Current draft: June 10, 2022

First draft: February 1, 2012

An Empirical Assessment of Characteristics and Optimal Portfolios

Christopher G. Lamoureux111Department of Finance, The University of Arizona,
Eller College of Management, Tucson, 85721, 520–621–7488,
<EMAIL_ADDRESS>
and

Huacheng Zhang222Department of Finance, Risk, and Banking, Nottingham
University Business School, UK<EMAIL_ADDRESS>The current version
of this paper may be downloaded from lamfin.arizona.edu/rsch.html .

Abstract

We show that overfitting plagues optimal portfolios obtained by linking
weights directly to characteristics. We effectively mitigate overfitting
(regularize) by optimizing a more concave loss function than the investor’s
utility function. Significant certainty equivalent gains over benchmarks
require at least three characteristics: momentum, size, and residual
volatility. Out-of-sample utility gains are due to characteristic
complementarities, and depend on investor risk aversion. For example,
conditioning on momentum relieves the overfitting bias inherent in the other
characteristics for our most risk-averse investor. Optimal portfolios’ returns
lie largely outside the span of traditional factors and move closer to the
market as risk-aversion increases.

Key Words: cross-section of stock returns; overfitting; stock characteristics;
optimal portfolios; out-of-sample evaluation

1\. Introduction

Brandt, Santa-Clara, and Valkanov (2009) introduce a brilliant algorithm to
optimize an in-sample utility function by linking portfolio weights directly
to measurable characteristics. They demonstrate that characteristics can be
used by power utility investors to build portfolios that have a higher
certainty equivalent than standard benchmarks. This approach eliminates the
need to specify a data generating process and estimate conditional moments of
the return distribution, traditionally the first step in portfolio
optimization. They also show that these optimized portfolios’ relative
performance degrades out of sample. We are motivated to expand the vast
literature that has looked at characteristics’ predictive content beyond the
mean-variance framework using this algorithm. In order to do that we first
have to document, understand, and manage the algorithm’s estimation risk,
which is known to plague optimal portfolio selection. That is, we have no
interest in the properties of in-sample optimal portfolios if these portfolios
have poor out-of-sample performance.

This paper’s first contribution is to extend Brandt, Santa-Clara, and Valkanov
(2009) to analyze estimation risk in the context of power utility investors
with varying degrees of relative risk aversion. We find that overfitting has a
first-order effect on portfolios selected using the algorithm, as is the case
with mean-variance optimization. For most characteristic combinations the
optimal portfolios do not dominate market benchmarks out of sample as a result
of overfitting. This overfitting problem arises despite the small number of
estimated coefficients as a result of the dimensionality of the characteristic
space and the infinite dimensionality of the parameter space, as the utility
function depends on all of the properties of the unspecified multivariate
conditional return distribution. Our first major result is that increasing the
curvature of the loss function used to obtain portfolio weights relative to
the utility function preserves benefits from conditioning on characteristics,
and mitigates overfitting.

In-sample optimal portfolios perform notoriously poorly out of sample. Most of
the work done to understand and mitigate this problem is in the context of
mean-variance optimization. Best and Grauer (1991) stress that this
deterioration results from optimization amplifying estimation errors. The
models that have been developed to manage estimation risk include constraining
weights and constraining estimators.111Jagannathan and Ma (2003) demonstrate
that there is a duality between constraining the weights and shrinking moment
estimators in the mean-variance optimization context. Kan and Zhou (2007)
derive the expected loss function from using sample (rather than true) moments
when returns are normally distributed. They show that estimation risk can be
diversified by holding a minimum variance portfolio in addition to the
estimated tangency portfolio. Recent Bayesian approaches establish a prior
using economic theory. Pástor (2000) and Pástor and Stambaugh (2000) use asset
pricing models to form the prior. MacKinlay and Pástor (2000) impose moment
restrictions according to a factor model. However, DeMiguel, Garlappi, and
Uppal (2009) show that in general, these economic and statistical tools that
have been developed to optimize in the presence of estimation error are
dominated by the $\frac{1}{N}$ rule.

Part of the motivation for linking portfolio weights to characteristics is
that portfolio selection is reduced to a single step. Traditionally portfolio
choice is split into two steps (Gennotte 1986), with the first step being
estimation of the conditional return distribution. In the mean-variance
context this entails obtaining consistent estimators of the conditional mean
and variance covariance matrix. Aït-Sahalia and Brandt (2001, p. 1299) refer
to conditional moment (or conditional likelihood) estimation as the “Achilles’
heel of conditional portfolio choice because although the moments are
predictable, this predictability is for some moments quite tenuous.” They
argue that by not specifying a likelihood (i.e., the conditional return
distribution) we avoid “introducing additional noise and potential
misspecifications through the intermediate, but unnecessary, estimation of the
return distribution.” Brandt, Santa-Clara, and Valkanov (2009, p. 3412) say
that linking optimized weights directly to characteristics “escapes the common
statistical problems of imprecise coefficient estimates and overfitting.”

Brandt, Santa-Clara, and Valkanov (2009, p. 3423) note that shrinkage is a
common approach to mitigating overfitting. Shrinkage via regularization
generally entails an auxiliary loss function with separate tuning parameters,
such as with a Lasso (L1-norm) penalty on the estimated coefficients. The
regularized loss function is maximized over the tuning parameters in a
(cross-) validation step. DeMiguel, Martín-Utrera, Nogales, and Uppal (2020)
use the direct optimization algorithm to maximize the Sharpe ratio (i.e., they
specify a quadratic utility function), with transactions costs. They impose an
L1-norm penalty, and demonstrate that it does better out of sample than a non-
regularized optimization. However, Kozak, Nagel, and Santosh (2020, p.274)
note that a Lasso-type L1-norm penalty has poor statistical properties when
the characteristics are correlated, and it lacks economic motivation.

Brandt, Santa-Clara, and Valkanov (2009) show that as investor risk aversion
increases the coefficients on their three characteristics tend to decrease in
absolute value. However, these effects are not uniform across the
characteristics, and there are sign changes. This suggests an alternative to
shrinkage which we implement. Rather than introduce a statistical penalty on
the sizes of the coefficients managed with a tuning parameter, we treat the
concavity of the loss function used on the sample as the tuning parameter,
disentangling the loss function optimized on the data from the investor’s
utility function. For example consider a power utility investor with
coefficient of relative risk aversion, $\gamma=2$. For this loss function, we
optimize in sample using power “utility” functions indexed by
$\gamma^{*}=2,3,4,\ldots$, and evaluate the sampling distribution of the
certainty equivalent returns. The characteristic set comprising: momentum, log
size, beta, residual volatility, and average same-month return is optimal for
this loss function. Using this set and $\gamma^{*}=\gamma=2$ generates an
optimal (in-sample) portfolio whose mean certainty equivalent return over the
528-month out-of-sample period is -42% per month. By contrast, when
$\gamma^{*}=3$, this statistic is 3% per month. We find similar results for
all levels of relative risk aversion. Using $\gamma^{*}=\gamma=5$ with the
optimal characteristic set generates an optimal portfolio with a 95% certainty
equivalent confidence interval of $(50\,,\,146)$ basis points per month, which
does not statistically dominate the benchmark. The comparable values for the
optimal portfolio when $\gamma=5$ and $\gamma^{*}=10$, in this case are
$(133\,,\,170)$ basis points per month.

This method of regularization, or shrinkage, affords a natural interpretation
of the tuning parameter, $\gamma^{*}$. Relative to $\gamma$, it measures the
increase in the costs of even moments in terms of odd moments needed to offset
overfitting. It also allows the degree of shrinkage to vary across loss
functions and characteristic sets, as well as allowing for sign changes. And
unlike the Lasso penalty its efficacy is not affected by high correlations
among the factors. The effectiveness of this method to mitigate overfitting is
an empirical question. If overfitting results from apparent spurious near-
arbitrage opportunities in the sample then our regularization of increasing
the concavity of the loss function used on the data would be ineffective. We
show that this is not the case. That is, the spurious sample patterns are the
result of outliers–largely exaggerations of a stationary data generating
process. Consequently, making even moments more costly in terms of odd moments
produces superior out-of-sample results.

Our out-of-sample model assessment is more robust to data mining and less
sensitive to outliers than in-sample inference. Because we optimize in-sample
and evaluate out-of-sample we use a long time series–our data start in 1955
and end in 2018. We contrast a rolling sample optimization design with an
updating design to ascertain whether there have been structural breaks in the
relationships between characteristics and returns over our sample period. We
find that the rolling design is subject to more severe overfitting–requiring a
bigger adjustment of $\gamma^{*}$ relative to $\gamma$ to regulate. This has
the effect of reducing the beneficial informational content of the
characteristics, and therefore the rolling design is never preferred to
updating. The fact that updating serves to mitigate overfitting is further
evidence that the overfitting results from “outliers,” and manifests largely
in exaggerated patterns rather than spurious arbitrage opportunities.

We extend the large literature on cross-sectional return predictability beyond
the mean-variance framework. Since power utility is a function of all of the
moments of the return distribution we extend analysis of characteristics’
predictive content to higher moments including skewness and kurtosis. Leland
(1999), Ferson (2013), Broadie, Chernov, and Johannes (2009), and Goetzmann,
Ingersoll, Spiegel, and Welch (2007) raise concerns about measuring
performance using linear measures such as the Sharpe ratio, or alpha from a
factor model when the return process is non-Gaussian. Harvey and Siddique
(2000) argue that conditional skewness in stock returns can explain deviations
from linear factor pricing models. Furthermore, Patton (2004) shows that there
is enough persistence in higher-order moments to generate significant utility
gains to a constant relative risk aversion (CRRA) investor in an asset
allocation problem. Motivating our extention of portfolio performance beyond
the first two moments, Barroso and Santa-Clara (2015) and Kadan and Liu (2014)
show that characteristic-based portfolios that generate a high alpha and
Sharpe ratio come at the cost of negative skewness.

Much of the recent research on characteristics’ predictive content for cross-
sectional stock returns raises data mining concerns. Giglio, Liao, and Xiu
(2021), Hou, Xue, and Zhang (2020), Harvey, Liu, and Zhu (2016), and Harvey
(2017) stress the multiple test nature of testing characteristics’ predictive
efficacy. Researchers have scoured the same data in an attempt to identify
characteristics to predict means, after accounting for known sources of
systematic variance. Another statistical concern is the temporal nature of
findings of predictability. Linnainmaa and Roberts (2018) show that most
accounting-based anomalies are not robust to out-of-sample testing. McLean and
Pontiff (2016) argue that characteristics’ predictive content vanishes once
the relationship is published in our academic literature. On the theoretical
front, Martin and Nagel (2020) model agents’ learning about cross-sectional
predictive relationships. In this model there is evidence of in-sample
predictability which is not present out of sample. We address data mining on
two fronts. First we use a different loss function than that used in most
earlier studies. As noted above, even predictive content that is robust to
data mining concerns of alpha or the Sharpe ratio may not be useful to a CRRA
utility function. Our resampled out-of-sample design addresses the temporality
concerns, including learning. Finally the effects of overfitting are also loss
function dependent. We compare models, evaluate predictive efficacy and
characteristics’ roles using confidence intervals on optimal portfolio out-of-
sample certainty equivalent returns.

We evaluate sampling distributions of the out-of-sample certainty equivalent
returns for power utility investors with varying degrees of risk aversion
across 48 different sets of characteristics. We measure the marginal economic
effect of each characteristic on the small sample distribution of the
certainty equivalent in the context of the complementary characteristics
(Cochrane 2011). We show that no single characteristic can be used by all
power utility investors to outperform the relevant market benchmark.222As a
matter of exposition, we will use the phrase “Portfolio A outperforms the
benchmark” to mean that the 2.5%ile of the sampling distribution of portfolio
A’s certainty equivalent return exceeds the 97.5%ile of the sampling
distribution of the benchmark portfolio’s certainty equivalent return. The
“market benchmark” refers to either the equally-weighted portfolio of all
stocks or the value-weighted portfolio of all sample stocks, depending on
which the investor prefers. Our second major result is that the gains from
linking portfolio weights to measurable characteristics derive mostly from
complementarities amongst the characteristics as they relate to the cross-
section of future returns. All power utility investors can outperform the
benchmarks using the characteristic triplet: momentum, log size, and residual
volatility. As risk tolerance rises, average same-month return is added to the
optimal characteristic set, and the most risk-tolerant investors we consider
also include beta. We find that the book-to-market ratio is never part of an
optimal characteristic set because it is redundant in the context of size and
residual volatility. In their mean-variance setting, DeMiguel, Martín-Utrera,
Nogales, and Uppal (2020) show that combining characteristics has a salutary
trading-diversification effect. This characteristic complementarity is very
different from the forms we find. We document that the predictive density of a
small stock with positive momentum, for example, is very different from that
of a small stock with negative momentum.

Since much of the literature on identifying predictive characteristics seeks
to identify the incremental power of a factor conditional on another set of
factors, complementarity amongst highly correlated characteristics has been
largely ignored. For example, Feng, Giglio, and Xiu (2020) note that the
month-of-the-year factor (Heston and Sadka 2008) has a significant alpha
benchmarked to the Fama-French 3-factor model. However, since it has a
correlation of 0.63 with the momentum factor, including momentum in the
benchmarking model renders the portfolio alpha statistically insignificant. We
show that for a power utility investor with $\gamma=2$, adding average same-
month return to the characteristic set: size, beta, residual volatility, and
momentum, leads to a statistically significant increase in the out-of-sample
certainty equivalent return. Conversely, if this investor excludes momentum,
and allows weights to depend on size, beta, residual volatility, and average
same-month return, the resulting optimal portfolio suffers from severe
overfitting and fails to dominate the benchmark out of sample. In contrast, a
power utility investor with $\gamma=9$, derives no incremental benefits from
including average same-month return in the set of characteristics used to tilt
weights away from the market portfolio.

Our third major result demonstrates how and why characteristics are used to
construct preferred portfolios and how risk aversion interacts with
characteristic use. Our power utility investors use characteristics both to
achieve some exposure to the Fama-French-Carhart (FFC) factors, and to shift
away from the market. For example, the optimal portfolio for the power utility
investor with $\gamma=5$ has a market beta of 0.6, and only 3% of the
portfolio’s variance comes from the market variance. This portfolio has a
loading of 1.0 on the size factor (SMB), and derives 5% of its variance from
this source. By contrast, these values for the benchmark (equally-weighted
portfolio) are 0.5 and 8%, respectively. Similarly the optimal portfolio’s
loading on the value factor (HML) is 2.0, with 17% variance attribution. The
benchmark’s HML exposures are 0.3 and 2%. The momentum factor (MOM) accounts
for 20% of the optimal portfolio’s variance, as this portfolio’s MOM loading
is 1.5. The equally-weighted benchmark has a momentum loading of -0.1, and it
derives less than 1% of its variance from this factor. Finally, this
portfolio’s alpha is 269 basis points per month, and 69% of the portfolio
variance lies outside the span of the four factors.

