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

Transcription

1 Current draft: March 31, 2017 First draft: February 1, 2012 An Empirical Assessment of Characteristics and Optimal Portfolios Christopher G. Lamoureu and Huacheng Zhang Key Words: Stock characteristics; optimal portfolios Department of Finance, The University of Arizona, Eller College of Management, Tucson, 85721, , Department of Finance, Southwest University of Finance and Economics, Changdu, China, We are grateful to Scott Cederburg, Kei Hirano, and Michael Weber. The current version of this paper and the accompanying Internet Appendi can be downloaded from lamfin.arizona.edu/rsch.html. The direct link to the Internet Appendi is:

2 Abstract We use Brandt, Santa-Clara, and Valkanov s (2009) seminal algorithm to assess the role of measurable stock characteristics in optimizing a conve loss function out-of-sample. Not only are characteristics useful in predicting cross-sectional differences in epected returns, they can be used to maimize such a non-linear objective function. Optimal portfolios tend to be positively skewed and fat-tailed. Their eposure to the market factor is significantly less than unity, and they have generally positive eposure to the momentum and value factors. Their eposure to the size factor decreases monotonically in risk-aversion. Optimal portfolios have high (but not maimal) Sharpe ratios, and significantly positive Fama, French, Carhart alphas. Alpha and the utility functions are quite disparate however, as those portfolio with the highest alphas are severely penalized by the objective function s conveity. 70% of the return variance of the most risk-tolerant investor s optimal portfolio is orthogonal to the four Fama, French, Carhart factors. This falls to 55% for the most risk-averse investor s optimal portfolio. As the dimension of the problem grows (in the number of characteristics), this algorithm suffers from estimation risk. We use out-of-sample cross-validation to identify the optimal set of characteristics as well as the optimal degree of conveity to use for a range of power utility functions. A general solution to the problem of in-sample overfitting is to select portfolios using a loss function that is more conve than the actual loss-function (used to evaluate the portfolios). We eamine complementarity and substitutability across characteristics. We find that the book-to-market ratio is redundant in the contet of size and market model residual standard deviation. The latter two characteristics fine-tune the portfolio s eposure to HML more accurately than the book-to-market ratio. Momentum is especially useful as a complement to size and residual standard deviation. A large stock with high momentum is more attractive than a generic large stock. In isolation beta appears to be unambiguously bad, but in combination with the market model residual standard deviation this effect vanishes. The bootstrap shows that there is a high level of imprecision in the relationships between vectors of characteristics and the distribution of (future) portfolio returns. While beta and average same-month return are effective in increasing epected utility for relatively risk-tolerant investors, they are dropped from the optimal set of characteristics for more risk-averse investors. Average same-month return in particular can amplify epected return. When the most risk-tolerant investor adds this characteristic to momentum, size, beta, and residual standard deviation, the optimal portfolio s epected return, Sharpe ratio, and certainty equivalent increase significantly.

3 1. Introduction Firm characteristics can predict future stock returns in the cross-section. The cross-section of epected stock returns also has a strong seasonal component. This paper analyzes these facts from the perspective of a risk-averse epected utility maimizing investor. To what etent does this cross-sectional predictability cause such an investor to tilt her optimal portfolio weights away from the market portfolio? This is an open question since research on the relationship between characteristics and future returns has focused almost eclusively on alpha that is, epected returns conditional on the Fama, French, Carhart factors. Many characteristic-based alpha generating strategies entail high volatility, fat tails, and/or negative skew (Barroso and Santa- Clara 2015, Kadan and Liu 2014). Lewellen (2015) notes that much of the empirical literature on characteristics and returns uses portfolios that are formed by sorting on one, and possibly two characteristics. As such, we have limited information about characteristics complementarities and substitutabilities. Because characteristics are not independent these effects are important. How do the complementarity and substitution effects among characteristics affect our interpretation of the nature and usefulness of the characteristics for a risk averse investor? We use Brandt, Santa-Clara, and Valkanov s (2009) portfolio selection algorithm to answer these questions. This algorithm uses the relationships between characteristics and portfolio moments to maimize in-sample epected utility. It does not specify a likelihood function and portfolio weights themselves are directly estimated. There is no first-step where distributional properties are estimated and then plugged into an objective function. Despite its parsimonious design it is subject to estimation risk. Before we use the algorithm as an empirical tool to eamine the nature of the data we use out-of-sample cross-validation to measure and manage estimation risk. This gives the econometrician three choices. First is the subset of characteristics used. Our data contain seven measurable characteristic variables: momentum, book-to-market ratio, log size, market model beta, market model residual standard deviation, five-year average same-month return, and last year same-month return. We consider 47 subsets of these seven characteristics, including each as a singleton. The second choice is the loss function. We use 14 loss functions: all are power utility functions with eponent γ, with γ = {2, 3,..., 13, 16, 22}. This yields 658 (in-sample) optimal portfolios. Finally we use both a rolling and an updating protocol to obtain optimal portfolios. To assess estimation risk we separate the γ used to obtain the optimal portfolio from the value of gamma in the investor s utility function. To convey and maintain this separation in the tet we refer to the eponent in the loss function to estimate optimal portfolio weights in-sample as γ, and the eponent in the loss function used to measure the utility of the out-of-sample portfolio 1

