and loss ﬁgures effectively and potentiallyrealised respectively. The computation of the drawdowns (i.e. the differences between themaximum proﬁt potentially realised to date and the proﬁt or loss realised to date) arecomputed in columns M and N. Lastly, the trading summary statistics are computed incells I2 to I6.The second worksheet (“Graph”) contains the graph showing the open proﬁt and loss.The third worksheet (“Backup”) contains the raw data, i.e. columns A to D contain theforex data from Reuters (at 22:00 GMT) with the bid and ask quotes.Cells C3 and C4 allow the user to simulate the performance of different scenariosdepending on the EMA operator and the volatility threshold value respectively selected.The summary results of the trading strategy are displayed in cells I2:I6.Figure 6.4 displays the daily proﬁt and loss trajectories for the two hedging strategies analysed in this chapter. During the ﬁrst period (until mid-1997), the EMA withoutvolatility ﬁlter dominates the other strategy. As of mid-1997, the volatility ﬁlter strategyclearly outperforms in terms of proﬁt and risk. The chart also shows that the equitycurve of the volatility ﬁlter strategy is less volatile, thus leading to lower risk. The ﬁgurealso suggests a somewhat poor performance for the EMA strategy without volatilityﬁlter.
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Figure 6.4
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Daily cumulative proﬁt and loss curves
6.5 CONCLUSIONIn this chapter we have studied the behaviour of a Markovian switching volatility modelfor the USD/DEM daily foreign exchange series.The analysis of the residuals shows that the heteroskedasticity of the initial dataset hasbeen removed although the residuals series still presents high excessive kurtosis and failsto pass two standard normality tests.When applied within a hedging framework, the ﬁltered probabilities computed withthe Markovian switching model enable us to improve the performance of trend-followingsystems both in terms of risk and absolute proﬁts. This empirical result denotes thatforeign exchange markets do not follow trends in a systematic way and that volatilitymodelling could be one interesting approach to classify market trend dynamics.As a further extension, an economic interpretation of these results should be considered.The existence of long-lasting trending and jagged periods in very liquid markets are issuesthat deserve more attention.
REFERENCESBaillie, R. and T. Bollerslev (1989), “Intra-Day and Inter-Market Volatility in Exchange Rates”,Review of Economic Studies, 58, 565–85.Bera, A. K. and M. L. Higgins (1993), “ARCH Models: Properties, Estimation and Testing”, Journal of Economic Surveys, 7, 305–66.Bollerslev, T. (1986), “Generalized Autoregressive Conditional Heteroskedasticity”, Journal ofEconometrics, 31, 307–27.Bollerslev, T., R. Chou and K. Kroner (1992), “ARCH Modeling in Finance: A Review of theTheory and Empirical Evidence”, Journal of Econometrics, 52, 5–59.
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Bollerslev, T., R. Engle and D. Nelson (1993), “ARCH Models”, in R. F. Engle and D. McFadden(eds), Handbook of Econometrics, Vol. 4, North-Holland, Amsterdam.Campbell, J. Y., A. W. Lo and A. C. MacKinlay (1997), The Econometrics of Financial Markets,Princeton University Press, Princeton, NJ.Chesnay, F. and E. Jondeau (2001), “Does Correlation Between Stock Returns Really IncreaseDuring Turbulent Periods”, Economic Notes, 30 (1), 53–80.Dueker, M. J. (1997), “Markov Switching in GARCH Processes and Mean-Reverting Stock MarketVolatility”, Journal of Business & Economic