Market beta is flat in $\gamma$, so that the percent of portfolio variance
attributed to the market factor increases in risk aversion. By contrast, the
optimal portfolio’s loading on the momentum factor decreases in risk aversion,
whereas the percent of variance attributed to this factor is constant across
$\gamma$. The loading on SMB and relative importance of this factor
monotonically decrease in $\gamma$. The optimal portfolio for the power
utility investor with $\gamma=2$ has a significantly positive SMB loading, the
optimal portfolio for the investor with $\gamma=5$ has an insignificant SMB
loading, and the loading on SMB of the optimal portfolio for the investor with
$\gamma=9$ is significantly negative. The value factor’s appeal increases in
risk aversion. It accounts for 37% of the portfolio variance of the optimal
portfolio of the investor with $\gamma=9$, and 20% of the optimal portfolio of
the investor with $\gamma=2$. The four factors account for 32%, 42%, and 54%
of optimal portfolio variance as $\gamma$ increases from 2 to 5 to 9.

These decompositions shed further light on the characteristic complementarity.
For the power utility investor with $\gamma=2$, adding average same-month
return to the characteristics: momentum, size, beta, and residual volatility
significantly reduces the portfolio’s exposure to the momentum factor, with no
change in exposure to the other three factors.

Figure 1 shows the out-of-sample return distribution of the optimal
characteristic-based portfolio for the power utility investor with $\gamma=9$
and the value-weighted (market) portfolio. The optimal portfolio is tilted
toward stocks with positive momentum, smaller market capitalizations, and
lower residual volatility (and away from stocks at the other end of the
spectrum along these dimensions). The optimal portfolio is symmetric, whereas
the benchmark has a negative skew. The optimal portfolio is significantly more
leptokurtotic than the market. The optimal portfolio’s mean return is 176
basis points per month, the market’s mean return is 106 basis points per
month. The optimal portfolio’s return interquartile range of 4.6 (percent per
month) is significantly less than the benchmark’s 5.2%. However the optimal
portfolio’s return standard deviation (4.5%) is not significantly different
from the benchmark’s 4.3%. This lack of consistency in the comparisons of
measures of the distributions’ scales highlights the importance of not
assuming that portfolio returns are normally distributed. In general, the
standard deviation is not a good measure of scale for a skewed and/or
leptokurtotic distribution. The mean (95%ile confidence interval) of the
optimal portfolio’s certainty equivalent return (to this investor) is 62
$\left(44\,,\,77\right)$ basis points per month. These statistics for the
benchmark are 10 $\left(4\,,\,15\right)$.

This complementarity of correlated characteristics together with the
positioning of portfolios outside the span of the Fama-French-Carhart factors,
our management of estimation risk, and the behavior of portfolios as risk
aversion increases are all consistent with the arguments in Kozak, Nagel, and
Santosh (2018) and the findings of Kozak, Nagel, and Santosh (2020), who only
consider the first two moments of the return distribution. A multivariate set
of correlated characteristics identifies tradeoffs in the data of odd for even
moments. A relatively risk tolerant investor wants exposure to a factor that
looks like a month-of-the-year factor, provided that it is also adjusted for
momentum and residual volatility. By comparison, a very risk averse investor
seeks refuge in the market factor. While Brandt, Santa-Clara, and Valkanov
(2009) show that with a single characteristic an optimized portfolio resembles
a characteristic-based long-short portfolio, when there are multiple
characteristics the resulting portfolio is not spanned by the univariate
characteristic-based factors.

2\. Portfolio Selection

2.1 Algorithm

In Brandt, Santa-Clara, and Valkanov’s (2009) algorithm, the vector $\theta$
is estimated to maximize a concave loss function over $T$ periods:

$\displaystyle\max_{\theta}\sum\limits_{t=0}^{T-1}\frac{(1+r_{p,t+1})^{1-\gamma^{*}}}{1-\gamma^{*}}\left(\frac{1}{T}\right)$
(1)

by allowing portfolio weights to depend on observable stock characteristics:

$\displaystyle
r_{p,t+1}=\sum\limits_{i=1}^{N_{t}}\left(\overline{\omega}_{i,t}+\frac{1}{N_{t}}\theta^{{}^{\prime}}x_{i,t}\right)\cdot
r_{i,t+1}\,,$ (2)

where: $x_{i,t}$ is the $K$-vector of zero-mean characteristics on firm $i$,
measurable at time $t$; $\overline{\omega}_{i,t}$ 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 $t$.333We 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). Unlike Brandt, Santa-Clara, and Valkanov or DeMiguel, Martín-
Utrera, Nogales, and Uppal (2020), we do not identify $\gamma^{*}$, the
parameter used to generate a feasible portfolio strategy in (1) with the
relevant statistical loss function. Instead, we consider a statistical loss
function (alternatively “an investor”) that takes the same form as (1) indexed
by $\gamma$ (the curvature of the statistical loss function which is pre-
determined and fixed). From this perspective, $\gamma^{*}$ is a choice
variable–a tool to manage estimation risk.

Our empirical design fixes $\gamma$, the curvature of the statistical loss
function. Our interest is in out-of-sample statistical comparisons across
portfolios generated by various feasible portfolio rules–from the perspective
of this nonlinear statistical loss function. An eligible portfolio rule
consists of a subset of characteristics, the in-sample period construction
protocol, and the curvature of the in-sample loss function, $\gamma^{*}$. By
definition, the optimal in-sample portfolio is obtained using all
characteristics and by setting $\gamma^{*}$ equal to $\gamma$. Whether this is
also true out of sample is an empirical question. If there is a positive
relationship between an optimal portfolio’s in-sample even-order moments and
its noise (i.e., higher out-of-sample even-order moments) then using a more
concave loss function than the investor’s utility function (i.e.,
$\gamma^{*}>\gamma$), may generate portfolios that are preferred to those
obtained by constraining $\gamma^{*}$ to equal $\gamma$.

2.2 Data and specifications

Our sample spans January 1955 through December 2018. Our out-of-sample period
comprises the 44 years 1975 – 2018. We use the following characteristics:
momentum (M), the book-to-market ratio (V), log size (S), beta ($\beta$),
market model residual standard deviation ($\sigma_{\epsilon}$), last year’s
same-month return, ($r_{t-12}),$ and the average same-month return over the
preceding five years, ($\overline{r}$). Momentum is the stock’s compounded
return from month $t-13$ though $t-2$. Equity market capitalization is the
market value of the company’s outstanding shares (aggregated across all 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 equity market capitalization, the book-to-
market ratio is the natural log of $B$. Size is equity market capitalization.
We estimate beta and the residual standard deviation by regressing monthly
returns from months $t-60$ through $t-1$ on the CRSP value-weighted index.
Thus the stocks in the January 1960 sample have no missing data from January
1955 through December 1959.444Our focus on out-of-sample statistics militates
a long time-series. Starting in 1955 constrains our choice of characteristics.
We use characteristics that can be directly computed from market prices, and
the book value from the CRSP-Compustat merged file.

We obtain all characteristic and return data from the merged CRSP–Compustat
file on WRDS. To be eligible for inclusion in the sample in month $t$, a stock
must have no missing returns in the CRSP database for the previous 60 months
(in order to construct the characteristics in month $t$), 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 Consumer Price Index from the Federal
Reserve (FRED)555We extracted the series CPIAUCSL: consumer price index for
all urban consumers: all items. 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 $99.1 million in December 2018. 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.666Green, Hand, and Zhang (2017) and Hou, Xue, and
Zhang (2020) highlight the importance of removing microcap stocks from studies
of return predictability. 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 389 (exclusively New York Stock Exchange) stocks in the final sample
in January 1960. The sample size jumps in August, 1967, from 674 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
1,420 stocks) when Nasdaq stocks are eligible to enter our sample. The maximum
number of stocks is 2,291 in April, 2006. There are 1,848 stocks in our sample
in August 2008 and 1,617 stocks on the last date in our sample, December 2018.

We normalize and standardize the characteristics so that the mean is zero and
the standard deviation is one. Inspection of (2) shows that optimal portfolio
weights will thereby sum to unity for any value of $\theta$. This also means
that the characteristics are observationally equivalent to shrinkage values.
For example, let $\beta$ be a stock’s OLS beta. Consider a prior-weighted
beta, such as $\beta^{S}=.5\cdot\beta+.5\cdot 1$. The normalized $\beta^{S}$
are identical to the normalized $\beta$. This implicit shrinkage mitigates the
usual concerns about outliers in characteristic space so we do not winsorize
the characteristics. A single observation ($\Psi_{i,t}$) 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 $i$, $i=1,\ldots,N_{t}$. Importantly,
$\Psi_{i,t}$ also includes the market capitalization of stock $i$ at $t-1$,
since the passive portfolio at $t$ (i.e., the portfolio when the
$\theta$-vector is zero) is the market-weighted portfolio at $t-1$.

Because financial firms’ leverage is potentially different from that of non-
financial firms we use two alternative treatments of the book-to-market
ratio.777This issue is discussed in Barber and Lyon (1997). The proportion of
financial stocks in our sample changes dramatically over the period from 1.5%
in January 1960 to 11.9% in January 1978, to 21.6% in July 1994, and ending at
19.9% in December 2018. In the first, we treat the book-to-market ratio the
same as other characteristics: we standardize it using all stocks, and allow
it to tilt their weights away from value-weighting. In the second treatment we
exclude financial firms from this characteristic’s standardization, and set
their standardized book-to-market ratio to zero so that their weights are
invariant with respect to this characteristic. This alternative treatment
means that there are two versions of every characteristic set that includes
the book-to-market ratio. We refer to the book-to-market ratio under the
treatment that exempts financial firms as $V_{-f}$.

Excluding the alternative treatment of the book-to-market ratio, there are 127
possible sets of our 7 characteristic variables. We consider 48 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). 23 of these 48 include the
book-to-market ratio, and each of these sets has two alternative treatments,
which means that we use 71 unique sets of characteristics/treatments. The base
48 sets, maximized over the book-to-market treatment are reported in Table 1.
We consider 14 values of $\gamma^{*}$: $2,\ldots,13,$ 16, 22.

The first in-sample period comprises the 180 months of January 1960 through
December 1974\. We use this sample to solve for the $\theta$-vector which we
use to construct the out-of-sample feasible portfolio returns over the next 12
months (January through December, 1975). We estimate a new $\theta$-vector at
the beginning of each out-of-sample year. The out-of-sample period consists of
the 528 months from January 1975 through December 2018. We obtain the
$\theta-$vector using all available data through the end of year $T-1$ for
$T=1,\cdots,44$ 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 $\theta$-vector). Since the portfolio is defined by
$\theta$, 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 2018, so the last
optimization/$\theta$ estimation (applied to the 12 months of 2018, out of
sample) uses 696 months under the updating protocol, and the 180 months from
2003 through 2017 under the rolling protocol. This means that we optimize
expected utility (and estimate the $\theta-$vector) from each of the 1,988
cases (71 characteristic/treatment sets $\times$ 14 values of $\gamma^{*}$
$\times$ 2 protocols) 44 times–at the end of each in-sample year.

2.3 Empirical design

The possibility that investors are learning about the data generating process,
the fact that optimal portfolio selection amplifies estimation error, and the
possibility of intertemporal changes in the conditional return distribution
argue for analyzing optimal portfolios on an out-of-sample basis. We use a
resampled–out-of-sample procedure to construct sampling distributions of the
functions of interest, such as certainty equivalent, portfolio loading on
factors, portfolio skewness, etc. The data in month $t$ consists of the
$N_{t}$-row vector $\Psi_{t}$, with elements $\Psi_{i,t}$, for
$i=1,\dots,N_{t}$. We stack these $T$ row-vectors (of varying lengths) to form
the array $\Psi$. This is not a traditional panel since stock $i$ at time $t$
is different from stock $i$ at any other time, and as noted, the number of
columns is different for each row.

Our resampling is nonparametric cross-sectional bootstrap (see Kapetanios
2008), motivated by the single period optimization problem with out-of-sample
inference. It is nonparametric since we draw from the raw data. It assumes
that returns are temporally independent conditional on $\Psi$. As such,
$\Psi^{*}_{t}$ is a cross sectional resample from $\Psi_{t}$, for months
$t=1,\dots,T$. Consistency holds under $N-$asymptotics (as the number of
stocks in each period increases), since we can view $\theta$ as a GMM
estimator. Preserving the original time series in all resamples allows us to
evaluate the effects of structural instability on out-of-sample portfolios,
and also maintains the cyclical (month of the year) and serial dependence
patterns inherent in the design of $\Psi_{t}$. A resampled draw for month $t$
resamples (with replacement) an $N_{t}$-row vector from $\Psi_{t}$. Thus each
pseudosample consists of the same number of observations in each period as the
original sample, we preserve the calendar structure of the original data, and
we maintain the same in-sample estimation, out-of-sample evaluation periods as
in the data. We also maintain the temporal structure of the investment
opportunity set. In the original data $\Psi_{t}$ includes the raw values of
the characteristics as well as the market weight of the stock at $t-1$. Once
we draw a resampled cross-section from $\Psi_{t}$we normalize (to 0-mean
characteristics), standardize (to characteristics with unit variances), and
construct the value-weighted market weights for each resampled stock. Thus our
resampling design preserves the integrity of the investment opportunity set at
each $t$. This critically includes the relationship between stocks’
characteristics and sizes. We also maintain the integrity of the seasonality
in the data. The month of January, for example occurs only once in any
12-month cycle, and its timing is deterministic. The matrix of characteristics
at $t$ is not independent of the size of each stock at $t$. We manage concerns
about unmodeled temporal dependencies by maintaining the out-of-sample test
design with each bootstrapped sample.

We also stress that we are not relying on resampling to evaluate the
algorithm. We use out-of-sample validation. In each of our pseudosamples time
plays the same role as it does in generating the data. We use resampling to
construct sampling distributions of the out-of-sample properties, such as
certainty equivalent and market exposure, of the optimal portfolio rules.

