Models for expected returns with statistical factors

Cueto, José Manuel; Grané, Aurea; Cascos, Ignacio Article Models for expected returns with statistical factors Journal of Risk and Financial Management Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Cueto, José Manuel; Grané, Aurea; Cascos, Ignacio (2020) : Models for expected returns with statistical factors, Journal of Risk and Financial Management, ISSN 1911-8074, MDPI, Basel, Vol. 13, Iss. 12, pp. 1-17, https://doi.org/10.3390/jrfm13120314 This Version is available at: https://hdl.handle.net/10419/239400 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Journal of Risk and Financial Management Article Models for Expected Returns with Statistical Factors José Manuel Cueto , Aurea Grané and Ignacio Cascos * Department of Statistics, Universidad Carlos III de Madrid, 28903 Getafe, Spain; [email protected] (J.M.C.); [email protected] (A.G.) *Correspondence: [email protected] Received: 3 November 2020; Accepted: 3 December 2020; Published: 8 December 2020   Abstract: In this paper, we propose multifactor models for the pan-European Equity Market using a block-bootstrap method and compare the results with those of traditional inferential techniques. The new factors are built from statistical measurements on stock prices—in particular, coefficient of variation, skewness, and kurtosis. Data come from Reuters, correspond to nearly 2000 EU companies, and span from January 2008 to February 2018. Regarding methodology, we propose a non-parametric resampling procedure that accounts for time dependency in order to test the validity of the model and the significance of the parameters involved. We compare our bootstrap-based inferential results with classical proposals (based on F-statistics). Methods under assessment are time-series regression, cross-sectional regression, and the Fama–MacBeth procedure. The main findings indicate that the two factors that better improve the Capital Asset Pricing Model with regard to the adjusted R2 in the time-series regressions are the skewness and the coefficient of variation. For this reason, a model including those two factors together with the market is thoroughly studied. We also observe that our block-bootstrap methodology seems to be more conservative with the null of the GRS test than classical procedures. Keywords: asset pricing; Big Data; bootstrap; cross-sectional regression; factor models; time series 1. Introduction Understanding why and how certain assets go up in price while others go down is a major concern for both Industry and Academia. There is an overwhelming number of research articles regarding asset pricing; however, for the moment, no model has been able to explain the behaviour of the Stock Market in a fully satisfactory manner. Some of the classical proposed models rely on financial measures, such as the Price-to-Book ratio, Market Capitalisation, or Profitability to predict future asset returns (Fama and French 1992). Others also incorporate macroor industry-related measures, such as interest rate levels (Viale et al. 2009), or oil price (Ramos et al. 2017). Another part of the literature focuses on higher order statistical moments of returns (Elyasiani et al. 2020). Lately, such measures are combined with others involving psychological factors, such as Momentum, in order to build more sophisticated models (Carhart 1997). Markowitz (1952) determined that rational investors use diversification to optimize the profitability of their portfolios. This implies that the return required by an investor does not depend on the risk of a certain stock, because part of that risk is diversifiable. The only important factor is, thus, non-diversifiable risk. Markowitz assumes that investors are risk-averse, so when they choose a portfolio they tend to minimize the variance of the return of the portfolio for a given expected return and maximize the expected return for a given variance. The Capital Asset Pricing Model, CAPM, (see Lintner 1965;Mossin 1966;Sharpe 1964) represents the expected profitability of the i-th stock, E(Ri), in the following manner: E(Ri) = rf+βi(E(rm)−rf), J. Risk Financial Manag. 2020,13, 314; doi:10.3390/jrfm13120314 www.mdpi.com/journal/jrfm J. Risk Financial Manag. 