Equity index replication with standard and robust regression estimators

OR Spektrum (2000) 22: 525–543 c Springer-Verlag 2000


Equity index replication with standard and robust regression estimators Tracking eines Performance-Indexes mittels klassischer und robuster Regressionssch¨atzer ¨ Gunter Bamberg1 and Niklas Wagner2, 1 2

Lehrstuhl f¨ur Statistik, Universit¨at Augsburg, Universit¨atsstraße 16, 86135 Augsburg, Germany (e-mail: [email protected]) Haas School of Business, University of California in Berkeley, Berkeley, CA 94720, USA (e-mail: [email protected])

Received: August 3, 1999 / Accepted: March 20, 2000

Abstract. Approximate equity index replication, based on a linear regression setting, is critically reviewed. It is shown that tracking a performance index like the German DAX necessarily leads to violations of basic assumptions of the classical regression model. Violations occur even if the model is formulated in terms of stock price levels. When the model is based on discretely or continuously compounded returns the situation is more critical. Due to these violations, the optimality properties of the regression estimators are generally weak. In the time series context, outliers in financial time series may additionally affect the standard least squares estimator. Despite of these critical points, it is argued that regression techniques may still provide a useful tool for index replication. With respect to outliers, robust estimators can potentially provide an alternative to least squares. Hence, apart from least squares, a non-redescending and a redescending robust estimator is fitted. The empirical results for the DAX are obtained with a subset portfolio containing the most heavily weighted index members. Compared to a naive weighting scheme, the results document that least squares estimation highly improves out-of-sample replication performance. Typically, the use of robust estimators does not show replication improvements. However, when substantial market movements are present in the sample, superior replication can be obtained. Zusammenfassung. Ein Performance-Index wie der DAX kann nicht durch zeitkonstante Gewichtung von Kursen dargestellt werden. Dasselbe gilt f¨ur bereinigte Kurse, falls der Beobachtungszeitraum mindestens einen Verkettungstermin
The author is grateful for grants from the Deutsche Forschungsgemeinschaft (DFG). Correspondence to: G. Bamberg

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umfasst. Analoge (negative) Aussagen gelten auch f¨ur diskrete Index- bzw. Aktienrenditen. G¨anzlich unm¨oglich wird eine lineare Darstellung bei Verwendung von stetigen Renditen. Im ersten Teil der Arbeit wird aufgezeigt, daß sich diese Probleme bei Regressionsmodellen zur n¨aherungsweisen Nachbildung des DAX in Verletzungen der Pr¨amissen des klassischen Regressionsmodells niederschlagen. Der zweite Teil enth¨alt einen empirischen Vergleich verschiedener TrackingProzeduren: naive Gleichgewichtung, Kleinst-Quadrate-Sch¨atzung und robuste Sch¨atzung. Key words: Approximate equity index replication – Linear regression – Robust estimation – Non-linear estimation – Tracking error ¨ Schlusselw¨ orter: Performance-Index – N¨aherungsweise Aktienindex-Nachbildung – Lineare Regression – Robuste Sch¨atzverfahren

1 Introduction Equity index replication is a standard application of computational methods in financial decision making. The goal is to select a portfolio which, according to some performance measure, optimally replicates the behavior of a given index. The problem is considered when an exogenously given set of stocks is a subset of the set of index members and possibly contains non-index members. A procedure of approximate index replication may be desirable for constructing portfolios under transactions cost or liquidity constraints. Hedging strategies and the use of index derivatives frequently involve the construction of index portfolios. Alternatively, index replication methods will be applied when the performance of a portfolio manager is measured relative to some given benchmark index. Since equity indices are linear aggregates of single stock prices, linear regression analysis seems to be a natural numerical candidate for approximate index replication. In an ideal regression setting, where the disturbance terms obey the classical assumptions and additionally follow a normal distribution, the least squares (LS)estimator is known to be efficient in the class of all unbiased estimators. Considering the subclass of linear estimators, the Gauss/Markov-theorem states that the least squares criterion generates the unique best linear unbiased estimator (BLUE) under any set of uncorrelated disturbance terms with identical variance. Hence, with nonnormal disturbances, non-linear estimators potentially provide higher efficiency than least squares. A common non-linear estimator is based on the least absolute value (LAV)criterion. This procedure is more robust in the sense that its estimates are characterized by a lower sensitivity towards extreme sample observations. Note however, that minimizing least absolute deviations implies a substantial loss in efficiency relative to LS, when deviations from normality are small. Alternative non-linear regression estimators as outlined in Huber (1981, 1996) and Hampel, Ronchetti, Rousseeuw, and Stahel (1986) can maintain a high degree of efficiency with normally distributed disturbances. On the other hand, as compared to LS, they are