Once we have a pseudosample draw we obtain a $\theta$-vector by maximizing (1)
at the end of year 15, for a given feasible portfolio rule. We use this
$\theta$-vector to construct the out-of-sample portfolio returns in each of
the next 12 months. We repeat the optimization and out-of-sample portfolio
return construction at the end of each year $16,\dots,59$. Thus we obtain 44
$\theta$-vectors and 528 monthly feasible portfolio returns from each
pseudosample. We take 10,000 pseudosamples to construct the sampling
distributions for each feasible portfolio rule.

We use the empirical Monte Carlo small sample distributions of the certainty
equivalent return to select the optimal feasible portfolio rule. As
econometricians we view each of the pseudosamples as being equally likely. We
consider that Portfolio A statistically significantly dominates Portfolio B if
the 2.5%ile value of the sampling distribution of A’s certainty equivalent
return (the lower bound of the 95% statistical confidence interval) is greater
than the 97.5%ile value of the sampling distribution of B’s certainty
equivalent return. Therefore we rank portfolio rules using the 2.5%ile value
of the out-of-sample certainty equivalent return’s sampling distribution. So
the portfolio that we identify as optimal for a given loss function, indexed
by $\gamma$ is maximally dominant, and not dominated by any other portfolio.
However, as stressed by Lewellen, Nagel, and Shanken (2010 p. 185) the
sampling error in stock returns is large relative to mean returns so we report
confidence intervals of all of our functions of interest in order “to make
sampling issues more transparent.”

3\. Algorithm efficacy: Noise and estimation risk

Table 1 shows the sampling distribution of the out-of-sample certainty
equivalent returns on the optimal portfolios under each of the three loss
functions, conditional on each of the 48 characteristic sets, maximized over
the dimensions of $\gamma^{*}$, sample construction (rolling or updating), and
book-to-market treatment (when relevant). The global optimal portfolios (i.e.,
maximally dominant over all feasible portfolio rules) are emboldened. The
higher of the two benchmarks–equally-weighted and value-weighted portfolios of
sample stocks–for each investor are also provided for comparison. This table
shows that restricting $\gamma^{*}$ to equal $\gamma$ leads to poor out-of-
sample performance, that this effect and the optimal characteristic set vary
across $\gamma$, and that the updating sample construction protocol is
preferred to rolling.

Panel A contains results for the $\gamma$ 2 loss function, where the global
optimal characteristic set consists of momentum, log size, beta, average same-
month return, and residual standard deviation.888As the only statistical loss
functions that we consider are power utility functions, we will write
$\gamma\,k$ loss function to mean “power utility function with coefficient of
relative risk aversion, $\gamma=k$.” We will use the phrases $\gamma\,k$ loss
function and $\gamma\,k$ investor interchangeably. The globally optimal
portfolio is achieved with $\gamma^{*}=3$, updating sample construction, and
characteristic set: momentum, log size, beta, average same-month return and
residual volatility. This portfolio’s mean (95% confidence interval) certainty
equivalent (basis points per month) 302 $(257\,,\,352)$, dominating the
equally-weighted benchmark with its 95%ile certainty equivalent confidence
interval of $(101\,,\,105)$. Seven of the 48 characteristic sets generate
optimal portfolios that are not statistically dominated by the global optimal.
For all of these seven, the optimal $\gamma^{*}$ is higher than 3; only one
(that drops beta from the optimal set) has four characteristics. The others
all have five, six, or seven characteristic variables. Fixing the
characteristic set for this loss function, the rolling protocol is only
preferred to updating when the book-to-market ratio is included in the
characteristic set. But there are characteristic sets that include the book-
to-market ratio where updating outperforms rolling.

Panel B shows the optimal portfolios from the perspective of the $\gamma$ 5
loss function. In this case the global optimal portfolio is achieved with
characteristics: momentum, log size, average same-month return, and residual
standard deviation; $\gamma^{*}=10$; and updating sample construction. The
mean (95% confidence interval) certainty equivalent (basis points per month)
on this portfolio is 152 $(133\,,\,170)$. This portfolio dominates this loss
function’s equally-weighted benchmark with its 95%ile certainty equivalent
confidence interval of $(57\,,\,62)$. For this loss function there are nine
alternative characteristic sets that generate portfolios that are not
statistically dominated by the global optimal portfolio. Only one of these
sets has three characteristics, all others have four or more. As was the case
for the less concave loss function in Panel A, rolling sample construction is
only preferred to updating when the characteristic set includes the book-to-
market ratio.

Panel C reports the results from the perspective of the power utility function
with $\gamma=9$. Here the global optimal portfolio is achieved with
$\gamma^{*}=16$; characteristic set: momentum, log size, and residual standard
deviation; and the updating sample construction. The global optimal
portfolio’s mean (95% confidence interval) certainty equivalent (basis points
per month) is 62 $(44\,,\,77)$, which dominates the value-weighted benchmark
with its 95%ile certainty equivalent of $(4\,,\,15)$. For this loss function
21 of the 48 optimized portfolios’ 95% certainty equivalent confidence
intervals include zero; 14 of the 48 portfolios dominate the benchmark and are
not statistically dominated by the global optimal portfolio. These include two
which condition on only a single characteristic: either beta or residual
volatility.

For each loss function, Table 1 Panel D isolates the characteristic set
containing the book-to-market ratio which produces the highest 2.5%ile
certainty equivalent return for each of the three loss functions. This panel
shows the effects of treating financial firms differently as well as sample
construction. In all three cases the choice of sample construction is more
important than the treatment of financial stocks. And for these characteristic
sets, updating always outperforms rolling for all three loss functions and for
either treatment of financial firms.

Table 2 shows the effect of varying the curvature of the in-sample loss
function $\gamma^{*}$ and the in-sample construction protocol on out-of-sample
portfolio certainty equivalent returns for the globally optimal characteristic
set and another characteristic set for each of the three loss functions. This
table shows the importance of using $\gamma^{*}>\gamma$ as a tool to mitigate
estimation risk. It also shows that once $\gamma^{*}$ reaches a critical level
above the loss function’s curvature of $\gamma$, further increases in
$\gamma^{*}$ have fairly small effects on the resulting portfolio’s certainty
equivalent returns.

Table 2 Panel A shows the effects of changing $\gamma^{*}$ from $2,\ldots,12$
for the $\gamma$ 2 loss function and the optimal characteristic set along with
the characteristic set that includes all characteristics except momentum. This
case provides an example of the well-documented problem of the poor out-of-
sample properties of in-sample optimal portfolios. For the globally optimal
characteristic set restricting $\gamma^{*}$ to equal $\gamma$ creates a
feasible portfolio rule whose mean certainty equivalent is -42%, with 95%
confidence interval, $\left(-100\%\,,\,3\%\right)$ per month.999The certainty
equivalent in a single draw is -100% when at least one of the 528 monthly
returns in the out-of-sample period is less than -100%. Increasing
$\gamma^{*}$ holding all else constant shows that much of the estimation risk
in this context is the result of overfitting. Increasing the cost of even
moments–expressed in terms of odd moments–has a dramatic salutary effect.
Fixing $\gamma^{*}$ there are 142 optimal portfolios (across the
characteristic sets and sample formation protocol). Restricting
$\gamma^{*}=\gamma=2$ causes 94 of these 142 portfolios’ 2.5%ile out-of-sample
certainty equivalent return to be -100%.

Overfitting is more severe when the book-to-market ratio replaces momentum in
the characteristic set. For this characteristic set, with either updating or
rolling, $\gamma^{*}$ must exceed 5 in order for the 2.5%ile of the certainty
equivalent’s sampling distribution to exceed -100%. The maximally dominant
portfolio with this characteristic set is obtained with $\gamma^{*}=8$ under
the updating protocol and $\gamma^{*}=9$ under the rolling protocol. Both of
these portfolios are statistically significantly dominated by the global
optimal portfolio for the loss function indexed with $\gamma=2$.

Table 2 Panel B shows the out-of-sample certainty equivalent returns of
optimal portfolios from the perspective of the $\gamma$ 5 loss function. Under
the updating protocol this loss function is less sensitive to estimation risk
than is the $\gamma$ 2 loss function. Further, estimation risk is more severe
under the rolling protocol. Table 1 showed that this investors’ optimal
characteristic set consists of momentum, log size, average same-month return,
and residual volatility. The updating protocol is optimal, and the optimal
value of $\gamma^{*}$ is 10. As is the case with the less concave loss
function, the optimal value of $\gamma^{*}$ varies with the characteristic set
and protocol. In general, a higher value of $\gamma^{*}$ is needed to mitigate
overfitting under the rolling protocol than under the updating protocol. Here
restricting $\gamma^{*}=\gamma$ results in a 2.5%ile certainty equivalent
return of -100% in 18 of the 71 cases with the updating protocol, and 83 of
the 142 cases (under both protocols) out-of-sample certainty equivalent 95%ile
confidence intervals include zero. Eight of these 71 portfolios statistically
dominate the benchmark, and 10 are not statistically dominated by the global
optimal portfolio.

As in Table 1, Table 2 Panel C shows that the optimal portfolio for the power
utility loss function with $\gamma=9$ is achieved with feasible portfolio rule
comprising the updating protocol, $\gamma^{*}=16$, and characteristic set:
momentum, log size, and residual volatility. The 2.5%ile of the out-of-sample
certainty equivalent’s sampling distribution is less than zero in 112 of the
142 optimal in-sample portfolios generated when $\gamma^{*}=\gamma=9$.
Imposing this constraint produces eight portfolios that statistically dominate
the value-weighted index. Whereas, Table 1 Panel C shows that 15 (of the 48)
characteristic sets produce portfolios that dominate this benchmark under the
updating protocol with $\gamma^{*}$ unconstrained.

In general, for all three loss functions overfitting is more severe under the
rolling protocol than the updating protocol. One manifestation of this is that
the optimal value of $\gamma^{*}$ under the former is greater than under the
latter. For example for the $\gamma$ 2 loss function the best rolling
portfolio is achieved using all seven characteristics and $\gamma^{*}=6$. For
the $\gamma$ 5 investor the optimal rolling portfolio is achieved with
$\gamma^{*}=13$. Panel C shows that when the characteristic set includes
average same-month return, there is no value of $\gamma^{*}$ that produces a
positive 2.5%ile certainty equivalent for the power utility function with
$\gamma=9$ under the rolling protocol. This is not the case with the updating
protocol.

Table 2 also shows that the effects of overfitting are not monotonic in
$\gamma$. These effects are more pronounced for the $\gamma$ 2 investor and
the $\gamma$ 9 investor than for the $\gamma$ 5 investor. In fact, under the
updating sample formation, using characteristics: momentum, size, beta,
average same-month return, and residual volatility, restricting
$\gamma^{*}=5=\gamma$ produces a portfolio that statistically dominates this
investor’s preferred market benchmark.

All panels in both Tables 1 and 2 highlight the fact that the sampling
variations of the certainty equivalent returns are large. The global optimal
portfolio does not dominate any of the other portfolios in Table 2, for both
the $\gamma$ 5 loss function (Panel B) and the $\gamma$ 9 loss function (Panel
C). By contrast, the global optimal portfolio for the $\gamma$ 2 loss function
(Panel A) statistically dominates 29 of the other 43 portfolios in this panel.
These large sampling variances highlight the usefulness of using the 2.5%ile
of the (out-of-sample) certainty equivalent returns’ sampling distribution to
rank the portfolio rules.

These three loss functions are representative of others. The optimal
characteristic set for power utility loss functions with $\gamma=3$ and
$\gamma=4$ is the same as for the $\gamma$ 2 loss function. The $\gamma$ 3
loss function is maximized with $\gamma^{*}=5$, and the $\gamma$ 4 loss
function is maximized with $\gamma^{*}=7.$ Similarly, the $\gamma$ 6 and
$\gamma$ 7 loss functions have the same optimal characteristic set as the
$\gamma$ 5 loss function. The global out-of-sample maximum for both of these
loss functions is achieved with $\gamma^{*}=12$, under the updating protocol.
Both the $\gamma$ 8 and $\gamma$ 10 loss functions’ global optimal portfolios
are formed using the same characteristic set as the $\gamma$ 9 loss function,
and $\gamma^{*}$ of 16. All loss functions prefer updating to rolling. The
out-of-sample certainty equivalent return’s 95% confidence interval of the
optimal portfolio for the power utility function with $\gamma=10$ is
$\left(22.5\,,\,62.9\right)$ basis points per month. An investor with these
preferences would choose to invest in Treasury bills over any equity position.

As noted in Section 2.2, we have Monte Carlo out-of-sample results for 1,988
portfolios. The 48 optimal portfolios in Table 1 have the highest 2.5%ile
certainty equivalent across the dimensions of $\gamma^{*}$, protocol (rolling
or updating), and treatment of the book-to-market ratio. Since we consider 14
integer values of $\gamma^{*}$, each of these portfolios is the optimal of 56
portfolios for the 23 characteristic sets that include the book-to-market
ratio, or 28 for the 25 characteristic sets that do not. Even though we do not
estimate conditional moments, overfitting degrades the out-of-sample
performance of the optimal portfolios. This is especially severe for the
least- and most- concave loss functions that we consider. Using a loss
function in-sample that places a higher penalty on extreme losses than the
actual (out-of-sample) loss function mitigates the unwelcome effects of noise
on portfolio weights.

4\. How characteristics affect optimal portfolios

In this section we evaluate properties of the global optimal portfolio
returns, and compare these to the benchmarks. We also isolate the effect of
each characteristic on the sampling distributions of various portfolio
properties. To this end, we keep $\gamma^{*}$ and the protocols fixed at their
optimal values for each of the three loss functions indexed by $\gamma$, and
vary the characteristic set. We start with the optimal set for each loss
function, and remove each characteristic in turn. We examine the effect of
each characteristic on certainty equivalent returns and unconditional
portfolio moments in Section 4.1. We assess the complementarities between the
characteristics by examining the effects of removing an optimal characteristic
on the $\theta$ coefficients that link the remaining characteristics to
portfolio weights in Section 4.2. In Section 4.3 we examine the role of each
characteristic in increasing or decreasing portfolio exposure to the Fama-
French-Carhart factors. We conclude this section by summarizing across these
dimensions at the characteristic level.

