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Optimized Fundamental Portfolios
Matthew Lyle and Teri Lombardi Yohn∗
February 14, 2019
Abstract
Equity portfolio construction consists of two stages: generating beliefs about thefuture performance of available stocks and allocating wealth across the stocks tomaximize the expected return subject to a specified risk tolerance. Two streams ofprior research have addressed each portfolio construction stage independently. Fun-damental analysis aids in the first stage by identifying accounting ratios that predictfuture stock returns, but provides little insight with respect to creating portfolios.Portfolio optimization aids in the second stage by determining weights to place onstocks to build a portfolio that maximizes expected returns subject to a specifiedrisk tolerance, but there is little empirical evidence suggesting that it is helpful toinvestors. We use a fundamentals-based returns model in conjunction with classicmean-variance portfolio optimization and find that portfolio optimization combinedwith fundamental analysis offers substantial improvements in portfolio performanceover either fundamental analysis or portfolio optimization alone. Long-only mean-variance optimized fundamental portfolios produce CAPM alphas of over 3.2% perquarter and 5-factor alphas of over 2.2% per quarter, with high Sharpe and Infor-mation ratios. The relative gains to investors from combining fundamental analysiswith portfolio optimization are even more pronounced when small capitalizationfirms are eliminated from the investment set.
JEL: G12, G14, G17
Keywords: Fundamental Analysis, Portfolio Optimization, Return Prediction
∗Lyle ([email protected]) is an Associate Professor at the Kellogg School of Man-agement, and Yohn ([email protected]) is Visiting Professor at the Kellogg School ofManagement and Professor of Accounting at the Kelley School of Business. We appreciate helpful sug-gestions and comments from Larry Brown, Ron Dye, Jeremiah Green, Bob Korajczyk, Bob McDonald,Steve Penman, Beverly Walther and workshop participants at the Kellogg School of Management andthe Fox School of Business. A special thanks to Rishabh Aggarwal, who provided invaluable researchassistance. We are grateful for the funding of this research by the Kellogg School of Management, Lylethanks The Accounting Research Center at Kellogg for funding provided through the E&Y Live andRevsine Research Fellowships.
1. INTRODUCTION
[T]here are still many “miles to go” before the gains promised by optimal portfoliochoice can actually be realized out of sample.
DeMiguel, Garlappi, and Uppal (2009, 1915)
The equity investor’s fundamental problem is to build a stock portfolio that maximizes
its expected return subject to some constraint (e.g., risk tolerance). In his seminal paper,
Markowitz (1952) argues that the process of constructing a portfolio consists of two stages.
The first stage involves generating beliefs about the future performance of available stocks.
The second stage uses the beliefs about future stock performance to allocate wealth
across the stocks in order to maximize the expected return of the portfolio subject to the
given constraints. Two independent streams of prior research, fundamental analysis and
portfolio optimization, address the two stages of the investor’s problem independently.
In this study, we connect both stages in one analysis and provide large sample evidence
of substantial gains to investors of doing so.
Fundamental analysis focuses on the first stage of the investor’s problem, belief gener-
ation, by using fundamentals (e.g., book-to-market and return on equity) to help predict
the ranking of future stock returns. Fundamental analysis and the ability of financial
ratios to predict the cross-sectional ranking of future returns can be dated at least as
far back as Benjamin and Dodd (1934). The fundamental ratios identified as useful for
predicting the ranking of future returns have also become prominent in empirical as-
set pricing (e.g., Fama and French 1992, 1993, 2015). While there remains considerable
debate as to why fundamental signals are able to predict the ranking of future stock
returns, a large body of evidence documents that the predictability exists over multiple
time periods and across countries, suggesting that the predictability is unlikely to be due
to random chance (e.g., Basu, 1977; Abarbanell and Bushee, 1998; Sloan, 1996; Bradshaw
et al., 2006; Novy-Marx, 2013; and Asness et al., 2017). However, fundamental analysis
1
provides limited usefulness for building portfolios because it is not clear how a risk-averse
investor might use this information to construct portfolios that simultaneously consider
both risk and reward. The research on fundamental analysis generally ranks the avail-
able stocks based on fundamentals and then equal or value weights groups of stocks to
create a portfolio. This is of limited usefulness for risk-averse investors because it does
not allocate weights according to risk preferences.
Portfolio optimization focuses on the second stage, wealth allocation, by providing
economically intuitive and mathematically rigorous rules for constructing portfolios of
risky assets to maximize returns subject to a specified risk tolerance. However, despite
the theoretical appeal, there is little evidence that even classic mean-variance portfolio
optimization, proposed by Markowitz (1952), is useful in practice. Prior research finds
that the allocation weights generated from portfolio optimization are unstable and lead
to poor portfolio performance. A common explanation for this poor performance is
that stock return moments, particularly the mean, are notoriously difficult to estimate
(DeMiguel et al., 2009; Jagannathan and Ma, 2003; Merton, 1980; Michaud, 1989).
There have been two proposed solutions to this problem. The first is to completely
disregard expected return estimates and to rely exclusively on variance estimates within
mean-variance optimization, resulting in so called “minimum variance portfolios.” Mini-
mum variance portfolios have been shown to generate higher Sharpe ratios than optimized
portfolios that incorporate historical returns as expected returns and non-optimized (i.e.,
equal or value weighted) portfolios that ignore expected returns (Engle et al., 2017; Jorion,
1985, 1986, 1991). The second proposed solution is to use alternative estimation tech-
niques that do not incorporate estimated moments or the use of mean-variance portfolio
optimization. Brandt et al. (2009) propose a novel methodology that combines a power
utility function with firm characteristics (i.e., firm size, book-to-market, and momen-
tum) and solves for portfolio weights via non-linear estimation. Hand and Green (2011)
extend Brandt et al. (2009) by incorporating accounting-based characteristics (i.e., accru-
als, change in earnings, and asset growth) and show that accounting-based fundamental
2
signals enhance portfolio performance over price-based signals.
In this study, we propose an alternative solution which leverages recent innovations
in fundamental analysis research to estimate expected returns directly to form mean-
variance optimized portfolios. Specifically, research on fundamental analysis has pro-
gressed beyond using fundamentals to rank stocks, and has instead quantified the relation
between fundamentals and stock returns by using accounting-based valuation models to
directly infer expected returns (e.g., Gebhardt et al., 2001; Gode and Mohanram, 2003;
Frankel and Lee, 1998). Recent research also shows that fundamentals can be used to gen-
erate unbiased time-varying estimates of expected returns for the cross-section of stocks
in the U.S. (Lyle et al., 2013; Lyle and Wang, 2015) and worldwide (Chattopadhyay et al.,
2018). This ability to estimate fundamentals-based expected returns, when coupled with
innovations in multivariate variance estimation (e.g., Engle et al. 2017; Ledoit and Wolf
2017), provides the two key inputs required for mean-variance optimization. Therefore,
this approach does not disregard expected returns in mean-variance optimization, but
rather exploits the insights from fundamental analysis research to improve estimates of
expected returns.
We first examine the performance of fundamental portfolios without optimization.
Specifically, we use quantifiable inputs of return moments which are constructed by esti-
mating a parsimonious fundamentals-based model that includes book-to-market, return
on equity, and two additional variables that capture growth in book value: growth in net
operating assets (Sloan, 1996; Fairfield et al., 2003; Cooper et al., 2008) and growth in
financing (Bradshaw et al., 2006; Cooper et al., 2008). We use this fundamentals-based
model and examine the performance of an equal weighted (hereafter, EW) portfolio and
a value weighted (hereafter, VW) portfolio of the top decile of stocks based on expected
future returns from the model. These portfolios represent our fundamental portfolios
without optimization.
We also examine the performance of optimized portfolios that do not incorporate
fundamentals-based expected returns. We examine the performance of minimum-variance
3
portfolios that do not incorporate expected returns (hereafter, MV), portfolios that incor-
porate historical average-based expected returns and either mean-variance optimization
with a target expected return (hereafter, MVT) or maximization of the Sharpe ratio
(hereafter, MS). We also examine the performance of Brandt et al. (2009) portfolio op-
timization (hereafter, BSV) using the price-based characteristics in Brandt et al. (2009),
the accounting-based characteristics in Hand and Green (2011), or historical average-
based expected returns as the characteristic. These portfolios represent our optimized
portfolios without incorporating fundamentals-based expected returns.
Finally, we examine the performance of portfolios which are optimized where the
return predictions from the fundamentals-based model are used directly as inputs into
the portfolio optimizer. We use the fundamentals-based model of expected returns with
MVT optimization, MS optimization, and BSV optimization. These portfolios represent
our optimized fundamental portfolios.
We compare the performance of the fundamental portfolios without optimization,
optimized portfolios without fundamentals-based expected returns, and optimized fun-
damental portfolios. We examine “long-only” portfolios because taking short positions is
often not feasible and even when feasible, implementation costs are often very high (e.g.,
Beneish et al., 2015).
We find that combining fundamental analysis with portfolio optimization results in
complementary gains to both. Despite the evidence in prior studies of limited to no gains
from employing standard portfolio optimization techniques, we find that portfolio opti-
mization can provide large gains to investors, but only when used with a fundamentals-
based model to estimate expected returns. Combining fundamental analysis with mean-
variance portfolio optimization yields higher out-of-sample Sharpe ratios, Information
ratios, factor alphas, and average mean-variance utilities, over strategies of employing
fundamental analysis or portfolio optimization alone.
Long-only fundamental portfolios using mean-variance optimization (MVT and MS)
results in substantial portfolio performance improvements over non-optimized fundamen-
4
tal portfolios, whereas BSV optimization yields no improvement. MS optimization yield
quarterly Sharpe and Information ratios of 0.473 and 0.522, respectively, which represent
11 (89) and 16 (427) percentage increases over the respective ratios of the non-optimized
EW (VW) fundamental portfolios. The fundamental portfolios using MS optimization
produce CAPM alphas of over 3.2% and 5-factor alphas of over 2.3% per quarter over our
sample period, and are generally higher in magnitude and statistical significance than
EW benchmark portfolios. We also estimate the risk aversion coefficient that would be
required by a mean-variance investor to be indifferent to optimized fundamental port-
folios relative to EW fundamental portfolios. Our estimates tend to range from zero to
one, indicating that virtually any risk-averse investor would be better off with optimized
fundamental portfolios.
The relative increase in these key performance metrics for the optimized versus non-
optimized fundamental portfolios is even more pronounced when we eliminate small stocks
from our sample, which, when considering that we form long-only portfolios, indicates
our results are not driven by investing in illiquid stocks or from taking short positions.
We also find that the gains to the optimized versus non-optimized fundamental portfolios
hold over multiple time periods and tend to be increasing over time, even after well-known
academic research which highlights the predictive ability of financial ratios was published.
Collectively, these results suggest that portfolio optimization dramatically improves the
performance of the fundamentals-based investment strategies.
We also find substantial gains from combining fundamentals with portfolio optimiza-
tion over portfolio optimization alone. Consistent with prior research, we find that MVT,
MS, and BSV optimized portfolios using historical average stock returns to estimate
expected returns all yield low Sharpe ratios, negative Information ratios, and zero or
negative alphas. Thus, our results suggest that, unlike fundamentals-based investment
strategies, portfolio optimization on its own yields essentially no gains when using his-
torical returns to estimate expected returns, the common approach employed in prior
5
studies.1 While the MV portfolios, which ignore expected returns, yield higher Sharpe
and Information ratios than the portfolios optimized using historical average returns to
estimate expected returns, the fundamental portfolios using MS optimization outperform
the MV portfolios. These results suggest that portfolio optimization combined with fun-
damental analysis offers substantial benefits to investors in terms of portfolio performance
over fundamentals-based strategies alone and over portfolio optimization alone.
This study provides important contributions to both practice and the research on
fundamental analysis and portfolio optimization. Fundamental analysis is aimed at iden-
tifying stocks that are likely to experience higher future returns but provides little in-
sight with respect to creating portfolios. Our study provides an implementable method
of developing portfolios that improve the performance of fundamental analysis. Simi-
larly, portfolio optimization provides theoretical arguments for optimizing portfolios, but
there is little empirical evidence to date suggesting that it results in superior portfolio
performance. Our findings suggest that portfolio optimization, when combined with fun-
damental analysis, can help investors realize “the gains promised by optimal portfolio
choice” and highlight that combining the findings from seemingly independent fields of
research can help to achieve these gains.
