An Empirical Assessment of Characteristics and Optimal Portfolios. Christopher G. Lamoureux and Huacheng Zhang. Abstract

Transcription

1 Current draft: November 18, 2018 First draft: February 1, 2012 An Empirical Assessment of Characteristics and Optimal Portfolios Christopher G. Lamoureux and Huacheng Zhang Key Words: Cross-section of stock returns; stock characteristics; optimal portfolios Abstract We empirically link measurable characteristics to CRRA investors first-order conditions. We mitigate overfitting with a loss function that is more concave than the investor s utility function. Most of the utility gains derive from complementarities amongst the characteristics. Residual volatility and size complement one-another in-sample, small stocks become more attractive when we can also underweight high residual volatility stocks. Adding momentum to the other characteristics lowers portfolio sampling variance out-of-sample reducing overfitting. Characteristics allow risk averse investors to optimally shift portfolio variance outside the span of the traditional factors. As risk aversion increases investors take refuge in the value and market factors although over 40% of the the most risk averse investor s optimal portfolio return variance is orthogonal to these factors. Department of Finance, The University of Arizona, Eller College of Management, Tucson, 85721, , lamoureu@ .arizona.edu. Institute of Financial Studies, Southwestern University of Finance and Economics, Chengdu, China, zhanghuacheng1@gmail.com. We are grateful to Michael Brennan, Scott Cederburg, Kei Hirano, Zhongzhi Song, Ross Valkanov, and Jialing Yu. As well as seminar participants at: Southwestern University of Finance and Economics, the University of Arizona, the University of Western Ontario, and the China International Conference. The current version of this paper may be downloaded from lamfin.arizona.edu/rsch.html.

2 1. Introduction Firm characteristics can predict future stock returns in the cross-section. The cross-section of expected stock returns may also have a seasonal component. This paper analyzes the optimal implications of these facts to a risk-averse expected utility maximizing investor. The efficacy of characteristics in this context is an open question since research on the relationship between characteristics and future returns has focused almost exclusively on alpha: expected return conditional on the Fama-French-Carhart factors. Our alternative loss function is motivated by the fact that many characteristic-based alpha generating strategies entail high volatility, fat tails, and/or negative skew (Barroso and Santa-Clara 2015, Kadan and Liu 2014). We build on the structure of Brandt, Santa-Clara and Valkanov (2009) (BSV) to analyze their finding that risk averse investors can significantly increase expected utility by allowing portfolio weights to depend on measurable characteristics. Most of the gains from linking portfolio weights to characteristics derives from complementarities amongst the characteristics. By conditioning on both size and momentum, small stocks that are small because they have done poorly are avoided, and large stocks that have grown relative to peers are neither aggressively shorted nor overweighted a result that cannot be achieved by using either characteristic in isolation. Before assessing the role of characteristics in forming optimal portfolios we confront the problem of estimation risk inherent in selecting optimal portfolios using sample data. While the BSV algorithm is parsimonious and does not require a first stage estimation of inputs (e.g., moments), estimation risk is severe and can be mitigated by obtaining portfolio weights using a criterion function that has more curvature than the investor s utility function. For example, an exponential utility investor with coefficient of relative risk aversion, γ, of 2 reduces estimation risk significantly by selecting portfolios using a more concave exponential loss function, with γ, = 3. Similarly, an exponential utility investor with a γ of 9 reduces estimation risk significantly by selecting portfolios using an exponential loss function with γ = Characteristics also interact with estimation risk. Overfitting problems increase in the number of characteristics used, although including momentum in the characteristic set mitigates overfitting amongst the other characteristics. Our results on estimation risk complement the well-known results in the literature that optimizing over portfolio weights by treating sample estimates as population parameters produces portfolios which are dominated out-of-sample by naive portfolios. DeMiguel, Garlappi, and Uppal (2009) provide a summary of many of these results. Jagannathan and Ma (2003) show that 1 To be clear: we assume an investor with an exponential utility function defined by the coefficient γ. The investor and the econometrician know this utility function: it is a primitive condition (and time-invariant). But the choice of which loss function to use in selecting optimal portfolios using sample data is not constrained by the investor s γ. To distinguish these two constructs in the text we use γ to describe the investor s utility function, and γ to represent the coefficient used in obtaining optimal portfolios. 1