2020,13, 314 2 of 17 where rf is the risk-free rate, E(rm)−rf is the Market Risk Premium, and βi the sensitivity of expected excess asset’s return associated with the i-th asset. CAPM is used to predict stock profitabilities, not portfolio profitabilities. According to Kristjanpoller and Liberona (2010), the best strategy under the assumptions of CAPM is a passive one: buy the Market portfolio in the long run. This conclusion is even stronger if we take into account transaction costs. Even though CAPM is well-formulated on a theoretical level, the assumptions behind the model limit its practical application. For instance, CAPM’s β is static and only applies for 1 year. Indeed, Lintner (1965) and Miller and Scholes (1972) tested the model with NYSE stocks obtaining certain inconsistencies, which led to the definition of multifactor models. Multifactor models should better explain the behaviour of assets’ returns. The identification of these factors is a key issue in the process of building the model. The seminal example of multifactor models is the one by Fama and French (1992). The asset’s returns are explained by a linear model that depends on the size of the company (market capitalization), the price-to-book ratio, and, as in CAPM, the market risk premium: E(Ri) = rf+βi(E(rm)−rf) + βi1SMB +βi2HML, where SMB is the difference between the return of a portfolio of small companies and one of big companies (small minus big), HML is the difference between the return of a portfolio of companies with high price-to-book ratios and one of companies with low price-to-book ratios (high minus low), while rf and E(rm) are as before. Adding the two factors to the regressions results in large increases in the coefficient of determination. According to Fama and French (1993), the market factor alone produces only two (of 25) R2 values greater than 0.9; in the three-factor regressions, R2 values greater than 0.9 are usual (21 of 25). Inferential procedures about multifactor models strongly rely on assumptions regarding the data: for instance, factors being uncorrelated over time, normally distributed errors which are i.i.d. over time and independent of the factors, and so forth. When these assumptions are not satisfied, classical estimators may be biased. To circumvent such a drawback, non-parametric techniques can be used. Efron (1972) developed the Bootstrap methodology as a resampling technique to approximate the distribution of test statistics. In the Asset Pricing literature, Kosowski et al. (2006) proposed a bootstrap procedure by which they resample from the returns of mutual funds in order to test that all portfolios’ returns are explained by the factors. Fama and French (2010) modified the process to jointly resample both fund and explanatory returns. This additional step is important, as it takes into account possible correlation. However, none of these procedures consider dependency over time. Grané and Veiga (2008) explored the block bootstrap for computing the unconditional distribution of returns. These authors found huge differences in the minimum capital risk requirement estimates when using conditional approaches (such as GARCH-type models and stochastic volatility models) specially for long positions and larger investment horizons. The objectives of this paper are: (1) to propose a multifactor model based on naive statistical factors, (2) develop non-parametric resampling techniques that account for time dependency in order to test the validity of the model and the significance of the parameters involved, and (3) to compare bootstrap-based inferential techniques with the classical proposals. These procedures are tested on a real dataset of assets’ returns of over 2000 European companies extracted from Reuters, and span from January 2008 to February 2018. Specifically, the statistical factors that we consider are the coefficients of variation, skewness, and kurtosis of the stock prices. With the coefficient of variation, we aim to capture the stock volatility, while thirdand fourth-order moments can also be strongly related to subsequent returns (see Chang et al. 2013;Conrad et al. 2013;Harvey and Siddique 2000) . In the traditional framework, investors have specific preferences regarding the first (mean) and second (variance) moments of returns. Our intuition suggests that investors should also have preference for high-CV stocks or lottery stocks J. Risk Financial Manag. 