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significantly less sensitive with respect to deviations from the normal model and hence are called ’robust’ estimators. In an early study on index replication with regression techniques, Alderson and Zivney (1989) use least squares in order to estimate portfolio weights. Others, including Rudolf, Wolter, and Zimmermann (1999), propose using LAV-estimation of portfolio weights as an alternative to least squares. Also recently, the application of robust estimation techniques in finance has gained increasing attention. Marcos Duarte and de Melo Mendes (1998) discuss robust estimators for the construction of equity index portfolios. Sakata and White (1998) propose robust techniques for estimating the level of time-varying variance. Both papers emphasize possibly severe effects of a small number of extreme observations on the estimation results. In this paper, we argue that violations of the classical regression assumptions in the index replication context are inevitable and will generally weaken both the optimality properties of the LS-estimator as well the asymptotic optimality properties of the robust alternatives. When an empirical viewpoint is chosen, regression analysis may still remain a useful tool as long as its sensitivity with respect to those violations is quite low. Focusing on the distributional assumption for the model disturbances, the question arises whether normality can serve as an appropriate distributional approximation. Here, robust estimators may provide a relevant alternative to least squares estimation. In the empirical part of the paper discretely as well as continuously compounded returns are applied. Whereas the former are appropriate due to the linearity assumption given by the regression model, the latter are commonly used in empirical investigations since the assumption of normality seems, if at all, more suitable for continuously compounded returns. We then focus on the robustness of least squares with respect to deviations from normality under continuous compounding. If certain sample observations cause the LS-estimator to give misleading information about the optimal ex ante portfolio composition, reducing their influence on the estimator should enable the decision maker to improve replication performance. The remainder of this paper is organized as follows. Chapter 2 reviews the classical linear regression model in the index replication context. Chapter 3 presents maximum likelihood type estimation and four specific estimators. In Chapter 4 an empirical investigation of replication portfolios for the German DAX-index is carried out. The relevance of robust techniques is inferred by comparing the performance of different estimators. The paper ends with a brief conclusion in Chapter 5. 2 A model for index replication Assume that the decision maker faces uncertainty and wants to infer information about the optimal portfolio weights from historical market behavior. His or her task is to choose a parameter vector x = (xi )N ×1 which contains variables related to the units of stock i in the replication portfolio. The replicating stocks are given by a set A= / ∅ with |A| = N . Without loss of generality, the index is calculated from a set B= / ∅ of index member stocks j with |B| = M . A typical case of index replication is the task of approximating the index with (i) a proper subset of index members,

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A ⊂ B. Besides of that, (ii) replication with non-index members only, A ∩ B = ∅ or (iii) replication with index and non-index members, can be considered.1 2.1 The linear regression model Consider the application of the linear regression model for index replication. Let the explained variable in period t be yt and the column vector of explanatory variables be wt = (wt,i )1×N . Given the unrestricted parameter vector x = (xi )N ×1 , the standard model without constant term is yt = wt x + t .

(1)

The xi are the true and time independent regression parameters which are to be estimated for each stock in the replication portfolio. Imposing the classical regression assumptions, the disturbance terms t obey: E(t ) = 0,

(2)

E(2t ) = const.,

(3)

Cov(t ; t−k ) = 0,

k= / 0,

(4)

Cov(wt,i ; t ) = 0,

∀i ∈ A.

(5)

Actually, the unbiasedness property of the LS-estimator in the case of stochastic explanatory variables requires conditions which are considerably stronger than assumption (5) (see Sch¨onfeld, 1971, Chap. 8). In the following, a brief introduction of the performance index concept which underlies the German DAX is given. Thereafter, two index replication specifications within the linear model are critically presented. 2.1.1 The construction of a performance index The replication strategy in this paper focuses on a capitalization weighted performance index. This index concept is usually appealing from an economic standpoint: The index return can be achieved by performing a buy and hold portfolio strategy under reinvestment of cash proceeds. It therefore represents the outcome of a naive, low cost investment strategy. The German DAX is an example of an index constructed along these lines. For a discussion see e.g. Stehle, Huber, and Maier (1996) or Wagner (1998, Chap. 8) and the literature given therein. Assume that individual stock prices Pt,j , j ∈ B, are observed at the end of trading day t. The end of day equity index Bt is calculated based on a set of index Note that the case (iv) A = B, which is assumed to be trivial here, in practice may not give perfect replication results due to transactions costs, violated reinvestment assumptions and imperfect divisibility of single stock investments. Dorfleitner (1999) provides an algorithm facing the latter problem. 1

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weights q = (qj )M ×1 . The non-normalized weights q are constant within two linkage time points, where the DAX is calculated with annual linkages. During a time period of constant weights q, the performance index concept requires the + + introduction of recursive stock specific correction factors ct,j = ct−1,j ·Pt,j /(Pt,j − + Dt,j ) where initially c0,j = 1. These factors are calculated from prices Pt,j which include day t claims Dt,j such as a dividends or rights offer values. The calculation procedure assumes that the claim value Dt,j is reinvested in stock j, where the day + t stock price ex claim equals its theoretical counterpart Pt,j − Dt,j . With the column vector of adjusted stock prices st,B = (St,j )1×M = (ct,j Pt,j )1×M and some normalizing linkage factor kt we may write the following index formula: Bt = kt · st,B q.