3 constraining weights to be non-negative in standard mean-variance analysis has salutary effects. They demonstrate that there is a duality between constraining the weights and shrinking moment estimators in this context. Our analysis is purely empirical there is no statistical model of why conditioning on characteristics may generate portfolios that are more attractive than standard alternatives to risk-averse investors. Decoupling the curvature of the loss function used to select portfolios from the investor s utility function provides an additional degree of freedom that can be used to manage the overfitting that occurs with portfolio optimization. All of our statistical analysis is conducted out-of-sample, after first adjusting for estimation risk. We also compare a rolling protocol using only the most recent 180 months of data with an updating protocol where the entire history is used, for example 660 months in our last year s optimization. Updating is preferred to rolling, but the difference is not large. This suggests that the nature of the relationships between characteristics and future portfolio returns is not gradually evolving over time, and also that 180 months is a sufficient sample size to estimate these relationships reliably. We compare portfolios using their out-of-sample certainty equivalent returns. Thus our statistical loss function is based on expected utility. As such we do not take a stand on the nature and sources of systematic risk. By contrast using alpha as the metric for evaluating characteristics predictive content implies that we know the sources of systematic risk. Furthermore, alpha is not an appropriate measure of performance when the objective function in nonlinear or when portfolio returns are not normally distributed (Leland 1999; Ferson 2013; and Broadie, Chernov, and Johannes 2009). We explore why and how optimal characteristic-based portfolios are preferred to indices and equally-weighted portfolios. Because we do not impose any parametric structure on the ultimate loss function, we consider the first four moments of the return distribution, as well as loadings on the four Fama-French-Carhart factors. Over 60% of the γ 2 investor s optimal portfolio return variance is orthogonal to the four Fama-French-Carhart factors. Average same-month return has the largest incremental effect on this orthogonalization, but even when it is removed from the optimal characteristic set, orthogonality remains more than 40%. This paper provides an empirical link between measurable characteristics and the first-order conditions of CRRA investors. We examine how investors use these characteristics to build portfolios that dominate market indices within the metric of expected utility. 2 Investor preferences vis-à-vis the factors vary across the factors. These nonlinear effects are not detectable using alpha- 2 Kozak, Nagel, and Santosh (2018a) refer to factor models, such as the Fama-French-Carhart four factor span that we use as a frame in this paper, as reduced form models since they lack a theoretical foundation that relates investor beliefs and preferences to the economy s stochastic discount factor. Here we empirically link preferences to stock characteristics under the belief structure that past is prologue correcting for overfitting. 2

4 based metrics. The risk tolerant, γ 2 investor s optimal portfolio has only a 2% correlation with the market. As risk aversion increases, investors seek refuge in the value and market factors, whereas optimal exposure to momentum is invariant to risk aversion. The γ 2 investor loads positively on SMB, the γ 5 investor has no SMB exposure, and the γ 9 investor has a significant negative loading on this size factor. Nevertheless, all of these investors have a significant negative coefficient linking portfolio weights to the characteristic log size; highlighting the complex complementarities between the characteristics. All investors significantly down-weight stocks with higher residual standard deviations. In itself, this results in portfolios with large positive exposure to the value factor, HML, and large negative exposure to SMB. Conditioning on both log size and residual standard deviation amplifies the HML exposure. SMB exposure depends on the relative sizes of the coefficients linking portfolio weights to these characteristics. Investors optimally use the characteristic momentum to gain positive exposure to the momentum factor. This characteristic is especially important since conditioning on size and the book-to-market ratio generate portfolios with negative loadings on the momentum factor. Conditioning on the characteristic momentum also significantly shifts the portfolio away from the span of the four Fama-French-Carhart factors. Investors optimally use the volatility characteristics beta and residual standard deviation to reduce exposure to the market factor, gain positive exposure to HML, negative exposure to SMB, and to shift portfolio away from the span of the factors. The characteristic average same month return allows the most risk averse investor to increase the portfolio expected return largely by shifting the portfolio (variance) away from the span of the factors. Although conditioning only on this characteristic produces significant negative SMB and HML loadings, and a significantly higher beta than the market. No investor optimally uses the characteristic book-to-market ratio. It is redundant, adding no useful information beyond what is contained in the other characteristics especially log size and residual standard deviation. Redundancy is penalized since adding even a redundant characteristic results in overfitting in-sample, which manifests in higher sampling variation out-of-sample. Thus, while conditioning on the book-to-market ratio adds to in-sample utility it increases overfitting, and does not survive out-of-sample optimization. The large gains in expected utility afforded by conditioning on characteristics are achieved only by using them jointly. There is a vast body of research that links observable characteristics to future stock returns. Much of this was univariate in nature, and much of it has been shown to be ephemeral (McLean and Pontiff 2016), and not immune to data mining concerns (Linnainmaa and Roberts 2018; and Hou, Xue, and Zhang 2017). Cochrane (2011), Lewellen (2015), and Freyberger, Neuhierl, and Weber (2018) highlight the fact that most of the empirical analysis on 3

5 characteristics does not address the multidimensional aspect of characteristics. This is especially important in light of several recent studies that show decreasing predictive content of individual characteristics. Kogan and Papanikolaou (2013) argue that firms with higher idiosyncratic volatility have higher growth opportunities, and lower risk premia. This hypothesis can explain why a portfolio of low volatility stocks has a high alpha since their growth opportunity factor is not spanned by the Fama-French-Carhart factors. It does not necessarily imply our finding that risk averse investors (at all levels of γ) optimally shift away from high idiosyncratic volatility stocks provided that this shift complements changed exposures to other characteristics especially size and average same-month return. Our results suggest that there are latent factors and that the Fama-French- Carhart factors: HML, MOM, and even the market are not priced risk factors in the traditional sense. Leverage via short selling is a critical feature of all optimal portfolios. Even relatively risktolerant investors use characteristics to build significant leverage. For example, the γ 5 investor optimally uses 150% leverage to increase the portfolio mean and standard deviation beyond those of market benchmarks. The γ 9 investor, whose certainty equivalent return on the market is only 3 basis points per month, optimally uses characteristics to build a portfolio whose mean return is significantly higher than the market s, while its interquartile range is significantly less, and the standard deviation not significantly different. The Sharpe ratio of the γ 9 investor s optimal portfolio is significantly higher than the market portfolio, as is its kurtosis. While the market portfolio has a significant negative skew, this portfolio is symmetric. Our statistical analysis is computationally intensive. We analyze out-of-sample results from a non-linear loss function, which requires several hundred thousand non-linear optimizations for each combination of characteristic set, protocol, and γ. This restricts the number of characteristics that we can consider. 3 This (higher dimensional loss function) differentiates this paper from some of the recent literature on the link between characteristics and future returns. Freyberger, Neuhierl, and Weber (2018) examine expected returns. Kozak, Nagel, and Santosh (2018b) point out that this focus on risk premia is not tantamount to risk prices as characteristics that are correlated with pricing factors can explain cross-sectional variations in expected returns. DeMiguel, Martín-Utrera, Nogales, and Uppal (2018) also consider optimal portfolios, but they use a mean-variance loss function, which affords parametric structure at the cost of some generality. We rely on their results which suggest that there is a lot of redundancy in the plethora of characteristics and focus on those that have been directly linked to prospective factors. Ex- 3 Generating the out-of-sample bootstrap results for a typical case takes 26 hours of CPU time on a 3.7 GHz Xeon processor. We consider 658 cases. However, since the bootstrap draws are independent (unlike in Markov Chain Monte Carlo, for example), the algorithm is easily parallelizable. 4