4 returns as gamma. We conduct cross-validation by evaluating the value of the investor s epected utility on the out-of-sample returns of each of the eligible 658 optimal portfolios. We consider a portfolio eligible if the γ used to obtain it is not less than gamma in the investor s utility function. We consider 9 investors indeed by gamma = {2, 3,..., 10}. We construct sampling distributions of all of the variables of interest, including the utility function, using the bootstrap. We find the the optimal adjustment for estimation risk entails using a value of γ to obtain portfolio weights that is strictly greater than the investor s gamma. To tease out the nature of this estimation risk we compare updating and rolling protocols to select portfolios. We find that the updating protocol that is keeping all of the historical record, as far back as 56 years dominates the rolling protocol which discards all but the most recent 180 months of data. This suggests that the estimation risk inherent in this portfolio selection algorithm is not due to slowly evolving relations between (cross-)moments and characteristics. Instead it resembles the conventional overfitting that plagues nonparametric estimation. Our bootstrapped out-of-sample cross-validation means that our analysis is not normative since we make statistical choices based on out-of-sample results. Instead we use portfolio selection as a descriptive analytical tool to evaluate the relationships between characteristics and portfolio returns. 1 We find important complementarities and substitution effects in the relationships of the characteristics with future portfolio returns. For eample, investors prefer low beta to high beta stocks when beta is the only characteristic available. However, when we add residual standard deviation to beta allowing weights to depend on both then high beta stocks become attractive to risk-averse investors with relatively high risk tolerance. 2 We also find that the book-to-market ratio is redundant in the contet of the other characteristics. This result is anticipated by Gerakos and Linnainmaa (2016, p. 2) who demonstrate that there is a significant disconnect between book-to-market ratio and the value premium. Estimation risk is eacerbated when either separately or jointly, the book-to-market ratio and the two variance characteristics are combined with same-month returns. By contrast, adding momentum to other characteristics generally reduces estimation risk. We use the characteristics to optimize utility. This is a different loss function than traditional studies that look at mean returns and often focus on alpha. In addition to ignoring higher order 1 We also cannot claim normative implications since we ignore transactions costs and assume costless shortselling. These assumptions are common in the inferential literature eamining the relationships between stock returns and characteristics wherein this paper fits. 2 The bad beta result would be consistent with Frazzini and Pedersen (2015) who argue that leverageconstrained investors bid up the prices of high beta stocks, which therefore have lower epected returns, ceteris paribus. Liu, Stambaugh, and Yuan (2017) also find that the bad beta effect is linked to the positive correlation between beta and idiosyncratic volatility. 2

5 moments, the use of alpha as a benchmark does not address the question of optimal eposures to the empirical factors. It imbues these empirical factors with importance that they may not merit, and treats eposure to each of the factors equally. By contrast, using epected utility as the statistical loss function makes no assumptions about factor eposures. We eamine the properties of the optimal portfolios for each of our nine investors including their Fama, French, Carhart factor loadings. We decompose the sources of the portfolio return variances to isolate the importance of each of the factors. In addition we study the optimal portfolios alphas, Sharpe ratios, means, variances, as well as nonparametric measures of location, scale, skewness, and kurtosis. We seek to understand how the various characteristics affect these portfolio measures, as well as how they are affected by the conveity of the loss function (i.e., how they are affected by the investor s gamma). The relationships between gamma and the four factor loadings and variance decompositions suggests that the four factors affect risk and return differently from one another and do not span the systematic risk in the market. The loading on the market is flat across gamma, and portfolio variance declines monotonically in gamma, so that the percent of portfolio variance eplained by the market factor increases from 3% for the gamma 2 investor s optimal portfolio to 34% for the gamma 10 investor s optimal portfolio. The loading on SMB is significantly positive for the gamma 2 investor s optimal portfolio insignificantly different from zero for the gamma 5 and gamma 6 investors optimal portfolios, and significantly negative for the gamma 10 investor s optimal portfolio. SMB does not account for more than 5% of the return variance of any optimal portfolio. Optimal portfolio loadings on HML decline from 1.9 for the least risk-averse investor s optimal portfolio to 1.0 for the most risk-averse investor s optimal portfolio. HML accounts for 15% of the return variance of the gamma 2 investor s optimal portfolio and 41% of that of the gamma 10 investor. Optimal portfolio loadings on MOM decline from 1.4 for the least risk-averse investor s optimal portfolio to 0.4 for the most risk-averse investor s optimal portfolio. Over this gamma range the percentage of portfolio return variance eplained by this factor decreases from 21 to 16%. Bootstrapping highlights the lack of precision in the role for characteristics. Much of this is due to the complementarity between the characteristics as we show that adding characteristics generally decreases the precision of the coefficients that relate portfolio weights to characteristics. Furthermore estimation risk tends to increase in the dimensionality of the set of characteristics used. Nevertheless the benefits of using characteristics to tilt portfolio weights transcend this imprecision. We eamine the month-of-the-year properties of the optimal portfolios. Our use of lagged same-month return as a characteristic calls attention to the possibility that optimal portfolios 3