2020,13, 314 3 of 17 (Kumar 2009) and right-skewed portfolios (in fact, skewness could explain the excess returns found in portfolios of low-size and high book-to-markets in traditional models). The main findings are that the two factors that better complement the market (only factor at the CAPM) are the skewness and the coefficient of variation. This is consistent with previous studies both for the US and the European markets that have found the existence of risk premia for volatility and skewness (Elyasiani et al. 2020). With regard to the time-series regressions, the conclusions drawn from the newly proposed bootstrap inferential techniques are somehow coherent with the ones obtained from classical inference procedures. Indeed, since the used data do not fulfill all the classical distributional requirements, the bootstrap conclusions tend to be less strict with departures from the benchmark described at the null hypotheses. In particular, our bootstrap inferences suggest that all adjusted time-series regression models have intercepts that are jointly equal to zero, which is not the case for the classical method. With respect to the specific models estimated for each portfolio, those with high loading on the coefficient of variation (respectively, skewness) have a higher coefficient of variation (resp. skewness) coefficient, while market β s are all close to 0.5. The cross-section model built from the estimated coefficients presents some deficiency, since the coefficient of variation does not contribute significantly to it, while the market and skewness do. The underlying reason might be some multicollinearity problem. However, the performance of the model seems reasonably good in spite of the period analyzed, that comprises the European debt crisis. The remainder of this paper is organized as follows. In Section 2we present the materials and methods. Specifically, in Section 2.1 we discuss our data and the construction of portfolio returns, while Section 2.2 explains the methodology, first the classical (Time-Series, Cross-Sectional, and Fama-MacBeth) and second, the resampling technique developed for the analysis. Section 3reports the results of the analysis and compares different methodologies. We close the paper with some conclusions in Section 4. 2. Materials & Methods 2.1. Data and Portfolio Returns 2.1.1. Data We start with 2012 European companies that have been selected from all EU countries. We build up from the idea that a European multifactor asset pricing model exists and that its factors are equally priced across the European capital markets (Morelli 2010). As is usual in the Factor literature, we exclude Financials as they usually have high leverage ratios, affecting several financial ratios. We extract Reuters prices at the end of each month from January 2008 to February 2018 for these companies. We apply several filters to the data: first, we delete all the datapoints where price is not available; second, we apply a transaction filter, excluding all companies that do not show transactions for a whole semester; third, we exclude companies with no Market Cap or P/B info for more than two consecutive years. Finally, we exclude companies with non-positive equity at the end of any year. After all these filters have been passed, we end up with 1913 companies. Next, we calculate monthly returns for all the companies and estimate the Risk-Free Rate ( rf ) with the two-year German Bond yield and Market ( rm ) through the SP600. Monthly returns were calculated with the natural logs of the market price divided by the market price in the previous month. Table 1contains some descriptive statistics by year. In particular, it provides the number of firms, the number of considered months in each period, and price statistics. J. Risk Financial Manag. 2020,13, 314 4 of 17 Table 1. Descriptive statistics of the data by year. Year Companies Months Mean SD Min Median Max 2009 1631 12 69.55 1619.75 0.01 5.59 92,316.88 2010 1643 12 47.39 765.65 0.01 6.48 42,907.84 2011 1672 12 36.20 516.56 0.00 6.50 26,316.80 2012 1706 12 23.77 120.80 0.00 5.48 4203.77 2013 1729 12 24.98 109.47 0.00 5.98 3400.01 2014 1737 12 28.37 123.28 0.00 7.20 3789.00 2015 1789 12 30.98 139.74 0.00 7.25 3999.00 2016 1833 12 32.39 164.73 0.01 7.20 6000.00 2017 1864 12 38.99 195.06 0.00 8.74 6149.00 2018 1878 2 41.34 204.65 0.00 9.08 6000.00 We can observe that the average price decreases until 2012 (from 69.55 in 2009 to 23.77 in 2012) and then increases until 2018 (41.34). This behaviour is related to the financial crisis. In terms of standard deviation, we can observe a similar pattern. 