2. Fundamental Analysis and Portfolio Optimization
The basic idea behind fundamental analysis is to estimate an “intrinsic value” which
can then be compared to market valuations to cross-sectionally rank stocks based on
expected future stock returns. Almost all fundamental analysis starts with a form of the
residual income formula, which allows valuations to be expressed in terms of accounting
variables. While the residual income formula is identical to the dividend discount formula,
the formulation offers insight into the determinants of valuations: book values, expected
profitability, expected growth in book values, and discount rates. A large literature on
1We also examined if using factor models to estimate expected returns improved the performancerelative to a simple historical average. We found that using these estimates results in portfolios that alsoperform very poorly.
6
fundamental analysis has shown that current accounting variables such as profitability
and growth in operating assets and financing are predictive of future profitability and
growth (Sloan, 1996; Fairfield et al., 2003; Cooper et al., 2008), which in turn implies
that intrinsic values can be written as a function of these fundamental variables as well
as discount rates. Fundamental analysis research demonstrates that cross-sectionally
ranking firms based on these variables predicts the ranking of future stock returns.
While there is robust evidence that fundamentals are able to predict the ranking of
future returns, there is considerable debate as to what drives this predictability. Some
argue that the fundamental variables capture firm’s differential risk characteristics and
that the future stock returns reflect this differential risk (Fama and French, 1992). Others
argue that the ability of fundamental signals to predict the cross-section of future stock
returns is driven by investor behavioral or cognitive biases such that investors tend to
display preferences for certain stocks that may not be justified by the fundamentals
(Frankel and Lee, 1998). Biases such as investor sentiment toward certain types of stocks
(e.g., glamour stocks), a recency bias, over-confidence, earnings fixation, and limited
attention (e.g., Sloan, 1996; Hirshleifer et al., 2009 ) have been offered as drivers of
the predictability of stock returns. Regardless of the underlying mechanism, the ability
of fundamentals to predict the cross-sectional ranking of firm’s future stock returns is
strongly supported in empirical data and has been shown to be robust across time periods
and countries.
While prior research shows that fundamentals can be used to predict the ranking of
future stock future returns, it provides little insight into how to form optimal portfo-
lios based on the analysis. The standard approach in the academic literature is to form
either equal or value weighted portfolios from groups of stocks ranked by the financial
ratio of interest and determine if there exists differences across portfolio returns. While
informative in academic settings, this approach imposes significant challenges for an in-
vestor because it is not clear how investor risk tolerance can be accommodated in forming
portfolios and allocating wealth among stocks.
7
The proposed solution in financial economics is to use a mathematical program to de-
termine weights that generate an optimal portfolio, simultaneously incorporating investor
beliefs about future stock returns as well as risk tolerance. Portfolio optimization pro-
vides economically intuitive and mathematically rigorous rules for constructing portfolios
of risky assets to maximize returns subject to a specified risk tolerance. However, there is
little evidence that even classic mean-variance portfolio optimization (Markowitz, 1952) is
useful in practice. Prior research finds that the allocation weights generated from portfolio
optimization are unstable and lead to poor portfolio performance. A common explanation
for this poor performance is that expected returns are difficult to estimate (DeMiguel
et al., 2009; Jagannathan and Ma, 2003; Merton, 1980; Michaud, 1989), and research
shows that completely disregarding expected return estimates within mean-variance op-
timization yields better performance than optimized portfolios that incorporate historical
returns as expected returns and non-optimized (equal or value weighted) portfolios that
ignore expected returns (Engle et al., 2017; Jorion, 1985, 1986, 1991). Thus, prior re-
search suggests that portfolio optimization may be of use to investors only when beliefs
about expected returns are completely disregarded.
Given that poor quality estimates of expected returns appear to drive the poor perfor-
mance of optimized portfolios and given that fundamental analysis is focused on predict-
ing expected returns, we examine whether tying together recent innovations in fundamen-
tal analysis and portfolio optimization provides gains to investors. In what follows below
we outline how fundamental analysis and optimal portfolio theory can be combined.
2.1. Fundamentals and Returns
Lyle and Wang (2015) use a log-linear approximation to show that expected firm log
stock returns can be expressed as a linear combination of the book-to-market ratio, bmt,
and expectations about future return on equity, Et[roet+1]:
Et[rt+1] = α0 + α1bmt + α2Et[roet+1]. (1)
8
The key coefficients, α1 and α2, are both predicted to be positive. Lyle and Wang
(2015) implement the model by using lagged roet as a simple proxy for Et[roet+1]. We
expand the Lyle and Wang (2015) implementation by incorporating insights from prior
financial statement analysis research which shows that future profitability is a function of
not only lagged roet, but also variables that measure growth. We use growth in net op-
erating assets, got, and growth in financing, gft, as our proxies for growth given the prior
fundamental analysis research of the relation between growth and future profitability and
returns (Fairfield et al., 2003; Cooper et al., 2008) and the relation between financing and
future returns (Bradshaw et al., 2006; Cooper et al., 2008). We conducted a formal model
selection test using a LASSO selection algorithm to test if each of the variables included
in the model are incrementally informative. The results of the LASSO selection algorithm
(untabulated) confirm that including all four variables yields the most informative model.
Therefore, our expected return on equity model takes the form:
Et[roet+1] = γ0 + γ1roet + γ2got + γ3gf t. (2)
Substitution of (2) into (1) gives a stock return equation of the the form:
rt+1 = A0 + A1bmt + A2roet + A3got + A4gft + εt+1, (3)
where the expected return is given by Et[rt+1] = A0 + A1bmt + A2roet + A3got + A4gft,
and εt+1 represents an unpredictable noise term. The parsimonious linear structure of
(3) allows for a straightforward connection to mean-variance portfolio optimization as
outlined in the next section.2
2In our analysis we use “simple” returns and roe as opposed to logs, as in Lyle and Wang (2015).The use of simple returns follows from a first order Taylor approximation that exp(rt+1)− 1 ≈ rt+1.
9
2.2. Portfolio Optimization and Fundamentals
Virtually any investor faces the challenge of how to allocate wealth such that the
expected return on wealth is maximized given the investor’s risk tolerance. In his semi-
nal paper, Markowitz (1952) provides a mathematically elegant approach to solving the
investor’s problem, which can be summarized as a constrained optimization program:
maxωiN
i=1
Et[rP,t+1], (4)
s.t. Vt[rp,t+1] ≤ Ω, (5)
rP,t+1 =N∑i=1
ωi × ri,t+1, (6)
N∑i=1
ωi = 1. (7)
Here rP,t+1 represents the future time t + 1 return on the portfolio, which consists of a
combination of N assets each with the ith return, ri,t+1, and a portfolio weight, ωi, where
i ∈ N . The approach is appealing as it is simple and is able to capture the straightforward
intuition that investors consider both expected returns, Et[rP,t+1], expected risk, which
is captured by variance,Vt[rp,t+1], and risk tolerance, Ω, when constructing a portfolio.
Conceptually, applying the program is trivial and weights can easily be generated using
numerous software packages since all that is required as inputs are expected returns and
a covariance matrix.3
To tie the expected returns from fundamental analysis with portfolio optimization,
we substitute the fundamental analysis equation of (3) into the portfolio optimization of
equation (4). This substitution allows us to write the optimization program in terms of
3For this study, we used the software package Matlab, and specifically it’s built-in function “quad-prog” to solve the optimization problem. However, several popular open source software packages,including R and Python, have similar capabilities.
10
fundamentals:
maxωiN
i=1
N∑i=1
ωi(Ai,0 + Ai,1bmi,t + Ai,2roei,t + Ai,3goi,t + Ai,4gfi,t), (8)
s.t. Vt[N∑i=1
ωiεi,t+1] ≤ Ω, (9)
N∑i=1
ωi = 1. (10)
3. Data and Sample Selection
Our data are from standard sources: CRSP and Compustat. Our full sample time
period is from 1991-2015. We use the period 1991-1995 as an initial model estimation
period and 1996-2015 as the out-of-sample test period. Focusing the out-of-sample tests
on this recent time period allows us to more easily assess the gains that an investor could
have generated in periods that follow the publication of several academic papers that
document the predictability of stock returns based on the variables that we use in our
model (e.g., Bradshaw et al., 2006; Fama and French, 1992; Sloan, 1996; Fairfield et al.,
2003; Cooper et al., 2008).
At the end of each month, prior to portfolio construction, we remove penny stocks,
stocks with negative book values, and stocks that have less than three years of historical
stock return data. These criteria ensure we can reasonably estimate stock return volatility
and pairwise correlations. We also remove observations that have “outlier” values of
the book-to-market ratio, return on equity, growth in net operating assets, or growth
in financing. Given the limitations of windorization at detecting and addressing outliers
(e.g., Leone et al., 2017), we use the Minimum Covariance Determinant (MCD) algorithm
to identify outliers as it represents a robust algorithm that can formally detect outliers in
multivariate data (Rousseeuw and Driessen, 1999). In addition to these filters, we also, as
is common in the literature, remove financial and regulated firms from the sample since
the accounting for these types of firms is systematically different from other firms. The
11
risk-free rates and factor portfolios that are used in our empirical tests are downloaded
from Ken French’s data library.4
3.1. Expected Returns and Model Estimation
To generate expected returns, we must estimate equation (3). Prior research has es-
timated models both cross-sectionally (e.g., Chattopadhyay et al., 2018; Lewellen, 2015;
Lyle et al., 2013) and by industry (e.g., Lyle and Wang, 2015). However, cross-sectional
estimation assumes that every firm in the sample has an identical slope coefficient,
whereas industry definitions tend to be exceptionally noisy and can lead to worse es-
timates for prediction than simple cross-sectional estimation (e.g., Fairfield et al., 2009).
In light of this, we estimate the model monthly by using five years of rolling historical
data using three forms of estimation: 1) cross-sectional, 2) by industry (using the Fama
and French 48 industry classifications), and 3) by size decile. Our choice for estimating
within size deciles is motivated by the fact that it represents an easy to measure charac-
teristic, that the predictability of future returns has been shown to vary systematically
with size, and that similar sized firms tend to comove (e.g., Fama and French, 1992).
In untabulated analyses, we find expected return estimates from both industry and size-
based estimation dominated those based on cross-sectionally estimating parameters and
that size-based estimation provided the highest level of significance in terms of resultant
expected return measures. We chose size-based estimation of expected returns based on
this analysis.5
In our estimation, we update firm fundamentals, bmt, roet, got, and gft, quarterly
at the end of the month in which they are reported according to Compustat to ensure
that the fundamentals have been publicly disclosed. If the reporting date is missing
in Compustat we assume that the information is public three months after the firm’s
fiscal quarter. bmt is book value of equity scaled by market value of equity from the4These data can be downloaded from http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data library.html5We also formally tested for systematic variation in estimated coefficients for both industry and
size-based estimation methods. The null of no variation across coefficients could not be rejected forindustry-based estimation, but it was rejected for size-based estimation.
12
Compustat quarterly files, roet, got, and gft, are earnings before extraordinary items,
the change in net operating assets, and the change in financial assets, respectively, each
scaled by lagged quarterly book value. To avoid potential issues with outliers, we cross-
sectionally standardize each of the predictor variables using Blom’s normal score method.
The coefficients A0, A1, A2, A3, A4 are estimated by regressing one-month ahead stock
returns on the fundamentals within each size decile. This estimation yields an expected
stock return estimate for each firm i of the form:
Et[ri,t+1] = µi,t = Aj,0 + Aj,1bmi,t + Aj,2roei,t + Aj,3goi,t + Aj,4gfi,t, (11)
where j denotes the jth size decile at time t for which i is a member.
We use this estimate of expected returns in our mean-variance optimizer. Since mean-
variance optimization also requires estimates of a covariance matrix, we use the recently
developed Ledoit and Wolf (2017) covariance estimator. Covariance estimates are up-
dated each month, using three years of historical monthly stock return data. The Ledoit
and Wolf (2017) covariance estimator represents a non-linear shrinkage estimator that
dominates traditional linear estimators and was constructed for implementation in mean-
variance portfolio optimization.
3.2. Implementing Portfolio Optimization
In our empirical tests, we examine four versions of portfolio optimization: 1) minu-
mum variance (MV), 2) minimum variance with a target expected return (MVT), 3) the
maximum Sharpe Ratio (MS) and 4) the approach of Brandt et al. (2009) (BSV).