6 eposures to risk factors may vary across the 12 months of the year. We summarize this by looking at the bias in alpha from the Fama, French, Carhart regressions that is due to assuming that the factor loadings are the same across the 12 months. We decompose this bias into riskpremium eposure and volatility eposure. Referring to the optimal portfolio for the gamma 2 investor (with mean alpha of 296 basis points per month), the mean calendar bias is 46 basis points per month. The largest portion of this bias is due to volatility timing of the SMB factor. 3 That is the portfolio loads more heavily on SMB in months when SMB has higher volatility than in other months. The remainder of the paper is organized as follows. We describe the data and empirical approach in Section 2. Summary results concerning the algorithm and estimation risk are provided in Section 3. We report the results linking characteristics to optimal portfolios in Section 4. Section 5 eamines the properties of the optimal portfolios in detail. Section 6 considers the possibility that ignoring month-of-the-year factor timing may lead to a bias in estimating alpha. Section 7 concludes the paper. 2. Portfolio Selection 2.1 Algorithm In Brandt, Santa-Clara, and Valkanov s (2009) algorithm, investors choose the vector θ in order to maimize average utility over T periods: T 1 (1 + r p,t+1 ) 1 γ ma θ 1 γ t=0 ( ) 1 T by allowing the weights to depend on observable stock characteristics: N t ) r p,t+1 = (ω i,t + 1Nt θ i,t r i,t+1 (2) i=1 Where: 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. 4 Since this loss function is asymmetric in portfolio return it is relevant for an institutional money manager who wants to avoid large losses in a single month. (1) 2.2 Data and specifications An observation of stock i at time t consists of the return in month t + 1 and the set of characteristics that are measurable at time t. We use the following characteristics: momentum 3 We use the measures of volatility timing and factor eposure timing from Boguth, Carlson, Fisher, and Simutin (2011). 4 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). 4

7 (M), book-to-market (V), log size (S), beta (β), market model residual standard deviation (σ ɛ ), and r t 12, and r t 12j, for j = 1,..., 5. Momentum is measured as the 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 log of B. Size is the natural logarithm 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 inde. 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 ecluded from the sample. This ecludes stocks with market capitalization less than $11.5 million in January 1960, and $93.3 million in December We net eclude 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. Figure 1 shows the sample size used each month. There are 395 (eclusively New York Stock Echange-listed) stocks in the sample in January There is a jump in the sample size in August, 1967 (from 675 to 875) when the American Stock Echange stocks are eligible for inclusion in our sample. The largest jump is in January 1978 (from 1,000 to 1,419 stocks) when Nasdaq stocks enter our sample. The maimum 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 eample, let β be a stock s OLS beta, were we to use shrinkage betas, such as β S =.5 β +.5 1, 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. A model is defined by the (sub-)set of characteristics used in portfolio selection. We consider 47 models which include each of the si characteristics as a singleton, and all seven variables 5

8 together. We consider 14 values of γ, which means that we evaluate 658 unique portfolios. Empirical analysis of the portfolios selected using the epected utility optimizing algorithm must be conducted out-of-sample. We use both a rolling and updating protocol. The first out-ofsample month is January At this point we have estimated the θ-vector using 180 months of data. As in Brandt, Santa-Clara, and Valkanov (2009), we use this θ-vector construct the optimal portfolios in each of the net 12 months. We add the 12 months of 1975 to the original 180 months under the updating protocol, whereas we drop the first 12 months in the last sample (i.e., the 12 months of 1960) under the rolling protocol to estimate the θ-vector to use in forming the optimal portfolios in 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 of under the rolling protocol). This means that we optimize utility (and estimate the θ vector) from each of the 658 cases 41 times at the end of each in-sample period. The out-of-sample period comprises the 41 years The bootstrap Our interest is in statistical comparisons across various models. We use the bootstrap to construct sampling distributions of the functions of interest in this paper, such as certainty equivalent, portfolio loading on factors, portfolio skew, etc. The bootstrap also shows that there is a small sample bias in the θ estimates, and other functions of θ. Our bootstrap is designed as follows. The data in month t in our sample consist of the N t vectors Ψ i,t, for i = 1,..., N t and t = 1,..., 672. A bootstrap draw resamples (with replacement) N t vectors from Ψ i,t in all 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. We take 10,000 bootstrap samples to estimate the sampling distributions for all 658 unique portfolios Algorithm efficacy and estimation risk Table 1 reports the model with the highest bootstrap 2.5%ile certainty equivalent (across the 47 specifications) for each of the nine investors, for her own and all higher γ values used to estimate θ. The first row under each γ value is the optimal model under the updating protocol and the second row is optimal model from the rolling protocol. This table also reports the bootstrap and sample values of the certainty equivalent for both the equal- and value-weighted 5 The computational burden is non-trivial. A typical bootstrap takes 26 hours of CPU time on a 3.7 GHz Xeon processor. However, since the bootstrap draws are independent (unlike in Markov Chain Monte Carlo, for eample), the algorithm is easily parallelizable. 6