2.1.2. Statistical Factors We calculated different measures for each stock in the sample, namely: 1. Coefficient of variation (CV) of prices for each year and company: CV =sn ¯ x, where s2 n=1 n∑n i=1(xi−¯ x)2 and ¯ x is the sample mean. CV represents the relative spread for positive random variables and it can take values in (0, +∞). 2. Skewness (Skew) of prices for each year and company: Skew = 1 n∑n i=1(xi−¯ x)3 s3 n Positive values indicate that the distribution is positively skewed, that is, the right tail is longer than the left one, while the contrary occurs for negative values. When both tails are similar, the skewness is roughly 0. 3. Excess Kurtosis (Kurt) of prices for each year and company: Kurt = 1 n∑n i=1(xi−¯ x)4 s4 n−3 measures how fat the tails of a distribution are in comparison to those of a normal random variable. Positive (negative) values indicate heavier (thinner) tails than those of a normal random variable. In general, factors are positively correlated, except for Kurt, which is negatively correlated with Market and CV. As can be seen in Table 2, Market and CV on one side and CV and Kurt on the other are highly correlated. This could lead to multicolineality in models built with these factors and, thus, to larger confidence intervals. Table 2. Correlation among the factors. Market CV Kurt Skew Market 1.0000 0.5970 −0.4210 0.1770 CV 0.5970 1.0000 −0.5450 0.2930 Kurt −0.4210 −0.5450 1.0000 0.1650 Skew 0.1770 0.2930 0.1650 1.0000 J. Risk Financial Manag. 2020,13, 314 5 of 17 2.1.3. Portfolio Returns Next, we calculate percentiles for each of these measures and assign them to portfolios accordingly following (Fama and French 1992). There are two portfolios for each measure, except for Skew, where we calculate three. • Stocks with low CV will be included in portfolios 1-y-z, while stocks with high CV will be included in portfolios 2-y-z. • Stocks with low Kurt will be included in portfolios x-1-z, while stocks with high Kurt will be included in portfolios x-2-z. • Stocks with low Skew (less than the 30-th percentile, P30) will be included in portfolios x-y-1, while stocks with high Skew (greater than P70) will be included in portfolios x-y-3. The rest will be included in portfolios x-y-2. The resulting portfolios are summarized in Table 3. Portfolios are updated yearly based on the previous year’s measurements (2008 data are only used to build the starting portfolios). We proceed to calculate average monthly returns for the stocks included in each portfolio. Additionally, we calculate each factor as the excess return of the higher portfolio in each category minus the return of the lower portfolio. All returns are calculated for equally weighted portfolios at t+ 1. In Figure 1, we have plotted the cumulative returns of all the portfolios. Table 3. Description of portfolios. Portfolio CV Kurt Skew (1) 1-1-1 Low Low Low (2) 1-1-2 Low Low Medium (3) 1-1-3 Low Low High (4) 1-2-1 Low High Low (5) 1-2-2 Low High Medium (6) 1-2-3 Low High High (7) 2-1-1 High Low Low (8) 2-1-2 High Low Medium (9) 2-1-3 High Low High (10) 2-2-1 High High Low (11) 2-2-2 High High Medium (12) 2-2-3 High High High Figure 1. Portfolios’ cumulative returns. J. Risk Financial Manag. 2020,13, 314 6 of 17 The three portfolios with the best performance are 1-1-2, 1-2-1, and 2-1-1. The first two share the feature of having low CV. There is one portfolio that ends with a negative cumulative return (2-2-3); it includes stocks with high CV, high Kurt, and high Skew. Portfolios 2-2-1 and 2-2-2 (high CV and high Kurt) are also among the worst performers. 2.2. Methodology 2.2.1. Classical Methodologies Linear Factor Models have gained great popularity lately, and this has led to a vast amount of literature on estimating and testing such models. Typically, these models take the form of the following equation: E(Ri) = αi+β0iλ. The expected return for the i -th portfolio has a linear relationship with a set of factors, λ . The simplest of such models is the one-factor model developed by Jensen (1968) and based on CAPM, where asset risk premiums are linear functions of market risk premium and the systematic risk of the asset ( βi ), that is: E(Ri) = αi+βi(E(rm)−rf), where α is referred to as the Jensen’s Alpha of the asset and is commonly used as a performance measure. In general, we will