Optimal Fundamental Investing Matthew Lyle and Teri Lombardi Yohn * September 23, 2019 Abstract We show how fundamental analysis can be seamlessly integrated with mean-variance portfolio optimization to construct stock portfolios. We find that fundamental anal- ysis combined with mean-variance portfolio optimization yields substantial improve- ments in portfolio performance relative to portfolios constructed using alternative approaches examined in prior research. Both long-short and long-only optimal fun- damental portfolios produce large out-of-sample CAPM and factor alphas, with high Sharpe and Information ratios. The results are persistent through time and remain when small capitalization firms are eliminated from the investment set. We therefore demonstrate how to combine fundamental analysis and portfolio optimiza- tion to jointly exploit the benefits of each. 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 of Management and Professor of Accounting at the Kelley School of Business. We appreciate helpful sugges- tions and comments from Larry Brown, Charles Lee, Ron Dye, Ravi Jagannathan, Jeremiah Green, Bob Korajczyk, Bob McDonald, Steve Penman, Beverly Walther and workshop participants at the Kellogg School of Management, Fox School of Business, and Stanford Graduate School of Business. A special thanks to Rishabh Aggarwal, who provided invaluable research assistance. We are grateful for the fund- ing of this research by the Kellogg School of Management, Lyle thanks The Accounting Research Center at Kellogg for funding provided through the E&Y Live and Revsine Research Fellowships.
Transcript
Optimal Fundamental Investing
Matthew Lyle and Teri Lombardi Yohn∗
September 23, 2019
Abstract
We show how fundamental analysis can be seamlessly integrated with mean-varianceportfolio optimization to construct stock portfolios. We find that fundamental anal-ysis combined with mean-variance portfolio optimization yields substantial improve-ments in portfolio performance relative to portfolios constructed using alternativeapproaches examined in prior research. Both long-short and long-only optimal fun-damental portfolios produce large out-of-sample CAPM and factor alphas, withhigh Sharpe and Information ratios. The results are persistent through time andremain when small capitalization firms are eliminated from the investment set. Wetherefore demonstrate how to combine fundamental analysis and portfolio optimiza-tion to jointly exploit the benefits of each.
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 sugges-tions and comments from Larry Brown, Charles Lee, Ron Dye, Ravi Jagannathan, Jeremiah Green, BobKorajczyk, Bob McDonald, Steve Penman, Beverly Walther and workshop participants at the KelloggSchool of Management, Fox School of Business, and Stanford Graduate School of Business. A specialthanks to Rishabh Aggarwal, who provided invaluable research assistance. We are grateful for the fund-ing of this research by the Kellogg School of Management, Lyle thanks The Accounting Research Centerat Kellogg for funding provided through the E&Y Live and Revsine Research Fellowships.
1. INTRODUCTION
Fundamental analysis uses accounting-based characteristics and ratios to estimate a
stock’s intrinsic value. The notion underlying fundamental analysis is that the difference
between a stock’s price and intrinsic value is predictive of the direction of the stock’s future
returns as the price converges to intrinsic value. Many valuation and accounting ratios,
such as book-to-market and profitability, have been shown to be useful for identifying
stocks whose future returns are likely to increase or decrease. Using fundamental analysis
for stock selection can be dated at least as far back as Benjamin and Dodd (1934),
and its ability to provide information about firm’s future stock returns has been shown
to be robust across time periods and countries. Despite the documented usefulness of
fundamental analysis for stock selection, it remains unclear how it can be employed to
construct stock portfolios.1
In his seminal paper, Markowitz (1952) argues that constructing a stock portfolio
consists of generating beliefs about the future performance of individual stocks and then
allocating wealth across the stocks to maximize the expected return of the portfolio sub-
ject to a given constraint (e.g., risk tolerance). While portfolio optimization has been
documented to improve portfolio performance (Engle et al., 2017; Jorion, 1985, 1986), its
usefulness has been limited due to poor quality estimates of expected returns (DeMiguel
et al., 2009; Jagannathan and Ma, 2003; Michaud, 1989). The research shows that com-
pletely disregarding expected return estimates and constructing minimum variance port-
folios (e.g., Jagannathan and Ma, 2003; Ledoit and Wolf, 2017) yields better performance
than optimized portfolios that incorporate expected returns. Thus, prior research sug-
gests that mean-variance portfolio optimization may be of use to investors only when
beliefs about expected returns are completely disregarded.
1The standard approach in the academic literature is to form either equal or value weighted portfoliosfrom groups of stocks ranked by the financial ratio of interest and to determine if there exists differencesin future stock returns across portfolios. Because the goal of these tests is to document systematic associ-ations between fundamental ratios and future returns, the literature provides insight into predictability,but not how an investor can or “should” form portfolios.