9 portfolios constructed from the eligible securities, for all investors. This table is ecerpted from Tables IA-1 (pp. 2 3) through IA-90 (pp ), 6 which report the bootstrap distributions and sample estimates of the certainty equivalent for each of the nine investors, respectively, using each value of γ and all 47 models under the updating protocol. That combination (of protocol, γ and characteristic set) that produces the maimum 2.5%ile value of the certainty equivalent is highlighted in bold face. The gamma 2 investor s median certainty equivalent for the value- (equally-weighted) inde is 106 (86) basis points per month. When this investor uses her own γ, the model with the highest 2.5%ile certainty equivalent comprises momentum, book-to-market, and size, using the updating protocol. The gamma 2 investor does not prefer this portfolio to the equally-weighted inde, on a statistical basis. Using this loss function, one portfolio statistically significantly dominates another when its 2.5%ile certainty equivalent value eceeds the alternative portfolio s 97.5% certainty equivalent. A glance at Table IA-1 (pp. 2 3) shows the nature and severity of the estimation risk problem in this case. Twenty-one of the forty-seven portfolios, including all 11 portfolios selected using five or more characteristics, have a 2.5%ile certainty equivalent of -100%. This means that in at least 250 of the bootstrap samples, the portfolio lost at least 100% of its value in at least one month in the out-of-sample period. This is also the case with all models that include last year s same-month return regardless of the number of characteristics. Table 1 also shows that estimation risk is more severe under the rolling protocol than the updating protocol. Using γ = 2, the 25%ile certainty equivalent of all 47 models is -100% for the gamma 2 investor, when the rolling protocol is used. Table 1 shows that the portfolio with the highest 2.5%ile certainty equivalent for the gamma 2 investor results from using γ = 3; characteristic set: momentum, log size, beta, residual standard deviation, and 5-year average same-month return; and the updating protocol to estimate θ. This portfolio s bootstrap median certainty equivalent is more than three times higher than that of the equally-weighted inde, and the bootstrap 2.5%ile is 288 basis points per month. Table IA-2 (pp. 4-5) makes clear that using γ = 3 is not a panacea for the gamma 2 investor, as ten of the 47 models produce a 2.5%ile certainty equivalent of -100%, including most that contain last year s same-month return. Tables IA-2 (pp. 4-5) IA-14 (pp ) also demonstrate that the (out-of-sample) loss function is fairly flat under the updating protocol, provided that γ is at least 3 and the characteristics: momentum, log size, residual standard deviation, and average same-month return are included in the characteristic set. Using γ = 3, si additional models are both statistically preferred to the benchmark, and not statistically dominated by the optimal. 6 Tables starting with IA are collected in this paper s Internet appendi. Page numbers for this appendi (pdf page numbers) are also provided. 7

10 For eample, adding book-to-market to the characteristic set lowers the 2.5%ile (mean) certainty equivalent by 9 (8) basis points per month, but increases the median and 97.5%ile by 1 and 4 basis points per month, respectively. The optimal model significantly dominates all (25) of the sets of characteristics involving three or fewer characteristics. Tables IA-1 (pp. 4-5) IA-14 (pp ) also show that for the gamma 2 investor using the optimal set of characteristics, the certainty equivalent peaks at γ = 3. The decline in certainty equivalent as γ increases beyond three is gradual. The certainty equivalents of portfolios selected using the optimal characteristics and γ values between three and eight are not significantly different from the globally optimal case. Increasing γ from three to four, for eample, reduces the mean certainty equivalent return by 15 basis points. For γ values of 8 and higher, the optimal portfolio is statistically dominated by the optimal portfolio selected using γ = 3. Using γ = 3 with the rolling protocol does not produce any optimal portfolios that allow the gamma 2 investor to dominate the benchmarks. In fact, Table 1 shows that for this investor the optimal result under the rolling protocol requires the use of γ = 5, and all 7 characteristics. This portfolio also statistically dominates the benchmarks, and is not statistically dominated by the global optimum portfolio for the gamma 2 investor (from the updating protocol), as its 95%ile sampling interval certainty equivalent return is [248, 366] basis points per month. The nature of the optimal portfolios and estimation risk for the gamma 3 and gamma 4 investors is very similar to that of the gamma 2 investor. The equally-weighted inde statistically dominates the value-weighted inde for these investors. The gamma 3 investor s optimal portfolio is achieved using γ = 5 and the gamma 4 investor optimally uses γ = 7. The optimal characteristic set for these two investors is the same as for the gamma 2 investor: momentum, log size, beta, residual standard deviation, and average same-month return. The proportional gains in certainty equivalent returns from allowing portfolio weights to depend on characteristics for these investors are also similar to that of the gamma 2 investor. The gamma 4 investor increases certainty equivalent mean return from 77 basis points per month under the 1 N rule to 199 basis points per month using these five characteristics to tilt portfolio weights away from the market portfolio. As with the gamma 2 investor, the rolling protocol has more estimation risk than updating. The gamma 4 investor for eample has to increase γ to 9 to maimally reduce estimation risk under the rolling protocol. The optimal portfolio in this case also significantly dominates the equally-weighted inde and is not dominated by the global optimal portfolio. The gamma 5 and gamma 6 investors are statistically indifferent between the equally-weighted and value-weighted indices and the gamma 7 investor prefers the value-weighted inde. These three investors optimally use the same characteristic set under the updating protocol: momentum, 8