(6)

Note that the simple linear structure (6) will be violated if the periods t under consideration include at least one linkage time point. This is because of (i) a possible replacement of member companies and since (ii) the correction factors ci,t of the index member companies are reset to unity. Both occurrences cannot be described by correction factors depending only on the history of the stock under consideration. Therefore, the structure (6) is only seemingly linear. 2.1.2 Model specification based on prices A rather crude approach would be to explain the index Bt through unadjusted stock prices Pt,j . However, the disadvantage is obvious: Too much and even systematic noise must be absorbed by the disturbance terms t in model (1), thus tending to violate at least assumptions (2) and (3). As Bt is based on adjusted prices, a more reasonable assumption is to replicate the index through adjusted prices, Bt = st,A x + t ,

(7)

where st,A = (St,i )1×N = (ct,i Pt,i )1×N , i ∈ A is the column vector of adjusted replicating portfolio stock prices. The comparison with the index formula (6) shows that x roughly corresponds to kt q. Hence, over a longer observation period (including at least one linkage point of time), x cannot be time invariant. 2.1.3 Model specification based on returns As an alternative to price levels, it may be convenient to specify a regression model based on returns. With discretely compounded returns measured over the interval (t−1; t], the choice of x depends on a sample of index returns rt,B = (Bt /Bt−1 )−1 and stock returns rt,j = (ct,j Pt,j /ct−1,j Pt−1,j ) − 1.2 Based on the index formula (6) and a constant linkage factor kt = k, the index return can be written as: 2

If the latter returns are calculated under the index formula assumption of Section 2.1.1, i.e. stock prices ex claim equal their theoretical counterpart, they are identical to the standard return definition.

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rt,B =

M (ct,j Pt,j − ct−1,j Pt−1,j )qj Σj=1 Bt − Bt−1 = . M c Bt−1 Σj=1 t−1,j Pt−1,j qj

Since (ct,j Pt,j − ct−1,j Pt−1,j ) = rt,j ct−1,j Pt−1,j , we can write: M
ct−1,j Pt−1,j qj rt,B = rt,j M . Σj=1 ct−1,j Pt−1,j qj j=1 The second factor in the above summation denotes end of period t − 1 capitalization weights. Defining end of period t capitalization weights by bt,j =

ct,j Pt,j qj st,B q

(8)

and collecting them in the vector bt = (bt,j )M ×1 it follows: rt,B = rt,B bt−1 .

(9)

Here, the column vector of index member returns is denoted by rt,B = (rt,j )1×M . In equation (9), the discrete index return rt,B is a linear aggregation of period t index member returns rt,j , weighted by beginning of period t weights (which equal end of period t − 1 weights). A corresponding index return approximation is now given by the model rt,B = rt,A x + t ,

(10)

where rt,A = (rt,i )1×N is the column vector of replication portfolio stock returns. 2.2 Violation of the classical model assumptions When applying the linear model, the question arises whether the model specifications (7) and (10) each hold together with the classical assumptions (2) to (5). Unfortunately, the answer is no. In the following, we concentrate on the question of (i) time-variation in capitalization weights and the application of model (10), (ii) the use of continuously compounded returns in model (10) and (iii) correlation between the explanatory variables and the disturbance term. It turns out that the latter issue is a source of misspecification in model (7) and (10). 2.2.1 Variation in portfolio weights As shown in the derivation of return model (9), the calculation of the index return rt,B will be based on beginning of period weights. When the underlying index is capitalization weighted as in the index formula (6), the weights bt,j given by equation (8) will change over time. It is straightforward to show that the following intuitive relation (see e.g. Wagner, 1998, p. 260) bt,j = bt−1,j

1 + rt,j , 1 + rt,B

(11)

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holds. Starting with a set of weights at the end of period t − 1, weights at the end of period t depend on the stocks returns rt,j and the end of period weights bt−1,j . By recursion it follows bt,j = b0,j

t−1  k=0

1 + rt−k,j . 1 + rt−k,B

(12)

Hence, bt−1 in model (9) is given conditional on all previous returns rt−k,B , 1 ≤ k ≤ t − 1 and a set of initial weights b0 . In principle, the latter relation makes it possible to define a model where the weights bt−1,j in equation (9) are substituted by the corresponding right hand side of equation (12). We then arrive at a constant weighting scheme b0 and an index approximation can be estimated with model (10). This approach is based on adjusting the explanatory variables rt,i . If on the other hand this adjustment is not carried out, variation of the capitalization weights used in the calculation of the index will not be strictly compatible with the regression model: The period t returns rt,A = rt,A x of the replication portfolio represent a constant weights portfolio strategy which implies rebalancing after each trading period t. Even in the case A = B, with time varying weights bt there is no constant weights vector x which can perfectly replicate the index return. The regression equation is: rt,B bt−1 = rt,B x + t . To summarize, it is emphasized that the assumption of a constant weights vector x in model (10) is not compatible with a buy and hold portfolio strategy. The aim of the model analysis is to select a set of constant weights which—under some average performance measure—provides a preferable index replication in a given sample. Note further that relation (11) shows that large differences in cross-sectional returns cause larger changes in capitalization weights. In other words, higher correlation between stock returns will tend to reduce the deviation caused by time variation in portfolio weights and vice versa. 2.2.2 Continuous compounding Unfortunately, discrete returns are not well-suited for the convenient approximate assumption of being drawn from a normal distribution as they have the lower bound c −1. Therefore, continuously compounded returns, rt,j = ln(ct,j Pt,j /ct−1,j Pt−1,j ), are commonly used in capital market investigations, especially when time series properties are studied. For a given discrete stock return rt,i ∈ (−1; ∞), the difference between both return measures is given by:3 δ(rt,i ) = rt,i − ln(1 + rt,i ). With continuous compounding and δt = [δ(rt,i )]1×N , the regression model (10) can be rewritten as Discrete returns r are larger or equal to continuous returns, where equality results for a zero return. For negative returns, absolute continuous returns are larger than absolute discrete returns. The opposite is true for positive returns. Obviously, the nonnegative difference δ(r) is
k rk non-linear, where expanding the logarithm in a Taylor series yields: δ(r) = ∞ k=2 (−1) k . For example, the difference for a discrete return of 1, 5, 10, 25, −1, −5, −10 or −25 percent is 0.0050, 0.12, 0.47, 2.7, 0.0050, 0.13, 0.54 or 3.8 percent, respectively. 3