4.1 Certainty equivalents and unconditional moments

Table 3 contains sampling distribution properties of: certainty equivalent
returns, unconditional portfolio moments, and other portfolio properties for
benchmarks and optimal portfolios from the perspectives of each of the three
loss functions. Certainty equivalent returns are reported in basis points per
month. The mean return, median return, return standard deviation, and
interquartile return range are all reported in percentage (per month) terms.
The robust measures of skewness ($S_{4}$) and kurtosis ($K_{3}$) are
recommended by Kim and White (2003):

$\displaystyle S_{4}=\frac{\mu-r_{.5}}{\sigma}$ (3)

and

$\displaystyle
K_{3}=\frac{\overline{r}^{+}_{.95}-\overline{r}^{-}_{.05}}{\overline{r}^{+}_{.5}-\overline{r}^{-}_{.5}}-2.63$
(4)

where: $\mu$ is the mean portfolio return over the 528-month out-of-sample
period, $\sigma$ is the return standard deviation, $\overline{r}^{+}_{.95}$ is
the mean of the highest 5% of returns, $\overline{r}^{-}_{.05}$ is the mean of
the smallest 5% of returns, $\overline{r}^{+}_{.5}$ is the mean of the top
half of returns, and $\overline{r}^{-}_{.5}$ is the mean of the bottom half of
returns; and $r_{.5}$ is the median out-of-sample return.101010As per Kim and
White (2003), in our pseudosamples $S_{4}$, 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.

Panel A shows that the $\gamma$ 2 loss function strictly prefers the equally-
weighted portfolio to the value-weighted portfolio of sample stocks. The
$\gamma$ 5 loss function is indifferent between the two statistically. An
investor whose utility function matches the $\gamma$ 9 loss function would
prefer the value-weighted portfolio, but is so averse to risk that she would
prefer the risk-free asset (offering an average of 37 basis points per month)
to the value-weighted portfolio, whose 95%ile certainty equivalent return
confidence interval is only $\left(4\,,\,15\right)$ basis points per month.
Both benchmarks are more leptokurtic than the normal distribution, and both
are significantly negatively skewed.

Panel B shows eight portfolios from the $\gamma$ 2 loss function’s
perspective, starting with this loss function’s global optimal portfolio. This
portfolio has a 2.5%ile certainty equivalent of 258 basis points per
month–which statistically dominates the equally-weighted benchmark. Table 1
shows the optimal value of $\gamma^{*}$ is 3 in this case, so we fix
$\gamma^{*}=3$ to evaluate the comparative statics relating to the
characteristics of the other seven portfolios in this panel.

The 95%ile confidence interval of the $\gamma$ 2 loss function’s global
optimal portfolio’s annualized Sharpe ratio is $\left(1.07\,,\,1.27\right),$
which is significantly larger than the equally-weighted portfolio’s
$\left(0.61\,,\,0.64\right)$. This portfolio’s skewness and kurtosis are not
significantly different from the benchmark, although unlike the benchmark,
this portfolio is symmetric. This portfolio uses leverage to exploit the
conditional cross-sectional return distribution, and to take on significantly
more risk than the benchmark.

Dropping momentum from the optimal characteristic set increases the sampling
standard deviation of the resulting portfolio rule’s out-of-sample monthly
certainty equivalent return from 106.1 to 3,366.8 basis points, and the range
of its 95% confidence interval from 94 to 10,279 basis points. Unlike the
optimal portfolio, this portfolio does not dominate the benchmark, despite the
fact that none of the moments or properties shown in Table 3 differ
significantly from the global optimal portfolio. The (asymmetric) loss
function amplifies the effect that removing momentum has on the sampling
distribution of the minimum out-of-sample return. In 2.5% of the 10,000
pseudosamples the minimum optimum portfolio monthly return is less than
$-115$%, and in half of the samples the minimum return is less than $-79$%. By
contrast the 2.5%ile minimum return from the global optimal portfolio rule is
$-68$%. This confirms the evidence from Table 2 that adding momentum to the
other four optimal characteristics reduces overfitting. This complementary
property of momentum can only be appreciated out of sample, and with a non-
linear loss function. We will explore this aspect of momentum in more detail
as we further analyze the nature of these portfolios in the next subsections.

Dropping log size from the optimal portfolio characteristic set significantly
lowers the certainty equivalent return to $\left(170\,,\,248\right)$, the
Sharpe ratio to $\left(0.83\,,\,1.03\right)$, and the portfolio median return
from $\left(426\,,\,619\right)$ to $\left(253\,,\,402\right)$ basis points per
month. Unlike the globally optimal portfolio, this portfolio is significantly
more leptokurtic than the benchmark. It is also the only portfolio in this
table that is positively skewed. Size is the only characteristic whose
marginal addition to the others results in a significant increase in the
median out-of-sample return. In contrast with momentum, removing log size from
the characteristic set has virtually no effect on the resulting optimal
portfolio rule’s out-of-sample minimum monthly return.

On the margin, adding residual volatility to the other four optimal
conditioning characteristics also significantly increases the certainty
equivalent return. Adding this characteristic does not significantly alter any
of the first four moments, but the Sharpe ratio is significantly increased
from $\left(0.84\,,\,1.01\right)$. Removing either log size or residual
volatility from the optimal characteristic set constrains this risk tolerant,
$\gamma$ 2 loss function’s ability to take advantage of the conditional cross-
section. By contrast, beta is not as important in constructing the conditional
distribution. Removing beta from the optimal characteristic set does not
significantly reduce certainty equivalent return, and (unlike the removal of
momentum), does not generate a portfolio which fails to dominate the
benchmark.

Adding average same-month return to the other four optimal characteristics
significantly increases portfolio leverage from $\left(417\,,\,522\%\right)$
to $\left(541\,,\,699\%\right)$, while increasing the 2.5%ile of the minimum
monthly return from -86% to -68%. The marginal effects of average same-month
return include a significantly higher out-of-sample portfolio expected return
and Sharpe ratio. It has the largest marginal effect on reducing kurtosis,
although this effect is not statistically significant.

Adding the book-to-market ratio to the global optimal characteristic set
lowers the 2.5%ile certainty equivalent return from 258 to 247 basis points
per month, but has no statistically significant effects on the unconditional
moments or portfolio leverage.111111In this context there is virtually no
difference between the protocol that exempts financial stocks from
conditioning on the book-to-market characteristic and the protocol that does
not differentiate financial stocks in this way. The 95%ile certainty
equivalent confidence interval for the former case is
$\left(246.2\,,\,353.4\right)$, using $\gamma^{*}=3$. This is consistent with
the results shown in Table 2 Panel D for this loss function. In that case we
optimize over $\gamma^{*}$ as well. For this set of characteristics the
optimal $\gamma^{*}=4$. Here we use $\gamma^{*}=3$ for the purpose of
comparative statics. The certainty equivalent effect is due to the slightly
higher kurtosis and lower minimum return. This suggests that the book-to-
market ratio is redundant in the context of the five optimal characteristics.

Substituting the book-to-market ratio for log size in the context of the other
four optimal characteristics extenuates in-sample overfitting, as the width of
the out-of-sample certainty equivalent return’s 95% confidence interval
increases from 94 to 127 basis points. This effect is the result of an
increase in the sampling variance of the minimum return, whose 2.5%ile drops
to -86% with this characteristic set.

Table 3 Panel C provides basic unconditional out-of-sample properties of 8
portfolios from the perspective of the $\gamma$ 5 loss function. As Table 1
shows, this loss function’s optimal (out-of-sample) portfolio is formed using
the characteristic quartet: momentum, log size, average same-month return, and
residual volatility; $\gamma^{*}=10$, and the updating protocol. Accordingly,
all of the portfolios in Panel B are generated using $\gamma^{*}=10$. Here the
optimal portfolio has an out-of-sample certainty equivalent return of
$\left(133\,,\,170\right)$ basis points per month and Sharpe ratio of
$\left(1.15\,,\,1.32\right)$, which are more than twice those of the
benchmark. This portfolio is much less levered than the $\gamma$ 2 loss
function’s optimal portfolio, with an average investment of 244% of equity.
This portfolio’s mean, median, standard deviation, and kurtosis are all
significantly higher than the benchmark’s. However, its monthly return
interquartile range $\left(557\,,\,695\right)$ basis points per month is not
significantly higher than the benchmark’s $\left(590\,,\,619\right)$.
Interquartile range is a robust measure of scale that is not distorted by the
distributions’ leptokurtosis.

For the $\gamma$ 5 loss function, removing either momentum or average same-
month return does not significantly lower the certainty equivalent return.
However, because conditioning on momentum–in conjunction with the other
characteristics–reduces overfitting, the portfolio formed without momentum
does not significantly dominate the benchmark. Dropping momentum increases the
width of the certainty equivalent return’s 95% confidence interval from 38 to
90 basis points per month. Removing average same month return lowers leverage
significantly from $\left(123\%\,,\,168\%\right)$ to
$\left(96\%\,,\,121\%\right)$, but otherwise has no significant effects on any
of the moments or properties reported in Table 3 Panel C, for the $\gamma$ 5
loss function.

Removing log size from the optimal characteristic set for the $\gamma$ 5 loss
function results in a significant drop in certainty equivalent return–which is
due to a significant drop in the first moment, and not the second moment. This
is also reflected in the resulting portfolio’s significantly lower Sharpe
ratio $\left(0.87\,,\,1.03\right)$. This suggests that log size is
particularly useful in the context of momentum and residual volatility at
boosting expected return. The residual standard deviation is the only
characteristic for the $\gamma$ 5 loss function whose removal not only
significantly lowers the certainty equivalent return (to
$\left(58\,,\,73\right)$ basis points per month), but also results in a
portfolio that does not dominate the benchmarks. We will return to the role of
residual volatility in this case in Section 4.3

Adding the book-to-market ratio to the four optimal characteristics for the
$\gamma$ 5 loss function does not improve the portfolio in any way. Adding
this characteristic has little effect on the resulting portfolio’s out-of-
sample properties. Relative to the optimal for this loss function the minimum
monthly return is lowered, manifesting in a 10 basis point per month reduction
in the out-of-sample certainty equivalent return’s 2.5%ile sampling value.
Similarly, adding beta to the optimal characteristic set does nothing to
improve the portfolio, and has almost no discernible effect on any of the out-
of-sample properties shown in Panel C.

The $\gamma$ 9 loss function’s optimal portfolio, and related portfolios
selected using $\gamma^{*}=16$ are reported in Table 3 Panel D. While this
loss function prefers the riskless asset to the value-weighted portfolio of
sample stocks, cross-sectional predictability is reliable enough to lever 1.77
times, conditioning portfolio weights on momentum, log size, and residual
volatility. This loss function’s optimal portfolio return standard deviation
of $\left(433\,,\,477\right)$ basis points per month is not statistically
different from the value-weighted portfolio’s $\left(427\,,\,437\right)$,
while its expected return $\left(164\,,\,189\right)$ and median return
$\left(170\,,\,208\right)$ are significantly higher than this benchmark’s
$\left(102\,,\,110\right)$ and $\left(121\,,\,144\right)$, respectively.
Figure 1 shows the densities of these two portfolios, which are easy to
compare visually because of their similar scales. The figure and the summary
statistics show that the optimal portfolio is significantly more leptokurtic
than the benchmark, and has a significantly lower interquartile range
$\left(4.33\,,\,5.02\right)$ than the benchmark $\left(5.04\,,\,5.42\right)$.
Together these features suggest that the portfolio is so different from the
Gaussian distribution that the standard deviation is not a sufficient
statistic for the scale. That is, the optimal portfolio’s departures from
normality imply that the standard deviation is not a proper measure of the
portfolio’s riskiness from the perspective of the $\gamma$ 9 power utility
function.

Figure 1 and Table 3 also show that unlike the benchmark, the optimal
portfolio is not negatively skewed. The area in the figure between -10% and 0%
(monthly return) is clearly higher under the benchmark, whereas the opposite
is true between 2.5% and 10%. The difference is especially large between -6%
and -8%. The $\gamma$ 9 loss function’s heightened asymmetry precludes the
portfolio from attaining the Sharpe ratio $\left(0.99\,,\,1.14\right)$ of the
more risk tolerant loss functions (the $\gamma$ 5 optimal portfolio’s Sharpe
ratio is $\left(1.15\,,\,1.32\right)$).

For this loss function and optimal characteristic set removing momentum does
not mitigate overfitting, as was the case with the flatter loss functions and
higher dimensional optimal characteristic spaces. Although dropping momentum
does result in a lower 2.5%ile minimum monthly return–despite the reduced
dimensionality of the parameter space and significantly lower leverage. The
portfolio mean, median, and standard deviation are all significantly lower
with momentum removed from the optimal set, and $\gamma^{*}=16$.

Removing log size from the optimal characteristic triplet for this loss
function does not significantly lower the certainty equivalent return, but
leverage and the Sharpe ratio are significantly lowered. As was the case for
the $\gamma$ 5 loss function, removing residual volatility significantly
adversely affects every desirable attribute of the optimal portfolio. In fact,
for this loss function the optimal portfolio conditional only on momentum and
size is dominated by the riskless asset.

As with the other two loss functions, adding the book-to-market ratio to the
optimal characteristic set adds only estimation risk, so that it is redundant
in this context. A utility function with $\gamma=9$ is too risk-averse to
benefit from the boost to portfolio expected return provided by the average
same-month return. Adding beta and average same-month return to the optimal
characteristic triplet, using $\gamma^{*}=16$ amplifies the role of in-sample
noise. The width of the certainty equivalent 95% confidence interval increases
from 33 to 77 basis points, and the resulting portfolio does not dominate the
value-weighted benchmark or the risk-free rate.

4.2 Coefficients linking portfolio weights to characteristics

Empirical designs with linear loss functions evaluate a characteristic’s
efficacy by the statistical significance of the coefficient on that
characteristic (e.g., DeMiguel, Martín-Utrera, Nogales, and Uppal 2020; and
Freyberger, Neuhierl, and Weber 2020). This is not the case with our empirical
design. We are interested in the size and sign of the coefficients from
alternative specifications in order to understand how characteristics affect
portfolio properties and how they complement, or substitute for, one another.
However, the statistical significance of a $\theta$ coefficient is neither a
necessary nor sufficient condition for a characteristic’s efficacy. This is
because we select the optimal model based on out-of-sample certainty
equivalent, not in-sample properties of $\theta$. Table 4 reports the sampling
distributions of the $\theta$ coefficients that link portfolio weights to
characteristics for the 22 portfolios in Table 3. For each feasible portfolio
rule there are 44 (annual) estimates of the $\theta$-vector. We average these
across the 44 years; the table reports the small sample properties of these
annual averages. All three globally optimal portfolios have significantly
positive coefficients on momentum, and significantly negative coefficients on
size and residual volatility. The sense in which momentum complements the
other characteristics is evident in the comparative statics in this table. For
all three loss functions removing momentum does not significantly alter any of
the $\theta$ coefficients on the remaining characteristics. This is consistent
with the conclusion from the portfolio comparisons that the complementarity
between momentum and the other optimal characteristics arises primarily out of
sample.

