility model describes quite well thetime series and the cross sectional behaviour of the German yield curve between 1972and 1998. The empirical analysis supports the expectations theory with constant termpremiums. Thus, an average term premium structure can be calculated and short-terminterest rate expectations can be derived from the adjusted forward rate curve. The nonobservable factors that explain the German yield curve are extracted using a Kalmanﬁlter technique. Following the conjecture that these factors capture, respectively, theexpected short-term real interest rate and the expected inﬂation rate, alternative methodsof identifying one of the factors with the inﬂation process are discussed. The ﬁrst factor(real interest rate) is more important for explaining interest rate movements at the shortend of the yield curve. The second factor (inﬂation) carries greater weight in explainingthe longer end of the yield curve. These ﬁndings are of interest to risk managers foranalysing the shape of the yield curve under different scenarios and to policy makersfor assessing the impact of ﬁscal and monetary policies. The Matlab codes necessary toreproduce the results are presented in great detail.
3.1 INTRODUCTIONThe identiﬁcation of the factors that determine the time-series and cross-section behaviourof the term structure of interest rates is a recurrent topic in the ﬁnance literature. Eventhough the interest rates for different maturities typically exhibit high correlations, theseare far from being perfectly correlated. The main conclusion from the literature is thatthe yield curve is determined by different factors, usually described by its level, slopeand curvature. Nevertheless, the identiﬁcation of the yield curve factors is a controversialsubject that has several empirical and practical implications, namely for the managementof portfolios of ﬁxed income securities, inﬂuencing the investment and hedging strategieswithin the context of portfolios of ﬁxed income securities.∗
This chapter is based on Cassola and Luı́s (2003) and on the PhD thesis of the second author (Luı́s, 2001).
Applied Quantitative Methods for Trading and Investment. 2003 John Wiley & Sons, Ltd ISBN: 0-470-84885-5
Edited by C.L. Dunis, J. Laws and P. Naı̈m
72
Applied Quantitative Methods for Trading and Investment
Moreover, a better knowledge of these factors would facilitate the task of risk managers,in particular for Value-at-Risk (VaR) purposes. In fact by simulating different paths forthe factors the behaviour of the yield curve under different scenarios could be traced andalternative empirical distributions for interest rates could be retrieved (see e.g. Bolder(2001)). The analysis of the determining factors of the yield curve is also relevant forpolicy makers, namely in assessing the impact of monetary and ﬁscal policies (see, forinstance, Fleming and Remolona (1998) and Bliss (1997)).In this chapter, the German yield curve is analysed, given its relevance in the international bond markets and its informational content about future macroeconomic developments in one of the major European Union economies. Two databases are used in thischapter, the ﬁles “datgerse.txt” and “datgersb.txt”, which are both included on the CDRom. The ﬁrst database comprises monthly averages of nine daily spot rates for maturitiesof 1 and 3 months and 1, 2, 3, 4, 5, 7 and 10 years, between January 1986 and December1998. The spot rates were estimated using the Nelson and Siegel (1987) and Svensson(1994) smoothing techniques from raw market data on euro–Deutschemark short-terminterest rates and par yields of German government bonds.1 The smoothed estimates wereused to construct one-period forward rates and perform forward rate regressions.2As illustrated in Figure 3.1, the German end-month interest rates ﬂuctuated markedly inthe period from January 1986 to December 1998 (between 3% and 10%). These interestrate moves corresponded to signiﬁcant shifts in the shape of the yield curve (Figure 3.2).In fact, during this period the yield curve moved from a ﬂat pattern to a positively slopedcurve. The sharp deceleration of the German economic growth between 1992 and 1993 wasaccompanied by a negatively sloped yield curve. Finally, at the end of the sample period,the yield curve returned to an almost null slope, achieving the lowest levels in the sample.141210
%
8642
1Y7Y
1m4Y
3m5Y
2Y10Y
Figure 3.1
German yield curve: 1986–1998
1998
1997
1996
1995
1994
1993
1992
1991
1990
1989
1988
1987
1986
0
3Y
1A TSP 4.3 routine was written for the estimation of spot rates. A likelihood ratio test was adopted forchoosing between the Nelson and Siegel (1987) and the Svensson (1994) methods, in each sample day, asdescribed in Cassola and Luı́s (1996).2On this issue the research assistance of Fátima Silva, then afﬁliated with the Research Department of theBanco de Portugal, is acknowledged.
Modelling the Term Structure of Interest Rates
73
1210
%
8642
10
9
8
7
6
5
4
3
2
1
0
0Term to Maturity (Years)Jan-86
May- 88
Aug- 92
Figure 3.2
German yield curves
Jul - 96
Dec - 98
The second database for Germany covers a longer period, between September 1972and December 1998.3 However this sample includes only spot rates for annual maturitiesbetween 1 and 10 years, excluding the 9-year maturity (Figure 3.3). For the overlappingperiod, interest rates in the two data sets are very similar for equivalent maturities, whichsuggests robustness of market data and smoothing techniques.The analysis assumes that bond yields are functions of several macroeconomic andﬁnancial variables, observable or latent. In afﬁne models, parameters are linear in boththe maturity of the assets and the factors, which makes these models easier to implement.4Additionally, compared to principal component analysis, afﬁne models have the advantageof allowing for correlation among the factors, providing several outputs that are relevantfor ﬁnancial market participants and policy makers, such as forward rates and the structureof term premia.Afﬁne models are the result of four decades of evolution in asset pricing theory. Inthe context of yield curve modelling these can be seen as developments of the one-factormodels by Vasicek (1977) and Cox et al. (1985a), where the short-term interest rate wasthe single factor.5 Multifactor models were developed because of the discrepancy betweenthe yield curve implied by the theory and the observed time-series properties of bondyields, namely the fact that the observed curves are substantially more concave thanimplied by the theory (see, for instance, Backus et al. (1998)).63We are grateful to Manfred Kremer, then afﬁliated with the Research Department of the Bundesbank, forproviding the data.4See Campbell et al. (1