11 log size, residual standard deviation, and average same-month return: beta is dropped from the set used by more risk-tolerant investors. For these three investors the rolling protocol is strictly dominated by the updating protocol, since all of the optimal portfolios generated by the former are significantly dominated by those from the latter. The magnitude of the utility gain for these investors is similar to that of the less risk-averse group. For eample the gamma 7 investor s mean certainty equivalent increases from 34 to 116 basis points per month switching from the value-weighted inde to the optimal portfolio, which is obtained using γ = 13. The gamma 8, gamma 9, and gamma 10 investors comprise the third set of investors. All three of these investors optimally use γ = 16 (since the grid of γ used to obtain portfolios is not continuous), and the optimal set of characteristics is: momentum, log size, and residual standard deviation. For these investors the rolling protocol is also optimized using these three characteristics and γ = 16 (with very similar results using γ = 13). The optimal rolling portfolios are not significantly dominated by the optimal portfolios produced under the updating protocol for the gamma 8 and gamma 9 investors. None of the portfolios from the rolling protocol are preferred to the value-weighted inde for the gamma 10 investor. Overall the relationship between the updating and rolling protocols means that slowly evolving relationships between the (multivariate) return distribution and the characteristics is not driving estimation risk in the contet 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 contet 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. Rolling has more estimation risk in the sense that the optimal γ is strictly higher than the optimal γ under the updating protocol. This implies that 15 years of data (180 months) is sufficient to measure the relationships between the characteristics and future returns in a portfolio contet. Table IA-1 (pp. 2-3) IA-14 (pp ) provide additional evidence about the relationships between characteristics and estimation risk from the perspective of the gamma 2 investor. Including momentum in the set of characteristics is important to reduce estimation risk. For eample when the book-to-market ratio is substituted for momentum in the optimal characteristic set, the mean certainty equivalent drops from 336 to 209 basis points per month, and tellingly, the 2.5%ile 9

12 drops from 288 to -10,000. Adding last year s same-month return to this set of characteristics further eacerbates estimation risk. Dropping momentum from the optimal set of characteristics, leaving log size, beta, residual standard deviation and average same-month returns results in an optimal portfolio whose 2.5%ile certainty equivalent return is -100% when γ = 3 is used to estimate θ. The optimal portfolio using all (si) characteristic variables ecluding momentum has a mean certainty equivalent of -6,045 basis points per month, when γ = 3. The mean certainty equivalent using this set of characteristics is also negative for the gamma 2 investor, using a γ value as high as 5 to select the optimal portfolio. Estimation risk increases in the dimensionality of the characteristic set, with the eception that adding momentum always mitigates estimation risk. In what follows we will consider how momentum affects the factor eposure and moments of the optimal portfolios to consider its effect on estimation risk in more detail. It is never optimal to include last-year s same-month return in the characteristic set. This is because in the contet of the five-year average same-month return this is very noisy and results in increased estimation risk. 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 eposed to the HML factor, but instead that size and residual standard deviation provide enough fleibility to allow optimal eposure to the value factor. In other words the book-to-market ratio is redundant and adds estimation risk. The estimation risk in this contet then is conventional overfitting. This inference is also consistent with the fact that the dimensionality of the optimal model shrinks as investor risk aversion increases. Those investors with gamma values higher than 5 eschew beta, and those with gamma values higher than 7 also drop the average same-month return from the optimal characteristic set. The benefits delivered by these characteristics to lower gamma investors are offset by estimation risk for the more risk-averse investors. These results suggest that θ shrinkage or even holding a larger portion of assets in cash and/or the market are also not optimal. In most cases, the adjustment involved with using a higher γ than in the actual utility function is to lower the eposure to the characteristic (i.e., θ shrinks in absolute value), but this is not the case when characteristics are optimally removed from the set used to estimate θ, or when the sign of θ is a function of γ. We will revisit this as well when we evaluate the θ coefficients below. 10