Statistics, 15, 26–34.Engel, C. (1994), “Can the Markov Switching Model Forecast Exchange Rates?”, Journal of International Economics, 36, 151–65.Engel, C. and J. D. Hamilton (1990), “Long Swings in the Dollar: Are They in the Data and DoMarkets Know It?”, American Economic Review, 80, 689–713.Engle, R. (1982), “Autoregressive Conditional Heteroskedasticity with Estimates of the Variancesof U.K. Inﬂation”, Econometrica, 50, 987–1008.Engle, R. (1995), ARCH Selected Readings, Oxford University Press, Oxford.Ghysels, E., C. Gouriéroux and J. Jasiak (1998), “High Frequency Financial Time Series Data:Some Stylized Facts and Models of Stochastic Volatility”, Chapter III.7 in C. Dunis and B. Zhou(eds), Nonlinear Modelling of High Frequency Financial Time Series, John Wiley, Chichester.Goldfeld, S. M. and E. Quandt (1973), “A Markov Model for Switching Regressions”, Journal ofEconometrics, 1, 3–16.Goldfeld, S. M. and E. Quandt (1975), “Estimation in a Disequilibrium Model and the Value ofInformation”, Journal of Econometrics, 3, 325–48.Goodhart, C. and L. Figliuoli (1991), “Every Minute Counts in Financial Markets”, Journal ofInternational Money and Finance, 10, 23–52.Goodwin, T. H. (1993), “Business-Cycle Analysis with a Markov Switching Model”, Journal ofBusiness & Economic Statistics, 11, 331–39.Gouriéroux, C. (1997), ARCH Models and Financial Applications, Springer-Verlag, New York.Gray, S. F. (1996), “Modeling the Conditional Distribution of Interest Rates as a Regime-SwitchingProcess”, Journal of Financial Economics, 42, 27–62.Guillaume, D., M. Dacorogna, R. Davé, U. Müller, R. Olsen and O. Pictet (1997), “From the Bird’sEye to the Microscope: A Survey of New Stylized Facts of the Intra-Daily Foreign ExchangeMarkets”, Finance and Stochastics, 1, 95–129.Hamilton, J. D. (1988), “Rational-Expectations Econometric Analysis of Changes in Regimes: AnInvestigation of the Term Structure of Interest Rates”, Journal of Economic Dynamics and Control, 12, 385–423.Hamilton, J. D. (1989), “A New Approach to the Economic Analysis of Nonstationary Timeseriesand the Business Cycle”, Econometrica, 57, 357–84.Hamilton, J. D. (1994), Time Series Analysis, Princeton University Press, Princeton, NJ.Hamilton, J. D. and G. Lin (1996), “Stock Market Volatility and the Business Cycle”, Journal ofApplied Econometrics, 11, 573–93.Kim, C. J. and C. R. Nelson (1998), “Business Cycle Turning Points, a New Coincident Index,and Tests of Duration Dependence Based on a Dynamic Factor Model with Regime Switching”,Review of Economics and Statistics, 80, 188–201.Lam, P. S. (1990), “The Hamilton Model with a General Autoregressive Component”, Journal ofMonetary Economics, 26, 409–32.Müller, U. (1995), “Specially Weighted Moving Averages with Repeated Application of the EMAOperator”, Internal Paper, O&A Research Group.Müller, U., M. Dacorogna, D. Davé, R. Olsen, O. Pictet and J. Von Weizsäcker (1997), “Volatilitiesof Different Time Resolutions – Analyzing the Dynamics of Market Components”, Journal ofEmpirical Finance, 4 (2 & 3), 213–40.Quandt, R. E. (1958), “The Estimation of Parameters of a Linear Regression System Obeying TwoSeparate Regimes”, Journal of the American Statistical Association, 53, 873–80.Ramchand, L. and R. Susmel (1998), “Volatility and Cross Correlation Across Major Stock Markets”, Journal of Empirical Finance, 5, 397–416.
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Schnidrig, R. and D. Würtz (1995), “Investigation of the Volatility and Autocorrelation Function ofthe USD/DEM Exchange Rate on Operational Time Scales’, Proceedings of the High FrequencyData in Finance Conference (HFDF-I), Zürich.