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Because fundamental analysis provides information about expected returns and be-
cause portfolio optimization provides a solution for allocating wealth across stocks given
expected returns, it seems logical that linking the two streams of research would be ben-
eficial to both. However, fundamental analysis has not been used in a Markowitz mean-
variance portfolio optimization setting because the analysis has not generated expected
returns or stock return covariances of individual stocks. In this study, we analytically
demonstrate how fundamental analysis can be used to estimate expected returns and
covariances and seemlessly integrated with mean-variance portfolio optimization to con-
struct stock portfolios. We then empirically test whether optimal fundamental portfolios,
which combine fundamental analysis with mean-variance portfolio optimization, yield im-
provements in performance over portfolios formed using alternative approaches examined
in prior research.
Our model captures the key ideas underlying fundamental analysis, while remaining
tractable enough for straightforward estimation. The model is a new and practical ex-
tension of research that uses valuation models to either generate expected stock return
estimates (e.g., Gebhardt et al. 2001; Chattopadhyay et al. 2018; Lyle et al. 2013; Lyle
and Wang 2015) or to explain and predict index returns (e.g., Lee et al., 1999). We
assume, as is common in the literature (e.g., Frankel and Lee, 1998), that a fundamental
investor uses the residual income formula to estimate intrinsic value. We also assume
that a fundamental investor expects the difference between the stock’s intrinsic value
and share prices to follow a simple AR(1) process, with an unconditional mean of zero,
as in Johannesson and Ohlson (2019). These assumptions imply that stock returns are
a linear combination of firm size, book-to-market, expected earnings yield, and unpre-
dictable noise. This parsimonious linear structure allows us to uniquely incorporate both
cross-sectional and time-series information to estimate expected returns and a covari-
ance matrix within a single valuation setting, which allows us to integrate fundamental
analysis with mean-variance portfolio optimization.
We examine the performance of four portfolios that systematically incorporate dif-
2
ferent information in expected returns and covariances for portfolio construction. The
first portfolios (EW portfolios) are “naive” equal weighted portfolios that ignore both
expected returns and covariances. The second portfolios (MV portfolios) are minimum
variance portfolios that disregard expected return estimates and rely exclusively on co-
variance estimates. The third portfolios (ER portfolios) disregard covariance estimates
and rely exclusively on expected return estimates. The fourth portfolios (MS portfolios)
maximize the Sharpe ratio using both expected returns and covariance matrices. For
each construction strategy, we examine expected returns and covariances based on the
fundamentals-based model. For comparison, and as in prior research, we also examine
portfolios that use historical average stock returns to estimate expected returns and co-
variances. We examine both “long-short” and “long-only” portfolios because taking short
positions is often not feasible and, even when feasible, implementation costs are often high
(Beneish et al., 2015).
Using fundamentals-based expected returns and covariances, we find, consistent with
prior research on portfolio optimization, that portfolios that incorporate information
about covariances (i.e., MV portfolios) outperform those that use a naive equal weighted
strategy (i.e., EW portfolios). However, unlike the prior research, we show that portfolios
that incorporate fundamentals-based estimates of expected returns (i.e., ER portfolios)
generally outperform the MV portfolios. Our key finding is that our optimal fundamental
portfolios (i.e., fundamentals-based MS portfolios) yield the highest out-of-sample Sharpe
and Information ratios as well as factor alphas of all the portfolios we examine. From 1981-
2017 (our full out-of-sample period), long-only optimal fundamental portfolios, which are
rebalanced monthly, have average future returns of 1.92%, a Sharpe ratio of 0.367, an
Information ratio of 0.395, and five factor (Fama and French, 2015) alphas of 0.888%.
Long-short optimal fundamental portfolios generate even higher performance metrics than
the long-only portfolios, with a monthly Sharpe ratio of 0.524 and an Information ratio
of 0.523, and outperform the alternative portfolios by an even greater margin. Consistent
with the findings of DeMiguel et al. (2009), we find that portfolios optimized incorporating
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historical-based stock returns tend to be the worst performing portfolios. These findings
suggest that integrating mean-variance optimization and fundamentals-based expected
returns and covariances delivers substantial improvements to portfolio performance.
The results hold over different time periods, even after well-known academic research
which highlights the predictive ability of financial ratios was published. The results
also hold when portfolios are limited to include only large stocks. This latter result is
important since Hou et al. (2018) find that most return-predictive signals documented in
prior studies do not significantly predict returns when small firms are removed from the
sample. The robustness of our findings to the exclusion of small stocks suggests that our
results are unlikely driven by small illiquid stocks or an unreplicable anomaly.