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c rt,B + δ(rt,B ) = (rt + δt ) x + t .

This is equivalent to c rt,B = rc t x + t + ∆t ,

(13)

where ∆t ≡ δt x − δ(rt,B ). From the latter equation it is obvious that a regression with continuously compounded returns will cause violations of the classical assumptions: Since the equality ∆t = 0 will typically only hold if all returns rt,i are equal to zero, the term t + ∆t in model (13) will not have a zero expectation. This violates assumption (2). Furthermore, the magnitude of ∆t will generally depend on the magnitude of the absolute stock returns. Hence, model (13) will also cause a violation of the uncorrelatedness assumption (5). 2.2.3 Correlation between the explanatory variables and the disturbance term An important characteristic of model equation (7) in Sect. 2.1.2 and equation (10) in Sect. 2.1.3 is that stock prices enter as explained as well as explanatory variables. Note that unbiasedness in the classical model is based on assumption (5). The regression equation (1) in matrix notation with y =(yt )T ×1 , W =(wt,i )T ×N and  =(t )T ×1 is y = Wx + .  looses its When  and W are correlated in the above model, the LS-estimator x unbiasedness property, since its expectation is given by the well-known expression:   E( x) = x + E (W W)−1 W  . To see whether uncorrelatedness is plausible, consider first the model specification (7) based on prices. With t = Bt −st,A x, the bilinearity property of the covariance implies that assumption (5) is equivalent to requiring Cov(St,i ; t ) = Cov(St,i ; Bt ) − Cov(St,i ; st,A x) = 0,

∀i ∈ A.

(14)

The same reasoning holds for model (10). With t = rt,B bt−1 − rt,A x, assumption (5) becomes Cov(rt,i ; t ) = Cov(rt,i ; rt,B bt−1 ) − Cov(rt,i ; rt,A x) = 0,

∀i ∈ A.

(15)

Stock prices St,i (returns rt,i ) will typically have different covariances with the price (return) of the index and the replication portfolio, respectively. Therefore, assumption (5) will generally not hold. Note that the severity of the violation of assumption (5) will depend on the selection of the set of replicating stocks A: For A ∩ B = A the covariance terms in (14) and (15) typically have similar values, especially when A is a set of heavily weighted index members. For A∩B = / A, however, the difference of the covariances will tend to be of larger magnitude, as the index and the replication portfolio tend to have a lower correlation.

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2.2.4 Concluding remarks on the violations of the classical model assumptions The above paragraphs show that the assumptions of the classical model will be violated in the index replication context. A main reason is that explained and explanatory variables are correlated returns. Besides, the regression parameters are not given as time invariant constants: Only in a time period between two index linkages, the index is calculated with a (non-normalized) constant weights vector. When the regression model is based on discretely compounded returns, the index return is based on a normalized weights vector that varies with daily stock prices. Under continuously compounded returns, there is actually no weights vector from which the index return can be calculated. The index replication task is therefore comparable to a ”shot on a running deer”. As a result from these cross-sectional model specification issues, we have to conclude that the unbiasedness (and hence consistency) property of the estimator will be generally weak. Facing the time series assumptions for the disturbance terms of the model, the classical assumptions (3) and (4) will typically also face violations with financial time series data. Possible autocorrelation and time-varying variance results in a weakened efficiency property of the least squares estimator. In the presence of departures from normality, an additional loss in efficiency is to be expected. In the rest of the paper, we address the corresponding pragmatic question: Applying the linear model in a setting where the theoretical optimality properties of least squares and robust estimators are weak, can robustification provide improved replication results?

3 Parameter estimation Parameter estimation generally relies on a function of sample realizations. In the regression based index replication case, weight parameters are estimated by realized differences between the index and a linear combination of the explanatory variables. For model (1), these differences are given by the model residuals: et = yt − wt x,

t = 1, ..., T.

(16)

3.1 Estimation procedure In a general regression setting, with introducing a scale parameter σ > 0, the aggregate loss function is Q(x, σ) =

T


ρ ((yt − wt x)/σ) .