APPENDIX A: GAUSS CODE FOR MAXIMUM LIKELIHOODFOR VARIANCE SWITCHING MODELSThe code is made up of a core program (MLS.PRG) that contains sub-procedures (i) proc1that computes the maximum likelihood theta (θ ) and (ii) proc2 that computes the ﬁlteredand smoothed probabilities.The comments are framed with the following signs: /* comment */.MLS.PRG/* MLS.PRG Maximum likelihood code for model with changing variances *//* Part I : load the libraries and source files *//* ------------------------------------------------------------------- */library maxlik,pgraph;#include maxlik.ext;maxset;graphset;#include maxprtm.src;/* external source file see details here below */#include smooth.src;/* Part II : load the log returns data vector rt *//* ------------------------------------------------------------------- */open fin=filename for read;x=readr(fin, nb of obs in the rt vector);T=rows(x);dat=x[1:T-1,1];rt=100*x[1:T-1,2];/* Part III : initialisation of parameters *//* ------------------------------------------------------------------- */A=0.0;S1=0.6;S2=0.4;p=0.8; p=-ln((1-p)/p);q=0.2; q=-ln((1-q)/q);b0=A|S1|S2|p|q;let b0={0.0011.360.602.003.00}; b0=b0’;x=rt;T=rows(x);output file=out.out reset;output file=out.out off;
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/* Part IV : this procedure computes the vector of likelihood evaluated theta (θ ) *//* ----------------------------------------------------------------------------- */proc (1)=switch(theta,Rt);local A,S1,S2,p,q,PrSt,St,j,ft,aux,p0,x,BP,mu,sig,K,auxo,rho;local l,maxco,auxn, fRtSt, PrStRtM, PrStRt, fRt, t, BigT, PStat,const;A=theta[1];S1=ABS(theta[2]);S2=ABS(theta[3]);x=theta[4];p=exp(x)/(1+exp(x));x=theta[5];q=exp(x)/(1+exp(x));BP=(p∼(1-p))|((1-q)∼q);mu=A;sig=S1|S2;K=2;BigT=rows(rt);rho=(1-q)/(2-p-q);Pstat=rho∼(1-rho);const=1/sqrt(2*Pi);fRtSt=const*(1./sig’).*exp(-0.5*( ((Rt-mu)./sig’)ˆ2 ));PrStRtM=Pstat|zeros(BigT-1,K);PrStRt=zeros(BigT,K);fRt=zeros(BigT,1);t=2;do until t>=BigT+1;aux=fRtSt[t-1,.].*PrStRtM[t-1,.];PrStRt[t-1,.]=aux/(sumc(aux’)’);PrStRtM[t,.]=PrStRt[t-1,.]*BP;t=t+1;endo;fRt[1,.]=fRtSt[1,.]*Pstat’;fRt[2:BigT]=sumc( (fRtSt[2:BigT,.].*PrStRtM[2:BigT,.])’ );retp(ln(fRt[1:BigT]));endp;/* Part V : maximum likelihood Evaluation(BHHH optimisation algorithm) *//* ------------------------------------------------------------------- */max Algorithm=5; /* 5= BHHH Algorithm */max CovPar=2;/* Heteroskedastic-consistent */max GradMethod=0;max LineSearch=5; /* 5= BHHH Algorithm */{th,f,g,h,retcode}=maxlik(x,0,&switch,b0);output file=out.out reset;call maxprtm(th,f,g,h,retcode,2);/* Part VI : call the routine smooth.src that computes the filtered probabilities(fpe) and the smooth probabilities (spe) *//* -------------------------------------------------------------------------- */{fpe,spe}=smooth(th[1]|th,x);output file=out.out off;
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/* Part VII : this procedure computes filtered & smoothed probability estimates *//* --------------------------------------------------------------------- */proc (2)=smooth(th,y);local mu1,mu2,p,q,s1,s2,rho,pa,pfx,its,p1,ind,thn,yxx,fk,t,spe,pax,qax,n;/* Data initialisation *//* ------------------------------------------------------------------- */mu1 = th[1];mu2 = th[2];s1 = abs(th[3]);s2 = abs(th[4]);p = th[5]; p=exp(p)/(1+exp(p));q = th[6]; q=exp(q)/(1+exp(q));n = rows(y);rho = (1-q)/(2-p-q);pa = rho|(1 - rho);p1 = zeros(4,1);/* pax=filtered probas *//* ------------------------------------------------------------------- */pax = zeros(n,4);(S t|S t-1)=(1,1)∼(2,1)∼(1,2)∼(2,2)∼(S t=1)∼(S t-1=1);/* qax smoothed probas, same structure as pax *//* ------------------------------------------------------------------- */pfx = zeros(n,1);
@ likelihoods from filter @