consider that a model works when the expected value of α is zero. The model is extendable, mainly, by including additional factors as independent variables. Feng et al. (2020) have made a great effort to systematically evaluate the contribution to asset pricing of any new factor, above and beyond what a high-dimensional set of existing factors explains. Several procedures to estimate the parameters α , β , and λ have been developed in the literature. After reviewing two analytic procedures which assume a certain behaviour of the underlying data (Time-Series Regression and Cross-Sectional Regression), we propose a Block Bootstrap method that works even if the previous assumptions are violated. The setup of the experiment includes N assets (portfolios of equities), K factors ( K= 1 for CAPM), and T time periods. The factors are excess returns. We use portfolios instead of stocks because: (1) the errors of α and β are higher for individual stocks as their volatility is higher, and (2) portfolios have more stable characteristics, while stocks vary over time. All the statistics below are derived from the classical assumptions that returns are independent over time, correlated across assets, and (for finite sample results) normally distributed. Time-Series Regression (TS) A time-series regression was run for each portfolio i=1, . . . , Nas Ri t=αi+β0ift+ei t,t=1, 2, . . . , T. The estimates of αi and βi and their standard errors are those of the time-series regression (OLS), while λ is estimated as ˆ λ=1 T∑T t=1ft=f . Assuming factors are uncorrelated over time, its standard error is estimated as ˆ σ(ˆ λ) = ˆ σ(ft)/√T. To assess the ability of a model to explain excess returns, we used the GRS test (see Gibbons et al. 1989) whose null hypothesis is H0:α1=. . . =αN= 0. Under the assumption that eare normally distributed, the test statistic is: T−N−K N[1+f0Σ−1 ff]−1ˆ α0Σ−1ˆ α∼FN,T−N−K, (1) where Σfis the covariance matrix of the factors and Σis the residual covariance matrix. J. Risk Financial Manag. 2020,13, 314 7 of 17 Cross-Sectional Regression (CS) and Fama-MacBeth (FM) Procedure This is a two-step procedure: • First, a Time-Series regression is run to obtain the sensitivity of the expected excess return of each of the portfolios. For each i=1, . . . , Nestimate βiin the model Ri t=ai+β0ift+ei t,t=1, 2, . . . , T. Note that the intercept here is denoted by ainstead of α. •Run a Cross-Sectional regression to get λ E(Ri) = ˆ β0iλ+αi, for i=1, . . . , N, where the estimates for ˆ βi were obtained in the TS regression at the first step. In conclusion, ˆ λ represents the slope coefficients in the CS regression, while ˆ αare the residuals in the CS regression. Fama and MacBeth (1973) developed a simplified process. After running the TS regression, they run a CS regression at each time-period to get λ. Ri t=ˆ β0iλt+αit, for i=1, . . . , N. With FM, it is easy to do models in which β change over time. Additionally, it can be used when there is a big cross-section where elements are correlated among them (but not across time). Our estimates for λ and α are the averages across time. Note that if the β are the same over time, λand αestimates are identical to those in the CS regression. ˆ λ=1 T T ∑ t=1 ˆ λtˆ αi=1 T T ∑ t=1 ˆ αit with the following variance for λ: ˆ σ2 λ=1 T2 T ∑ t=1 (ˆ λt−ˆ λ)2. If the estimated β are important determinants of average returns, then the risk premiums should be statistically significant. The Cross-Section standard error formulas are a little bit better than the ones presented before because they include the error of estimating β (see Shanken 1992, for the so-called Shanken correction), although the difference may be very small in practice. 2.2.2. Resampling Techniques: Bootstrap A naive bootstrap procedure for factor models was proposed by Kosowski et al. (2006) and later modified by Fama and French (2010) to sample fund residuals and factor returns jointly in order to preserve cross-correlation among the different portfolios (iid-bootstrapping pairs). This second procedure was used by Sørensen (2009) in order to determine if a certain fund generates alpha (skill). However, in this paper, we are interested in determining whether the model works (all α are jointly zero). We propose a block-bootstrapping pairs scheme to preserve the time correlation of the data of B bootstrap samples of block size b. First Step Estimate benchmark regression models, one for each portfolio. For each portfolio, we save α , β , their corresponding t-statistics, residuals, the estimates of risk factors, and the GRS statistic, according to the time-series regression: Re i,t=αi+ K ∑ j=1 ˆ βi,jfj,t+ei,t, J. Risk Financial Manag. 