Several of the variables implied by our fundamentals-based return model (e.g., size
and book-to-market) are similar to those characteristics used by Fama and French (1993,
2015) to construct factor portfolios. Despite this, we find that our optimal fundamental
portfolios cannot be replicated by the Fama and French factor portfolios. This is not sur-
prising given that Fama and French portfolios are constructed by simultaneously longing
and shorting stocks that have high or low characteristics (e.g., book-to-market or size)
with portfolio weights being determined by market capitalization (i.e., value weighted).
In contrast, our optimal fundamental portfolio weights are optimized based on expected
returns and covariances.
Our study is also related to Brandt et al. (2009) who propose a novel methodology
that is meant to overcome some of the challenges of mean-variance optimization. Brandt
et al. (2009) combine a power utility function with firm characteristics and solve for
portfolio weights via non-linear estimation. Hand and Green (2011) extend Brandt et al.
(2009) by incorporating accounting-based characteristics and show that accounting-based
fundamental signals enhance portfolio performance over price-based signals. We employ
the Brandt et al. (2009) methodology using the same fundamental variables in our es-
timation for mean-variance optimization. We find that while the Brandt et al. (2009)
methodology does offer benefits to investors relative to naive portfolio construction, the
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performance does not match that of our optimal fundamental portfolios.
This study provides important contributions to both practice and research on funda-
mental analysis and portfolio optimization. Fundamental analysis is aimed at identifying
stocks that are likely to experience higher or lower future returns, but provides little
insight with respect to creating portfolios. Similarly, portfolio optimization provides the-
oretical arguments for optimizing portfolios, but poor quality expected return estimates
limit its usefulness in constructing portfolios in practice. We demonstrate how funda-
mental analysis can be combined with portfolio optimization and empirically document
the substantial gains to portfolio performance from doing so.
2. Fundamental Analysis, Stock Returns, and Portfolio Optimization
In this section we formally outline the investment problem from the point of view of a
fundamental investor who initially performs fundamental analysis to derive an “intrinsic
value” estimate of a stock, which we call Vt, that is currently trading at a price Pt. We
show how the fundamental investor can use differences between Vt and Pt to construct an
optimal portfolio. This setting can be considered a formal extension of Lee et al. (1999)
who argue that if Pt deviates from, but tends to converge to intrinsic value over time,
then high quality estimates of intrinsic value will have return prediction power. While
Lee et al. (1999) examine the ability of different valuation models to explain and predict
index returns, our study is focused on constructing portfolios from the cross-section of
stocks.
2.1. Fundamental Investing and Stock Returns
Fundamental investing requires the assertion that, at some point, the stock’s price
will converge to its intrinsic value. We operationalize this notion by assuming that the
difference between the market price and intrinsic value for firm i, Pi,t − Vi,t, follows an
AR(1) process, with an unconditional mean of zero, as in Johannesson and Ohlson (2019):
5
Pi,t+1 − Vi,t+1 = ωi(Pi,t − Vi,t) + εi,t+1, (1)
where ωi ∈ (0, 1) represents a persistence parameter and εi,t+1 is a mean-zero vari-
ance one noise term, which represents a shock to expected convergence. Assuming fun-
damental investors price an equity by discounting cash flows/dividends, then Vi,t =∑∞j=1 EFt [R−ji Di,t+j], where EFt [·] represents the time t expectation operator given fun-
damental investors’ beliefs and Di,t+j the dividends to be paid at time t+ j. Implicit in
this assumption is that EFt [Vi,t+1 + Di,t+1] = RiVi,t, where Ri > 1 is the gross discount
rate used in the valuation model. This in turn implies that fundamental investors believe
that future stock returns take the standard form of an expectation plus shocks:
Pi,t+1 +Di,t+1
Pi,t= Vi,tPi,t
Ri + ωi(1−Vi,tPi,t
)︸ ︷︷ ︸Expected Return
+ (Vi,t+1 +Di,t+1)−RiVi,tPi,t︸ ︷︷ ︸
Fundamentals Shock
+ εi,t+1
Pi,t︸ ︷︷ ︸Convergence Shock
. (2)
Fundamental investors’ beliefs about expected returns are, naturally, increasing in
the discount rate used in valuation, Ri, but this is modified by the value-to-price ratio,
Vi,t/Pi,t, which acts like a multiplier, where higher value-to-price ratios increase the ex-
pected returns. The persistence in the deviation between intrinsic value and price, ωi, has
an intuitive effect. For high value-to-price ratios, a high persistence parameter implies
a lower expected return because it is expected to take longer for the fundamental in-
vestor to capture gains from convergence between intrinsic value and price.2 Ultimately,
expected returns are maximized from a fundamental investor’s perspective if intrinsic
value-to-price ratios are high and expected to converge quickly.