6 tending what can be gleaned from mean-variance analysis, we find that investors do not use characteristics to build portfolios with positive skew. Although market benchmarks have a significant negative skew and optimal portfolios are symmetric. As risk aversion increases, optimal portfolios shift mass from the flanks of the return distribution to the tails. Whereas the standard deviation of the γ 9 investor s optimal portfolio is not significantly different from the market, its interquartile range is significantly lower. Since we find that (average) same-month return is a useful characteristic we explore the possibility that exposures to the Fama-French-Carhart factors vary across the 12 months in the year. We analyze this by considering the timing bias in portfolio alphas estimated under the assumption that factor exposures are constant across the 12 months. Heston and Sadka (2008) revive a fact that Jegadeesh (1990) unearthed: stocks that with high (low) returns last May, for example, are predictably more likely to have high (low) returns this May. Keloharju, Linnainmaa, and Nyberg (2016) suggest that this may reflect the cumulative effects of many factors that have a monthly seasonal pattern. Adding this characteristic to momentum, beta, size, and residual volatility significantly increases the portfolio certainty equivalent return for relatively risk-tolerant investors. This is achieved by a statistically significant reduction in the percentage of portfolio variance spanned by the four Fama-French-Carhart factors (from 53 to 30%), along with a significantly higher mean return and Sharpe ratio. Adding average same-month return does not significantly alter the higher moments of the resulting optimal portfolio return distribution. In spite of this, many of the recent papers on factor selection do not include same-month return as a prospective characteristic Portfolio Selection 2.1 Algorithm In Brandt, Santa-Clara, and Valkanov s (2009) algorithm, the vector θ is estimated to maximize a concave loss function over T periods: T 1 (1 + r p,t+1 ) 1 γ max θ 1 γ t=0 ( ) 1 T by allowing portfolio weights to depend on observable stock characteristics: N t ) r p,t+1 = (ω i,t + 1Nt θ x i,t r i,t+1, (2) i=1 4 Feng, Giglio, and Xiu (2017), Freyberger, Neuhierl, and Weber (2018), and DeMiguel, Martín-Utrera, Nogales, and Uppal (2018) do not include this characteristic in their analysis. Kozak, Nagel, and Santosh (2018b) do include five-year average same month return (as we do), and find that it has one of the four largest t statistics in their Table 1, p. 29. (1) 5

7 where: x i,t is the K-vector of characteristics on firm i, measurable at time t; ω i,t is the weight of stock i in the (value-weighted) market portfolio at time t; and N t is the number of stocks in the sample at time t. 5 A difference between our use of this algorithm and the presentation in Brandt, Santa-Clara, and Valkanov is that we do not identify the γ used to generate a viable portfolio strategy in Equation (1) with a specific investor s preferences. Instead, we consider an investor whose utility function takes the same functional form as (1) indexed by γ (the investor s risk-aversion coefficient which is pre-determined and fixed). This investor treats γ as a choice variable as a means of managing estimation risk or overfitting. 2.2 Data and specifications We use the following characteristics: momentum (M), book-to-market (V), log size (S), beta (β), market model residual standard deviation (σ ɛ ), last year s same-month return, (r t 12 ), and the average same-month return over the preceding five years, (r t 12j ). Momentum is measured as the stock s compounded return from month t 13 though t 2. Market capitalization is the market value of all of a company s outstanding shares (aggregated across all classes for companies with multiple share classes) at time t 2. Book value is obtained from the Compustat database for the most recent fiscal year-end between t 6 and t 18. Letting B be one plus the ratio of book value to market capitalization, the book-to-market ratio is the natural log of B. Size is the natural log of equity market capitalization. Beta and the residual standard deviation are obtained by regressing monthly returns from months t 60 through t 1 on the CRSP value-weighted index. All characteristic and return data is drawn from the merged CRSP Compustat file on WRDS. To be eligible for inclusion in the sample in month t, the stock must have no missing returns in the CRSP database for the previous 60 months, and it must have a positive book value in the Compustat database for a fiscal year-end between t 6 and t 18. We obtain the US GDP deflator from the Federal Reserve (FRED) and use this to construct a minimum size criterion of $50 million in January 1990 dollars. Stocks whose market capitalization is less than this inflation-adjusted size criterion are excluded from the sample. This excludes stocks with market capitalization less than $11.5 million in January 1960, and $93.3 million in December We next exclude the smallest 10% of stocks that meet all inclusion criteria prior to January 1978, when the first Nasdaq stocks enter the sample, and the smallest 20% afterwards. If the stock return is missing in month t, we look to the CRSP delisting return. If that is missing, we substitute 30% for NYSE and AMEX listed stocks and 50% for Nasdaq stocks. There are 395 (exclusively New York Stock Exchange) stocks in the sample in January We condition only on information that is available to investors at the time the portfolios are formed. This avoids the overconditioning bias analyzed by Boguth, Carlson, Fisher, and Simutin (2011). 6