3.2.1. Mean-Variance Optimization
The standard mean-variance representation described above can be equivalently writ-
ten as a minimization problem, and each of the optimizations MV, MVT, and MS follow
from the following quadratic program:
13
minωiN
i=1
Vt[rP,t+1], (12)
s.t. Et[rp,t+1] ≥ r. (13)
where MV solves the program ignoring constraint (13). MVT solves the program
directly, where we set r to the expected return of a benchmark non-optimized portfolio.
MS involves solving the program for a continuum of values for r and building a mean-
variance frontier. The MS portfolio that we implement in our empirical tests represents
the portfolio that has the highest ex ante expected return over expected standard devi-
ation on the frontier. In all cases, we impose the constraint that ∑Ni=1 ωi = 1 and that
ωi ≥ 0. To ensure that any one stock does not overly influence a portfolio, our main
results are based on constraining each stock to have no more weight than one percent;
however, when we varied this constraint to be up to five percent in untabulated analyses,
we found that our main results hold.
3.2.2. BSV Optimization
Brandt et al. (2009) propose a method to incorporate firm-level characteristics by
utilizing the following estimation approach:
maxθ
1t
t−1∑j=0
(1 + rp,j+1)1−γ
1− γ , (14)
s.t. ωi,j = ωi,j + 1Nj
θxi,j,
where ωi,j is the firm i’s market capitalization weight at time period j, Nj is the
number of firms in the portfolio, θ is a vector of parameters to be estimated, and xi,j
a vector of firm characteristics. Portfolio weights at time t are then given by ωi,t =
ωi,t + 1Nθxi,t. Using our estimate of expected returns from (11) as the firm characteristic
14
gives ωi,t = ωi,t + 1Nθµi,t.
Like Brandt et al. (2009) and Hand and Green (2011), we set γ = 5 and estimate
the parameter θ using equation (14) with 5 years of rolling historical data. As in our
mean-variance optimization, we ensure the weights are non-negative and sum to one.
4. Empirical Results
Table 1 provides the descriptive statistics. Panel A shows that the mean (median)
firm size is $4.9 ($.62) billion, book-to-market is 0.60 (0.51), and quarterly return on
equity is 2.34% (2.48%). The correlations in Panel B suggest that future returns are
positively correlated with book-to-market, return on equity, and growth in financing, and
negatively correlated with growth in net operating assets and size, consistent with prior
research (e.g., Fama and French, 1992; Fairfield et al., 2003).
Table 2 presents the results of regressing future stock returns on expected return
estimates, µt. We show the results of using historical average returns, HIST, and the
results of using expected returns from the fundamentals-based model, FUND. HIST uses
the rolling historical monthly average stock return over the prior 36 months. FUND is
calculated as in equation (11).
The table reports results for predicting monthly returns in columns (1) and (2) and the
results for predicting quarterly returns in columns (3) and (4). We examine a quarterly
holding period because this requires less frequent re-balancing of portfolios than a monthly
holding period. Results are similar across both time periods. The historical average
model, HIST, does not predict returns; in fact, it has a negative relation with future
stock returns. The fundamentals-based model, FUND, does predict out-of-sample stock
returns, with significant positive coefficients on µt.
Table 3 presents performance metrics (Panel A) and characteristics (Panel B) of port-
folios that are not optimized. The stocks in the portfolio are either equal weighted (EW)
or value weighted (VW). Columns (1) and (2) report the performance of EW and VW
portfolios, respectively, of all available stocks. Columns (3) and (4) report the perfor-
15
mance of EW and VW portfolios, respectively, of the top decile of available stocks based
on the expected return using HIST. Columns (5) and (6) report the performance of EW
and VW portfolios, respectively, of the top decile of stocks based on the expected return
using FUND. The table allows us to assess the potential gains from non-optimized port-
folios that incorporate expected returns from the fundamentals-based model relative to
portfolios that do not incorporate fundamentals.
To assess the performance of the portfolios, we present the Sharpe Ratio, which is
calculated as the sample mean portfolio return less the risk-free rate divided by the
sample standard deviation of the portfolio. We also present the Information Ratio, which
is calculated as the intercept of the market model divided by the of the residual from the
market model. To ensure that the portfolios are not merely reproducing the returns of
commonly used factor portfolios, we also report alphas from the CAPM, the Fama and
French (1993) 3-factor (FF3), the Fama and French (1993) 4-factor (FF4), and the Fama
and French (2015) 5-factor (FF5) benchmarks. The latter models are based on similar
characteristics to those used in our fundamentals-based expected return model and are
formed to explain returns of portfolio constructed from those characteristics. We also
report average and excess stock returns for each portfolio.
The EW and VW portfolios of all firms in our sample, reported in columns (1) and
(2), respectively, represent easy-to-implement strategies as they do not require any esti-
mation or analysis. The EW portfolio of all stocks yields a Sharpe (Information) ratio of
0.256 (0.151) while the VW portfolio yields a Sharpe (Information) ratio of 0.214 (0.106).
The results using HIST, reported in columns (3) and (4), suggest that using historical
returns as an estimate of expected returns results in, not surprisingly, very poorly per-
forming portfolios. The Sharpe ratio of 0.110 for the EW portfolios and 0.139 for the
VW portfolios are much lower than those using all available stocks and ignoring expected
returns. In addition, the Information ratios for the HIST portfolios are negative.
Columns (5) and (6) report the results for FUND, and show that both the EW and
VW portfolios using expected returns from the fundamentals-based model dominate the
16
respective portfolios that do not incorporate the expected returns based on fundamentals.
The Sharpe (Information) ratios are 0.426 (0.451) for the EW portfolios and 0.250 (0.122)
for the VW portfolios. We note that the superior performance of the EW portfolios over
the VW portfolios is likely attributable to the VW portfolios being dominated by a few
large firms. The superior performance of the fundamental portfolios is consistent with
the findings in the prior literature that fundamental analysis is useful in predicting the
cross-section of stock returns. We next turn to whether portfolio optimization improves
portfolio performance.
Table 4 presents performance metrics of portfolios that are optimized but do not
incorporate expected returns from fundamental analysis. The table reports the perfor-
mance metrics of portfolios formed using MV, MVT, and MS optimization in Panel A
and BSV optimization in Panel B. In Panel A, column (1) reports the performance of
minimum-variance (MV) portfolios in which expected returns are ignored. Columns (2)
and (3) report the results for all available stocks using HIST-based expected returns with
MVT and MS portfolio optimization, respectively. Columns (4) and (5) report the results
for the top decile of stocks based on the HIST-based expected returns with MVT and
MS portfolio optimization, respectively.
The superior performance of MV reported in column (1) over the HIST portfolios in
columns (2) through (5) replicate the findings in prior studies that portfolio optimization
that does not incorporate estimates of expected returns yields superior performance over
portfolios optimized using the historical average of stock returns as the expected returns.
Specifically, MV portfolios yield a Sharpe ratio of 0.290 and an Information ratio of 0.255,
which are much higher than those for portfolios using HIST expected returns for the full
sample with either optimization method. The MV portfolios also yield positive CAPM
and Fama-French 3-factor alphas, while portfolios using HIST-based expected returns
with MVT and MS optimization yield negative alphas. The portfolios of the top decile
of stocks based on HIST expected returns also yield low Sharpe ratios and negative In-
formation ratios using either MVT or MS optimization. Overall, the results show that
17
combining portfolio optimization with historical average returns results in poor portfolio
performance, with low Sharpe ratios, and generally negative Information ratios and al-
phas. If we compare the performance of the portfolios in columns (1) through (5) with
the performance of the non-optimized EW and VW portfolios in Table 3, the results are
consistent with prior studies that portfolio optimization provides some gains to investors
over an EW or VW strategy only if the mean return estimate is ignored. However, the
EW fundamental portfolios yield superior performance over all the optimized portfolios,
including the MV portfolios.
In Panel B, we report the performance of portfolios that incorporate price-based char-
acteristics (PRICE) as in Brandt et al. (2009) in column (1), accounting-based character-
istics (ACCT) as in Hand and Green (2011) in column (2), and historical average-based
expected returns (HIST) in column (3) using BSV optimization. We find that using the
PRICE and ACCT characteristics yields superior portfolio performance relative to us-
ing HIST-based expected returns as the characteristic.6 The Sharpe (Information) ratio
for the portfolios formed using PRICE and ACCT characteristics are 0.277 (0.184) and
0.282 (0.185), respectively, which are higher than the Sharpe (Information) ratio of 0.207
(0.061) for portfolios formed using HIST-based expected returns as the characteristic.
Consistent with Hand and Green (2011), we also find that the Sharpe (0.282) and In-
formation (0.185) ratios for portfolios using the ACCT characteristics are higher than
the Sharpe (0.277) and Information (0.184) ratios for portfolios formed using PRICE
characteristics.
In addition, the portfolios formed using the ACCT and PRICE characteristics with
BSV optimization outperform the portfolios reported in columns (2) through (5) in Panel
A which are formed using HIST-based expected returns with mean-variance MVT or MS6The Sharpe and Information ratios are lower than those reported in Brandt et al. (2009). The
difference can be attributed to our later time period and differences in sample size. In untabulatedanalyses, we formed portfolios over the time period examined in and formed portfolios using the samemethodology as Brandt et al. (2009) and find similar results to those reported in Brandt et al. (2009).However, that the Brandt et al. (2009) method does not provide gains to investors over equal weightedfundamentals-based portfolios. The Sharpe ratios, Information ratios, and alphas for the portfoliosformed using the Brandt et al. (2009) method are lower than the EW fundamentals-based portfoliosduring that time period.
18
optimization. However, the ACCT and PRICE portfolios with BSV optimization yield
lower Sharpe and Information ratios relative to the Sharpe ratio of 0.290 and Information
ratio of 0.255 for the MV portfolios reported in column (1) of Panel A. In addition, if
we compare the performance of the BSV portfolios with the performance of the non-
optimized EW and VW portfolios in (1) and (2) Table 3, there are gains to investors over
an EW or VW strategy. However, the EW fundamental portfolios yield superior perfor-
mance over all the portfolios using BSV optimization. These findings suggest that port-
folios optimized using MV, MVT, MS, or BSV optimization without fundamentals-based
expected returns yield lower performance portfolios than non-optimized EW portfolios
using fundamentals-based expected returns.
Table 5 presents the performance metrics for optimized fundamental portfolios which
incorporate expected returns from the fundamentals model, FUND, with either MVT,
MS, or BSV portfolio optimization. Panel A presents the portfolio performance metrics.
Panel B provides portfolio characteristics. We present the performance metrics for non-
optimized (EW) fundamental portfolios as a benchmark for comparison. Columns (1),
(2), and (3) present the performance of the optimized fundamental portfolios for the full
sample of available stocks. Columns (4), (5), and (6) present the performance of the
optimized fundamental portfolios after constraining the available set of stocks to those in
the highest decile of stocks based on the expected returns from the fundamentals-based
model. Optimizing using the entire sample allows the optimized fundamental portfolios to
differ from the non-optimized (EW) fundamental portfolios both in terms of the portfolio
weights and in terms of the firms included in the portfolio. Optimizing within the top
decile provides insight into the extent to which the improvement in performance from
optimization (over EW portfolios) is due to the portfolio weights, given that the firms
included in the portfolio are held fixed.
As with Table 4, our assessment of performance is primarily based on the Sharpe
and Information ratios. We report results of statistical tests of whether the Sharpe
(Information) ratio for each portfolio is significantly higher than that of the benchmark
19
EW top decile fundamental portfolio, designated as *, **, and *** for a significantly
higher ratio at the 10%, 5%, and 1% significance level, respectively.7 We also include a
third metric, λ∗, which captures the level of risk aversion required for a mean-variance
investor to be indifferent to the equal weighted fundamental portfolio.8 λ∗ = 0 indicates
that even a risk-neutral investor is better off with the portfolio of interest relative to an
equal weighted portfolio, while λ = ∞ indicates that no investor of any risk aversion
level is better off with the portfolio of interest relative to an equal weighted portfolio. A
common risk aversion value assumed in asset pricing is λ = 10. This value is also often
used in practice when performing mean-variance optimization. If we take this value as
representative of the average investor, then a λ∗ value of less than 10 indicates that the
average investor is better off with the portfolio of interest.