2020,13, 314 8 of 17 where Kis the number of factors and index iis used to denote the i-th portfolio. Second Step Produce a set of simulation runs (in our case, B = 10,000) which is the same for every portfolio to preserve the cross-correlation of the returns. Draw a vector ˜ Ts of T/b independent components from a discrete uniform distribution on { 1, 2, . . . , T−b+ 1 } . Each component represents the initial time-point for a block of size b. Third Step Use the simulated time indices to build a new series of α -free portfolio returns, which has the property that α is zero. If the factor model adequately describes returns, then the expected value of the intercept is zero. ˜ Re i=f(˜ Ts)ˆ βi+ei˜ Ts. Fourth Step Run the time-Series factor model regression on the artificially constructed returns. Obtain ˆ α , ˆ β and the corresponding confidence intervals. Finally, generate B = 10,000 samples of a GRS Statistic, calculate different percentiles from the bootstrapped distribution, and compare them to the original GRS statistic. Regarding cross-sectional regression, we have used β s and average returns from each bootstrapped sample to determine the significance of the risk factors’ estimates ( λ s). Given that β s are estimated, confidence intervals for λ s present a lot of bias. We have decided to use a reverse bootstrap percentile interval to determine the significance of the factors. The reverse percentile bootstrap interval is not transformation-respecting, neither is it very asymptotically efficient—but it is at least unaffected by bootstrap bias. To be consistent, we have also applied this type of interval to all the bootstrapped distributions. In the empirical application, we have set the number of items per block b= 2 months. We noticed that when we set the number of months of the block to b= 1 (iid-bootstrapping pairs as in Fama and French 2010;Sørensen 2009), the confidence intervals shrink. The process is developed through parallel computing, using several cores to improve calculation time. 3. Results 3.1. Time-Series Regressions In this section, we present the results for the Time-Series regressions of five models, considering several combinations of the statistical factors described before. Other combinations have been considered, but were outperformed by the ones shown below. 3.1.1. Model 1: CAPM In Table 4we present the results of the time-series estimation using only the Market Factor. The dependent variables are the returns of the the 12 portfolios. J. Risk Financial Manag. 2020,13, 314 15 of 17 Table A1. Results for Time-Series estimation of Model 5 (Market, CV and Skew). Estimates t-Statistic Bootstrap CI (2.5%,97.5%) Portfolio Alpha Market CV Skew Alpha Market CV Skew Alpha Market CV Skew Adj. R2 1-1-1 0.001 0.441 0.306 0.175 0.639 9.815 3.889 1.381 −0.001,0.005 0.348,0.563 0.171,0.417 −0.063,0.485 0.702 1-1-2 0.003 0.522 0.234 0.427 2.282 15.469 3.962 4.476 0.003,0.009 0.455,0.593 0.115,0.351 0.257,0.608 0.842 1-1-3 0.002 0.540 0.140 0.651 1.252 10.347 1.528 4.420 0.001,0.009 0.425,0.666 −0.043,0.304 0.349,0.991 0.692 1-2-1 0.003 0.470 0.243 0.216 1.838 12.706 3.749 2.070 0.002,0.008 0.387,0.554 0.102,0.370 0.022,0.428 0.779 1-2-2 0.002 0.558 0.161 0.416 1.080 14.183 2.341 3.743 0.000,0.006 0.460,0.654 0.020,0.277 0.187,0.643 0.797 1-2-3 0.003 0.442 0.140 0.730 1.853 11.680 2.113 6.832 0.002,0.008 0.365,0.526 −0.015,0.255 0.522,0.959 0.769 2-1-1 0.003 0.484 1.267 −0.196 1.913 10.782 16.143 −1.551 0.003,0.010 0.387,0.575 1.125,1.403 −0.442,0.055 0.894 2-1-2 0.003 0.539 1.175 0.187 2.002 14.727 18.352 1.813 0.003,0.008 0.465,0.607 1.022,1.303 −0.022,0.402 0.930 2-1-3 0.005 0.404 1.276 0.840 2.412 7.795 14.082 5.749 0.005,0.013 0.308,0.514 1.094,1.445 0.528,1.176 0.873 2-2-1 0.001 0.427 1.242 −0.298 0.613 10.021 16.643 −2.476 −0.001,0.005 0.339,0.523 1.083,1.394 −0.479,−0.065 0.891 2-2-2 0.001 0.344 1.299 0.446 0.561 5.058 10.897 2.320 −0.002,0.008 0.196,0.498 1.067,1.487 0.141,0.844 0.772 2-2-3 −0.001 0.489 1.034 1.242 −0.598 11.200 13.550 10.084 −0.005,0.001 0.391,0.595 0.900,1.159 1.009,1.510 0.906 Notes. 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