/* Calculate probability weighted likelihoods for each obs *//* ------------------------------------------------------------------- */yxx = (1/s1)*exp(-0.5* ((y-mu1)/s1)ˆ2 ) ∼ (1/s2)*exp(-0.5* ((y-mu2)/s2)ˆ2 );yxx = (p*yxx[.,1])∼((1-p)*yxx[.,2])∼((1-q)*yxx[.,1])∼(q*yxx[.,2]);/* Next call basic filter, store results in pax, pfx *//* ------------------------------------------------------------------- */its = 1;do until its > n;p1[1] = pa[1]*yxx[its,1];p1[2] = pa[1]*yxx[its,2];p1[3] = pa[2]*yxx[its,3];p1[4] = pa[2]*yxx[its,4];pfx[its] = sumc(p1);p1 = p1/pfx[its,1];pax[its,.] = p1’;pa[1,1] = p1[1,1] + p1[3,1];pa[2,1] = p1[2,1] + p1[4,1];its = its+1;endo;/* Smoothed probability estimate *//* ------------------------------------------------------------------- */spe = pax[1,.]∼pax[1,.];spe[1,2]=0; spe[1,4]=0; spe[1,5]=0; spe[1,7]=0;t = 2;do until t > n;spe = (((yxx[t,1]*spe[.,1:4])+(yxx[t,3]*spe[.,5:8]))∼((yxx[t,2]*spe[.,1:4])+(yxx[t,4]*spe[.,5:8])))/pfx[t,1];
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spe = spe | (pax[t,.] ∼ pax[t,.]);spe[t,2]=0; spe[t,4]=0; spe[t,5]=0; spe[t,7]=0;t=t+1;endo;spe = spe[.,1:4] + spe[.,5:8];/* Calculate filtered and smoothed probs that st=1 (col. 5) and st-1 =1 (col. 6) *//* ------------------------------------------------------------------- */pax=[pr(st=1|st-1=1) ∼ pr(st=2|st-1=1) ∼ pr(st=1|st-1=2) ∼ pr(st=2|st-1=2)]pax = pax ∼ (pax[.,1] + pax[.,3]) ∼ (pax[.,1] + pax[.,2]);qax = spe ∼ (spe[.,1] + spe[.,3]) ∼ (spe[.,1] + spe[.,2]);retp(pax,qax);endp;
7
Quantitative Equity Investment Managementwith Time-Varying Factor Sensitivities∗YVES BENTZ
ABSTRACTFactor models are widely used in modern investment management. They enable investment managers, quantitative traders and risk managers to model co-movements amongassets in an efﬁcient way by concentrating the correlation structure of asset returns into asmall number of factors. Because the factor sensitivities can be estimated by regressiontechniques these factors can be used to model the asset returns. Unfortunately, the correlation structure is not constant but evolves in time and so do the factor sensitivities. Asa result, the sensitivity estimates have to be constantly updated in order to keep up withthe changes.This chapter describes three methods for estimating time-varying factor sensitivities.The methods are compared and numerous examples are provided. The ﬁrst method, basedon rolling regressions, is the most popular but also the least accurate. We show thatthis method can suffer from serious biases when the sensitivities change over time. Thesecond method is based on a weighted regression approach which overcomes some of thelimitations of the ﬁrst method by giving more importance to recent observations. Finally,a Kalman ﬁlter-based stochastic parameter regression model is described that optimallyestimates non-stationary factor exposures. The three methods have been implemented inthe software provided on the CD-Rom so that readers can use and compare them withtheir own data and applications.
7.1 INTRODUCTIONAre you satisﬁed with the accuracy of your factor sensitivity estimates? If not, perhapsthe following situation will sound familiar. . . After days of careful analysis, John hadconstructed a long–short portfolio of stocks. John’s boss, however, felt uncomfortableabout the position as he feared that the expected outperformance, i.e. the alpha, may taketime to materialise and be perturbed by unwanted risk exposures. John updated the riskmodel with the latest estimates of factor sensitivities and ran the optimiser in order toimmunise the position against these exposures. After monitoring the proﬁt and loss over∗The information presented and opinions expressed herein are solely those of the author and do not necessarilyrepresent those of Credit Suisse First Boston.