Table 5, Panel A shows that the performance of the fundamental portfolios is signif-
icantly improved with MVT or MS optimization relative to non-optimized fundamental
portfolios, however BSV optimization reduces portfolio performance. Specifically, ap-
plying MVT or MS portfolio optimization for the full sample of stocks or for the top
decile of stocks results in higher Sharpe and Information ratios than the EW fundamen-
tal portfolios, whereas these metrics are lower with BSV optimization. The MVT and
MS optimized top decile fundamental portfolios yields the highest performance with λ∗’s
that are less than one, indicating almost any mean-variance investor would be better
off. The Sharpe ratios for the top decile fundamental portfolios with both MVT and MS
optimization are 0.473, which are significantly higher than the Sharpe ratio of 0.426 for
the top decile fundamental EW portfolios. They are also higher than the Sharpe ratio of
0.447 (.0456) for the full sample fundamental portfolios using MVT (MS) optimization.
Unlike mean-variance optimization, the BSV approach actually performs worse when the
7Significance levels for Sharpe and Information ratios are calculated by simultaneously estimatingthe sample moments of each series via GMM and testing if the ratio of the optimized portfolio is largerthan the EW portfolio. Significance levels are based on heteroskedasticity consistent standard errors anda Newey-West correction with three lags.
8Specifically, a mean-variance investor has an expected utility function of the form Et[U(rp,t+1)] =Et[rp,t+1] − λ
2Vt[rp,t+1]. Given two portfolios, an equal weighted portfolio, rEW,t+1, and an optimizedportfolio, rO,t+1, then λ∗ is the λ such that Et[U(rEW,t+1)] = Et[U(rO,t+1)].
20
sample is constrained to the top decile.
Similarly, the Information ratios are highest for the top decile fundamental portfolios
with MS or MVT optimization. Specifically, the Information ratios are 0.520 for the
top decile fundamental portfolios with MVT optimization and 0.522 with MS optimiza-
tion. In comparison, the Information ratios are 0.451 for the EW top decile fundamental
portfolios and 0.476 (0.494) for the full sample fundamental portfolios with MVT (MS)
optimization. Importantly, the returns to the top decile fundamental portfolios with MS
or MVT optimization are not driven by common portfolio factors as we also find that
optimization leads to higher portfolio alphas. Specifically, the top decile fundamental
portfolios yield a CAPM alpha of 3.19% per quarter with MVT optimization and 3.2%
per quarter with MS optimization. These returns are compared to 3.02% for the EW
top decile fundamental portfolios and 2.80% (2.89%) for the full sample fundamental
portfolios with MVT (MS) optimization.
Collectively, these results suggest that mean-variance portfolio optimization provides
substantial gains to investors when combined with fundamental analysis as it is able
to exploit the considerable research has been devoted to estimating the first and second
movements of stock returns. BSV optimization does not match the performance of mean-
variance optimization in our setting because BSV was devoted to constructing portfolios
without direct estimation of return moments, and thus does not fully leverage the value
provided by these estimates.
Panel B reports the characteristics of the portfolios in terms of size, bmt, roet, got,
gft, portfolio turnover, and the number of stocks included in the portfolio. Focusing on
column (5), fundamental portfolios with MS optimization tend to consist of larger firms
and have higher bmt, roet, lower got and higher gft than the EW fundamental portfolios.
Portfolio turnover is modestly higher for the optimized fundamental portfolios and we
note that the improved portfolio performance is achieved with a smaller number of stocks.
21
4.1. Excluding Small Stocks
Given the importance of stock liquidity for portfolio construction, Table 6 presents the
results for a sample that excludes the smallest 20 percent of stocks at portfolio formation.
While the portfolios in previous analyses excluded penny stocks, excluding the smallest
20 percent of stocks further removes stocks for which investors are more likely to face
liquidity issues and higher transactions costs. Consistent with the results in Table 5,
we find that top decile fundamental portfolios with MVT or MS optimization yield the
highest Sharpe ratios, Information ratios, and alphas, as well as the lowest λ∗. Specifically,
the Sharpe ratio is 0.435 for top decile fundamental portfolios with MVT optimization
and 0.436 with MS optimization. These compare to 0.277 for the top decile fundamental
portfolios with BSV optimization and 0.389 for the EW top decile fundamental portfolios,
and to 0.396, .0420, and 0.346 for the full sample fundamental portfolios with MVT, MS,
and BSV optimization, respectively.
Similarly, the Information ratios of 0.462 (0.466) for the top decile fundamental port-
folios with MVT (MS) optimization are higher than those for the top decile fundamental
portfolios with BSV optimization, no optimization (i.e, EW), and those for the full sam-
ple fundamental portfolios with any of the three optimization methods. Specifically,
the Information ratio is 0.390 for the top decile EW fundamental portfolios and. 0.178
for the top decile fundamental portfolios with BSV optimization; and 0.386, .0439, and
0.337 for the full sample fundamental portfolios with MVT, MS, and BSV optimization,
respectively. The Sharpe and Information ratios are statistically higher for the top decile
fundamental portfolios with MVT and MS optimization relative to the EW top decile
fundamental portfolios.
We also that find that the fundamental portfolios with MVT and MS optimization
lead to higher relative portfolio alphas after excluding small stocks. The top decile
fundamental portfolios with MVT and MS optimization yield the highest CAPM, Fama-
French 3-factor alphas, and Fama-French 4-factor alphas. For example, the top decile
22
fundamental portfolios yield a CAPM alpha of 2.78% per quarter with MVT optimization
and 2.80% per quarter with MS optimization. These alpha are compared to 2.53% for
EW top decile fundamental portfolios and 1.28% for the top decile fundamental portfolios
with BSV optimization, and 2.25%, 2.51%, and 1.80% for the full sample fundamental
portfolios with MVT, MS, and BSV optimization, respectively. These results suggest that
MVT and MS portfolio optimization provide substantial gains to investors when combined
with fundamental analysis even after excluding small stocks from the investment set.
In panel B, the top decile fundamental portfolios with MVT and MS optimization
include smaller firms with higher book-to-market ratios and lower growth in net operating
assets. The portfolios also have a smaller number of stocks than the portfolios with no
optimization.
4.2. Quality of Expected Returns
A curious result that emerges in Table 5 and Table 6 is that optimizing over the entire
cross-section of firms produces portfolios that perform less well than optimizing within
the top decile of expected returns. This result holds for both MVT and MS optimization.
An investigation of our expected return measure provides an explanation for these pat-
terns. Untabulated tests of expected return estimates across rankings of expected returns
show that the predictive power is lowest among firms with the lowest ranked expected
returns and that standard errors tend to be the highest among these firms. Specifically,
the standard errors of predictability from our expected return estimates are systemat-
ically higher within the lower deciles of expected returns. When we include the entire
cross-section of firms in the optimization routine, the optimizer falsely assumes that the
expected return estimates are uniformly precise in the cross-section. Including firms with
on average less precise estimates than those in the benchmark portfolio in the investable
set results in portfolios that perform marginally worse than when we restrict the sample.
We explore this issue more in Section 4.4 and show that when we optimize within a fixed
set of firms based on expected returns, the optimized portfolios within that investment
23
set outperform an EW portfolio.
4.3. Over Time Analysis
Our analysis thus far has focused on summary performance measures over the the
entire 20-year period from 1996 to 2015. In Table 7, we report the portfolio performance
metrics for overlapping 10-year periods and non-overlapping 5-year periods to provide
insight into the performance of fundamental portfolios over time. For brevity, we report
only the performance of portfolios using MS optimization of the top decile of stocks over
time. Panel A reports the performance of optimized fundamental portfolios for rolling
10-year periods. Panel B reports the results of the optimized fundamental portfolios
for independent 5-year periods. We also indicate whether the Sharpe and Information
ratios for the optimized fundamental portfolios are significantly higher than those for the
non-optimized (EW) fundamental portfolios for the period.
The results in Panel A suggest significant gains from optimized fundamental portfolios
relative to non-optimized fundamental portfolios exist in the majority of the periods.
The results also suggest that the performance of the optimized fundamental portfolios
declines in the more recent time periods. The Sharpe ratios decline starting in 2004
and the Information ratios decline starting in 2000. The alphas also decline over time,
with the lowest performance of the optimized fundamental portfolios occurring in the last
decade. However, there are significant gains from optimized fundamental portfolios over
non-optimized fundamental portfolios even in the last period.
Figure (1) provides graphical analysis of the excess Sharpe ratio from using optimized
fundamental portfolios over non-optimized fundamental portfolios and over the market
portfolio (SPDR). We present the SPDR portfolio as a benchmark because it is a low-
cost and easy-to-implement portfolio that serves as an alternative investment option. The
figure shows the cumulative Sharpe ratio gains of the top decile fundamental portfolio
with MS optimization relative to the EW top decile fundamental portfolio and relative
to the SPDR. The gains are almost monotonically increasing over time, and are actually
24
increasing over the latter part of the sample (2006-2015).
Panel A of Table 7 also shows stable benefits of portfolio optimization as quantified
by λ∗ over time that persist and are more pronounced over time. To inform about the
cumulative gains to investors over time, Figure (2) provides graphical evidence of gains
to investors over alternative portfolios over time. The figure shows the utility gains for
a mean-variance investor of the top decile fundamental portfolios with MS optimization
relative to the EW top decile fundamental portfolios and the SPDR portfolios. We allow
investor risk aversion, λ, to take on three different values, 5, 10, 15. The figure shows
that the utility gains to a risk-averse investor are substantial relative to the benchmark
portfolios, and are increasing in the level of risk aversion. The figure also shows that the
gains are generally stable over time, with the exception of years in which stock valuations
were more likely to diverge from fundamental values such as during the internet stock
boom in 2000 and the financial crisis in 2008.
Because Panel A reports the moving average performance of the portfolios over 10-
year periods, the periods presented are not independent. To provide more insight into the
portfolio performance over time, Panel B reports the performance of the optimized top
decile fundamental portfolio for 5-year non overlapping periods. The results suggest that
the declining performance in the most recent decade reported in Panel A is attributable to
the 2006-2010 period, which includes the financial crisis. The Sharpe ratio is significantly
higher in the 2011-2015 period and the utility gains are the highest in this most recent
time period.
Overall, the results reported in Table 7 and Figures 1 and 2 suggest that the perfor-
mance of optimized top decile fundamental portfolios is relatively stable over time and
that the benefits to optimized versus non-optimized fundamental portfolios are also stable
over time.
25
4.4. Varying the Set of Investable Stocks
Our main benchmark has been an equal weighted portfolio based on the top decile of
stocks ranked by our fundamentals-based model. This is a common portfolio formation
strategy in academia; however, the choice of the top decile is somewhat arbitrary. To
provide insight into whether focusing on the top decile, relative to other cutoffs, impacts
our results, we report the results for each decile of stocks based on the expected returns
from the fundamentals-based model. That is, we form an equal weighted portfolio of the
stocks in each decile and an optimized fundamental portfolio of the stocks in each decile.
Table 8 reports the Sharpe and Information ratios for the EW fundamental portfolios, the
fundamental portfolios with maximum Sharpe ratio (MS) optimization, and the difference
in the performance of the optimized and non-optimized portfolios for each decile.
The performance of the non-optimized and optimized fundamental portfolios increases
in the deciles. The relative gains from optimization also increase across the deciles. Given
that the standard errors of predictability from our expected return estimates are system-
atically higher within the lower deciles of expected returns, this result is not surprising
and highlights that the gains from optimization are diminished relative to a naive EW
portfolio as expected return estimates become noisier.
In the previous analyses, we limit the number of stocks to those within each decile.
To provide insight into how varying the number of stocks included in the portfolio affects
the performance, we vary the number of investable stocks (N) from 30 to 300. An
investable universe of N stocks consists of the top N stocks based on the expected return
estimates using the fundamentals-based model. For each set of N investable stocks, we
then construct a fundamental portfolio with MS optimization and an EW portfolio of the
same stocks.9 We chose 30 as our lower bound because Fisher and Lorie (1970) argue
that a reasonably diversified portfolio can be constructed with 30 stocks. We arbitrarily
chose 300 as our upper bound because this was roughly three times the number of stocks9For number of investable stocks, N less than 100, upper bound on the individual stock weight of
one percent is infeasible. For the purpose of Figure 3, we allow the upper bound on the individual stockweight to be 1/30 for a consistent comparison with the EW portfolio
26
held in the top decile MS optimized portfolio. Figure 3 shows the impact that varying
the number of investable stocks has on portfolio performance. The top figure presents
the Sharpe ratio across sets of investable stocks. In all cases, the optimized fundamental
portfolios dominate the EW fundamental portfolios and the gains tend to increase as the
number of investable stocks increases.