13 4. Characteristics and portfolios 4.1 Characteristics in isolation Tables IA-1 (pp. 2-3) IA-90 (pp ) show that momentum, book-to-market, and log size never serve as a singleton characteristic to produce significantly higher certainty equivalent for any investor, using any γ value. The most risk-tolerant investors (those whose coefficient of relative risk aversion is two and three) can use the average same-month return by itself to create portfolios with significantly higher certainty equivalent than the equally-weighted benchmark. Investors with mid-levels of risk aversion (those with coefficients of relative risk aversion between five and eight) can significantly increase certainty equivalent above the benchmark by using the residual standard deviation in isolation. Investors with high risk aversion (those with coefficients of relative risk aversion between seven and nine) can significantly increase certainty equivalent above the benchmark by using beta in isolation. Therefore, most and in some cases all of the (out-of-sample) utility gains from using characteristics to affect portfolio weights is derived from using them in combination. 4.2 Substitutions and complementarities Whereas momentum and log size used in isolation never generate a portfolio that is significantly preferred to the benchmark, both of these characteristics along with the residual standard deviation are used in the optimal characteristic set for all nine investors. Therefore the salutary effects of the characteristics for risk-averse utility optimizing investors is primarily due to complementary effects between the characteristics. This fact belies traditional analysis of the role of characteristics and returns, which has focused mainly on the relationship between characteristics and epected returns, and has generally analyzed these effects one characteristic at a time (Lewellen 2016). Another important difference between this approach and much of the literature is that this analysis has to be out-of-sample (since the model places no testable restrictions on the data generating process), (Lewellen, Nagel, and Shanken 2010). Tables IA-91 (p. 182) through IA-250 (p. 341) report bootstrap properties for θ coefficients on each of the seven characteristic variables for all portfolios produced under the updating protocol, from all of the combinations of characteristic sets that contain that variable, for all 14 γ values. Table 2 shows the sampling distributions of the average of the 41 annual θ coefficients on the (seven) portfolios that are optimal for the nine investors. Figures 2 6 show bootstrap properties of the estimated θ coefficients by year (41 years: 1974 through 2014, used to form the out-ofsample portfolios in the following year). We show each of the five characteristic variables that enter the optimal set for the gamma 2, 3, and 4 investors, when they are used in isolation(top 11

14 panel), and in the optimal portfolio using γ = 3 (lower panel). Figure 2 shows the bootstrap properties of the θ coefficients on momentum. When momentum is used in isolation, there appears to be a structural break after year 25, so we split the results temporally at this point in the tables. Comparing the two panels in Figure 2 it is clear that the sampling distribution of the estimated θ is much tighter when momentum is the only characteristic. This is also evident in Table IA-91 (p. 182) as the standard deviation of the momentum θ in the second period (Years 26-41) is 0.18, when momentum is the only characteristic used, and 0.40 when momentum is used along with: log size, beta, residual standard deviation, and average same-month returns (Table IA-109, p. 200). Also the imprecision in the latter case is such that there is no evidence of a structural break in the momentum θ coefficient in the optimal model for the gamma 2 investor. The figures also report the sample estimate of θ in the contet of the parameter s bootstrap distribution. These coefficients are biased (away from zero). This bias is etenuated by dimensionality of the sample space, and in most cases appears to get worse over time. 7 These biases support our use of bootstrap means instead of the sample estimates for our statistical analyses. These biases also eplain the differences between the bootstrap mean certainty equivalent and the sample value, shown in Table 1 as well as Tables IA-1 IA-90. The sample certainty equivalent for the gamma 2 investor s optimal portfolio is basis points per month, above the bootstrap 75%ile value. The preceding section established that momentum is part of the optimal characteristic set for all nine investors. For the gamma 8, 9, and 10 investors the mean θ coefficient on momentum is 1.54 in and 0.91 in The difference between these is statistically significant. These momentum θ coefficients are not significantly different from the optimal θ values for the gamma 6 and 7 investors, but they are significantly smaller than the optimal θ for those investors whose gamma is less than 6. Figure 2 shows that the set of characteristics, {log size, β, residual standard deviation, and average same-month return} is complementary to momentum when γ = 3. The θ coefficient on momentum is larger when it is part of the optimal set than when it is used as a singleton characteristic. Over the final 16 years in the sample this difference is significant. Table IA-109 (p. 200) shows that the 95% sampling interval of the average momentum θ over the period , when used in isolation is [2.62, 3.32]. Whereas when momentum is part of the optimal set this interval is [3.87, 5.44]. Since the periods used to evaluate portfolio properties and performance 7 This suggests that a bias adjustment technique such as bootstrap aggregation (or bagging ) might improve matters in terms of selecting optimal portfolios. Small-scale eperimentation shows that both bagging and bragging 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. 12