8 There is a jump in the sample size in August, 1967 (from 675 to 875) when stocks listed on the American Stock Exchange are eligible for inclusion in our sample. The largest jump in sample size is in January 1978 (from 1,000 to 1,419 stocks) when Nasdaq stocks enter our sample. The maximum number of stocks is 2,291 in April, There are 1,728 stocks in our sample in the last month, December We normalize and standardize the characteristics ensuring that optimal portfolio weights will sum to unity for any value of θ. This also means that the characteristics are observationally equivalent to shrinkage values. For example, let β be a stock s OLS beta. Consider a priorweighted beta, such as β S =.5 β The normalized β S are identical to the normalized β. A single observation (Ψ i,t ) comprises stock i s return in month t, r i,t, as well as the vector of characteristics, measurable at month t 1, for stock i, i = 1,..., N t. We consider 47 characteristic sets, including each of the six characteristics as a singleton, and all seven variables together (last year s same-month return is only used in conjunction with the five-year average same-month return). We consider 14 values of γ : 2,..., 13, 16, 22. The out-of-sample period comprises the 41 years We obtain the θ vector using all available data through the end of year T for T = 1,, 41 under the updating protocol. Under the rolling protocol we use only the previous 180 months at the end of each year (so in Year 1, the two protocols produce the same θ-vector). This θ-vector is used to construct the out-of-sample portfolio in each of the next 12 months. (Since the portfolio is defined by θ, there is no restriction on the available set of stocks in each of the out-of-sample months.) The last out-of-sample year is 2015, so the last optimization/θ estimation (applied to the 12 months of 2015, out-of-sample) uses 660 months under the updating protocol, and the 180 months from 2000 through 2014 under the rolling protocol. This means that we optimize expected utility (and estimate the θ vector) from each of the 658 cases 41 times at the end of each in-sample period, under both protocols (a total of 1,316 cases). 2.3 Empirical design The sensitivity of expected utility to in-sample overfitting in portfolio selection suggests that the statistical analysis of optimal portfolios be conducted on an out-of-sample basis. Our empirical design fixes the investor. This investor evaluates the algorithm with a particular set of tuning parameters (protocol, set of characteristics, and γ ) as a feasible portfolio rule. Our interest is in statistical comparisons across portfolios generated by various sets of tuning parameters from a specific risk-averse investor s perspective. We use a bootstrapping-out-of-sample procedure to construct sampling distributions of the functions of interest, such as certainty equivalent, portfolio loading on factors, portfolio skew, etc. 7

9 The data in month t consist of the N t vectors Ψ i,t, for i = 1,..., N t and t = 1,..., T = 672. A bootstrap draw resamples (with replacement) N t vectors from Ψ i,t in all T months. Thus each bootstrapped sample consists of the same number of observations in each period as the original sample, and the calendar structure of the original data is preserved. The latter is important because we consider the possibility that the 12 months of the year are important characteristics. Once we have a bootstrap sample we maximize (1) at the end of each year: 16,..., 56, for a set of tuning parameters. In this way we have 41 feasible portfolio rules which we use to construct the out-of-sample optimal portfolio returns in each of the next 12 months, yielding 492 monthly feasible portfolio returns. We take 10,000 bootstrap samples to estimate the sampling distributions for all 1,316 unique portfolios. Brandt, Santa-Clara, and Valkanov (2009) show that their algorithm can be cast as a gmm estimator (i.e., the solution of the first-order condition). Since it is also a utility function it is just-identified, and the θ vector is pivotal and asymptotically normal. Therefore the bootstrap distribution of θ is valid. The bootstrap produces non-parametric Monte Carlo sampling distributions of out-of-sample optimal portfolio properties. These distributions are standard and valid since the portfolios are feasible. The portfolios are formed based only on information that was available at the time of formation under each draw. Since this is a bootstrapped nonparametric Monte Carlo there is no look ahead bias in terms of simulation moments. (The draws used to obtain the portfolio weights at time t depend only on observable data to time t. The same is true of the optimization.) Since the θ vector is pivotal, the bootstrap affords measurement of the bias in the sample θ estimates (MacKinnon 2002). This bias increases in the sample size and dimensionality of the characteristic space. 3. Algorithm efficacy and estimation risk Panel A of Table 1 shows sampling distribution properties of the bootstrapped out-of-sample portfolio certainty equivalent returns (in basis points per month) for an investor with an exponential utility function with γ = 2. The choice variables are the protocol (updating and rolling), characteristic set, and concavity of the loss function (γ ). The Sample column shows the certainty equivalent for the out-of-sample returns of optimal portfolios in the original data. The median certainty equivalent for the value- and equally weighted indices are 88 and 106 basis points per month respectively. We will consider that Portfolio A is statistically preferred to (dominates) Portfolio B if A s 2.5%ile (out-of-sample) certainty equivalent return exceeds B s 97.5%ile certainty equivalent return. Thus, for this investor the equally weighted index statistically dominates the value-weighted index. In recognition of estimation risk, we rank portfolios by their 2.5%ile certainty equivalents. 8