(17)

t=1

The loss function ρ(ut ) : R −→ R+ is assumed to be convex, non-decreasing for ut > 0 and symmetric with ρ(ut ) = ρ(−ut ). The argument ut can be interpreted as a standardized residual, ut = et /σ. M-estimates (or maximum likelihood type estimates) of regression are defined as

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( x, σ ) = arg min Q(x, σ), (x,σ)∈Z

(18)

where: Z = {(x, σ) : x ∈ RN , σ > 0}. Assume that ρ(ut ) is continuously differentiable with dρ(ut )/dut = ψ(ut ). Since ρ(ut ) is convex, necessary and sufficient conditions for a local minimum now follow from the first order conditions of the problem. Setting the partial derivatives of Q(x, σ) equal to zero under the scale parameter restriction σ > 0, the resulting estimating equations are determined by the ψ-function. Using the price model (7) of Sect. 2.1.2, the vector x contains units of shares. When short sales are not allowed, or more generally, when certain upper or lower boundaries for holdings in a single stock are relevant, the choice of x may be restricted to a set S ⊆ RN such that Z = {(x, σ) : x ∈ S, σ > 0}. In the index replication model (10) of Sect. 2.1.3 the vector x contains portfolio weights which have to sum to unity. The set of admissible solutions to the minimization problem (18) is then given by: Z = {(x, σ) : x ∈ S, x 1 = 1 , σ > 0}.4

3.2 Alternative estimators The basic idea of robust estimation is to reduce the estimator’s sensitivity towards deviations from normality while preserving a high degree of efficiency. This is achieved by implementing ψ-functions that give an at least approximately linear weight to small deviations |ut | as with the least squares criterion and downweight large deviations |ut | as done by the least absolute value criterion. In the following we outline LS, LAV and two M-estimators. For a detailed survey of approaches to M-estimation refer to Hampel, Ronchetti, Rousseeuw, and Stahel (1986). An overview of robust estimation with reference to numerical issues is also provided in Venables and Ripley (1999). 3.2.1 Least squares The least squares estimator is characterized by a linear ψ-function ψLS (ut ) = ut

(19)

and quadratic loss ρLS (ut ) = 4

1 2 u . 2 t

Note that the requirement that the portfolio weights have to sum to unity does not alter the optimality properties of the LS-estimator. Under quite general conditions within the classical model, LS is a BLUE even when the regression parameters are chosen subject to a linear restriction. For details refer to Sch¨onfeld and Werner (1986).

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3.2.2 Least absolute value An alternative to the least squares criterion is least absolute value, which minimizes absolute rather then squared deviations. It follows ψLAV (ut ) = sgn(ut ),

(20)

ρLAV (ut ) = |ut | , where ρLAV is not continuously differentiable. Hence, note that the LAV-estimator is not defined by first order conditions depending on the ψ-function. 3.2.3 Huber M-estimates of regression Following the proposals of Huber, various functional forms for ψ(ut ) have been applied in the computation of robust M-estimators. The most commonly used function is the Huber ψ-function given by  ψH (ut ) =

if |ut | ≤ c ut c sgn(ut ) if |ut | > c

(21)

and some constant cutoff point c with 0 < c < ∞. A corresponding loss function is 1 2 1 u + β if |ut | ≤ c ρH (ut ) = 2 t 2 1 1 c |ut | − 2 c + 2 β if |ut | > c, where β = E[ψH (Z)2 ] with Z ∼ N (0; 1). The additive β-term in the loss function makes the estimate of the scale parameter consistent under the normal model (see Huber (1973) and Huber (1981)). 3.2.4 Tukey redescending M-estimates of regression When redescending ψ-functions are chosen, ψ(ut ) approaches zero for |ut | → ∞. This implies that the loss function is bounded, i.e. ρ(ut ) does not exceed a fixed value for any given deviation ut . A common redescending ψ-function is Tukey’s biweight  ψT (ut ) =

ut (d2 − u2t )2 0

if |ut | ≤ d if |ut | > d,

with 0 < d < ∞. A corresponding loss function is given by  1 4 2 1 2 4 1 6 if |ut | ≤ d 2 d ut − 2 d ut + 6 ut ρT (ut ) = 1 6 d if |ut | > d. 6

(22)

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4 Empirical investigation As outlined in Chapter 2, the application of the classical linear model for index replication will generally be subject to misspecification. In the empirical investigation we now address the question of the robustness of the model with respect to departures from normally distributed disturbances. We follow the commonly used approach of regressing returns. In this setting, our results are subject to comparable sources of misspecification as given in related investigations. Due to the estimators’ weak optimality properties, the evaluation of the replication performance will be based on in-sample and out-of-sample tracking error. The latter will be defined as the difference between index and replication portfolio return.