The bottom figure reports the Sharpe ratio per number of stocks in the investable set
and the Sharpe ratio per number of stocks invested in the portfolio. The results in Panel
B of Table 5 suggest that the optimized portfolios tend to invest in a smaller number of
stocks relative to the EW top decile portfolios. This implies that optimized portfolios
tend to select stocks that collectively yield higher Sharpe ratios (and other performance
metrics) per invested stock. The bottom graph visualizes this. Regardless of the number
of investable stocks, the portfolio Sharpe ratio per stock invested is higher for optimized
portfolios and ratio of the portfolio Sharpe ratio to the number of invested stocks for the
optimized portfolio dominates the ratio for the EW portfolio regardless of the number of
investable stocks.
5. Conclusion
Constructing an investment portfolio generally consists of two activities: predicting
stock returns and creating an optimized portfolio of stocks based on those predictions and
investors’ risk tolerance. Fundamental analysis focuses on the first activity by predicting
stock returns based on financial ratios, whereas portfolio optimization focuses on the
second activity by mathematically determining the allocation of wealth to maximize
expected returns for a specified risk tolerance. Prior research has generally considered
each activity independently. Our study provides initial large sample evidence of potential
gains to investors of combining fundamental analysis and portfolio optimization.
We use a fundamentals-based model of expected returns that relies on the notion that
high book-to-market stocks with high expected future profitability have higher expected
returns. Our fundamentals-based future return model includes book-to-market, return
27
on equity, growth in net operating assets, and growth in financing. We find that using
fundamentals to estimate expected future stock returns as in input to the portfolio op-
timization yields substantial gains to investors, in terms of out-of-sample Sharpe ratios,
Information ratios, factor alphas, and mean-variance utilities over strategies of employing
fundamental analysis or portfolio optimization alone. Long-only optimized fundamental
portfolios produce CAPM alphas of over 3% per quarter and 5-factor alphas of over
2.3% per quarter, with high Sharpe and Information ratios. A mean-variance investor
with a risk aversion parameter of 1 is better off combining fundamentals with portfo-
lio optimization than investing with fundamentals alone, suggesting that virtually any
risk-averse investor would be better off. Gains to investors over naive strategies are even
more pronounced when small capitalization firms are eliminated from the investment
space. These gains are also present in recent decades, well after well-known academic
research was published which highlighted the predictive content of financial ratios.
Our findings contribute to fundamental analysis research and practice by demonstrat-
ing the gains to combining the analysis with portfolio optimization. In addition, in con-
trast to the prior portfolio optimization research that documents limited to no investment
gains to employing standard portfolio optimization techniques, we find that portfolio op-
timization can provide large gains to investors, but only when used in conjunction with
fundamental analysis.
28
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505–525.
31
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32
Figu
re1:
Shar
peR
atio
Gai
nsF
igur
e1
disp
lays
the
Shar
peR
atio
gain
sfo
rth
em
axim
umSh
arpe
Rat
io(M
S)po
rtfo
lios
for
the
stoc
ksin
the
top
deci
leba
sed
onth
eex
pect
edre
turn
esti
mat
es,
calc
ulat
edus
ing
the
fund
amen
tals
mod
el,r
elat
ive
toth
eeq
ually
wei
ghte
d(E
W)
port
folio
sof
the
stoc
ksin
the
top
deci
leba
sed
onex
pect
edre
turn
esti
mat
esfr
omth
efu
ndam
enta
lsm
odel
and
the
Mar
ket
(SP
DR
)po
rtfo
liofo
rth
eti
me
peri
od19
96-
2015
(upp
erpa
nel)
and
2006
-201
5(l
ower
pane
l).
Shar
peR
atio
Gai
nsar
ede
fined
asth
ecu
mul
ativ
esu
mof
the
diffe
renc
ein
the
cond
itio
nalS
harp
era
tio
ofth
etw
opo
rtfo
lios
whe
reth
eco
ndit
iona
lSha
rpe
Rat
ioof
apo
rtfo
lioat
tim
et
isca
lcul
ated
asth
era
tio
ofth
eco
ndit
iona
lmea
nto
the
cond
itio
nals
tand
ard
devi
atio
nof
the
port
folio
.T
hepo
rtfo
lioex
cess
retu
rn(d
efine
das
port
folio
raw
retu
rnle
ssth
eri
skfr
eera
te)
seri
esis
assu
med
tofo
llow
anA
R(1
)pr
oces
sw
ith
the
vari
ance
ofth
eer
ror
term
follo
win
gA
RC
H(1
)pr
oces
s.
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
Year
-200
20
40
60
80
10
0
12
0
Cumulative Excess Sharpe Ratio
Rela
tive to E
W P
ort
folio
Rela
tive to M
ark
et (S
PD
R)
Port
folio
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
Year
-505
10
15
20
25
30
35
40
45
Cumulative Excess Sharpe Ratio
33
Figu
re2:
Util
ityG
ains
Fig
ure
2di
spla
ysth
eU
tilit
yga
ins
for
the
max
imum
Shar
peR
atio
(MS)
port
folio
sfo
rth
est
ocks
inth
eto
pde
cile
base
don
the
expe
cted
retu
rnes
tim
ates
,ca
lcul
ated
usin
gth
efu
ndam
enta
lsm
odel
,rel
ativ
eto
the
equa
llyw
eigh
ted
(EW
)po
rtfo
lios
ofth
est
ocks
inth
eto
pde
cile
base
don
expe
cted
retu
rnes
tim
ates
usin
gth
efu
ndam
enta
lsm
odel
(Top
Pan
el)
and
Mar
ket
(SP
DR
)po
rtfo
lios
(Bot
tom
Pan
el)
for
diffe
rent
leve
lsof
inve
stor
’sri
sk-a
vers
ion
para
met
ers
(λ∈5,1
0,15)
for
the
tim
epe
riod
1996
-201
5.U
tilit
yG
ains
are
defin
edas
the
cum
ulat
ive
sum
ofth
edi
ffere
nce
inth
eco
ndit
iona
luti
lity
ofth
etw
opo
rtfo
lios
whe
reth
eco
ndit
iona
luti
lity
ofa
port
folio
atti
met
isca
lcul
ated
asth
eco
ndit
iona
lmea
nof
the
port
folio
less
the
cond
itio
nalv
aria
nce
tim
esha
lfth
eri
sk-a
vers
ion
para
met
er.
The
port
folio
exce
ssre
turn
(defi
ned
aspo
rtfo
liora
wre
turn
less
the
risk
free
rate
)se
ries
isas
sum
edto
follo
wan
AR
(1)
proc
ess
wit
hth
eva
rian
ceof
the
erro
rte
rmfo
llow
ing
AR
CH
(1)
proc
ess.
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
Yea
r
-500
50
10
0
15
0
20
0
25
0
30
0
Cumulative Excess Utility (%)
Re
lati
ve t
o E
W P
ort
foli
o
=5
=10
=15
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
Yea
r
-20
00
20
0
40
0
60
0
80
0
10
00
12
00
Cumulative Excess Utility (%)
Rela
tive t
o M
ark
et
(SP
DR
) P
ort
folio
34
Figu
re3:
Com
paris
onof
Shar
peR
atio
asa
Func
tion
ofIn
vest
able
Ass
ets
Fig
ure
3di
spla
ysth
epo
rtfo
lioSh
arpe
rati
oas
afu
ncti
onof
the
num
ber
ofas
sets
inth
ein
vest
able
univ
erse
for
the
port
folio
sba
sed
onth
eex
pect
edre
turn
esti
mat
esca
lcul
ated
usin
gth
efu
ndam
enta
lsm
odel
for
the
tim
epe
riod
1996
-20
15.
The
inve
stab
leun
iver
seofN
stoc
ksco
nsis
tsofN
top
stoc
ksba
sed
onth
eex
pect
edre
turn
esti
mat
esat
the
end
ofea
chm
onth
.M
axim
umSh
arpe
Rat
ioP
ortf
olio
repr
esen
tsth
eop
tim
alm
axim
umSh
arpe
Rat
io(M
S)po
rtfo
lios
ofN
stoc
ksan
dE
Wpo
rtfo
liore
pres
ents
the
equa
llyw
eigh
ted
port
folio
sof
the
sam
est
ocks
.T
heto
ppa
neld
ispl
ays
the
port
folio
Shar
peR
atio
whi
chis
the
sam
ple
mea
npo
rtfo
liore
turn
less
the
risk
free
rate
divi
ded
byth
esa
mpl
est
anda
rdde
viat
ion
ofth
epo
rtfo
lio.
The
bott
omP
anel
disp
lays
the
Shar
peR
atio
per
unit
num
ber
ofst
ocks
inth
ein
vest
able
univ
erse
(N)
and
the
Shar
peR
atio
per
unit
num
ber
ofst
ocks
inve
sted
inth
epo
rtfo
lio(N
In
v)
for
the
two
port
folio
s.
50
10
01
50
20
02
50
30
0
Num
ber
of
Investa
ble
Sto
cks (
N)
0.4
1
0.4
2
0.4
3
0.4
4
0.4
5
0.4
6
0.4
7
0.4
8
0.4
9
0.5
Sharpe Ratio
Maxim
um
Sharp
e R
atio P
ort
folio
EW
Port
folio
50
10
01
50
20
02
50
30
0
Num
ber
of
Investa
ble
Sto
cks (
N)
0
0.51
1.5
Sharpe Ratio Per-Unit Stock Invested (Investable)(x100)
35
Tables
Table 1: Descriptive StatisticsTable 1 presents descriptive statistics of key variables used in the analysis for the time period 1996 - 2015. Panel Aprovides summary statistics (the time-series averages of the cross-sectional mean, median, standard deviation, and selectpercentiles). Panel B provides the correlation matrix where lower and upper diagonals are Spearman and Pearson correla-tions respectively. rt+1 is the monthly return adjusted for delistings in percent, Sizet is the month-end market capitalizationin $billions, bmt is the book-to-market ratio, updated each quarter, roet is the quarterly earnings before extraordinarydivided by lagged book value, got is the quarterly changed in net operating assets divided by lagged book value, gft is thechange in financial assets divided by lagged book value.
(a) Panel A: Summary Statistics
10% 25% Median Mean 75% 90% StdDev
rt+1(%) -12.38 -5.80 0.40 1.14 7.00 14.97 12.86Sizet 0.05 0.17 0.62 4.92 2.32 8.40 20.00bmt 0.21 0.32 0.51 0.60 0.79 1.12 0.36roet(x100) -2.07 0.58 2.48 2.34 4.36 6.42 3.53got(x100) -6.32 -2.33 1.17 1.59 5.28 10.28 6.73gft(x100) -9.06 -3.76 0.68 0.74 5.20 10.57 7.99
(b) Panel B: Correlation Table
rt+1(%) -0.008 0.028 0.010 -0.028 0.018Sizet -0.022 -0.148 0.158 0.012 0.036bmt 0.014 -0.459 -0.377 -0.127 -0.111roet(x100) 0.037 0.340 -0.460 0.182 0.241got(x100) -0.028 0.056 -0.131 0.176 -0.411gft(x100) 0.029 0.093 -0.125 0.235 -0.396
36
Tabl
e2:
Ret
urn
Pred
ictio
nsTa
ble
2pr
esen
tsre
sult
sof
regr
essi
ons
offu
ture
real
ized
stoc
kre
turn
s,on
expe
cted
retu
rnpr
oxie
s,µ
t,u
sing
two
diffe
rent
esti
mat
ion
mod
els
from
1996−
2015
.In
colu
mns
(1)
and
(3),µ
tis
HIS
T,w
hich
isth
ero
lling
hist
oric
alm
onth
lyav
erag
est
ock
retu
rn(o
ver
the
prio
r36
mon
ths)
,in
colu
mns
(2)
and
(4),µ
tis
FU
ND
,whi
chis
the
fund
amen
tals
-bas
edm
odel
.T
hesl
ope
coeffi
cien
tsar
ees
tim
ated
usin
gFa
ma-
Mac
Bet
hre
gres
sion
s.t-
stat
isti
csar
ein
pare
nthe
ses
and
sign
ifica
nce
leve
lsof
1%,
5%,
and
10%
are
deno
ted
by,
***,
**,
and
*,re
spec
tive
ly.
Qua
rter
lyre
turn
test
sus
ea
Hod
rick
(199
2)co
rrec
tion
toac
coun
tfo
rov
erla
p.
Dep
ende
ntVa
riabl
e:M
onth
lyR
etur
nsD
epen
dent
Varia
ble:
Qua
rter
lyR
etur
ns
(1)
(2)
(3)
(4)
HIS
TFU
ND
HIS
TFU
ND
µt
-0.0
651*
*0.