10 The γ 2 investor s global optimal portfolio uses the updating protocol, γ = 3, and the characteristics: momentum, log size, beta, residual standard deviation, and average same-month return. None of the optimal characteristic-based portfolios generated using γ = 2 is preferred to the equally weighted index. Furthermore all of the portfolios that condition on five or more characteristics have a 2.5%ile certainty equivalent of -100%. That is, in at least 250 of the 10,000 bootstrapped samples, the return on the portfolio fell below -100% in at least one month in the out-of-sample period. This highlights the overfitting problem that plagues portfolio optimizers. Table 1 shows the certainty equivalent return for the portfolio selected using γ = 2 and the set of characteristics that produces the global optimal portfolio: its interquartile range is -10, basis points per month. While this is not statistically significantly different from the global optimal portfolio, it also does not dominate the indices, and highlights the use of the 2.5%ile of the distribution to rank the portfolios. 6 Table 1 also shows that while the use of γ = 3 is optimal, the cost of using a higher value of γ is relatively small. Using γ = 4, the updating protocol, and the optimal five characteristics produces a portfolio with a similar certainty equivalent distribution to the globally optimal portfolio. When the rolling protocol is used the optimal value of γ is 5, and the optimal characteristic set contains all seven characteristic variables. This portfolio s certainty equivalent is not significantly different from the global optimal, but its mean is 37 basis points per month lower, and its sampling standard deviation is almost three times larger than the global optimal portfolio s. Panel A shows the results of using each of the characteristics as a singleton in optimal portfolio selection. The only one that can be used in isolation to produce a portfolio that is significantly preferred to the indices is the (previous five-year) average same-month return. We evaluate the marginal effects of each characteristic by comparing the certainty equivalent returns of the optimal portfolio formed by excluding it from the optimal set. Removing momentum leads to a dramatic increase in overfitting. The certainty equivalent of the optimal portfolio formed from the other four characteristics is not significantly different from the global optimal portfolio, but the 2.5%ile is -100% per month, as there is a thirtyfold increase in the bootstrap standard deviation. Each of the three characteristics size, residual standard deviation, and same-month return add significantly to the certainty equivalent. The global optimal portfolio s 2.5%ile certainty equivalent is 288 basis points per month. When average same-month return is removed from the optimal characteristic set, the resultant portfolio s 97.5%ile certainty equivalent is 246 basis points per month. Removing beta from the optimal characteristic set has a more modest effect on the certainty equivalent 6 The fact that the bootstrap standard errors are so large when γ = γ suggests that a bias adjustment technique such as bootstrap aggregation (or bagging ) might improve matters in terms of selecting optimal portfolios. Experimentation with both bagging and bragging suggests that the methods produce small gains in certainty equivalent that are orders of magnitude smaller than the gains achieved with simply using a higher γ to select the optimal portfolio. The numerical demands of such techniques are also overwhelming. 9

11 distribution. Adding the book-to-market ratio to the optimal characteristic set has the adverse effects of lowering the mean of the certainty equivalent s sampling distribution and increasing its standard deviation. Nevertheless, the table shows that adding this characteristic does not lead to a statistically significant change in the certainty equivalent. Panel B of Table 1 reports certainty equivalent returns for the γ 5 investor. This investor views the two benchmarks as statistically equivalent, and is able to increase the mean certainty equivalent return from 61 to 164 basis points per month by maximizing over the three dimensions. In this case the global optimum uses: the updating protocol; γ = 9; and characteristic set: momentum, size, average same-month return, and residual standard deviation. Unlike for the γ 2 investor, the use of γ = γ produces an optimal portfolio that statistically dominates the benchmarks, although this portfolio is statistically dominated by the global optimal portfolio. The optimal portfolio produced under the rolling protocol for the γ 5 investor is obtained with γ = 11, and the three characteristics: momentum, size, and residual standard deviation. While this portfolio statistically dominates the benchmarks, it, in turn, is statistically dominated by the global optimal portfolio. The comparative statics in Panel B show that removing momentum from the optimal set increases the certainty equivalent s bootstrap standard deviation from 11 to 35 basis points per month. The resulting portfolio is not significantly different from the global optimal, nor does it statistically dominate the benchmarks. Removing any of the other three characteristics in turn results in a significant reduction in the certainty equivalent. The biggest single effect is removing residual standard deviation, which produces a portfolio that does not statistically dominate the benchmarks. The optimal portfolio using the same characteristic set as optimal for the more risk tolerant investor (i.e., adding beta to the characteristic set) has virtually no effect on the mean certainty equivalent, but the bootstrap standard deviation increases slightly resulting in a reduction in the 2.5%ile. Table 1, Panel B also shows that the only characteristic that the γ 5 investor can use as a singleton to beat the benchmark is the residual standard deviation. Most of the gains in utility are achieved through the complementary effects amongst the four characteristics. Panel C of Table 1 shows certainty equivalent returns for various portfolios from the perspective of the γ 9 investor. This investor prefers the value-weighted index to the equally-weighted index, and is indifferent between earning a risk-free return of 1% per year and holding the market portfolio. By using the updating protocol, γ = 16, and characteristics: momentum, size, and residual standard deviation, this investor can increase this to 7.5% per year, with 95%ile confidence band: [5.3%, 9.5%]. For this investor, the rolling protocol is optimized with the same γ and characteristic set as the updating protocol, and produces an optimal portfolio that also 10