15 are not the same as used to estimate θ, there are two types of complementarity that we can identify. The first problem with complementarity is that increasing the dimensionality of the characteristic space increases estimation risk, which highlights the importance of using out-ofsample data for inference. Complementarity can be amplifying adding a coefficient causes the θ on the incumbent characteristic(s) to increase in absolute value. Alternatively, one characteristic may complement another by refining the nature of the stocks appeal. For eample, consider momentum relative to the characteristic set {log size, β, residual standard deviation, and average same-month return}, when γ = 3. Adding momentum to this set of characteristics does not significantly change any of these characteristics θ coefficients. 8 Nevertheless, Table IA-2 shows that the addition of momentum increases the median certainty equivalent from 224 to 336 basis points per month, while reducing its sampling standard deviation from 3,374 to 107. For the gamma 2 investor using γ = 3, small stocks, stocks with high residual standard deviation, and stocks with high same-month returns are attractive, but basing weights on these characteristics entails high estimation risk. In many of the bootstrap samples, the optimal portfolios constructed from these characteristics only have out-of-sample months when the returns are less than -100%. Adding momentum mitigates the in-sample overfitting problem. This is eplored in more detail in Section 5.6 below. Figure 3 and Tables IA (pp ) report properties of the θ coefficient on log size. As with momentum, the figure shows that the sampling distributions are much less precise in the higher dimensional case. Bias is also worse in this contet, and at the end of the sample. This figure and table show that size is an amplifying complement to the other characteristics. In isolation, the 95% sampling interval of the θ coefficient on log size in the two periods are: [ 3.0, 1.7] and [ 2.5, 1.6], respectively. By contrast, in the optimal model for the gamma 2 investor, these ranges are: [ 13.7, 9.6] and [ 10.7, 8.3], respectively. Tables Tables IA show which characteristic(s) are most complementary to log size. Holding γ = 3, adding the book-to-market ratio, momentum, and same-month returns, individually or in combination have very little effect on the log size θ. On the other hand, both beta and residual standard deviation are amplifying complements to log size. Adding residual standard deviation to log size changes these 95%ile sampling intervals to: [ 9.2, 6.7] and [ 7.3, 5.7], respectively. Tables IA-166 IA-184 (pp ) and Figure 4 show properties of the θ coefficient on beta. When beta is the only characteristic (top panel in both Figure 4 and Table IA-166) the θ coefficient is significantly negative for all gamma values used to select the optimal portfolio in both subperiods. Further, there is no statistical difference between the coefficients as γ ranges 8 The sampling distributions of the θ coefficients on log size, β, residual standard deviation, and average samemonth return, with and without momentum are reported in Tables IA-154, IA-174, IA-200, IA-225; and IA-160, IA-179, IA-205, and IA-231, respectively. 13

16 from 2 through 22. Such a result is consistent with Frazzini and Pedersen (2016), who argue that high beta stocks are bid-up by borrowing-constrained investors, so that they offer lower epected rates of return after adjusting for risk. If this were indeed the case, then the appeal of low beta stocks in this setting would be independent of the investor s risk aversion, since we assume away borrowing constraints. This result is not robust, however. The portfolios formed by conditioning only on beta (and which have an average negative θ coefficient on beta) are dominated by the optimal portfolio for all investors. For the gamma 2, 3, and 4 investors these involve a statistically insignificant θ in the first 25 years and a significantly positive θ coefficient on beta in the period. For the more risk-averse investors with gamma values 5 and higher, these optimal portfolios place a 0 weight on beta. Table 2 shows that for the three most risk-tolerant investors, the average coefficient on beta is not statistically different from 0 over the full 41 year period. This is also evident in Figure 4. Despite this, investors with gamma values of 4 and lower are better off including beta in the set of characteristics. Tables IA-167 IA-184 show the effects of adding the other characteristics to beta. For eample with γ = 3, as noted, when beta is the only characteristic its θ coefficients in both subperiods are significantly negative. Adding the characteristic set {momentum, residual standard deviation, and average same-month return} makes the θ coefficient on beta significantly positive in both subperiods. Adding size to this set (which results in the optimal set for the gamma 2 investor) does not significantly alter the θ coefficient on standardized beta, although θ is now not significantly greater than zero in the first subperiod. These results suggest that beta is correlated with something that all investors do not like. Residual standard deviation is also correlated with this latent factor. Kogan and Papanikolaou (2013) argue that firms with higher idiosyncratic volatility have higher growth opportunities, and lower risk premia. When investors can condition only on beta they prefer low beta to high beta stocks as a way of avoiding this latent factor. When investors can also condition on the residual standard deviation, the residual standard deviation allows them to reduce their eposure to this latent factor, and then higher beta stocks are more attractive than low beta stocks. The fact that beta is dropped from the optimal set of characteristics when investor gamma is 5 and higher suggests that it has a lot of estimation risk, and that its appeal in-sample does not produce benefits out-of-sample. Figure 4 shows one reason for isolating the period , as the θ coefficient on beta when beta is used in isolation behaves differently over this period than in the preceding period. The figure also shows that the bias in the sample θ coefficient is larger when beta is combined with the four other characteristics than when it is used in isolation. Tables IA-185 IA-210 (pp ) and Figure 5 report the properties of the θ coefficient on the residual standard deviation. The figure shows that for the gamma 2 investor, the coefficient on 14