There are two sources of risk to the fundamental investor: 1) an unpredictable shock
from fundamentals, (Vi,t+1 +Di,t+1)−RiVi,t = (Vi,t+1 +Di,t+1)− EFt [Vi,t+1 +Di,t+1], and
2) an unpredictable shock related to the convergence between price and intrinsic value,
2For low valuation ratios ωi has the same effect if the fundamental investor considers shorting.
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εi,t+1. These shocks may be correlated, but for our purposes, their correlation makes
no difference for the investment decision. Therefore, we can represent returns more
compactly as
Ri,t+1 ≡Pi,t+1 +Di,t+1
Pi,t= ωi + Vi,t
Pi,t(Ri − ωi) + Ωi,tξi,t+1, (3)
where Ωi,t represents fundamental investor’s beliefs about the volatility of stock re-
turns and ξi,t+1 is a mean zero, unit variance noise term.
To draw a direct connection to fundamentals, it is useful to recast the dividend
discount formula in terms of residual income, such that Vi,t = ∑∞j=1 EFt [R−ji Di,t+j] =
Bi,t(1 + ∑∞j=1R
−ji EFt [xai,t+j]/Bi,t), where Bi,t is book value, xai,t+1 = xi,t+1 − (Ri − 1)Bi,t
represents t+ 1 “residual income” and xi,t+1 is t+ 1 accounting earnings. As such, valu-
ations are a function of book values and projections of future profitability, which in turn
implies that fundamental investor’s beliefs about stock returns must also be a function
of these variables,
Ri,t+1 = ωi + (Ri − ωi)Bi,t
Pi,t(1 +
∑∞j=1R
−ji EFt [xai,t+j]Bi,t
) + Ωi,tξi,t+1. (4)
For (4) to be implementable, we must make an assumption about the dynamics for
residual income. We assume that residual income follows a simple stochastic process
xai,t+1 = hi,t + zi,t+1, where hi,t+1 = hi + κi(hi,t − hi) + wi,t+1. Here zi,t+1 and wi,t+1 are
noise terms, hi,t represents the conditional mean of abnormal earnings, which follows an
AR(1) process with persistence κi and an unconditional mean of hi.3 If follows that stock
returns have the following functional form:
Ri,t+1 = ωi + αi,01Pi,t
+ αi,1Bi,t
Pi,t+ αi,2
EFt [xi,t+1]Pi,t
+ Ωi,tξi,t+1, (5)
3The assumption of auto-regressive abnormal earnings is common, see for example Christensen andFeltham (2012) and the references within.
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αi,0 = hi(Ri − ωi)(1− κi)(Ri − 1)(Ri − κi)
> 0, (6)
αi,1 = (Ri − ωi)(1− κi)Ri − κi
> 0, (7)
αi,2 = Ri − ωiRi − κi
> 0. (8)
The stock return equation above is similar to the partial equilibrium returns model
derived by Lyle et al. (2013), and highlights that a fundamental investor can use a com-
bination of three key variables (i.e., market value, Pi,t, book value, Bi,t, and projections
of future earnings, EFt [xi,t+1]) to predict stock returns. When we estimate (5), we must
use a proxy for expected earnings, EFt [xi,t+1], since they are not observable. We do so
based on prior financial statement analysis research which shows that future profitability
is a function of not only current earnings, but also variables that measure growth. We
use the change in net operating assets, ∆NOAi,t, and change in financing, ∆FINi,t, as
our proxies for growth given prior fundamental analysis research of the relation between
growth and future profitability (Fairfield et al., 2003; Bradshaw et al., 2006; Cooper et al.,
2008). After substituting these variables in the equation, we obtain an estimable stock
return equation that is linear in firm characteristics
Ri,t+1 = Ai,0 + Ai,11Pi,t
+ Ai,2Bi,t
Pi,t+ Ai,3
xi,tPi,t
+ Ai,4∆NOAi,t
Pi,t+ Ai,5
∆FINi,t
Pi,t+ Ωi,tξi,t+1.
(9)
The parsimonious linear structure of (9) allows for straightforward estimation of both
the conditional mean and covariance in a single setting at the firm-level, which provides
a direct connection to mean-variance portfolio optimization as outlined next.