With or without momentum, the algorithm puts more weight into stocks that are
relatively small, have relatively low residual volatility, and relatively high
average same-month average return based on the relationships between these
variables and future returns (in the sample). Table 3 shows that when momentum
is not part of the characteristic set these weights result in a portfolio with
such degraded out-of-sample properties that it is unacceptable to the $\gamma$
2 loss function using $\gamma^{*}=3$. However, if a stock with higher
weighting based on these four characteristics also has relatively low
momentum, then adding momentum to the characteristic set lowers the weight–
mitigating the effect of overfitting on portfolio formation. Similarly without
momentum in the characteristic set, the algorithm will heavily short
relatively large, high volatility stocks with low average same-month returns.
Allowing the algorithm to complement these weights by also weighting momentum
dramatically reduces the adverse effects of noise on the portfolio’s
performance (out of sample). Supplementing the predictive content of size,
volatility, and same-month return with momentum increases these
characteristics’ signal-to-noise ratio, resulting in improved out-of-sample
performance. This property of momentum is especially highlighted by the
asymmetric loss function.

In contrast to momentum, average same-month return is largely independent of
the other characteristics–both in- and out of-sample. For both the $\gamma$ 2
and $\gamma$ 5 loss functions removing average same-month return from the
respective optimal characteristic set does not induce a significant change in
any of the other $\theta$ coefficients. Similarly, adding beta and average
same-month return to the three optimal characteristics for the $\gamma$ 9 loss
function, with $\gamma^{*}=16$ (Panel C) has little effect on the other three
coefficients.

Beta is in the optimal characteristic set for only the least concave (most
risk tolerant) loss function. Removing beta from this optimal characteristic
set has no significant effects on the remaining $\theta$ coefficients. When
log size is removed, the coefficient on beta becomes strictly positive,
suggesting that high beta stocks with low residual volatility substitute for
small stocks (also with low residual volatility). Table 3 shows that this
substitution is less than satisfactory for the $\gamma$ 2 loss function, as
the certainty equivalent of the resulting portfolio is significantly less than
the optimal portfolio.

When residual volatility is removed from the optimal characteristic set for
the $\gamma$ 2 loss function the $\theta$ coefficient on beta becomes strictly
negative. There is something about high residual volatility stocks that all
investors want to avoid, and about low residual volatility stocks that all
risk averse investors want heightened exposure to. When this characteristic is
not available investors use beta as an imperfect proxy. Table 3, Panel B shows
that beta is a poor proxy because when it is the only volatility
characteristic both the $\gamma$ 2 loss function certainty equivalent return
and Sharpe ratio are significantly lower than the optimal portfolio’s (which
also conditions on residual volatility). Furthermore, whereas removing
residual volatility changes the sign of the $\theta$ coefficient on beta,
removing beta does not significantly affect the $\theta$ coefficient on
residual volatility (nor does it significantly change the certainty equivalent
return).

Panel B of Table 4 reports comparative statics on the $\theta$ coefficients
for optimal portfolios obtained with $\gamma^{*}=10$, starting with the global
optimal portfolio from the perspective of the $\gamma$ 5 loss function. In
this case the $\theta$ coefficient on log size is significantly negative, with
95% sampling confidence interval $\left(-5.2\,,\,-2.8\right)$. When we remove
residual volatility from the optimal characteristic set this coefficient
becomes insignificant, $\left(-0.1\,,\,0.8\right)$. Small stocks, per se, are
not attractive to this loss function. Small stocks with low residual
volatility are however.

Since the global optimal portfolio for the loss function indexed with
$\gamma=9$ uses three characteristics, Table 4 Panel C shows sampling
distribution properties of the $\theta$ coefficients on these characteristics
and related comparative statics. When residual volatility is removed from the
triplet the sign of the coefficient on size switches significantly from
negative $\left(-3.4\,,\,-1.4\right)$ to positive $\left(0.1\,,\,1.2\right)$.
And the coefficient on momentum switches from significantly positive
$\left(0.7\,,\,2.0\right)$ to insignificantly different from 0,
$\left(-0.1\,,\,1.4\right)$.

Table 4 Panel D reports the sampling standard deviations and correlations of
the $\theta$ estimators for the three optimal portfolios. These reflect in-
sample complementarities and substitutabilities. The largest correlations in
absolute value are the positive 0.92 and 0.91 correlations between the
coefficients on log size and residual volatility in the optimal portfolios for
the $\gamma$ 5 and $\gamma$ 9 loss functions, respectively. This is consistent
with the comparative statics results that show that for these loss functions,
small stocks are only appealing if they have low residual volatility. This
effect is dependent on the loss function, as the correlation between the
coefficients on log size and residual volatility is negative (-0.15) in the
case of the optimal portfolio for the $\gamma$ 2 loss function. The
substitution effect between beta and residual volatility for the $\gamma$ 2
loss function is seen in-sample in the -0.85 correlation between the $\theta$
estimators in this case. The in-sample correlation between size and momentum
is negative for all three loss functions and increasing in absolute value in
risk aversion. The same is true for the correlation between the $\theta$
coefficients on momentum and residual volatility.

All four $\theta$ coefficients on the globally optimal characteristics shared
by the $\gamma$ 2 and $\gamma$ 5 loss functions drop significantly in absolute
value as risk aversion increases. This is consistent with the hypothesis that
conditioning on these characteristics shifts portfolio exposure to priced
factors. This issue is explored further in the next subsection where we
evaluate the effects of characteristics on factor exposures.

Comparing Tables 3 and 4 shows the tension between in-sample statistical
significance of a characteristic’s $\theta$ coefficient and that
characteristic’s efficacy in the out-of-sample loss function. Beta is included
in the optimal characteristic set for the $\gamma$ 2 loss function. However
the 95% confidence interval on this $\theta$ coefficient is
$\left(-1.6\,,\,7.5\right)$. Table 3, Panel B shows that adding the book-to-
market ratio to the optimal characteristic set for this loss functions lowers
the 2.5%ile certainty equivalent by 10 basis points per month. However the 95%
confidence interval on this characteristic’s $\theta$ coefficient is
$\left(1.2\,,\,6.8\right)$.

4.3 Relationships between optimal portfolios and Fama-French-Carhart factors

Table 5 Panel A decomposes the two benchmark and 22 characteristic-based
portfolio returns (from Tables 3 and 4) using a linear projection of the 528
out-of-sample monthly excess returns on the four Fama-French-Carhart
factors.121212These four factors are: the excess return on the CRSP value-
weighted market index (MKT), the size factor (SMB), value factor (HML), and
momentum factor (MOM). We obtain these along with the riskfree rate from
Professor Kenneth French’s website at Dartmouth. We use the 528 monthly
returns in our out-of-sample period: January 1975 through December 2018. The
table reports the intercept from this regression ($\alpha$ in basis points per
month), along with the coefficients on each of the factors. The projection
decomposes the variance into 11 components with the orthogonal component being
the ratio of the residual variance to the total portfolio variance. The table
only provides covariance terms whose sampling means exceed 5% of total
variance, so the tabulated component means do not sum to unity. Panel B
contains the correlation / standard deviation matrix of the four factors over
the sample period.

The optimal portfolio for the $\gamma$ 2 loss function largely lies outside
the span of the four factors. Its mean alpha (95% confidence interval) is 269
$\left(201\,,\,348\right)$ basis points per month, and
$\left(61\,,\,75\right)$ percent of the (excess) return variance is not
spanned by the factors. The portfolio gets less than 10% of its variance from
each of the market and size factors. This portfolio has significantly more
exposure to the value $\left(12\%\,,\,21\%\right)$ and momentum factors
$\left(14\%\,,\,28\%\right)$ than the benchmarks.

Dropping momentum from the characteristic set for this loss function changes
the momentum factor loading from $\left(1.2\,,\,1.8\right)$ to
$\left(-0.5\,,\,-0.1\right)$. The momentum factor’s contribution to the
portfolio variance falls from $\left(14\%\,,\,28\%\right)$ to
$\left(0\%\,,\,3\%\right)$. Removing the characteristic momentum does not
shift exposures into the other three Fama-French-Carhart factors. The
percentage of variance that is orthogonal to these factors increases
significantly from $\left(61\%\,,\,75\%\right)$ to
$\left(76\%\,,\,86\%\right)$. This portfolio has the highest alpha of all the
tabulated portfolios. Yet, Table 3 shows that the mean (95% confidence
interval) of this portfolio’s out-of-sample certainty equivalent return is
$-11$% $\left(-100\%\,,\,2.8\%\right)$ per month. This portfolio with the
highest alpha does not statistically dominate the benchmark. This is a graphic
example of the importance of the loss function in assessing the predictive
content of characteristics.

Section 4.1 and Table 3 show that adding log size to the other optimal
characteristics for the $\gamma$ 2 loss function significantly increases the
portfolio’s certainty equivalent return, expected return, and Sharpe ratio.
Table 5 shows that this result is not the result of a significant increase in
the portfolio’s alpha. Instead, it is due to the effects that conditioning on
size have on the portfolio’s optimal SMB exposure. When log size is not in the
conditioning set the portfolio’s out-of-sample loading on SMB is significantly
negative $\left(-1.5\,,\,-0.9\right)$. This is primarily the result of the
negative coefficient on residual volatility for this characteristic set (as
shown in Table 4 Panel A). Adding size to the characteristic set shifts the
portfolio’s SMB loading to significantly positive $\left(0.6\,,\,1.4\right)$).
Similarly, when portfolio weights are not size-dependent, HML exposure is
statistically insignificant $\left(-0.1\,,\,0.4\right)$, and significantly
lower than that of the equally-weighted benchmark. Adding size to the
characteristic set raises the HML exposure to $\left(1.6\,,\,2.4\right)$).
Furthermore, adding log size to the other characteristics significantly
decreases the portfolio’s exposure to the market factor. This loss function
uses log size and residual volatility to create a portfolio that is largely
market-neutral.

Replacing the coefficient log size with the book-to-market ratio shows that
size is uniquely suited to zeroing-out the market. Table 5 shows that the
optimal portfolio obtained using the updating protocol, $\gamma^{*}=3$, and
characteristic set: momentum, book-to-market ratio, beta, average same-month
return, and residual volatility has the same HML exposure as the globally
optimal portfolio, however it inherits one-fourth of its out-of-sample
variance from the market–significantly more than the global optimal
portfolio’s 3%. Furthermore, substituting the book-to-market ratio for size
significantly decreases the SMB loading from $\left(0.6\,,\,1.4\right)$ to
$\left(-1.0\,,\,-0.3\right)$.

Consistent with the implications of Tables 3 and 4, removing beta from the
optimal characteristic set for the $\gamma$ 2 loss function has little effect
on the relationships between the factors and the out-of-sample returns of the
resulting portfolio.. Similarly, adding beta to the optimal characteristic set
for the $\gamma$ 5 loss function has little effect on the resulting
portfolio’s factor regression and variance decompositions.

Removing average same-month return significantly lowers the optimal
portfolio’s alpha–for optimal portfolios from both the $\gamma$ 2 and $\gamma$
5 loss functions. For the $\gamma$ 2 loss function the resulting portfolio has
significantly higher exposure to the momentum factor than the globally optimal
portfolio. HML becomes significantly more important for the $\gamma$ 5 loss
function. Nevertheless, $\left(45\,,\,51\right)$ percent of the variance of
the $\gamma$ 5 loss function’s optimal portfolio conditioned on the
characteristic triplet momentum, log size, and residual volatility (using
$\gamma^{*}=10$), is not explained by the Fama-French-Carhart factors. This
characteristic triplet is also the optimal set for the $\gamma$ 9 loss
function (using $\gamma^{*}=16$). Holding this set of characteristics fixed,
increasing $\gamma^{*}$ from 10 to 16 shifts the portfolio’s exposure away
from momentum to the market. The momentum factor’s loading decreases from
$\left(0.51\,,\,0.68\right)$ to $\left(0.36\,,\,0.48\right)$ with a
corresponding decrease in variance coverage from $\left(22\,,\,32\right)$
percent to $\left(12\,,\,21\right)$ percent. Movement into the market is
reflected by the increase in factor loading from $\left(0.45\,,\,0.56\right)$
to $\left(0.54\,,\,0.63\right)$ and variance coverage from
$\left(13\,,\,23\right)$ percent to $\left(26\,,\,40\right)$ percent.

Panel B of Table 1 and Panel C of Table 3 show that the $\gamma$ 5 loss
function can use residual volatility as a singleton to outperform the
benchmark. Table 4, Panel B shows that the $\theta$ average (95% confidence
interval) coefficient on residual volatility in this case is -2.3
$\left(-3.1\,,\,-1.5\right)$. Table 3, Panel C shows that this results in a
portfolio whose out-of-sample expected return is significantly lower than the
benchmark. Since this portfolio has a significantly higher certainty
equivalent than the benchmark, it follows that its standard deviation and
interquartile range are also significantly lower than the benchmark. It is
also significantly more leptokurtic than the benchmark. Table 5 provides
evidence about how this risk reduction in accomplished. This portfolio has a
mean (95% confidence interval) market beta of 0.76
$\left(0.73\,,\,0.79\right)$, and derives 43% $\left(46\%\,,\,49\%\right)$ of
its variance from the market. It has a significant negative loading on SMB,
and derives roughly one-fourth of its variance from this factor. The portfolio
has a mean loading of 0.4 on HML, and derives 10% of its variance from this
factor. Table 5 Panel B shows that the market and SML are positively
correlated, the opposite signed loadings means that the cross effect between
these two factor reduces this portfolios variance attributed to the two
factors in isolation by 22%. Similarly the negative correlation between the
market and HML coupled with loadings of the same sign reduce the net variance
contribution from these two factors by 13%. This means that 21% of the
portfolio’s variance lies outside the span of the FFC factors, and the
portfolio has an alpha of 29.9 $\left(22.4\,,\,37.4\right)$ basis points per
month.