4.1 Data and replication subset Our sample for the German equity index DAX covers daily returns in the period beginning January, 2, 1989 and ending December, 30, 1996. All returns are calculated from closing prices of the Frankfurter Wertpapierb¨orse as provided by Datastream. Returns are adjusted for cash dividends and rights offer values. The index replication is performed with a subset selection A ⊂ B which results from choosing the most heavily weighted index members available at the beginning of the overall sample period. The subset is fixed throughout the sample and contains 20 out of 30 stocks with the following ticker symbols: ALV, BAS, BAY, BHW, BMW, BVM, CBK, DAI, DEG, DBK, DRB, HOE, MAN, MMW, RWE, SIE, THY, VEB, VIA and VOW. The subset covers roughly between 85 and 90 percent of the capitalization of the index.5

4.2 Computation of the weights estimates The estimation problem (18) is solved iteratively where a solution to the first order  conditions within the set Z = {(x, σ) : x ∈ RN + , x 1 = 1 , σ > 0} is obtained by the Newton/Raphson-algorithm. The least squares estimator (19), the Huber M-estimator (21) and the redescending biweight estimator (22) are applied. Following the recommendations in the literature, a cutoff point c = 1.5 is chosen for Huber’s M-estimator and d = 1 is chosen for Tukey’s biweight (e.g. Huber, 1981; Morgenthaler, 1989; Mendes and Tyler, 1996). As with any non-linear estimation problem, a suitable start-solution is required. We choose an equally weighted vector as start-solution because it is uniquely defined and can be expected to be located near the optimal solution. Since Mestimators have weaker convergence properties as compared to LS, it is recommendable to use different alternative start-solutions for their calculation. The LS solution is therefore applied as a second start solution when convergence cannot be 5

Note that the replication improvements due to the application of the regression approach are typically high for portfolios with between ten and below thirty stocks. This is in line with the findings by Alderson and Zivney (1989).

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achieved with the equal weighted start solution. Furthermore, the Huber estimate is used as a start solution for redescending estimation if neither the equal weighted nor the LS solution provides convergence. Since the LS-estimator is scale invariant (i.e. with ρ = ρLS , σ is an overall scale parameter which can be extracted from the sum in (17)), a numerical solution to the estimation problem is obtained with an arbitrary fixed scale parameter such as σLS = 1. For the M-estimators, the estimation equations require the simultaneous calculation of a robust parameter vector and a scale parameter estimate. The initial scale parameter estimate σ 0 is calculated by the residuals et of the equal weighted start solution:      − med( e e ) (23) σ 0 = k med   , t, t = 1, ..., T. t t  t

t

The above expression measures the median of the residuals’ absolute deviations from their median or, in short, the ’Median Absolute Deviation’ (M AD). For initial M AD-scaling in regression a tuning constant k = 6 is frequently recommended (see e.g. Morgenthaler, 1989). The variation of the scale parameter in solving the estimation equations for the M-estimators is a noteworthy point: If σ is low, standardized residuals in (17) become large and will be downweighted. In this case, the Huber M-estimates become similar to the LAV-estimate. For redescending estimators many local minima will be found. If on the other hand σ is high, the residuals become small and the estimation approach resembles least squares. Therefore, depending on the estimated scale parameter, the M-estimator may not exist or converge to another estimate. The two cases are implosion of the scale parameter for σ → 0 and explosion for σ → ∞. Note further that, analogous to least squares, the Huber estimator is defined by a unique solution to the estimation equations. The redescending biweight, in contrast, may have various local solutions. Therefore, the use of estimators based on redescending ψ-functions is controversial in the literature. One possible approach to this problem is to restrict the estimator by setting a constraint on the aggregate loss function Q (Mendes and Tyler, 1996). 4.3 Measuring replication performance With the index replication model (10) based on returns, the tracking error given by the residual (16) is: et = rt,B − rt,A x. Referring to Section 2.2.1, note that the above equation defines the in-sample tracking error based on a constant weights vector. The out-of-sample tracking error is e t = rt,B − rt,A xt−1 , where xt is defined in analogy to bt in equation (8). In principle, any measure of dispersion of the tracking error can now be used to evaluate replication performance. With continuously compounded returns, a commonly used measure of replication performance is the standard deviation of tracking

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error. A reason for this is that, in this case cumulative deviations from the index return over a given time period are given by the sum of single tracking error deviations. Therefore, under a tracking error which has a constant, finite variance and is independently distributed in the time series context, the standard central limit theorem will ensure that the distribution of cumulative deviations will be asymptotically normal.6 When measuring index replication performance in the presence of outliers, the assumption of a constant, finite variance of the tracking error might be violated. Under quite weak large sample assumptions, the central limit theorem still applies even for unequal variances and normality of the cumulative deviations can again be established asymptotically.7 In small samples however, the variance of the largest outlier might dominate the variances of all other observations and inferences can yield misleading results. It is then advantageous to use a statistic for the standard deviation which is robust with respect to the assumption of a constant, finite variance of the tracking error. In order to give an evaluation of our empirical replication results, the standard deviation is estimated by two different statistics. The first statistic is the well-known square root of the sample variance. With e denoting the sample average of the et , the statistic is given by:  T 1
sST D = (et − e)2 . (24) T − 1 t=1 The second statistic is a robust alternative, the standardized M AD, whose realization is   1    med sM AD = −1 − med (e ) (25) e t t  , t Φ (3/4) t where Φ denotes the cumulative density function of the standard normal distribution. The premultiplied constant ([Φ−1 (3/4)]−1 ≈ 1.483) insures consistency of the statistic under normally distributed disturbances t (Huber, 1981, p. 107f.).8 6