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
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a few days, it became clear that the position was exposed to general market movements.In fact the alpha was dominated by a market beta.What went wrong? The optimiser? The trading strategy? Actually, neither of them. Thetrue underlying factor sensitivities had not been constant over the estimation period and,as a result, the OLS1 sensitivity estimates that John used for the risk model were seriouslymisleading.While factor sensitivities are known to vary over time, prevailing methods such asrolling regressions can be severely biased because they estimate past average factor exposures rather than forecast where these exposures are going to be in the future. In contrast,the adaptive procedure described in this chapter models and predicts the variations offactor sensitivities instead of merely smoothing past sensitivities. It can therefore be usedto take advantage of the dynamics driving the relationships between stock and factorreturns. As a result, risk models can be more accurate, investment strategies can be betterimmunised against unwanted risk, and risk–return proﬁles can be signiﬁcantly improved.While prevailing methods are often too simple to be correct, powerful adaptive procedures are often too complicated to be usable. This chapter presents in simple terms theessentials of dealing with time-varying factor sensitivities. It describes and compares thevarious methods of modelling time-varying risk exposures. Particular emphasis is givento an elegant, rich and powerful estimation method based on the Kalman ﬁlter.The software and spreadsheets supplied on the CD-Rom provide the reader with intuitive examples of the various procedures and show how most of the complexity can behanded over to the computer program. The reader may wish to use the estimation toolpack provided on their own data in order to estimate time-varying regression coefﬁcients.The software is provided for educational purposes only.7.1.1 Who should read this chapter and what are its possible applications?This chapter is aimed at all those using linear regression models with economic andﬁnancial time series. This includes areas such as investment analysis, quantitative trading,hedging, index tracking, investment performance attribution, style management and riskmeasurement. In particular, this chapter is targeted at everyone who has heard of adaptivemodels, stochastic parameter models or Kalman ﬁltering but could not or did not want toinvest the time and effort in order to implement such models.This chapter consists of four main sections. Section 7.2 is a short review of factor models and factor sensitivities. It introduces the notations used in the rest of the chapter. Thefollowing sections describe three different methods for estimating the factor exposures.The ﬁrst method, presented in Section 7.3, is based on a rolling regression procedureand uses OLS estimation. The procedure is straightforward and can be implemented ina simple spreadsheet. It does not require any complex estimation procedure and canuse the widely available linear regression models. The shortcomings of the method aredemonstrated. The second method, presented in Section 7.4, is based on weighted leastsquares estimation. The procedure rests on a slightly more complex set of equations butovercomes a number of weaknesses of the OLS procedure. The third method, presented inSection 7.5, consists of an adaptive procedure based on the Kalman ﬁlter. This stochastic1Ordinary least squares estimation is the usual linear regression estimation procedure used on rolling windowsto compute betas and other factor sensitivities.
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parameter regression model is shown to be the most accurate and robust procedure of thethree, yielding optimal estimates of factor sensitivities and modelling their time structure.Finally, Section 7.6 concludes the Chapter.
7.2 FACTOR SENSITIVITIES DEFINEDThe estimation and use of factor sensitivities play an important role in equity investmentmanagement. Investment diversiﬁcation, portfolio hedging, factor betting2 or immunisation,3 index tracking, performance attribution, style management all necessitate at somestage an accurate estimation of factor sensitivities. The factors can be the overall market(some broad stock market index), some industrial sector, some investment style grouping(index of growth stocks) or other variables that underlie the correlation structure of stockreturns such as macroeconomic factors (e.g. inﬂation, GDP growth), statistical factors(usually obtained by principal component analysis), or returns of other asset classes (e.g.crude oil, gold, interest rates, exchange rates).4The sensitivity of a stock5 to a factor is usually deﬁned as the expected stock returncorresponding to a unit change in the factor. If Y (t) is the return6 of the stock at timet and X(t) is the simultaneous return (or change) of the factor, then β in the followingequation can be viewed as the factor sensitivity:Y (t) = α + βX(t) + ε(t)
(7.1)
where α is a constant representing the average extra-factor performance of the stock andε(t) is a random variable with zero mean (by construction), constant variance and zerocovariance with X(t).If both the stock and the factor returns Y (t) and X(t) can be observed in the market(and they usually can), then α and β can be estimated by regression techniques. Oncethe β coefﬁcient has been estimated, it is possible to immunise the investment againstmovements in the factor by selling β amount of a tradable proxy for the factor for everyunit of the investment. For instance, if the factor is the UK stock market, one can sell βpounds of FTSE 100 index futures for each pound invested. The expected return of thehedged position would then be α.More generally, several factor sensitivities can be estimated simultaneously. In orderto estimate joint sensitivities, equation (7.1) is generalised to more than one factor:Y (t) = α +
N
βi Xi (t) + ε(t)
i=1
(7.2)
= X(t)β + ε(t)2For instance, an investor could construct a portfolio that would be sensitive to one factor only (e.g. defaultrisk) and immune to all other factors.3For instance, an investor could construct a portfolio that would not be sensitive to one factor (e.g. long-terminterest rates risk) but remain sensitive to all other factors.4Not all factors may be relevant for all factor models. For example, using macroeconomic factors makes littlesense when modelling daily or hourly returns.5Sensitivity estimation is obviously not restricted to stocks but can also apply to other assets, portfolios ofassets or investment strategies.6As factor sensitivities may not have a ﬂat term structure, β is a function of the horizon over which returns arecalculated, i.e. betas based on monthly, weekly, daily and hourly returns may all be different from each other.