These results are curious since as we have seen above (in Section 4.1),
removing residual volatility from the optimal characteristic set for the
$\gamma$ 5 loss function, significantly lowers the expected portfolio return
from 2.4 $\left(2.2\,,\,2.7\right)$, per cent per month to 1.4
$\left(1.4\,,\,1.5\right)$ per cent per month, with little change in the
portfolio returns’ interquartile range. Table 4, Panel C shows the in-sample
complementarities for residual volatility with $\gamma^{*}=10$. Removing
residual volatility significantly increases the $\theta$ coefficient on log
size from -4.0 $\left(-5.2\,,\,-2.8\right)$ to 0.4
$\left(-0.2\,,\,0.8\right)$. The loading on MOM, the momentum factor, drops
significantly from 0.6 $\left(0.5\,,\,0.7\right)$ to 0.3
$\left(0.2\,,\,0.4\right)$. The loading on HML drops significantly from 1.1
$\left(1.0\,,\,1.2\right)$ to -0.15 $\left(-0.2\,,\,-0.1\right)$. As discussed
above, the FFC factors account for less than 50% of the variance of the
optimal portfolio, with the market contributing 19%
$\left(13\%\,,\,25\%\right)$. However, restricting the coefficient on residual
volatility to 0 results in a portfolio for which the market accounts for 87%
$\left(83\%\,,\,91\%\right)$ of its variance. These complementarities show
that it is difficult to isolate residual volatility’s effects, as it is
closely linked to the size and value factors.

Table 4 shows that the $\theta$ coefficient linking log size to portfolio
weights is significantly negative $\left(-3.4\,,\,-1.4\right)$ in the optimal
portfolio rule for the $\gamma$ 9 investor. This tilts the portfolio toward
small stocks (and away from large-cap stocks). Yet Table 5 shows that this
portfolio has a significantly negative SMB loading
$\left(-0.3\,,\,-0.1\right)$. This follows from the fact that conditioning on
residual volatility tilts the portfolio toward large-cap stocks. Table 5 shows
that when this loss function conditions only on residual volatility and
momentum, the loading on SMB is significantly more negative than for the
global optimal portfolio, $\left(-0.7\,,\,-0.6\right)$. This loss function
uses log size to offset the excess exposure to large-cap stocks that results
from conditioning on residual volatility. This significantly increases the
portfolio’s expected return from $\left(1.2\,,\,1.4\right)$ to
$\left(1.6\,,\,1.9\right)$, per cent per month; alpha, from
$\left(16\,,\,31\right)$ to $\left(40\,,\,62\right)$ basis points per month;
and Sharpe ratio, from $\left(0.8\,,\,0.9\right)$ to
$\left(1.0\,,\,1.1\right)$.

Beta substitutes for log size as a means of increasing the loading on SMB when
the latter characteristic is removed from the optimal characteristic set for
the $\gamma$ 2 loss function. Optimally the coefficient on beta is
statistically insignificantly different from 0 (Table 4, Panel A). But when
size is removed from the optimal characteristic set, this becomes
significantly positive $\left(2.9\,,\,8.8\right)$. The portfolio dramatically
increases its exposure to the market portfolio, in order to offset the large
firm tilt that results from the negative coefficient on residual volatility.
And despite the tilt toward high beta stocks still has a significant negative
loading on SMB $\left(-1.5\,,\,-0.9\right)$. The efficacy of the
characteristic log size stems from its complementarity with residual
volatility–a phenomenon that is manifest for all three loss functions we
highlight.

Consider the use of leverage by the most risk-averse loss function. Despite
preferring the riskless asset to the market, this loss function’s optimal
portfolio uses 77% leverage to exploit predictable properties in the cross-
section. Green and Hollifield (1992) propose an ingenious argument as to why
well diversified portfolios may have extreme short positions. They point out
that if there is a single factor on which all stocks load positively (e.g., in
the CAPM or the pricing kernel), then minimizing portfolio variance might be
accomplished by finding a diversified portfolio with a zero loading on this
factor. The $\gamma$ 9 loss function provides some insight as to whether this
technique to reduce factor exposure appeals to highly risk-averse investors.
This portfolio has significantly more exposure to the market and HML factors
than the $\gamma$ 2 loss function’s optimal portfolio. The portfolio does use
short positions to reduce exposure to the market factor. However, the optimal
portfolio’s expected return, 182, $\left(169\,,\,196\right)$ and median
return, 196, $\left(176\,,\,217\right)$ (basis points per month) are
significantly higher than those of the market: 108, $\left(103\,,\,112\right)$
and 135, $\left(123\,,\,147\right)$, respectively. This most risk-averse loss
function finds exposure to the value factor and momentum factors desirable.
She uses characteristics to create portfolios with exposure to these factors,
as well as to factors lying outside the span of the Fama-French-Carhart
factors.

4.4 Characteristic comparative statics: Summaries

4.4.1 Momentum

All investors’ global optimal portfolios have a significant positive
coefficient on momentum. For the $\gamma$ 2 loss function momentum serves to
significantly reduce overfitting. Table 1, Panel A and Table 2, Panel A show
that the value of $\gamma^{*}$ required to overcome overfitting is highest for
the characteristic sets which do not include momentum. Table 3, Panel A shows
that when momentum is removed from the optimal conditioning set in this case
the 2.5%ile of the certainty equivalent sampling distribution drops from 2.6%
per month to to -100this behavior is unique across the characteristics. Table
4 Panel A shows that removing momentum from the optimal characteristic set
does not have a significant effect on the coefficients on the remaining
characteristics, consistent with the premise that the salutary effects of
momentum arise out-of-sample. This effect is also seen in Table 4, Panel D:
the in-sample correlations between the coefficient on momentum and that on
both residual standard deviation and average same-month return are close to
zero. This beneficial effect of reducing overfitting decreases in $\gamma$, as
the tilting toward, and dimensionality of the other characteristics
diminishes. Nevertheless, conditioning on momentum is the only tool investors
have to gain the desired exposure to the momentum factor which accounts for 15
- 20% of the return variance of all three loss functions’ optimal portfolios.

4.4.2 Log size

All optimal portfolio rules have a significant negative $\theta$ coefficient
on size. Adding size to the set of other optimal characteristics significantly
increases the Sharpe ratio in all three cases. Certainty equivalent increases
significantly for power utility investors indexed by both $\gamma=2$ and
$\gamma=5$. Expected return and alpha increase significantly for the $\gamma$
5 and $\gamma$ 9 loss functions. This effect is muted for the $\gamma$ 2 loss
function which substitutes high beta for small stocks. For the more risk-
averse loss functions adding size increases the SMB and HML loadings and
alpha, while also shifting the variance outside the span of the FFC factors.
Adding size to the other characteristics increases the $\theta$ coefficient on
residual volatility. The SMB loading absent size is significantly negative, so
adding size significantly reduces the effect of SMB on the portfolio. Without
size in the characteristic set, the portfolios of the $\gamma$ 5 and $\gamma$
9 loss functions derive 25% of their variance from SMB. Adding size shrinks
this to less than 4%. For the most risk averse investor we consider, Table 4,
Panel C shows that the $\theta$ coefficient on log size is significantly
negative, whereas Table 5, Panel A shows that this factor’s loading on SMB is
significantly negative. This arises because this investor wants a large stock
tilt, but significantly less than the tilt on this factor induced by shifting
toward low volatility stocks.

4.4.3 Residual volatility

All optimal portfolio rules have a significant negative $\theta$ coefficient
on residual volatility. For the power utility investor with $\gamma=2$, adding
this characteristic to momentum, log size, average same-month return, and beta
significantly increases resulting portfolio’s out-of-sample Sharpe ratio and
certainty equivalent return. This occurs with no significant changes in the
portfolio’s alpha or variance decomposition with respect to the Fama-French-
Carhart factors. The complementary set of characteristics for the more risk
averse investors, indexed with $\gamma=5$ and $\gamma=9$ does not include
beta, so in both of these cases adding residual volatility significantly
increases alpha and shifts the portfolio outside the span of the four factors.
In these two cases adding residual volatility significantly increases the
portfolio’s exposure to the momentum and value factors. This demonstrates the
complementarity between a measure of volatility and momentum and size. When
the investor cannot down-weight stocks with high residual volatility she does
not want exposure to the momentum factor. Exposure to momentum only increases
out-of-sample expected utility when the investor can put less weight in stocks
that had high momentum but that were also highly volatile. Table 5 shows that
the complementarity with log size is an in-sample effect: adding residual
volatility to the complementary set has a statistically significant on the
coefficient that weights size. By contrast, as noted above, the
complementarity with momentum is only observed out-of-sample, as adding
residual volatility does not significantly affect the weighting coefficient on
momentum. In this case it is conditioning on residual volatility that offsets
the (in-sample) overfitting that harms the momentum investor.

4.4.4 Average same-month return

While average same-month return is the only characteristic that can be used as
a singleton by our most risk-tolerant investor to outperform the benchmark its
predictive content is too risky for our most risk-averse loss functions to
exploit. Adding it to the other four characteristics for the $\gamma$ 2 loss
function has no effect on any of their $\theta$ coefficients, suggesting
limited in-sample complementarity. The resulting portfolio is less
leptokurtotic and has a higher minimum monthly return. While neither of these
effects is statistically significant each is amplified by the loss function.
This suggests that as is the case with momentum, conditioning on average same-
month return mitigates some of the estimation risk inherent in the other
characteristics. This manifests in the reduction in the width of the certainty
equivalent’s 95% confidence interval from 116 to 94 basis points, which when
scaled by the mean certainty equivalent is a drop from 63 to 32%.

4.4.5 Beta

Beta only enters the optimal characteristic set for our most risk-tolerant
loss functions, and has a minimal effect in the context of the other optimal
characteristics. Its complementarity to log size and residual volatility are
important in assessing their marginal contributions. Removing log size from
the optimal characteristic changes the average $\theta$ coefficient on beta
from insignificant to significantly positive, as the portfolio uses beta as an
imperfect inverse proxy for size. Similarly, removing residual volatility from
the optimal characteristic changes the average $\theta$ coefficient on beta to
significantly negative. Downweighting (residual) volatility is important for
all optimal portfolios. In the absence of residual volatility beta serves as
an imperfect substitute.131313This finding is consistent with Liu, Stambaugh,
and Yuan (2018) who show that the negative relationship between beta and
expected return is not robust to controlling for idiosyncratic volatility.

4.4.6 Book-to-market ratio

All of our optimal portfolios load significantly on the value factor (HML),
without conditioning on the book-to-market ratio which has been historically
associated with value. Although the book-to-market ratio has a positive,
statistically significant $\theta$ coefficient when added to the optimal
characteristic set for the $\gamma$ 2 and $\gamma$ 9 loss functions, its
marginal effects on the resulting portfolios are very small, the 2.5%ile
sampling value of the minimum monthly return is lowered, as is certainty
equivalent’s 2.5%ile sampling value. The desired positioning of the portfolio
relative to the FFC factors is achieved by residual volatility and size, in
combination. Conditional on that information the book-to-market ratio is
redundant, and adding it to optimal characteristic set increases sampling
variation, reflected in the width of out-of-sample certainty equivalent’s 95%
confidence interval. This result supports Kozak, Nagel, and Santosh’s (2020)
premise that characteristics and risk factors do not span the same space.

5\. Conclusions

An important goal of linking portfolio weights to characteristics, avoiding
the intermediate step of estimating conditional moments, is mitigation of
estimation risk, which is known to plague optimal portfolio selection.
However, we show that the benefits from conditioning on observable
characteristics are largely lost to overfitting when an agent obtains
portfolio weights by maximizing her own utility function on sample data.
Regularizing by ratcheting up risk aversion for the purpose of in-sample
utility maximization mitigates overfitting, and allows a risk averse investor
to profit from the predictive information contained in the characteristic and
return data. These out-of-sample utility gains derive from shifting into
stocks that are jointly small, have low residual volatility, and positive
momentum. The gains derive from complementarities across these
characteristics, as no one characteristic can be used to produce utility gains
by all loss functions.

The Fama-French-Carhart factors do not span the space of priced factors in the
market. The most risk tolerant investor’s optimal portfolio alpha is over 200
basis points per month. But this is not a free lunch. Increasing risk aversion
shifts the portfolio toward the four factors–significantly decreasing alpha.
Nevertheless, less than one-fourth of the variance of the optimal portfolio
for a power utility investor with relative risk aversion of five is driven by
the market. Furthermore more than half of this portfolio’s return variance is
unrelated to the four Fama-French-Carhart factors.

Whereas we have filtered our results through different loss functions, and
considered “out-of-sample” returns, our empirical approach is not normative.
Consistent with computer-intensive, machine-learning methods, we use the out-
of-sample period to select the optimal feasible portfolio rule and evaluate
performance. Because we are interested in comparing non-nested alternatives
fraught with estimation risk this is a design for inference not prescription.
This is also reflected in our empirical design which ignores trading costs and
assumes away short-selling constraints (although in the presence of either
static optimization is not appropriate).

How can these results be reconciled with equilibrium? Several of the
assumptions underlying our design are likely violated. First, we consider only
a static optimization problem. This is of course optimal if the investment
opportunity set does not change over time. That seems unlikely. Second, we
optimize only within the set of publicly traded equities. This can understate
the appeal of the market depending on how it correlates with other forms of
wealth (for example real estate and human capital). Another way to interpret
this is that the market portfolio, or even a convex combination of the Fama-
French-Carhart factors, is a poor proxy for the stochastic discount factor.

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Figure 1: Figure 1. Return densities The monthly out-of-sample return
densities of the value-weighted index (VWI) and global optimal portfolio for
the power utility loss function with $\gamma=9$ obtained (using
$\gamma^{*}=16$; allowing weights to depend on: momentum, log size, and
residual standard deviation); and updating sample construction. Both densities
are constructed using all returns from 10,000 pseudosamples of monthly returns
from the 44-year out-of-sample period, 1975 – 2018.

Table 1
Optimal $\gamma^{*}$ and protocol for each characteristic set:
Sampling properties of certainty equivalent returns

For each of the 48 characteristic sets, we report the value of $\gamma^{*}$
and protocol that generate the maximum 2.5%ile certainty equivalent for each
loss function, along with sampling statistics of the certainty equivalent. The
characteristic symbols are: M: momentum, V: book-to-market ratio, S: log size,
$\beta$: from lagged 60-month market model, $r_{t-12}$: last-year’s same-month
return, $\overline{r}$: average same-month return over the previous 5 years,
$\sigma_{\epsilon}$: standard deviation of lagged 60-month market model
residual. $V_{-f}$ refers to the treatment in which financial stocks’ weights
are not affected by their book-to-market ratios. We report the marginal
effects of this treatment in Panel D.