Two remarks are appropriate: (i) As outlined in Section 2.2.2, the tracking error under continuous compounding will include a difference term. (ii) In a model with discrete compounding  on the other hand, the cumulative return difference over the period [0; T ] is given by: Tt=1 (1 + et ). Hence, the central limit theorem does not apply. Apart from the the cumulative deviation itself, other measures of replication performance such as maxt |et | should be useful in this case. 7 This is the Lindeberg/Feller version of the central limit theorem, see for example Greene (1997, p. 122f.). The assumptions are that, in the limit, (i) the maximal variance does not dominate the sum of all variances and that (ii) the average variance exists. 8 In-sample sST D estimated over a long time horizon might sometimes seem to overestimate the ex-ante risk of deviating from the index over some relatively short future period. This however, should not obscure the fact that a large absolute tracking error due to some extreme event can possibly happen one period after the portfolio was set up. In a model context, additive outliers can be thought of as rare events occurring randomly over time. The risk of deviating from the index return is then given by a regular component which describes

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4.4 Replication results The replication methodology is as follows. Beginning with the first year of the sample period, the regression estimates are calculated (LS, M-Huber and M-Tukey). In-sample performance of the portfolio weights vectors is evaluated by the standard deviation estimates (24) and (25). Then, at the beginning of the succeeding sample year, the portfolios are constructed according to each estimated weights vector. The out-of-sample performance of the estimators is measured by (24) and (25) over a one year period. This procedure is repeated for the years 1989 up to 1996. The replication methodology implies in-sample estimation of constant portfolio weights and measuring out-of-sample performance of a buy and hold investment strategy having variable portfolio weights. As a reference to the outcomes of a naive control strategy, the results of an equally weighted portfolio are provided additionally (EW). The results for continuously compounded returns are summarized in Table 1. Before interpreting the replication results in Table 1, we point out that the application of continuous as compared to discrete compounding does neither have a substantial impact on parameter estimation nor replication performance in the given sample. Additional computation based on discretely compounded returns yielded results that were very similar to those reported in Table 1. Also, it is important to note that the statistics underlying (24) and (25) are based on the assumption of uncorrelated disturbance terms. In order to test the null hypothesis of uncorrelatedness, first order sample autocorrelation coefficients are given in parentheses in Table 1. Additionally, the realizations of the Ljung/Box-statistic Q(5) –referring to the alternative hypothesis that at least one of the first five autocorrelations is not equal to zero– are calculated. Note that the asymptotic distribution of Q(5) equals the χ2 (5)-distribution under the null. As the results for each single out-of-sample period show, there is only one significant negative and one significant positive first order autocorrelation realization in realized tracking errors. The significant Ljung/Box-statistics in 1995 are due to significantly negative second order autocorrelations. Hence, overall evidence of negative autocorrelation in disturbances is weak in the given sample (especially if the multiple testing problem is taken into account). A plausible economic reason for this observation is that the replication subset is only slightly more liquid than the index, so that stock prices in both portfolios nearly simultaneously react to new information. Autocorrelated disturbances do not provide a significant source of bias in the estimated variance of tracking error.9 Interpreting the performance measures (24) and (25), the following conclusions can now be drawn from Table 1: Estimating weights with linear regression improves in- and out-of-sample replication results as compared to the equally weighted portfolios (EW). The differences between in- and out-of-sample performance of the tracking behavior under the normal distribution and an outlier component. Differences between sM AD and sST D for a given sample, point out to these two components of tracking risk. 9 For smaller replication portfolios with highly capitalized stocks, the bias can be substantially positive, see Pope and Yadav (1994).

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Table 1. In- and out-of-sample performance measures of the different weights estimates in the years 1989 to 1996. First order autocorrelation coefficients and Ljung/Box-statistics for realized tracking errors given in brackets. ’*’ denotes significance at the 95 percent confidence level for each single out-of-sample period Regression estimate

EW

LS

M-Huber

M-Tukey

sample results

sMAD / sSTD % p.a.

sMAD / sSTD % p.a.

sMAD / sSTD % p.a.

sMAD / sSTD % p.a.

1989 in sample

2.10 / 3.17

1.28 / 2.15

1.06 / 2.41

0.94 / 2.66

1990 out of sample

1990 in sample

1991 out of sample

1991 in sample

1992 out of sample

1992 in sample

1993 out of sample

1993 in sample

1994 out of sample

1994 in sample

1995 out of sample

1995 in sample

1996 out of sample

2.37 / 2.55

1.87 / 1.98

1.56 / 1.88

1.36 / 1.68

(−0.12 / 6.79)

(−0.010 / 2.79)

(−0.089 / 6.70)

(−0.14* / 8.51)

2.63 / 2.74

1.64 / 1.65

1.34 / 1.57

1.28 / 1.55

1.70 / 2.04

1.25 / 1.46

1.15 / 1.41

0.99 / 1.24

(0.086 / 1.93)