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where Y (t) is the return of the investment at time t. Xi (t) are the returns of the ifactors. α is a constant representing the average extra-factor return of the investment. Itis the sensitivity coefﬁcient β0 to the constant factor X0 (t) = 1. βi is a parameter thatrepresents the joint sensitivity of the investment return to changes in factor i. ε(t) is arandom variable with zero mean (by construction), constant variance (by assumption) andzero covariance with the factor returns.The joint sensitivity coefﬁcients βi measure “clean” sensitivities, accounting for thesole effect of one variable Xi while controlling for the other effects. Hence, if the jointsensitivity of Y to X1 has a value of S, Y is expected to change by S when X1 changesby 1, if all the other variables X2 , . . . , Xn remain constant. βi is the partial derivative∂ Ŷ (Xi )/∂Xi of the expectation Ŷ of Y with respect to Xi .
7.3 OLS TO ESTIMATE FACTOR SENSITIVITIES: A SIMPLE,POPULAR BUT INACCURATE METHODEstimating factor sensitivities can be simple or complex, depending on the assumptionsmade about the relationships between stock and factor returns. The most popular methodwith practitioners and, unfortunately, also the least accurate is standard linear regression,also known as ordinary least squares (OLS) estimation. It is simple, easy to implement andwidely available in statistical software packages. OLS minimises the mean squared error(MSE) of the linear model described in equation (7.2). The problem has a simple closedform solution which consists of a matrix inversion and a few matrix multiplications.Equation (7.2) can be rewritten using a more concise matrix notation, as:Y = Xβ + ε
(7.3)
where Y is a T × 1 vector of the asset returns, X a T × (N + 1) vector of the factorreturns, β a (N + 1) × 1 vector of factor sensitivities and ε a T × 1 vector of randomvariables with zero mean, i.e.
Y1 Y2 Y=...,YT

1 X11 1 X21X=... ...1 XT 1
......Xtj...
X1N... ,... XT N

α β1 β=...,βN

ε1 ε2 ε=...εT
with expectation E(ε) = 0 and variance–covariance matrix σ 2 (ε) = σ 2 I. Here Yt is thetth observation of the investment return; Xti is the tth observation of the ith factor returnXi ; βi is the sensitivity of Y to factor Xi ; εt is the error term at time t.The expression of the OLS estimator β̂ of β can then be shown to be:β̂ = (X X)−1 X Y
(7.4)
and its estimated variance–covariance matrix is s 2 (β̂):s 2 (β̂) = MSE (X X)−1
(7.5)
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with MSE being the mean squared error of the regression model:MSE =
T1 (Y (t) − Ŷ (t))2T − 2 t=1
Ŷ (t) =
N
βi Xi (t)
(7.6)
i=0
OLS estimates of the factor sensitivities are unconditional, i.e. they are average sensitivities over the observation sample. Any variation of the betas over the sample isnot modelled and the resulting error is attributed to sampling, i.e. to the ε term inequation (7.3). Therefore, when using OLS, the