Panel A: $\gamma$ 2 loss function
---
Characteristic | Optimal | Protocol | Resampled
set / Benchmark | $\gamma^{*}$ |  | Out of sample
|  |  | 2.5%ile | Mean | 97.5%ile
Equally-weighted portfolio |  |  | 100.6 | 102.7 | 104.8
M, V, S, $\beta$, $r_{t-12}$, $\overline{r}$, $\sigma_{\epsilon}$ | 4 | Updating | 249.9 | 289.6 | 337.6
M, $V_{-f}$, S, $\beta$, $r_{t-12}$, $\overline{r}$ | 6 | Rolling | 197.8 | 232.9 | 268.5
M, V, S, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$ | 4 | Updating | 250.7 | 291.2 | 335.0
V, S, $\beta$, $r_{t-12}$, $\overline{r}$, $\sigma_{\epsilon}$ | 8 | Updating | 164.2 | 192.0 | 218.9
M, $V_{-f}$, S, $\beta$, $\sigma_{\epsilon}$ | 5 | Rolling | 184.3 | 212.2 | 242.1
M, $V_{-f}$, S, $r_{t-12}$, $\overline{r}$ | 6 | Rolling | 187.8 | 221.6 | 256.5
M, V, S, $\overline{r}$, $\sigma_{\epsilon}$ | 4 | Updating | 239.1 | 276.3 | 314.8
V, S, $\beta$, $r_{t-12}$, $\overline{r}$ | 9 | Rolling | 140.0 | 164.7 | 188.1
V, S, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$ | 6 | Updating | 169.2 | 203.1 | 235.4
M, $V_{-f}$, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$ | 4 | Updating | 199.5 | 236.9 | 282.2
M, S, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$ | 3 | Updating | 257.4 | 301.6 | 351.7
M, V, S, $\sigma_{\epsilon}$ | 5 | Updating | 181.8 | 201.0 | 221.1
M, $V_{-f}$, S, $\beta$ | 5 | Rolling | 153.5 | 172.6 | 193.2
V, S, $\beta$, $\sigma_{\epsilon}$ | 5 | Updating | 115.0 | 131.3 | 149.5
$V_{-f}$, S, $r_{t-12}$, $\overline{r}$ | 7 | Rolling | 115.2 | 136.8 | 157.1
$\beta$, $r_{t-12}$, $\overline{r}$, $\sigma_{\epsilon}$ | 8 | Updating | 126.1 | 144.8 | 165.0
M, $V_{-f}$, S, $\overline{r}$ | 4 | Updating | 178.3 | 207.8 | 238.0
M, S, $\beta$, $\overline{r}$ | 4 | Updating | 182.8 | 208.6 | 237.2
M, S, $\beta$, $\sigma_{\epsilon}$ | 5 | Updating | 182.5 | 205.6 | 231.1
S, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$ | 6 | Updating | 175.9 | 206.8 | 237.8
M, S, $\overline{r}$, $\sigma_{\epsilon}$ | 4 | Updating | 245.5 | 278.9 | 314.2
M, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$ | 4 | Updating | 178.4 | 210.5 | 246.6
M, $V_{-f}$, $\overline{r}$, $\sigma_{\epsilon}$ | 5 | Updating | 181.4 | 212.5 | 244.2
M, $r_{t-12}$, $\overline{r}$ | 5 | Updating | 122.7 | 137.2 | 153.3
M, $V_{-f}$, S | 4 | Rolling | 150.8 | 172.1 | 195.3
$V_{-f}$, $r_{t-12}$, $\overline{r}$ | 7 | Updating | 125.7 | 141.8 | 159.3
$r_{t-12}$, $\overline{r}$, $\sigma_{\epsilon}$ | 8 | Updating | 129.9 | 147.3 | 165.5
$V_{-f}$, S, $\sigma_{\epsilon}$ | 4 | Updating | 128.6 | 143.7 | 160.0

Table 1 Continued
Optimal $\gamma^{*}$ and protocol for each characteristic set:
Sampling properties of certainty equivalent returns

Panel A: $\gamma$ 2 loss function
---
Characteristic | Optimal | Protocol | Resampled
set /Benchmark | $\gamma^{*}$ |  | Out of sample
|  |  | 2.5%ile | Mean | 97.5%ile
M, S, $\sigma_{\epsilon}$ | 5 | Updating | 180.1 | 198.6 | 217.5
M, $V_{-f}$, $\sigma_{\epsilon}$ | 5 | Rolling | 152.7 | 172.5 | 194.2
S, $\overline{r}$, $\sigma_{\epsilon}$ | 6 | Updating | 184.4 | 215.1 | 244.1
M, S, $\overline{r}$ | 5 | Updating | 124.0 | 137.6 | 152.9
M, $\overline{r}$, $\sigma_{\epsilon}$ | 4 | Updating | 158.9 | 182.9 | 210.2
S, $\beta$, $\sigma_{\epsilon}$ | 4 | Updating | 122.9 | 138.9 | 156.2
$\beta$, $\sigma_{\epsilon}$ | 8 | Updating | 86.5 | 95.2 | 105.5
$V_{-f}$, $\sigma_{\epsilon}$ | 13 | Updating | 94.6 | 101.7 | 109.1
$V_{-f}$, S | 5 | Updating | 84.9 | 92.9 | 101.0
M, S | 5 | Updating | 100.9 | 108.9 | 117.4
M, $\sigma_{\epsilon}$ | 6 | Updating | 116.9 | 127.1 | 138.0
S, $\sigma_{\epsilon}$ | 3 | Updating | 134.9 | 149.9 | 165.4
M, $\overline{r}$ | 5 | Updating | 122.9 | 137.2 | 152.7
M | 4 | Updating | 95.9 | 103.1 | 110.7
$V_{-f}$ | 7 | Updating | 89.7 | 96.3 | 103.3
S | 3 | Updating | 88.9 | 94.6 | 100.4
$\beta$ | 13 | Updating | 81.9 | 87.7 | 93.6
$r_{t-12}$, $\overline{r}$ | 5 | Updating | 117.3 | 131.1 | 146.5
$\overline{r}$ | 4 | Updating | 117.3 | 133.4 | 151.6
$\sigma_{\epsilon}$ | 11 | Updating | 94.7 | 101.6 | 108.5
Panel B: $\gamma$ 5 loss function
Equally-weighted portfolio |  |  | 57.1 | 59.3 | 61.6
M, V, S, $\beta$, $r_{t-12}$, $\overline{r}$, $\sigma_{\epsilon}$ | 11 | Updating | 115.6 | 142.4 | 165.0
M, $V_{-f}$, S, $\beta$, $r_{t-12}$, $\overline{r}$ | 13 | Rolling | 94.7 | 129.2 | 152.7
M, V, S, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$ | 10 | Updating | 119.9 | 144.9 | 167.1
V, S, $\beta$, $r_{t-12}$, $\overline{r}$, $\sigma_{\epsilon}$ | 22 | Updating | 72.4 | 98.8 | 118.0
M, $V_{-f}$, S, $\beta$, $\sigma_{\epsilon}$ | 11 | Rolling | 107.4 | 122.1 | 137.3
M, $V_{-f}$, S, $r_{t-12}$, $\overline{r}$ | 13 | Rolling | 78.0 | 110.6 | 133.0
M, V, S, $\overline{r}$, $\sigma_{\epsilon}$ | 12 | Updating | 122.8 | 143.1 | 161.1
V, S, $\beta$, $r_{t-12}$, $\overline{r}$ | 22 | Updating | 64.4 | 86.1 | 102.2
V, S, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$ | 16 | Updating | 86.3 | 109.0 | 127.7
M, $V_{-f}$, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$ | 13 | Updating | 86.1 | 110.1 | 130.2
M, S, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$ | 9 | Updating | 130.9 | 153.0 | 174.5
M, $V_{-f}$, S, $\sigma_{\epsilon}$ | 11 | Updating | 111.8 | 124.0 | 136.2
M, $V_{-f}$, S, $\beta$ | 9 | Rolling | 98.4 | 110.6 | 123.1
V, S, $\beta$, $\sigma_{\epsilon}$ | 10 | Updating | 78.5 | 89.7 | 101.3
V, S, $r_{t-12}$, $\overline{r}$ | 16 | Updating | 39.2 | 55.7 | 68.3
$\beta$, $r_{t-12}$, $\overline{r}$, $\sigma_{\epsilon}$ | 22 | Updating | 62.2 | 77.6 | 90.8
M, $V_{-f}$, S, $\overline{r}$ | 10 | Updating | 79.6 | 95.6 | 112.5
M, S, $\beta$, $\overline{r}$ | 11 | Updating | 101.7 | 115.2 | 128.4

Table 1 Continued
Optimal $\gamma^{*}$ and protocol for each characteristic set:
Sampling properties of certainty equivalent returns

Panel B: $\gamma$ 5 loss function
---
Characteristic | Optimal | Protocol | Resampled
set | $\gamma^{*}$ |  | Out of sample
|  |  | 2.5%ile | Mean | 97.5%ile
M, S, $\beta$, $\sigma_{\epsilon}$ | 11 | Updating | 106.9 | 120.2 | 134.7
S, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$ | 16 | Updating | 85.3 | 108.5 | 126.3
M, S, $\overline{r}$, $\sigma_{\epsilon}$ | 10 | Updating | 132.7 | 151.9 | 170.2
M, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$ | 9 | Updating | 95.5 | 113.2 | 131.1
M, $V_{-f}$, $\overline{r}$, $\sigma_{\epsilon}$ | 16 | Updating | 88.0 | 107.0 | 123.5
M, $r_{t-12}$, $\overline{r}$ | 11 | Updating | 62.5 | 69.5 | 76.8
M, $V_{-f}$, S | 8 | Rolling | 87.2 | 99.6 | 112.5
V, $r_{t-12}$, $\overline{r}$ | 13 | Updating | 63.0 | 72.7 | 82.1
$r_{t-12}$, $\overline{r}$, $\sigma_{\epsilon}$ | 22 | Updating | 69.5 | 85.2 | 98.2
V, S, $\sigma_{\epsilon}$ | 9 | Updating | 89.2 | 100.1 | 111.0
M, S, $\sigma_{\epsilon}$ | 12 | Updating | 112.3 | 124.0 | 135.7
M, $V_{-f}$, $\sigma_{\epsilon}$ | 11 | Rolling | 96.4 | 109.0 | 121.7
S, $\overline{r}$, $\sigma_{\epsilon}$ | 16 | Updating | 93.6 | 117.3 | 135.7
M, S, $\overline{r}$ | 10 | Updating | 58.4 | 65.6 | 73.2
M, $\overline{r}$, $\sigma_{\epsilon}$ | 11 | Updating | 92.9 | 107.2 | 121.0
S, $\beta$, $\sigma_{\epsilon}$ | 9 | Updating | 81.2 | 91.3 | 101.5
$\beta$, $\sigma_{\epsilon}$ | 10 | Updating | 60.8 | 69.0 | 77.6
$V_{-f}$, $\sigma_{\epsilon}$ | 11 | Rolling | 60.5 | 69.2 | 77.7
$V_{-f}$, S | 9 | Updating | 46.4 | 53.5 | 60.7
M, S | 10 | Updating | 52.2 | 57.8 | 63.6
M, $\sigma_{\epsilon}$ | 12 | Updating | 81.1 | 89.5 | 98.1
S, $\sigma_{\epsilon}$ | 9 | Updating | 90.2 | 100.1 | 109.8
M, $\overline{r}$ | 11 | Updating | 62.2 | 68.8 | 75.8
M | 10 | Updating | 54.6 | 59.8 | 65.2
$V_{-f}$ | 11 | Updating | 55.6 | 61.6 | 67.5
S | 6 | Updating | 50.7 | 55.2 | 59.6
$\beta$ | 7 | Updating | 58.9 | 65.4 | 71.8
$r_{t-12}$, $\overline{r}$ | 12 | Updating | 62.9 | 69.2 | 75.9
$\overline{r}$ | 11 | Updating | 62.5 | 69.5 | 76.8
$\sigma_{\epsilon}$ | 12 | Updating | 66.5 | 74.1 | 81.8
Panel C: $\gamma$ 9 loss function
Value-weighted portfolio |  |  | 4.4 | 9.8 | 15.2
M, V, S, $\beta$, $r_{t-12}$, $\overline{r}$, $\sigma_{\epsilon}$ | 22 | Updating | -69.8 | 24.3 | 71.5
M, V, S, $\beta$, $r_{t-12}$, $\overline{r}$ | 22 | Updating | -86.7 | 5.5 | 50.4
M, V, S, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$ | 22 | Updating | -21.1 | 39.3 | 73.6
V, S, $\beta$, $r_{t-12}$, $\overline{r}$, $\sigma_{\epsilon}$ | 22 | Updating | -221.8 | -29.1 | 52.5
M, $V_{-f}$, S, $\beta$, $\sigma_{\epsilon}$ | 16 | Updating | 30.2 | 51.8 | 69.0
M, V, S, $r_{t-12}$, $\overline{r}$ | 13 | Updating | -56.2 | 3.1 | 30.5
M, V, S, $\overline{r}$, $\sigma_{\epsilon}$ | 16 | Updating | -17.0 | 54.9 | 89.7
V, S, $\beta$, $r_{t-12}$, $\overline{r}$ | 22 | Updating | -200.2 | -35.5 | 38.2

Table 1 Continued
Optimal $\gamma^{*}$ and protocol for each characteristic set:
Sampling properties of certainty equivalent returns