(0.049 / 7.46)

(0.0080 / 4.33)

(−0.066 / 4.85)

1.63 / 2.02

0.80 / 0.89

0.83 / 0.90

0.78 / 0.90

1.73 / 1.92

0.84 / 0.98

0.84 / 0.98

0.87 / 0.98

(0.060 / 7.04)

(−0.052 / 0.97)

(−0.049 / 0.88)

(−0.060 / 1.23)

1.55 / 1.91

0.78 / 0.82

0.77 / 0.82

0.79 / 0.82

1.94 / 1.95

0.71 / 0.82

0.72 / 0.82

0.72 / 0.81

(0.027 / 2.36)

(0.067 / 10.92)

(0.068 / 11.05)

(0.068 / 10.85)

1.73 / 1.89

0.80 / 0.80

0.81 / 0.80

0.79 / 0.80

1.97 / 1.94

0.93 / 0.97

0.93 / 0.97

0.93 / 0.97

(0.15* / 12.74)

(−0.0061 / 4.34)

(−0.0079 / 4.16)

(−0.0045 / 4.39)

1.65 / 1.78

0.90 / 0.95

0.97 / 0.97

0.86 / 0.95

1.50 / 1.67

0.95 / 1.24

0.99 / 1.26

0.94 / 1.23

(0.064 / 2.40)

(0.10 / 16.27*)

(0.095 / 15.86*)

(0.10 / 16.58*)

1.55 / 1.75

0.79 / 1.06

0.92 / 1.07

0.80 / 1.05

2.22 / 2.98

1.52 / 2.16

1.46 / 2.15

1.48 / 2.16

(0.053 / 5.90)

(0.015 / 4.73)

(0.013 / 5.58)

(0.012 / 4.58)

EW-portfolios within one sample year are due to the constant in-sample and variable out-of-sample weights, as mentioned above (Sect. 4.3). The results in Table 1 document that LS-estimation drives robust in-sample standard deviation below 1 percent annually in 5 out of 7 sample years. In these years, 1991 to 1995, the normal model obviously fits well, which results in good LS weights-estimates: The LS estimator keeps out-of-sample robust standard deviation of the tracking error below 1 percent when estimated for the years 1992 to 1995. Replication results get worse out-of-sample in 1996 since the heavily weighted stock SAP3 became a new index member on September 15, 1995. The robust estimators, on the other hand, show very slow convergence behavior starting with the equally weighted solution and provide nearly unchanged weights estimates

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when iterations start with the LS-solution. Therefore, it is not surprising that the replication results are very close to each other. Although M-estimates of regression do not guarantee superior results in general, they can prove their superiority with the return realizations in the years 1989 and 1990. The economic background of return contamination in these years can be assigned to the occurrence of the so called ’Mini Crash’ in October 1989 and the Golf War in summer 1990. Whereas a crash represents a typical additive outlier in the time series, the latter event was associated with days of volatile market behavior. It is obvious from their construction, that the different estimators will quite differently respond to these return realizations. As an illustration of the insample fit in 1989, the residuals et of each weights estimator are plotted in Fig. 1.

Fig. 1. In-sample fit of the different weights estimates for the replication portfolio in the year 1989. Observation 200 denotes the so-called Mini-Crash at October 16th when the index lost roughly 14 percent in value

As the out-of-sample results for the years 1990 and 1991 in Table 1 show, the replication performance of the linear LS-estimator is obviously improved when the loss associated with single large absolute residuals is downweighted. Here, the redescending variant of M-estimation achieves even better out of sample results than the Huber estimate. This gives some evidence for the hypothesis that those events do not provide any useful sample information at all. However, recalling the discussion in Section 4.2, note that these improvements come at a relatively high computational cost. Obviously, Table 1 further demonstrates that the robust in-sample standard deviations of the LS- and the M-residuals et provide information about the usefulness of fitting robust weights-estimates. The magnitude of sM AD as compared with sST D can be used as a guide for measuring possible in-sample contamination. In-

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terpreting the results as evidence for the hypothesis that tracking performance at an outlier date does not provide useful out-of-sample information, in-sample sM AD should serve as a criterion for choosing the appropriate ex-ante weights vector.

5 Conclusion Violations of the classical regression assumptions in the index replication context will generally weaken the optimality properties of the regression parameter estimators. For a given index replication problem, the empirical part of the paper could provide evidence that regression analysis may still remain a useful tool where the sensitivity to violations of the classical assumptions is quite low. As is well-known, markets can eventually show extreme behavior of market participants causing large absolute daily price movements as compared to average trading days. The relevance of such extreme events for parameter estimation has been a major point in the empirical study. The focus was therefore directed to the question of model robustness with respect to the distributional assumption of the disturbances. The results for the German market provide evidence for the hypothesis that, to some extent, there is contamination present in the sample which is relevant for the estimation of the ex-ante portfolio composition. When large price movements occur in the sample, robust estimates provide superior out-of-sample tracking performance. Assuming that time and computational effort play a minor role and especially if indications of contamination arise in-sample, it seems therefore appropriate to fit robust estimators additionally to least squares.

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