12 dominates the market and is not significantly different from the global optimal portfolio. Were this investor to use γ = γ, the optimal characteristic set would not include momentum, and instead would include the book-to-market ratio. Momentum s role in mitigating estimation risk is reduced in this case relative to the more risk-tolerant investors. This is partly because average same month return is not included in the optimal characteristic set. This was also suggested by the fact reported in Panel B, that for the γ 5 investor using γ = 5, the optimal characteristic set is: the book-to-market ratio, size, and residual standard deviation. The comparative statics in Panel C show that residual standard deviation is the only characteristic whose removal from the optimal set results in a significant reduction in certainty equivalent return. Nevertheless the γ 9 investor can not use any of the characteristics as a singleton to significantly increase expected utility. Overall the relationship between the updating and rolling protocols means that slowly evolving relationships between the (multivariate) return distribution and the characteristics are not driving estimation risk in the context of the optimal set of characteristics. Nor are there important structural breaks in these relationships. If there were, then the rolling protocol would yield portfolios that outperform those selected using the updating protocol. The rolling protocol s underperformance relative to updating also suggests that in the multivariate context there is no evidence of a diminution of the predictive content of the characteristics over time, as in McLean and Pontiff (2016). Although McLean and Pontiff evaluate long-short portfolios using one characteristic at a time, and we show below that the evidence is different for singleton characteristic sets than for multivariate sets of characteristics. Whereas updating is always preferred to rolling, generally similar results obtain under the two protocols. This implies that 15 years of data (180 months) is sufficient to measure the relationships between the characteristics and future returns in a portfolio context. Estimation risk tends to increase in the dimensionality of the characteristic set, with the exception that adding momentum always mitigates estimation risk. In what follows we will consider how momentum affects the factor exposure and moments of the optimal portfolios to consider its effect on estimation risk in more detail. It is never (globally) optimal to include last-year s same-month return in the characteristic set. This is because in the context of the five-year average same-month return last-year s samemonth return adds noise resulting in increased estimation risk. This noise more than offsets the value of any incremental information contained in this characteristic. The optimal portfolio also never conditions on the book-to-market ratio. We will show below that this is not because these portfolios are not exposed to the (value) HML factor, but instead that size and residual standard deviation provide enough flexibility to allow optimal exposure to the value factor. In other words, 11

13 as with last year s same-month return, whatever useful information the book-to-market ratio contains is more than offset by the estimation risk that it adds to the characteristic set. 4. Characteristics and portfolios Table 2, Panel A reports bootstrap properties for the θ coefficients on each characteristic for select portfolios produced under the updating protocol using γ = 3. Each bootstrap sample contains 41 sets of θ coefficients from the beginning of each out-of-sample year. This table averages these annual coefficients across the years. The first portfolio in the table is the global optimal portfolio for the γ 2 investor. As in Table 1, we remove each characteristic one-at-a-time to measure the effect on the coefficients of the remaining characteristics. The coefficients on momentum and average same-month return are positive, those on log size and residual standard deviation are negative, and the coefficient on beta is not significantly different from zero in the optimal portfolio for the γ 2 investor. Since the periods used to evaluate portfolio properties and performance are not the same as used to estimate θ, there are two types of complementarity that we can identify. First, complementarity can be amplifying adding a coefficient causes the θ on the incumbent characteristic(s) to increase in absolute value. Amplifying complementarity arises in sample. Alternatively, one characteristic may complement another by refining the nature of the stocks appeal. This complementarity is only evident out-of-sample. For example, consider momentum relative to the characteristics: log size, β, residual standard deviation, and average same-month return. Table 2 shows that when γ = 3, adding momentum to this set of characteristics does not significantly change any of these characteristics θ coefficients. Nevertheless, as we have seen, adding momentum increases the γ 2 investor s median certainty equivalent from 224 to 336 basis points per month, while reducing its sampling standard deviation from 3,374 to 107. Log size shows some complementarity with beta. The coefficient on beta is positive and significant when size is excluded from the characteristic set. Although removing beta has little effect on size s coefficient. The comparative statics relative to the γ 2 investor s optimal portfolio suggest very little in-sample complementarities or substitutabilities with respect to beta. Adding book-to-market to the optimal set has very little effect on the other θ characteristics, as well. However, amplifying complementarities become evident in comparisons between the optimal portfolio s θ values to those from portfolios formed with singleton characteristics. Consider beta as a singleton characteristic. In this case the coefficient is negative and significant. When combined with residual standard deviation, the coefficient on beta is insignificant. Size is complementary to residual standard deviation. In isolation the 95%ile band for the size θ is [ 3.5, 0.8]. When residual standard deviation is added this becomes [ 6.9, 3.6], and 12