Panel C: $\gamma$ 9 loss function
---
Characteristic | Optimal | Protocol | Resampled
set | $\gamma^{*}$ |  | Out of sample
|  |  | 2.5%ile | Mean | 97.5%ile
V, S, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$ | 22 | Updating | -62.6 | 23.2 | 65.6
M, $V_{-f}$, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$ | 22 | Updating | -72.6 | 5.9 | 46.9
M, S, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$ | 16 | Updating | 13.1 | 61.3 | 89.7
M, $V_{-f}$, S, $\sigma_{\epsilon}$ | 16 | Updating | 36.5 | 57.4 | 74.3
M, $V_{-f}$, S, $\beta$ | 12 | Rolling | 32.8 | 52.4 | 68.0
V, S, $\beta$, $\sigma_{\epsilon}$ | 13 | Updating | 21.9 | 41.6 | 57.3
V, S, $r_{t-12}$, $\overline{r}$ | 22 | Updating | -148.3 | -49.9 | -4.7
$\beta$, $r_{t-12}$, $\overline{r}$, $\sigma_{\epsilon}$ | 22 | Updating | -99.4 | -7.8 | 38.3
M, $V_{-f}$, S, $\overline{r}$ | 16 | Updating | -36.6 | 5.7 | 28.3
M, S, $\beta$, $\overline{r}$ | 16 | Updating | 13.0 | 46.2 | 67.1
M, S, $\beta$, $\sigma_{\epsilon}$ | 16 | Updating | 39.3 | 56.1 | 71.7
S, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$ | 22 | Updating | -68.8 | 21.2 | 64.9
M, S, $\overline{r}$, $\sigma_{\epsilon}$ | 16 | Updating | 19.2 | 68.6 | 95.9
M, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$ | 16 | Updating | 9.3 | 43.1 | 64.8
M, V, $\overline{r}$, $\sigma_{\epsilon}$ | 22 | Updating | -72.2 | 12.6 | 54.3
M, $r_{t-12}$, $\overline{r}$ | 16 | Updating | -1.5 | 6.1 | 13.3
M, $V_{-f}$, S | 11 | Rolling | 19.0 | 35.6 | 49.0
V, $r_{t-12}$, $\overline{r}$ | 16 | Updating | -25.2 | 1.1 | 15.8
$r_{t-12}$, $\overline{r}$, $\sigma_{\epsilon}$ | 22 | Updating | -94.2 | -0.8 | 44.8
V, S, $\sigma_{\epsilon}$ | 13 | Updating | 32.2 | 52.0 | 67.6
M, S, $\sigma_{\epsilon}$ | 16 | Updating | 44.2 | 62.2 | 77.4
M, $V_{-f}$, $\sigma_{\epsilon}$ | 13 | Rolling | 21.6 | 47.4 | 66.1
S, $\overline{r}$, $\sigma_{\epsilon}$ | 22 | Updating | -74.2 | 26.4 | 73.2
M, S, $\overline{r}$ | 22 | Updating | -8.5 | 0.4 | 8.9
M, $\overline{r}$, $\sigma_{\epsilon}$ | 16 | Updating | 11.7 | 45.6 | 66.3
S, $\beta$, $\sigma_{\epsilon}$ | 13 | Updating | 19.9 | 40.5 | 55.4
$\beta$, $\sigma_{\epsilon}$ | 12 | Updating | 11.5 | 27.8 | 40.8
V, $\sigma_{\epsilon}$ | 11 | Rolling | 3.7 | 21.2 | 34.5
V, S | 10 | Updating | -11.6 | -1.0 | 8.0
M, S | 13 | Updating | -4.4 | 2.6 | 9.3
M, $\sigma_{\epsilon}$ | 16 | Updating | 32.6 | 45.0 | 56.1
S, $\sigma_{\epsilon}$ | 12 | Updating | 27.2 | 48.1 | 63.4
M, $\overline{r}$ | 16 | Updating | -0.6 | 6.2 | 12.8
M | 13 | Updating | -1.9 | 4.4 | 10.5
$V_{-f}$ | 11 | Updating | 2.6 | 10.7 | 18.0
S | 10 | Updating | 1.4 | 7.2 | 12.8
$\beta$ | 6 | Updating | 21.5 | 31.2 | 39.9
$r_{t-12}$, $\overline{r}$ | 13 | Updating | 3.2 | 10.0 | 16.7
$\overline{r}$ | 16 | Updating | 6.9 | 12.8 | 18.6
$\sigma_{\epsilon}$ | 12 | Updating | 15.9 | 32.1 | 44.9

Table 1 Continued
Optimal $\gamma^{*}$ and protocol for each characteristic set:
Sampling properties of certainty equivalent returns

Panel D: Alternative protocols for the book-to-market ratio (V)
---
Protocols | Optimal | Resampled
| $\gamma^{*}$ | Out of sample
|  | 2.5%ile |  Mean | 97.5%ile
$\gamma$ 2 loss function
Characteristic set: M, V, S, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$:
Updating, V | 4 | 250.7 | 291.2 | 335.0
Updating, $V_{-f}$ | 4 | 249.6 | 290.5 | 333.8
Rolling, V | 7 | 205.3 | 245.1 | 286.2
Rolling, $V_{-f}$ | 8 | 201.4 | 235.1 | 269.8
$\gamma$ 5 loss function
Characteristic set: M, V, S, $\overline{r}$, $\sigma_{\epsilon}$:
Updating, V | 12 | 122.8 | 143.1 | 161.1
Updating, $V_{-f}$ | 12 | 119.9 | 142.4 | 161.3
Rolling, V | 16 | 76.4 | 122.3 | 148.7
Rolling, $V_{-f}$ | 16 | 70.4 | 126.1 | 154.6
$\gamma$ 9 loss function
Characteristic set: M, V, S, $\sigma_{\epsilon}$:
Updating, V | 16 | 33.0 | 55.5 | 72.5
Updating, $V_{-f}$ | 16 | 36.5 | 57.4 | 74.3
Rolling, V | 16 | 9.7 | 39.9 | 61.1
Rolling, $V_{-f}$ | 13 | 25.8 | 53.5 | 73.1

Table 2

Estimation Risk: Sampling distribution of out-of-sample certainty equivalent
returns (in basis points per month) –as portfolio rules vary

Sampling properties of the out-of-sample certainty equivalent returns of
various feasible portfolio rules. The utility function is isoelastic:
$E(U)=\sum\limits_{t=0}^{T-1}\frac{(1+r_{p,t+1})^{1-\gamma}}{1-\gamma}\left(\frac{1}{T}\right)$.
Panels A - C report results from the perspective of three different CRRA
utility functions (or “investors”) indexed by $\gamma$. Portfolios are
selected using loss functions indexed by $\gamma^{*}$ and the indicated sets
of characteristics. The first in-sample period is January 1960 - December
1974\. This is updated annually–and a new portfolio rule is determined at the
beginning of each out-of-sample year. The out-of-sample period consists of the
528 months from January 1975 through December 2018. The global optimal
portfolios for each loss function are highlighted in boldface. The rolling
protocol keeps only the most recent 180 months of data to estimate the
characteristic coefficients at the beginning of each year. Under the updating
protocol, the last in-sample period comprises 696 months of return data.
Characteristic abbreviations are: M - momentum; $V_{-f}$ \- book-to-market
ratio exempting financial stocks; S - log size; $\beta$ \- market model
regression coefficient estimated over previous 60 months; $r_{t-12}$ \- last
year’s same-month return; $\overline{r}$ \- average same-month return from
previous five years; $\sigma_{\epsilon}$ \- residual standard deviation from
the market model regression estimated over the previous 60 months.

Panel A: $\gamma$ 2 loss function
---
Characteristics: | V, S, $\beta$, $r_{t-12}$, $\overline{r}$, $\sigma_{\epsilon}$ |  | M, S, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$
$\gamma^{*}$ | Resampled |  | Resampled
Updating | 2.5%ile | Mean | 97.5%ile |  | 2.5%ile | Mean | 97.5%ile
2 | -10,000.0 | -9,446.3 | 0.2 |  | -10,000.0 | -4,212.2 | 295.8
3 | -10,000.0 | -6,383.9 | 227.3 |  | 257.4 | 301.6 | 351.7
4 | -10,000.0 | -1,804.8 | 250.2 |  | 255.1 | 293.8 | 336.4
5 | -10,000.0 | -111.3 | 247.0 |  | 240.5 | 275.0 | 313.4
6 | 114.5 | 178.0 | 238.8 |  | 226.2 | 256.8 | 291.3
7 | 158.4 | 196.2 | 229.2 |  | 214.2 | 241.0 | 272.8
8 | 164.2 | 192.0 | 218.9 |  | 202.8 | 228.3 | 256.4
9 | 162.5 | 186.5 | 211.4 |  | 193.3 | 217.0 | 243.3
10 | 160.3 | 181.0 | 203.5 |  | 185.2 | 207.2 | 232.2
11 | 156.5 | 176.0 | 197.2 |  | 177.9 | 198.7 | 222.0
12 | 152.6 | 171.2 | 191.1 |  | 171.7 | 191.3 | 213.6
Rolling |  |  |  |  |  |  |
2 | -10,000.0 | -8,693.4 | 17.7 |  | -10,000.0 | -6,641.3 | 176.9
3 | -10,000.0 | -5,253.9 | 245.3 |  | -10,000.0 | -443.6 | 336.8
4 | -10,000.0 | -2,039.6 | 276.0 |  | 143.6 | 172.6 | 344.8
5 | -10,000.0 | -424.1 | 271.5 |  | 223.7 | 269.5 | 325.2
6 | -42.0 | 78.5 | 261.5 |  | 224.6 | 263.2 | 303.3
7 | 139.2 | 185.5 | 249.5 |  | 215.6 | 249.1 | 284.6
8 | 159.7 | 196.9 | 238.0 |  | 205.0 | 235.2 | 267.2
9 | 163.5 | 196.2 | 227.8 |  | 195.7 | 223.4 | 253.0
10 | 161.3 | 190.1 | 218.6 |  | 188.3 | 213.0 | 240.4
11 | 158.2 | 184.3 | 210.6 |  | 179.7 | 203.5 | 229.0
12 | 154.7 | 178.7 | 203.4 |  | 173.5 | 195.5 | 219.5

Table 2 -Continued

Estimation Risk: Sampling distribution of out-of-sample certainty equivalent
returns (in basis points per month) –as portfolio rules vary

Panel B: $\gamma$ 5 loss function
---
Characteristics: | M, S, $\beta$, $\overline{r}$, $\sigma_{\epsilon}$ |  | M, S, $\overline{r}$, $\sigma_{\epsilon}$
$\gamma^{*}$ | Resampled |  | Resampled
Updating | 2.5%ile | Mean | 97.5%ile |  | 2.5%ile | Mean | 97.5%ile
5 | 72.3 | 121.5 | 158.6 |  | 50.1 | 108.5 | 145.7
6 | 111.0 | 144.4 | 172.9 |  | 98.4 | 136.5 | 162.8
7 | 125.9 | 152.5 | 177.4 |  | 120.7 | 148.2 | 170.9
8 | 130.3 | 154.2 | 176.7 |  | 129.8 | 152.6 | 173.3
9 | 130.9 | 153.0 | 174.5 |  | 132.2 | 153.1 | 172.5
10 | 129.8 | 150.6 | 171.1 |  | 132.7 | 151.9 | 170.2
11 | 127.5 | 147.3 | 167.5 |  | 131.9 | 150.1 | 168.0
Rolling |  |  |  |  |  |  |
5 | -1,190.3 | -57.7 | 135.2 |  | -2432.2 | -165.1 | 133.0
6 | -341.7 | 67.7 | 165.3 |  | -531.0 | 35.9 | 159.3
7 | -92.8 | 115.1 | 176.4 |  | -132.7 | 101.8 | 169.5
8 | 13.1 | 132.0 | 178.7 |  | -1.5 | 126.3 | 172.6
9 | 52.9 | 137.0 | 176.9 |  | 47.7 | 135.7 | 171.7
10 | 68.1 | 138.1 | 173.1 |  | 72.5 | 138.3 | 170.3
11 | 76.0 | 136.3 | 168.8 |  | 86.0 | 138.4 | 167.2
12 | 82.3 | 133.9 | 164.7 |  | 91.0 | 136.9 | 164.1
13 | 84.6 | 130.9 | 159.7 |  | 91.3 | 134.3 | 160.1
16 | 83.8 | 120.9 | 146.8 |  | 90.6 | 126.5 | 151.1
22 | 75.5 | 103.7 | 126.8 |  | 72.9 | 109.5 | 134.4
Panel C: $\gamma$ 9 loss function
Characteristics: | M, S, $\overline{r}$, $\sigma_{\epsilon}$ |  | M, S, $\sigma_{\epsilon}$
$\gamma^{*}$ | Resampled |  | Resampled
Updating | 2.5%ile | Mean | 97.5%ile |  | 2.5%ile | Mean | 97.5%ile
9 | -49.1 | 41.0 | 80.4 |  | -50.0 | 12.8 | 46.7
10 | -18.7 | 53.3 | 88.0 |  | -10.4 | 32.2 | 58.2
11 | -1.6 | 61.2 | 92.6 |  | 9.5 | 43.9 | 65.9
12 | 9.7 | 65.8 | 95.5 |  | 23.9 | 51.6 | 70.6
13 | 13.3 | 72.4 | 96.4 |  | 33.5 | 56.6 | 73.8
16 | 19.2 | 68.6 | 95.9 |  | 44.2 | 62.2 | 77.4
22 | 1.5 | 57.4 | 87.8 |  | 37.6 | 57.8 | 74.3
Rolling |  |  |  |  |  |  |
9 | -959.0 | -104.2 | 71.9 |  | -84.8 | 10.0 | 49.7
10 | -659.7 | -56.3 | 81.0 |  | -38.6 | 27.4 | 57.4
11 | -467.2 | -28.3 | 84.9 |  | -7.9 | 37.3 | 61.6
12 | -368.0 | -11.5 | 87.7 |  | 3.9 | 42.4 | 64.0
13 | -337.7 | -4.3 | 87.4 |  | 14.2 | 45.2 | 65.4
16 | -236.4 | 7.5 | 85.4 |  | 19.5 | 44.4 | 63.5
22 | -247.1 | -8.3 | 73.3 |  | -6.1 | 28.3 | 52.5

Table 3
Sampling distributions of out-of-sample certainty equivalent returns,
portfolio moments and properties

$\cal{CE}$i is the certainty equivalent return for a power utility function
with coefficient of risk aversion $\gamma=i$, in basis points per month.
Feasible portfolio rules are obtained by fixing $\gamma^{*}$, the set of
characteristics, and the updating protocol, and then maximizing the following
using $T$ months of sample data:
$\max_{\theta}\sum\limits_{t=0}^{T-1}\frac{(1+r_{p,t+1})^{1-\gamma^{*}}}{1-\gamma^{*}}\left(\frac{1}{T}\right)$
by allowing portfolio weights to depend on observable stock characteristics:
$r_{p,t+1}=\sum\limits_{i=1}^{N_{t}}\left(\overline{\omega}_{i,t}+\frac{1}{N_{t}}\theta^{{}^{\prime}}x_{i,t}\right)\cdot
r_{i,t+1}\,,$ where: $x_{i,t}$ is the $K$-vector of characteristics on firm
$i$, measurable at time $t$; $\overline{\omega}_{i,t}$ 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 $t$. ${\rm Intq\,Rng}=r_{.75}-r_{.25}$;
${\rm Skewness}=\frac{\mu-r_{.5}}{\sigma}$; ${\rm
Kurtosis}=\frac{\overline{r}^{+}_{.95}-\overline{r}^{-}_{.05}}{\overline{r}^{+}_{.5}-\overline{r}^{-}_{.5}}-2.63$;