872*
**-0
.056
8**
0.50
4***
(-2.
04)
(13.
06)
(-2.
14)
(7.1
1)
Adj
.R2
.005
5.0
051
.005
9.0
067
#O
bs.
418,
955
418,
955
418,
955
418,
955
37
Tabl
e3:
Port
folio
sw
ithou
tO
ptim
izat
ion
Tabl
e3
pres
ents
port
folio
met
rics
for
port
folio
sw
itho
utop
tim
izat
ion
whe
reth
eho
ldin
gpe
riod
isth
ree
mon
ths
(one
quar
ter)
.A
llSt
ocks
repr
esen
tsth
epo
rtfo
lios
that
incl
udes
the
enti
resa
mpl
eof
stoc
ks.
Top
Dec
ilere
pres
ents
the
port
folio
sth
atin
clud
esth
est
ocks
inth
eto
pde
cile
base
don
the
resp
ecti
veE
xpec
ted
Ret
urn
esti
mat
es.
HIS
Tre
pres
ents
the
expe
cted
retu
rnes
tim
ates
calc
ulat
edus
ing
rolli
nghi
stor
ical
mon
thly
aver
age
stoc
kre
turn
(ove
rth
epr
ior
36m
onth
s).
FU
ND
repr
esen
tsth
eex
pect
edre
turn
esti
mat
esca
lcul
ated
usin
gth
efu
ndam
enta
lsm
odel
.E
W(V
W)
repr
esen
tsan
equa
lly(v
alue
)w
eigh
ted
port
folio
ofth
est
ocks
.P
anel
Are
port
spo
rtfo
liope
rfor
man
cem
etri
csan
dP
anel
Bre
port
sth
ech
arac
teri
stic
sof
the
port
folio
sw
hich
are
calc
ulat
edas
the
wei
ghte
d-av
erag
eof
indi
vidu
alst
ock
char
acte
rist
ics.
The
Shar
peR
atio
isth
esa
mpl
em
ean
port
folio
retu
rnle
ssth
eri
skfr
eera
tedi
vide
dby
the
sam
ple
stan
dard
devi
atio
nof
the
port
folio
retu
rn.
The
Info
rmat
ion
Rat
iois
the
inte
rcep
tof
the
mar
ket
mod
eldi
vide
dby
the
ofth
ere
sidu
alfr
omth
em
arke
tm
odel
.C
AP
Mre
pres
ents
the
CA
PM
alph
a,F
F3,
FF
4,an
dF
F5,
resp
ecti
vely
repr
esen
tth
eFa
ma
and
Fren
chth
ree,
four
,and
five
fact
oral
pha’
s.R
awis
the
aver
age
real
ized
retu
rnof
the
port
folio
and
Exc
ess
isav
erag
ere
aliz
edre
turn
ofth
epo
rtfo
liole
ssth
eri
skfr
eera
te.
Size
isex
pres
sed
in$b
illio
ns.
Turn
over
isth
eav
erag
esu
mof
the
abso
lute
chan
gein
port
folio
wei
ghts
for
firmi,w
i,t,
from
one
peri
odto
the
anot
her( Tur
nover
t=∑ N t i=
1|w
i,t−w
i,t−
1|) an
dN
o.of
stoc
ksin
vest
edre
pres
ents
the
aver
age
num
ber
ofst
ocks
held
inth
epo
rtfo
lioea
chm
onth
.Si
gnifi
canc
ele
vels
of1%
,5%
,an
d10
%ar
ede
note
dby
,**
*,**
,an
d*,
resp
ecti
vely
and
are
base
don
two-
taile
dst
anda
rder
rors
wit
ha
Hod
rick
(199
2)co
rrec
tion
toac
coun
tfo
rov
erla
p.
(a)
Pane
lA:P
ortf
olio
Met
rics
Stoc
ksIn
clud
ed:
All
Stoc
ksTo
pD
ecile
Top
Dec
ileEx
pect
edR
etur
n:N
one
HIS
TFU
ND
Port
folio
Con
stru
ctio
n:EW
VW
EWV
WEW
VW
(1)
(2)
(3)
(4)
(5)
(6)
Metrics
Shar
peR
atio
0.25
60.
214
0.11
00.
139
0.42
60.
250
Info
rmat
ion
Rat
io0.
151
0.10
6-0
.148
-0.0
820.
451
0.12
2
Alpha’s
CA
PM0.
815
0.12
1-0
.914
-0.4
153.
023*
**0.
992*
FF3
0.47
7**
0.17
7-0
.752
0.30
12.
520*
**0.
895*
FF4
0.69
8***
0.18
5-0
.790
**0.
256
2.67
7***
1.04
2FF
50.
422
-0.0
217
0.14
21.
141*
2.46
7***
0.94
8*
Returns
Raw
3.30
7***
2.29
0***
2.26
32.
781*
5.58
7***
3.79
7***
Exce
ss2.
728*
**1.
710*
*1.
683
2.20
15.
008*
**3.
218*
**
38
Tabl
e3:
Port
folio
sw
ithou
tO
ptim
izat
ion,
Con
tinue
d
(b)
Pane
lB:P
ortf
olio
Cha
ract
erist
ics
Stoc
ksIn
clud
ed:
All
Stoc
ksTo
pD
ecile
Top
Dec
ileEx
pect
edR
etur
n:N
one
HIS
TFU
ND
Port
folio
Con
stru
ctio
n:EW
VW
EWV
WEW
VW
(1)
(2)
(3)
(4)
(5)
(6)
Sizet
4.91
988
.179
3.46
959
.877
1.00
024
.671
bmt
0.59
70.
344
0.38
40.
240
0.76
50.
558
roe t
(x10
0)2.
341
4.87
53.
110
5.12
53.
897
5.08
6go t
(x10
0)1.
593
2.08
32.
950
3.90
1-3
.379
-3.9
04gf t
(x10
0)0.
739
2.07
11.
855
4.21
36.
627
8.38
5Tu
rnov
er0.
120
0.10
40.
380
0.39
40.
903
1.13
2N
o.of
stoc
ksin
vest
ed17
4517
4517
417
417
417
4
39
Tabl
e4:
Opt
imiz
edPo
rtfo
lios
with
out
Fund
amen
tals-
Base
dEx
pect
edR
etur
nsTa
ble
4pr
esen
tspo
rtfo
liom
etri
csfo
rop
tim
ized
port
folio
sw
itho
utin
corp
orat
ing
expe
cted
retu
rns
base
don
fund
amen
talm
odel
.T
heho
ldin
gpe
riod
isth
ree
mon
ths
(one
quar
ter)
.A
llSt
ocks
repr
esen
tsth
epo
rtfo
lios
that
incl
udes
the
enti
resa
mpl
eof
stoc
ks.
Top
Dec
ilere
pres
ents
the
port
folio
sth
atin
clud
esth
est
ocks
inth
eto
pde
cile
base
don
the
resp
ecti
veE
xpec
ted
Ret
urn
esti
mat
es.
HIS
Tre
pres
ents
the
expe
cted
retu
rnes
tim
ates
calc
ulat
edus
ing
rolli
nghi
stor
ical
mon
thly
aver
age
stoc
kre
turn
(ove
rth
epr
ior
36m
onth
s).
MV
repr
esen
tsth
em
inim
umva
rian
cepo
rtfo
lio.
MV
Tre
pres
ents
the
min
imum
vari
ance
port
folio
subj
ect
toth
eex
pect
edre
turn
sof
the
port
folio
bein
ggr
eate
rth
anor
equa
lto
the
resp
ecti
veto
pde
cile
expe
cted
retu
rnpr
oxy
port
folio
usin
gm
ean-
vari
ance
opti
miz
atio
n.M
Sre
pres
ents
the
max
imum
Shar
peR
atio
port
folio
usin
gm
ean-
vari
ance
opti
miz
atio
n.B
SVre
pres
ents
apo
rtfo
lioop
tim
ized
follo
win
gth
eB
rand
tet
al.
(200
9)m
etho
dolo
gy.
PR
ICE
repr
esen
tspr
ice-
base
dch
arac
teri
stic
sas
inB
rand
tet
al.
(200
9),
AC
CT
repr
esen
tsac
coun
ting
-bas
edch
arac
tert
isti
csas
inH
and
and
Gre
en(2
011)
,and
HIS
Tre
pres
ents
hist
oric
alav
erag
e-ba
sed
expe
cted
retu
rns.
Pan
elA
repo
rts
the
port
folio
perf
orm
ance
met
rics
for
mea
n-va
rian
ceop
tim
izat
ion,
mea
n-va
rian
ceop
tim
ziat
ion
wit
hta
rget
and
max
imum
Shar
peR
atio
opti
miz
atio
n.P
anel
Bre
port
sth
epo
rtfo
liope
rfor
man
cem
etri
csfo
rth
epo
rtfo
lios
opti
miz
edfo
llow
ing
the
Bra
ndt
etal
.(2
009)
met
hodo
logy
.T
heSh
arpe
Rat
iois
the
sam
ple
mea
npo
rtfo
liore
turn
less
the
risk
free
rate
divi
ded
byth
esa
mpl
est
anda
rdde
viat
ion
ofth
epo
rtfo
lio.
The
Info
rmat
ion
Rat
iois
the
inte
rcep
tof
the
mar
ket
mod
eldi
vide
dby
the
ofth
ere
sidu
alfr
omth
em
arke
tm
odel
.C
AP
Mre
pres
ents
the
CA
PM
alph
a,F
F3,
FF
4,an
dF
F5,
resp
ecti
vely
repr
esen
tth
eFa
ma
and
Fren
chth
ree,
four
,and
five
fact
oral
pha’
s.R
awis
the
aver
age
real
ized
retu
rnof
the
port
folio
and
Exc
ess
isav
erag
ere
aliz
edre
turn
ofth
epo
rtfo
liole
ssth
eri
skfr
eera
te.
Sign
ifica
nce
leve
lsof
1%,5
%,a
nd10
%ar
ede
note
dby
,***
,**,
and
*,re
spec
tive
lyan
dar
eba
sed
ontw
o-ta
iled
stan
dard
erro
rsw
ith
aH
odri
ck(1
992)
corr
ecti
onto
acco
unt
for
over
lap.
(a)
Pane
lA:P
ortf
olio
Met
rics
Stoc
ksIn
clud
ed:
All
Stoc
ksA
llSt
ocks
Top
Dec
ileEx
pect
edR
etur
n:N
one
HIS
TH
IST
Port
folio
Con
stru
ctio
n:M
VM
VT
MS
MV
TM
S
(1)
(2)
(3)
(4)
(5)
Metrics
Shar
peR
atio
0.29
00.
121
0.12
80.
135
0.13
1In
form
atio
nR
atio
0.25
5-0
.123
-0.1
12-0
.101
-0.1
05
Alpha’s
CA
PM0.
900*
-0.5
89-0
.522
-0.4
82-0
.529
FF3
0.55
3*-0
.585
*-0
.519
-0.4
19-0
.465
FF4
0.52
7-0
.545
-0.4
17-0
.411
-0.4
99FF
50.
191
-0.1
29-0
.049
40.
255
0.26
3
Returns
Raw
2.72
1***
1.91
3*1.
992*
2.32
3*2.
284*
Exce
ss2.
142*
**1.
334
1.41
21.
744
1.70
5
40
Tabl
e4:
Opt
imiz
edPo
rtfo
lios
with
out
Fund
amen
tals-
Base
dEx
pect
edR
etur
ns,C
ontin
ued
(b)
Pane
lB:P
ortf
olio
Met
rics
-BSV
Stoc
ksIn
clud
ed:
All
Stoc
ksC
hara
cter
istic
s:PR
ICE
AC
CT
HIS
T
Port
folio
Con
stru
ctio
n:B
SVB
SVB
SV
(1)
(2)
(3)
Metrics
Shar
peR
atio
0.27
70.
282
0.20
7In
form
atio
nR
atio
0.18
40.
185
0.06
1
Alpha’s
CA
PM1.
162*
1.37
2**
0.12
8FF
30.
857*
**1.
075*
**0.
059
FF4
0.64
1**
1.24
0***
0.02
4FF
50.
961*
**1.
324*
**0.
046
Returns
Raw
3.69
9***
4.06
8***
2.27
8***
Exce
ss3.