14 in the optimal portfolio this is [ 14.1, 8.1] significant increases in both cases. The complementarity between size and residual standard deviation is symmetric. Residual standard deviation s θ coefficient s 95% band when it is used as a singleton is [ 3.1, 1.2]. Adding log size makes this [ 9.6, 5.4] which is not significantly different from this coefficient in the global optimum. By contrast, the coefficients on momentum and average same-month return in the cases where each is used as a singleton are not significantly different from these coefficients in the optimal portfolio. Table 2 Panel B shows the sampling correlation matrix (across the 10,000 bootstrap samples) of the average θ coefficients for the γ 2 investor s optimal portfolio. It shows the high (negative) correlation between the coefficients on beta and residual standard deviation, and to a lesser degree that between beta and log size. The coefficients on momentum and log size are also negatively correlated. Panels C and D of Table 2 report properties of the bootstrap distribution of the θ coefficients from the γ 5 and γ 9 investors global optimal portfolios, respectively. The optimal characteristic set for all three investors includes momentum, log size, and residual standard deviation. The absolute values of the θ coefficients that link portfolio weights to these three characteristics are monotonically decreasing in risk aversion. Nevertheless, the signs on all three are the same for the three investors. All are statistically significant. 5. Properties of optimal portfolios 5.1 Factor loadings and variance decompositions Table 3 contains the bootstrap distributions of the coefficients from a linear projection of returns on the Fama-French-Carhart factors and variance decomposition with respect to these four factors for benchmarks and various optimal portfolios from the updating protocol. Naturally, both benchmark portfolios loadings on the market portfolio are tightly distributed near The γ 2 investor s optimal portfolio has a significantly lower market loading: 0.5, [0.3, 0.8] than the benchmarks. Furthermore, whereas the market accounts for all of the return variance of the valueweighted index and over 80% of equally-weighted index variance, the market factor contributes less than 6% of this optimal portfolio s return variance. The table shows the effect on these metrics of removing one characteristic at a time from the γ 2 investor s optimal set, using γ = 3, as well as the direct effect of each characteristic when 7 The table does not report the six covariance terms in the variance decompositions. These are often non-trivial. For example, the covariance between the excess return on the market and the return on SMB accounts for some 14% of the variance of the equally-weighted benchmark. These covariance effects are often negative: the covariance between the excess return on the market and HML s return accounts for -8% of the variance of the equally-weighted portfolio. 13

15 used as a singleton. Momentum is directly linked to the momentum factor, and none of the other optimal characteristics alters exposure to this factor. Removing momentum from the optimal characteristic set lowers the the momentum factor s contribution to portfolio variance from 20% [13%, 28%] to 2% [0%, 3%], and the loading drops from 1.4 [1.2, 1.7] to -0.4 [ 0.6, 0.2]. Removing momentum from the optimal set has no effect on the loadings or variance effects of the other 3 factors. Nevertheless, the characteristic momentum is not entirely absorbed by the momentum factor. When used as a singleton with γ = 3, it shifts 17% [13%, 28%] of the portfolio variance outside the span of the four factors. This manifests in the mean return as a significant negative alpha of -16 [ 27, 3] basis points per month. Using momentum as a singleton also generates a portfolio with a significant negative loading on HML, whereas this is significantly positive in the γ 2 investor s optimal portfolio. Adding log size to the other four optimal characteristics shifts the portfolio away from the market factor. This shift manifests in both mean returns: the loading on the market drops from 1.4 [1.2, 1.5] to 0.5 [0.3, 0.8]; as well as the variance: the market s variance contribution falls from 22% [17%, 28%] to 3% [1%, 6%]. Without size in the characteristic set, using γ = 3, the portfolio has a significant negative loading on SMB of -1.2 [ 1.6, 0.9]. The γ 2 investor s global optimal portfolio has a significantly positive loading on SMB, 1.0 [0.6, 1.5]. Adding size to the other four optimal characteristics also shifts the portfolio toward the HML factor. Conditioning on the other four characteristics generates a portfolio with no significant exposure to HML, the γ 2 investor s global optimal portfolio has a significantly significant positive loading on HML of 1.9 [1.5, 2.3]. HML contributes 15% of this portfolio s variance, whereas excluding size from the characteristic set generates a portfolio that is unrelated to the value factor. Neither beta nor the residual standard deviation has any significant marginal effect on factor exposures in either the mean or variance. By contrast, average same-month return shifts the portfolio away from the four factors. Adding this characteristic to the other four with γ = 3 significantly increases portfolio alpha from 123 [82, 171] to 296 [223, 381] and significantly lowers the span of the fours factors from 53% [47%, 59%] to 30% [24%, 38%]. The only factor whose contribution is significantly lowered by adding average same-month return is momentum, from 38% [31%, 46%] to 20% [13%, 28%]. Adding the book-to-market ratio to the five optimal characteristics, using γ = 3 has no significant effects on the portfolio s mapping onto the space spanned by the four factors. When any of the 6 characteristics is used in isolation, the portfolio s correlation with the market drops significantly. Using average same-month return as a singleton characteristic increases the orthogonal portion of the portfolio returns from 1% of the value-weighted benchmark (i.e., the 0-θ portfolio) to 54% [47%, 59%]. The loadings on SMB and HML become significantly neg- 14