17 the residual standard deviation is uniformly negative. This characteristic is amplified by the other characteristics, especially size and beta. For eample, the average θ coefficient on the residual standard deviation using γ = 3 when residual standard deviation is the only characteristic is Adding momentum, log size, beta, and average same-month return decreases the bootstrap mean of this coefficient to The residual standard deviation, like momentum and log size is optimally used by all investors. Residual standard deviation s θ coefficient is significantly negative in all optimal portfolios. Figure 5 shows that as the case with the other characteristics the sample bias is etenuated by the model dimensionality and the increased sample size. Comparing Table IA-185 with Tables IA-186 through IA-210 shows that the θ coefficient on residual standard deviation is much more sensitive to investor risk aversion when it is used in combination with size and beta. Tables IA-211 IA-237 (pp ) and Figure 6 show properties of the θ coefficient on five-year average same-month return. The θ coefficient on this characteristic is significantly higher in the second subperiod in the optimal models for the gamma 3, 4, 5, 6, and 7 investors than in the model in which it is the only characteristic. The 95% sampling bands on the θ coefficient on r t 12j in the two subperiods for the optimal portfolio for the gamma 6 investor (using γ = 11) are: [1.7, 3.3] and [1.9, 3.0], respectively. When r t 12j is the only coefficient in the characteristic set, these bands are [1.2, 2.1] and [0.6, 1.2], respectively, when γ = 11. This means that in the second subperiod, for investors whose gamma eceeds 2, momentum, log size, and residual standard deviation are amplifying complements to same-month return. Figure 6 shows that when r t 12j is used in isolation the θ coefficient varies significantly from year to year (which is also apparent in significant differences between the two subperiods for this model). The lower panel of Figure 6 shows that this is not the case for the optimal model for the gamma 2 investor, largely because the sampling variances on these coefficients are so large. The incremental benefit of same-month return is statistically significant for risk-tolerant investors. From the perspective of the gamma 2 investor the highest certainty equivalent return using γ = 3 ecluding same-month return results from conditioning on: momentum, book-tomarket, log size, beta, and residual standard deviation. The 95%ile sampling band on the certainty equivalent is this case is [166, 256]. This is significantly lower than the optimal portfolio s (which substitutes lagged same-month return for the book-to-market ratio). Table 2 shows the relationship between the θ coefficients and investor gamma. In general θ shrinks in absolute value as risk aversion increases. All four of the gamma 5 investor s (using γ = 9) θ coefficients are significantly smaller in absolute value than those of the gamma 2 investor (using γ = 3). The optimal portfolio for the gamma 8, 9, and 10 investors does not condition on average same-month return but the three remaining θ coefficients are not significantly different 15

18 from the gamma 5 investor s optimal portfolio. Two of the seven characteristic variables, last-year s same-month return (whose θ coefficients are reported in Tables IA-238 IA-250, (pp ) and the book-to-market ratio are not used in any of the optimal portfolios. Tables IA-1 IA-90 show that for the most part this is because of redundancy and estimation risk, rather than direct harm. Adding both characteristic variables to the optimal set for the gamma 4 investor (using γ = 7, Table IA-31, pp ), results in an insignificant drop in the certainty equivalent; the 95%ile bands on the certainty equivalent decrease from [173, 226] to [157, 223] basis points per month. Adding a characteristic which is largely spanned by the other characteristics adds estimation risk, seen in the drop in the 2.5%ile of the certainty equivalent sampling distribution in these cases. 4.3 Marginal effects Table IA-17 (pp ) shows that the 95%ile sampling band of the certainty equivalent for the gamma 3 investor (using γ = 5) is [216, 284] basis points per month. When momentum is removed from the optimal set of characteristics this band becomes [ 36, 224], highlighting momentum s importance in reducing estimation risk. When log size is removed from the optimal characteristic set, the certainty equivalent drops significantly to [151, 209]. Removing beta from the optimal set results in the certainty equivalent falling to [208, 267], a statistically insignificant difference. When average same-month return is not included in the set of characteristics the 95%ile certainty equivalent band drops significantly to [141, 197]. Similarly ecluding residual standard deviation from the optimal characteristic set lowers the certainty equivalent significantly to: [154, 202]. Adding the book-to-market ratio to the optimal set lowers this certainty equivalent band to [207, 283], which is not significantly different from the optimal. Using all seven characteristics, which adds the book-to-market ratio and last year s same-month return to the optimal set changes this band to [202, 282]. Table IA-56 (pp ) shows the certainty equivalent distributions for the gamma 6 investor, using γ = 11. The 95%ile sampling band for the optimal portfolio s certainty equivalent is [115, 159] basis points per month. Removing momentum from the optimal characteristic set changes this to [7, 134]. Removing log size from the optimal set lowers this band to [78, 113]. Removing average same-month return from the optimal characteristic set changes the band to [94, 124], which is not significantly different. Removing the residual standard deviation results in the certainty equivalent band falling to [43, 59]. Adding beta to the optimal set has a small effect: the certainty equivalent band becomes [110, 159]. Adding the book-to-market ratio to the optimal set changes the certainty equivalent band to [98, 154]. Using all seven characteristics generate a 95%ile sampling band on certainty equivalent of [85, 154]. 16