120*
**3.
489*
**1.
699*
*
41
Tabl
e5:
Opt
imiz
edPo
rtfo
lios
with
Fund
amen
tals-
Base
dEx
pect
edR
etur
nsTa
ble
5pr
esen
tspo
rtfo
liom
etri
csfo
rpo
rtfo
lios
base
don
the
expe
cted
retu
rnes
tim
ates
calc
ulat
edus
ing
the
fund
amen
tals
mod
el,F
UN
D.T
heho
ldin
gpe
riod
isth
ree
mon
ths
(one
quar
ter)
.A
llSt
ocks
repr
esen
tsth
epo
rtfo
lios
that
incl
udes
the
enti
resa
mpl
eof
stoc
ksan
dTo
pD
ecile
repr
esen
tsth
epo
rtfo
lios
that
incl
udes
the
stoc
ksin
the
top
deci
leba
sed
onth
eex
pect
edre
turn
esti
mat
eF
UN
D.E
Wre
pres
ents
aneq
ually
wei
ghte
dpo
rtfo
lioof
the
stoc
ksin
the
top
deci
leba
sed
onex
pect
edre
turn
esti
mat
esfr
omth
efu
ndam
enta
lsm
odel
.M
VT
repr
esen
tsth
em
inim
umva
rian
cepo
rtfo
liosu
bjec
tto
the
expe
cted
retu
rns
ofth
epo
rtfo
liobe
ing
grea
ter
than
oreq
ualt
oth
ere
spec
tive
top
deci
leex
pect
edre
turn
prox
ypo
rtfo
lious
ing
mea
n-va
rian
ceop
tim
izat
ion.
MS
repr
esen
tsth
em
axim
umSh
arpe
Rat
iopo
rtfo
lious
ing
mea
n-va
rian
ceop
tim
izat
ion.
BSV
repr
esen
tsa
port
folio
opti
miz
edfo
llow
ing
the
Bra
ndt
etal
.(20
09)
met
hodo
logy
,wit
hth
eex
pect
edre
turn
esti
mat
eF
UN
Das
char
acte
rist
ic.
Pan
elA
repo
rts
the
port
folio
perf
orm
ance
met
rics
and
Pan
elB
repo
rts
the
char
acte
rist
ics
ofth
epo
rtfo
lios
whi
char
eca
lcul
ated
asth
ew
eigh
ted-
aver
age
ofin
divi
dual
stoc
kch
arac
teri
stic
s.T
heSh
arpe
Rat
iois
the
sam
ple
mea
npo
rtfo
liore
turn
less
the
risk
free
rate
divi
ded
byth
esa
mpl
est
anda
rdde
viat
ion
ofth
epo
rtfo
lio.
The
Info
rmat
ion
Rat
iois
the
inte
rcep
tof
the
mar
ket
mod
eldi
vide
dby
the
ofth
ere
sidu
alfr
omth
em
arke
tm
odel
.λ
∗re
pres
ents
the
leve
lof
risk
aver
sion
requ
ired
for
am
ean-
vari
ance
inve
stor
tobe
indi
ffere
ntto
the
EW
(Ben
chm
ark)
port
folio
.A
nin
vest
orw
ith
risk
-ave
rsio
npa
ram
eterλ>λ
∗in
dica
tes
that
inve
stor
wou
ldbe
wor
seoff
byin
vest
ing
inth
eE
W(B
ench
mar
k)po
rtfo
lio.
CA
PM
repr
esen
tsth
eC
AP
Mal
pha,
FF
3,F
F4,
and
FF
5,re
spec
tive
lyre
pres
ent
the
Fam
aan
dFr
ench
thre
e,fo
ur,a
ndfiv
efa
ctor
alph
a’s.
Raw
isth
eav
erag
ere
aliz
edre
turn
ofth
epo
rtfo
lioan
dE
xces
sis
aver
age
real
ized
retu
rnof
the
port
folio
less
the
risk
free
rate
.Si
zeis
expr
esse
din
$bill
ions
.Tu
rnov
eris
the
aver
age
sum
ofth
eab
solu
tech
ange
inpo
rtfo
liow
eigh
tsfo
rfir
mi,w
i,t,
from
one
peri
odto
the
anot
her( Tur
nover
t=∑ N t i=
1|w
i,t−w
i,t−
1|) an
dN
o.of
stoc
ksin
vest
edre
pres
ents
the
aver
age
num
ber
ofst
ocks
held
inth
epo
rtfo
lioea
chm
onth
.Si
gnifi
canc
ele
vels
of1%
,5%
,an
d10
%ar
ede
note
dby
,**
*,**
,an
d*,
resp
ecti
vely
.Si
gnifi
canc
ele
vels
ofth
eSh
arpe
and
Info
rmat
ion
Rat
ios
are
base
don
test
ing
ifth
era
tio
ofth
eop
tim
ized
port
folio
isla
rger
than
the
EW
(Ben
chm
ark)
port
folio
.Si
gnifi
canc
ele
vels
for
Alp
ha’s
and
Ret
urns
are
base
don
two-
taile
dst
anda
rder
rors
wit
ha
Hod
rick
(199
2)co
rrec
tion
toac
coun
tfo
rov
erla
p.
(a)
Pane
lA:P
ortf
olio
Met
rics
Stoc
ksIn
clud
ed:
Top
Dec
ileA
llSt
ocks
Top
Dec
ileEx
pect
edR
etur
n:FU
ND
FUN
DFU
ND
Port
folio
Con
stru
ctio
n:EW
MV
TM
SB
SVM
VT
MS
BSV
Ben
chm
ark
(1)
(2)
(3)
(4)
(5)
(6)
Metrics
Shar
peR
atio
0.42
60.
447
0.45
6**
0.37
50.
473*
**0.
473*
**0.
279
Info
rmat
ion
Rat
io0.
451
0.47
60.
495*
0.38
50.
520*
**0.
522*
**0.
176
λ∗
3.01
92.
556
9.12
00.
926
0.69
6∞
Alpha’s
CA
PM3.
023*
**2.
803*
**2.
888*
**2.
161*
**3.
192*
**3.
220*
**1.
397*
**FF
32.
520*
**2.
302*
**2.
384*
**1.
729*
**2.
656*
**2.
682*
**1.
180*
**FF
42.
677*
**2.
405*
**2.
568*
**1.
846*
**2.
782*
**2.
831*
**1.
279*
*FF
52.
467*
**1.
953*
**2.
050*
**1.
595*
**2.
274*
**2.
299*
**1.
283*
**
Returns
Raw
5.58
7***
4.95
6***
5.05
5***
4.63
5***
5.43
6***
5.47
9***
4.17
6***
Exce
ss5.
008*
**4.
377*
**4.
476*
**4.
056*
**4.
857*
**4.
900*
**3.
597*
**
42
Tabl
e5:
Opt
imiz
edPo
rtfo
lios
with
Fund
amen
tals-
Base
dEx
pect
edR
etur
ns,C
ontin
ued
(b)
Pane
lB:P
ortf
olio
Cha
ract
erist
ics
Stoc
ksIn
clud
ed:
Top
Dec
ileA
llSt
ocks
Top
Dec
ileEx
pect
edR
etur
n:FU
ND
FUN
DFU
ND
Port
folio
Con
stru
ctio
n:EW
MV
TM
SB
SVM
VT
MS
BSV
Ben
chm
ark
(1)
(2)
(3)
(4)
(5)
(6)
Sizet
1.00
01.
700
1.71
67.
801
1.17
61.
152
17.4
94bm
t0.
765
0.73
70.
751
0.69
00.
772
0.77
40.
653
roe t
(x10
0)3.
897
4.04
84.
105
3.69
54.
009
4.05
05.
179
go t
(x10
0)-3
.379
-3.2
40-3
.299
-1.8
87-3
.460
-3.5
85-4
.698
gf t
(x10
0)6.
627
6.40
36.
499
4.98
86.
529
6.66
89.
248
Turn
over
0.90
30.
868
0.87
80.
661
0.93
30.
926
1.13
9N
o.of
stoc
ksin
vest
ed17
438
837
292
212
612
611
1
43
Tabl
e6:
Opt
imiz
edPo
rtfo
lios
with
Fund
amen
tals-
Base
dEx
pect
edR
etur
nsEx
clud
ing
Smal
lSto
cks
Tabl
e6
pres
ents
port
folio
met
rics
for
port
folio
sba
sed
onth
eex
pect
edre
turn
esti
mat
esca
lcul
ated
usin
gth
efu
ndam
enta
lsm
odel
,F
UN
Dw
hen
the
inve
stab
leun
iver
seof
stoc
ksis
rest
rict
edto
top
80%
stoc
ksba
sed
onm
arke
tca
pita
lizat
ion
atti
met.
The
hold
ing
peri
odis
thre
em
onth
s(o
nequ
arte
r).
All
Stoc
ksre
pres
ents
the
port
folio
sth
atin
clud
esth
een
tire
sam
ple
ofst
ocks
and
Top
Dec
ilere
pres
ents
the
port
folio
sth
atin
clud
esth
est
ocks
inth
eto
pde
cile
base
don
the
expe
cted
retu
rnes
tim
ate
FU
ND
.EW
repr
esen
tsan
equa
llyw
eigh
ted
port
folio
ofth
est
ocks
inth
eto
pde
cile
base
don
expe
cted
retu
rnes
tim
ates
from
the
fund
amen
tals
mod
el.
MV
Tre
pres
ents
the
min
imum
vari
ance
port
folio
subj
ect
toth
eex
pect
edre
turn
sof
the
port
folio
bein
ggr
eate
rth
anor
equa
lto
the
resp
ecti
veto
pde
cile
expe
cted
retu
rnpr
oxy
port
folio
usin
gm
ean-
vari
ance
opti
miz
atio
n.M
Sre
pres
ents
the
max
imum
Shar
peR
atio
port
folio
usin
gm
ean-
vari
ance
opti
miz
atio
n.B
SVre
pres
ents
apo
rtfo
lioop
tim
ized
follo
win
gth
eB
rand
tet
al.(
2009
)m
etho
dolo
gy,w
ith
the
expe
cted
retu
rnes
tim
ate
FU
ND
asch
arac
teri
stic
.P
anel
Are
port
sth
epo
rtfo
liope
rfor
man
cem
etri
csan
dP
anel
Bre
port
sth
ech
arac
teri
stic
sof
the
port
folio
sw
hich
are
calc
ulat
edas
the
wei
ghte
d-av
erag
eof
indi
vidu
alst
ock
char
acte
rist
ics.
The
Shar
peR
atio
isth
esa
mpl
em
ean
port
folio
retu
rnle
ssth
eri
skfr
eera
tedi
vide
dby
the
sam
ple
stan
dard
devi
atio
nof
the
port
folio
.T
heIn
form
atio
nR
atio
isth
ein
terc
ept
ofth
em
arke
tm
odel
divi
ded
byth
eof
the
resi
dual
from
the
mar
ket
mod
el.λ
∗re
pres
ents
the
leve
lofr
isk
aver
sion
requ
ired
for
am
ean-
vari
ance
inve
stor
tobe
indi
ffere
ntto
the
EW
(Ben
chm
ark)
port
folio
.A
nin
vest
orw
ith
risk
-ave
rsio
npa
ram
eterλ>λ
∗in
dica
tes
that
inve
stor
wou
ldbe
wor
seoff
byin
vest
ing
inth
eE
W(B
ench
mar
k)po
rtfo
lio.
CA
PM
repr
esen
tsth
eC
AP
Mal
pha,
FF
3,F
F4,
and
FF
5,re
spec
tive
lyre
pres
ent
the
Fam
aan
dFr
ench
thre
e,fo
ur,a
ndfiv
efa
ctor
alph
a’s.
Raw
isth
eav
erag
ere
aliz
edre
turn
ofth
epo
rtfo
lioan
dE
xces
sis
aver
age
real
ized
retu
rnof
the
port
folio
less
the
risk
free
rate
.Si
zeis
expr
esse
din
$bill
ions
.Tu
rnov
eris
the
aver
age
sum
ofth
eab
solu
tech
ange
inpo
rtfo
liow
eigh
tsfo
rfir
mi,w
i,t,f
rom
one
peri
odto
the
anot
her( Tur
nover
t=∑ N t i=
1|w
i,t−w
i,t−
1|) an
dN
o.of
stoc
ksin
vest
edre
pres
ents
the
aver
age
num
ber
ofst
ocks
held
inth
epo
rtfo
lioea
chm
onth
.Si
gnifi
canc
ele
vels
of1%
,5%
,and
10%
are
deno
ted
by,*
**,*
*,an
d*,
resp
ecti
vely
.Si
gnifi
canc
ele
vels
ofth
eSh
arpe
and
Info
rmat
ion
Rat
ios
are
base
don
test
ing
ifth
era
tio
ofth
eop
tim
ized
port
folio
isla
rger
than
the
EW
(Ben
chm
ark)
port
folio
.Si
gnifi
canc
ele
vels
for
Alp
ha’s
and
Ret
urns
are
base
don
two-
taile
dst
anda
rder
rors
wit
ha
Hod
rick
(199
2)co
rrec
tion
toac
coun
tfo
rov
erla
p.