16 ative, whereas these are positive in the equally-weighted index as well as the γ 2 investor s global optimal portfolio. Conditioning on either of the variance characteristics significantly reduces the market loading and also results in a negative loading on SMB. Both SMB and HML are significant contributors to portfolio return variances in these cases. Conditioning on only the book-to-market ratio results in a positive loading on HML but also a negative loading on momentum. By contrast, conditioning only on the residual standard deviation generates a portfolio that loads positively on HML without loading negatively on momentum. Using residual standard deviation in isolation also shifts the portfolio significantly away from the market, and increases the portfolio s correlations with SMB and HML. Table 3 also shows that the span of the Fama-French-Carhart factors increases in investor risk aversion (γ). This is mostly concentrated in the market and HML factors. The Fama-French- Carhart factors span 55% [53%, 59%] of the γ 9 investor s optimal portfolio (using γ = 16). This increase relative to the γ 2 investor s optimal portfolio comes through reduction in the portfolio variance and increased relative exposure to the market (35%) and HML (41%). The factor loading on SMB is positive for the γ 2 investor, insignificantly different from zero for the γ 5 investor and significantly negative for the γ 9 investor. Both the momentum and HML factor loadings decline monotonically in γ. The loading on momentum drops significantly from 1.4 [1.2, 1.7] for the γ 2 investor to 0.6 [0.5, 0.7] for the γ 9 investor. The HML loading falls significantly from 1.9 [1.5, 2.3] to 1.0 [0.9, 1.1] as γ increases from 2 to 9. Overall our results on the inability of the Fama-French-Carhart factors to span the return space of the optimal portfolios is consistent with Kozak, Nagel, and Santosh (2018b, p. 40), that characteristics-sparse models do not adequately describe the cross-section of expected stock returns. 5.2 Unconditional return distributions Table 4 shows sampling distributions of portfolio features and moments for the benchmark portfolios, the optimal portfolios for the γ 2, γ 5 and γ 9 investors, as well as other portfolios selected using γ = 3. The γ 2 investor s optimal portfolio s mean return is 549, [465, 645] basis points per month, compared to the equally weighted benchmark s: 133, [131, 135] basis points. This portfolio s interquartile range is 1,648, [1, 416, 1, 917] basis points per month. The equally weighted portfolio has an interquartile range of 618, [603, 633] basis points. The optimal portfolio s (annualized) Sharpe ratio is 1.24, [1.14, 1.35], significantly higher than the equally weighted benchmark s of 0.63, [0.62, 0.65]. 8 In addition to mean, interquartile range, and Sharpe 8 The annualized Sharpe ratios are obtained by taking the ratio of the mean excess return (above the risk-free rate) to the product of the standard deviation and

17 ratio, the tables provide bootstrap properties of the median return, standard deviation, minimum return, percentage negative weights, and robust measures of skewness (S 4 ) and kurtosis (K 3 ): S 4 = µ r.5 σ (3) and K 3 = r+.95 r.05 r +.5 r (4) where: µ is the mean portfolio return over the 492-month out-of-sample period, σ is the return standard deviation, r +.95 is the mean of the highest 5% of returns, r.05 is the mean of the smallest 5% of returns, r +.5 is the mean of the top half of returns, and r.5 is the mean of the bottom half of returns; and r x is the observation corresponding to the x%ile of the return data. 9 Whereas the γ 2 investor s optimal portfolio return distribution is flatter than the equally weighted benchmark s, it is neither more skewed nor more leptokurtic than this benchmark; (although the benchmarks are both significantly negatively skewed and the optimal portfolios are not). Even the in-sample optimal returns of this portfolio are not skewed. The bootstrap mean (95%ile band) of the Pearson skewness coefficient, S 4 on the in-sample returns is 0.00 [ 0.04, 0, 04] Characteristics are not used to generate a positively skewed portfolio. Even the in-sample returns of this optimal portfolio are not skewed. 10 The γ 2 investor s optimal portfolio relies heavily on leverage to achieve a portfolio mean return that is five times higher than the value-weighted benchmark. The portfolio s total short position averages 519%, [444%, 606%]). For one bootstrap sample, we sum the negative weights in each of the 492 (out-of-sample) months. We average this across the 492 months. The table reports the sampling properties of this average across the 10,000 bootstrap samples. Comparing moments in this table highlights the non-linear effects of the utility function. For example, using γ = 3 and dropping momentum from the optimal characteristic set generates a portfolio whose mean return, interquartile range, and annualized Sharpe ratio are not significantly different from those of the γ 2 investor s optimal portfolio. Nevertheless, Table 1 shows that this portfolio s certainty equivalent is: -11%, [ 100%, 3%], per month. Adding momentum to the other characteristics increases the mean minimum return from 80%, [ 116%, 48%] to 53%, 9 The measures of skewness and kurtosis are recommended and discussed in Kim and White (2003). In our bootstrap samples, S 4, the Pearson skewness coefficient is similar to, and more reliable than S 3, the Bowley skewness coefficient-integrated over the tail size. Similarly K 3, the Hogg coefficient, is more reliable than K 4, the Crow and Siddiqui parameter. 10 It is possible that there are differences between the shapes of the in-sample and out-of-sample return distributions, such that the lack of positive skewness reflects non-stationarity in portfolio third moments. That is not the case. Instead the optimal use of characteristics in this algorithm does not entail generating a positively skewed portfolio. 16