(a)
Pane
lA:P
ortf
olio
Met
rics
Stoc
ksIn
clud
ed:
Top
Dec
ileA
llSt
ocks
Top
Dec
ileEx
pect
edR
etur
n:FU
ND
FUN
DFU
ND
Port
folio
Con
stru
ctio
n:EW
MV
TM
SB
SVM
VT
MS
BSV
Ben
chm
ark
(1)
(2)
(3)
(4)
(5)
(6)
Metrics
Shar
peR
atio
0.38
90.
396
0.42
0**
0.34
60.
435*
**0.
436*
**0.
277
Info
rmat
ion
Rat
io0.
390
0.38
60.
439*
0.33
70.
462*
**0.
466*
**0.
178
λ∗
.3.
509
2.06
28.
305
0.17
40
∞
Alpha’s
CA
PM2.
525*
**2.
249*
**2.
510*
**1.
799*
**2.
780*
**2.
802*
**1.
286*
**FF
31.
978*
**1.
747*
**1.
953*
**1.
349*
**2.
180*
**2.
202*
**1.
040*
**FF
42.
138*
**1.
701*
**2.
194*
**1.
488*
**2.
399*
**2.
416*
**0.
914*
FF5
1.70
7***
1.15
8***
1.42
8***
1.07
8***
1.60
7***
1.66
3***
1.11
3***
Returns
Raw
5.06
8***
4.27
7***
4.68
0***
4.27
0***
5.04
4***
5.07
3***
3.88
4***
Exce
ss4.
488*
**3.
698*
**4.
101*
**3.
691*
**4.
464*
**4.
494*
**3.
305*
**
44
Tabl
e6:
Opt
imiz
edPo
rtfo
lios
with
Fund
amen
tals-
Base
dEx
pect
edR
etur
nsEx
clud
ing
Smal
lSto
cks,
Con
tinue
d
(b)
Pane
lB:P
ortf
olio
Cha
ract
erist
ics
Stoc
ksIn
clud
ed:
Top
Dec
ileA
llSt
ocks
Top
Dec
ileEx
pect
edR
etur
n:FU
ND
FUN
DFU
ND
Port
folio
Con
stru
ctio
n:EW
MV
TM
SB
SVM
VT
MS
BSV
Ben
chm
ark
(1)
(2)
(3)
(4)
(5)
(6)
Sizet
3.46
92.
706
2.85
08.
924
2.00
21.
987
19.3
97bm
t0.
384
0.59
90.
639
0.60
40.
664
0.66
50.
592
roe t
(x10
0)3.
110
3.97
64.
228
3.83
84.
128
4.15
45.
299
go t
(x10
0)2.
950
-3.5
98-3
.783
-2.1
91-3
.915
-4.0
14-4
.976
gf t
(x10
0)1.
855
5.96
66.
353
5.04
36.
452
6.54
29.
210
Turn
over
0.38
00.
857
0.89
80.
675
0.93
80.
932
1.10
5N
o.of
stoc
ksin
vest
ed14
021
620
273
611
311
288
45
Tabl
e7:
Port
folio
Perfo
rman
ceO
ver
Tim
eTa
ble
7pr
esen
tspo
rtfo
liom
etri
csov
erti
me
for
the
max
imum
Shar
peR
atio
(MS)
port
folio
sus
ing
mea
n-va
rian
ceop
tim
izat
ion
for
the
stoc
ksin
the
top
deci
leba
sed
onth
eex
pect
edre
turn
esti
mat
esca
lcul
ated
usin
gth
efu
ndam
enta
lsm
odel
.T
heho
ldin
gpe
riod
isth
ree
mon
ths
(one
quar
ter)
.P
anel
Are
port
spo
rtfo
liom
etri
csov
erro
lling
ten
year
peri
ods
and
Pan
elB
repo
rts
port
folio
met
rics
usin
gno
nov
erla
ppin
gfiv
eye
arpe
riod
s.T
heSh
arpe
Rat
iois
the
sam
ple
mea
npo
rtfo
liore
turn
less
the
risk
free
rate
divi
ded
byth
esa
mpl
est
anda
rdde
viat
ion
ofth
epo
rtfo
lio.
The
Info
rmat
ion
Rat
iois
the
inte
rcep
tof
the
mar
ket
mod
eldi
vide
dby
the
ofth
ere
sidu
alfr
omth
em
arke
tm
odel
.λ
∗re
pres
ents
the
leve
lofr
isk
aver
sion
requ
ired
for
am
ean-
vari
ance
inve
stor
tobe
indi
ffere
ntto
aneq
ually
wei
ghte
d(E
W)
port
folio
ofth
est
ocks
inth
eto
pde
cile
base
don
expe
cted
retu
rnes
tim
ates
from
the
fund
amen
tals
mod
el.
An
inve
stor
wit
hri
sk-a
vers
ion
para
met
erλ>λ
∗in
dica
tes
that
inve
stor
wou
ldbe
wor
seoff
byin
vest
ing
inth
eE
Wpo
rtfo
lio.
CA
PM
repr
esen
tsth
eC
AP
Mal
pha,
FF
3,F
F4,
and
FF
5,re
spec
tive
lyre
pres
ent
the
Fam
aan
dFr
ench
thre
e,fo
ur,a
ndfiv
efa
ctor
alph
a’s.
Raw
isth
eav
erag
ere
aliz
edre
turn
ofth
epo
rtfo
lioan
dE
xces
sis
aver
age
real
ized
retu
rnof
the
port
folio
less
the
risk
free
rate
.Si
gnifi
canc
ele
vels
of1%
,5%
,and
10%
are
deno
ted
by,*
**,*
*,an
d*,
resp
ecti
vely
.Si
gnifi
canc
ele
vels
ofth
eSh
arpe
and
Info
rmat
ion
Rat
ios
are
base
don
test
ing
ifth
era
tio
ofth
eop
tim
ized
port
folio
isla
rger
than
the
EW
port
folio
.Si
gnifi
canc
ele
vels
for
Alp
ha’s
and
Ret
urns
are
base
don
two-
taile
dst
anda
rder
rors
wit
ha
Hod
rick
(199
2)co
rrec
tion
toac
coun
tfo
rov
erla
p.
(a)
Port
folio
Met
rics:
MS
Opt
imiz
edFu
ndam
enta
lsO
ver
10-y
ear
Rol
ling
Win
dow
s
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
’96-
’05
’97-
’06
’98-
’07
’99-
’08
’00-
’09
’01-
’10
’02-
’11
’03-
’12
’04-
’13
’05-
’14
’06-
’15
Metrics
Shar
peR
atio
0.65
1***
0.63
0***
0.51
4**
0.35
10.
412*
**0.
452*
*0.
428*
*0.
479
0.44
8***
0.38
3*0.
332*
**In
form
atio
nR
atio
0.62
4*0.
626
0.62
20.
750
0.84
00.
796
0.72
00.
672
0.65
5**
0.46
3**
0.41
9***
λ∗
1.71
91.
855
1.66
21.
388
0.00
01.
262
1.03
62.
143
0.00
00.
892
0.00
0
Alpha’s
CA
PM4.
833*
**4.
724*
**4.
375*
**4.
796*
**5.
230*
**4.
596*
**3.
838*
**2.
958*
**2.
784*
**1.
784*
**1.
599*
**FF
32.
956*
**2.
799*
**2.
966*
**3.
158*
**3.
492*
**3.
225*
**3.
150*
**2.
564*
**2.
540*
**1.
676*
**1.
801*
**FF
43.
017*
**2.
765*
**2.
904*
**3.
143*
**3.
492*
**3.
194*
**3.
152*
**2.
596*
**2.
632*
**1.
777*
**1.
892*
**FF
52.
901*
**2.
842*
**2.
905*
**2.
674*
**2.
733*
**2.
462*
**2.
676*
**2.
667*
**2.
599*
**1.
947*
**2.
000*
**
Returns
Raw
6.98
8***
6.74
2***
5.72
4***
4.58
5***
5.54
5***
6.01
6***
5.68
5***
6.11
0***
5.54
9***
4.67
0***
3.97
0***
Exce
ss6.
093*
**5.
853*
**4.
861*
**3.
808*
**4.
884*
**5.
498*
**5.
244*
**5.
706*
**5.
170*
**4.
325*
**3.
707*
*
46
Tabl
e7:
Port
folio
Perfo
rman
ceO
ver
Tim
e,C
ontin
ued
(b)
Port
folio
Met
rics:
MS
Opt
imiz
edFu
ndam
enta
lsA
cros
s5-
year
Tim
epe
riods
(1)
(2)
(3)
(4)
’96-
’00
’01-
’05
’06-
’10
’11-
’15
Metrics
Shar
peR
atio
0.52
1*0.
784*
**0.
249
0.54
3***
Info
rmat
ion
Rat
io0.
281
1.17
00.
498
0.33
1***
λ∗
2.18
1.18
1.37
0.00
Alpha’s
CA
PM3.
035*
6.77
5***
2.46
7***
1.07
8*FF
32.
039*
**3.
656*
**1.
998*
**2.
524*
**FF
42.
697*
**3.
567*
**1.
909*
**2.
644*
**FF
52.
881*
**3.
330*
**3.
190*
**2.
345*
**
Returns
Raw
5.98
9***
7.98
6***
4.04
53.
896*
**Ex
cess
4.71
4***
7.47
1***
3.52
43.
890*
**
47
Tabl
e8:
Port
folio
Perfo
rman
ceO
ver
Diff
eren
tD
ecile
sTa
ble
8pr
esen
tspo
rtfo
liom
etri
csfo
rpo
rtfo
lios
base
don
the
expe
cted
retu
rnes
tim
ates
calc
ulat
edus
ing
the
fund
amen
tals
mod
elov
erdi
ffere
ntde
cile
s.T
heho
ldin
gpe
riod
isth
ree
mon
ths
(one
quar
ter)
.E
Wre
pres
ents
aneq
ually
wei
ghte
dpo
rtfo
lios.
MS
repr
esen
tsth
em
axim
umSh
arpe
Rat
iopo
rtfo
lious
ing
mea
n-va
rian
ceO
ptim
izat
ion.
The
Shar
peR
atio
isth
esa
mpl
em
ean
port
folio
retu
rnle
ssth
eri
skfr
eera
tedi
vide
dby
the
sam
ple
stan
dard
devi
atio
nof
the
port
folio
.T
heIn
form
atio
nR
atio
isth
ein
terc
ept
ofth
em
arke
tm
odel
divi
ded
byth
eof
the
resi
dual
from
the
mar
ket
mod
el.
Sign
ifica
nce
leve
lsof
1%,5
%,a
nd10
%ar
ede
note
dby
,***
,**,
and
*,re
spec
tive
ly.
Sign
ifica
nce
leve
lsof
the
Shar
pean
dIn
form
atio
nR
atio
sar
eba
sed
onte
stin
gif
the
rati
oof
the
MS
port
folio
isla
rger
than
the
EW
port
folio
.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
Bot
tom
Dec
ileTo
pD
ecile
EWSh
arpe
Rat
io0.
072
0.13
40.
191
0.22
50.
256
0.27
60.
285
0.31
60.
366
0.42
6In
form
atio
nR
atio
-0.2
07-0
.104
0.00
50.
083
0.15
90.
196
0.21
40.
277
0.35
80.
451
MS
Shar
peR
atio
0.06
60.
194
0.23
20.
289
0.31
30.
320
0.33
30.
362
0.41
70.
474
Info
rmat
ion
Rat
io-0
.203
0.05
50.
125
0.24
10.
289
0.30
30.
326
0.37
30.
461
0.52
2λ
∗1.
010
00
00.
520.
181.
071.
210.
69
MS-
EWSh
arpe
Rat
io-0
.006
0.06
***
0.04
1**
0.06
4***
0.05
7***
0.04
4**
0.04
8***
0.04
6***
0.05
1***
0.04
8***
Info
rmat
ion
Rat
io0.
004
0.15
9***
0.12
***
0.15
8***
0.13
***
0.10
7***
0.11
2***
0.09
6***
0.10
3***
0.07
1***
48