2.2. Optimal Portfolios
Here we outline our four main portfolio construction strategies. The fundamentals-
based return model, equation (9), implies that the expected return for the ith firm is given
by µi,t = Ai,0 +Ai,11Pi,t
+Ai,2Bi,tPi,t
+Ai,3xi,tPi,t
+Ai,4∆NOAi,t
Pi,t+Ai,5
∆FINi,tPi,t
and the covariance
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between the the ith and jth firm is given by cov(Ri,t+1 − µi,t, Rj,t+1 − µj,t). We first
show how portfolio weights are allocated across stocks assuming an investor uses mean-
variance optimization and accounts for cross-sectional differences in expected returns and
covariances. We call these MS portfolios because they correspond to maximum Sharpe
ratio portfolios. We then show that each of the other portfolio construction strategies
we study to construct minimum variance (MV), expected return only (ER), and equal
weighted (EW)) portfolios are all particular cases of MS portfolios. The portfolios allow
for short selling and impose no constraints on the weights allocated to each stock, except
that the weights must sum to one. In our empirical tests, when we impose constraints,
exact solutions are often not possible, so we use a quadratic program to solve for portfolio
weights numerically.4
2.2.1. Mean-Variance Portfolios
As is the standard formulation in mean-variance optimization, we assume that a
fundamental investor allocates wealth by solving the classic mean-variance optimization
program, where the investor minimizes portfolio variance for a given expected return, µP :
minωωTΣω, (10)
s.t. ωTµ = µp, (11)
ωT e = 1, (12)
where ω = (ω1, ω2, . . . , ωN)T is aN×1 vector of portfolio weights, µt = (µ1,t, µ2,t, . . . µN,t)T
is a N ×1 vector of expected returns, Σ is a N ×N covariance matrix, e is a N ×1 vector
of ones. Solving the program, and setting µp to the value that achieves the maximum
expected portfolio return per unit portfolio volatility yields
4For 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.
9
ω∗MS = Σ−1µ
eTΣ−1µ. (13)
Equation (13) represents portfolios constructed that take full advantage of both expected
returns and the covariance matrix. This corresponds to our MS portfolios.
2.2.2. Minimum Variance Portfolios
If investors disregard expected returns in constructing portfolios, then this is equiva-
lent to assuming that expected returns are cross-sectionally constant, i.e., µ = c × e for
some real constant c, and equation (13) becomes
ω∗MV = Σ−1e
eTΣ−1e. (14)
Equation (14) corresponds to our MV portfolios.
2.2.3. Expected Return Portfolios
If investors disregard cross-sectional differences in covariances in constructing portfo-
lios, then this is equivalent to assuming that covariances are cross-sectionally constant,
i.e., Σ = C× I where I is a N ×N identity matrix and C is a real constant, and equation
(13) becomes
ω∗ER = µ
eTµ. (15)
Equation (15) corresponds to our ER portfolios.
2.2.4. Equal Weighted Portfolios
If investors disregard cross-sectional differences in means and covariances in construct-
ing portfolios, then this is equivalent to assuming that both are cross-sectionally constant,
i.e., µ = c× e and Σ = C × I, and equation (13) becomes
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ω∗EW = 1N. (16)
Equation (16) corresponds to our EW portfolios.
3. Data and Estimation
3.1. Data
Our data are from standard sources: CRSP and Compustat. Our full sample time
period is from 1981-2017. We use the period 1976-1980 as an initial model estimation
period and 1981-2017 as the out-of-sample test period. Our initial estimation period
begins in 1976 because prior to this Compustat quarterly is not well populated. We also
examine more recent sub-periods to assess the gains that a fundamental 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 a negative book value, and stocks with less than five years of historical stock
return data. These criteria ensure we can reasonably estimate stock return volatility
and pairwise correlations (i.e., the covariance matrix). All expected return estimates are
winsorized at the 1 and 99% level. 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 risk-free rates
and factor portfolios used in our empirical tests are downloaded from Ken French’s data
library.5
5These data can be downloaded from http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data library.html
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3.2. Model Estimation for Mean-Variance Optimization
To generate expected returns, we must estimate equation (9). Prior research has
estimated 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 on each
fundamental, whereas industry definitions tend to be exceptionally noisy and can lead
to worse estimates for prediction than simple cross-sectional estimation (e.g., Fairfield
et al., 2009). Given the lack of guidance on how to group firms for estimation, we use the
model directly to form groups. We do this by first estimating the model cross-sectionally
within our training period to obtain an initial estimate of expected returns. We then
form deciles by ranking these initial expected returns and re-estimate the model in our
training period within each of these deciles.
In our estimation, we update predictor variables, 1Pt
, BtPt
, xtPt
,∆NOAi,tPi,t
and ∆FINi,tPi,t
, quar-
terly 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 miss-
ing in Compustat, we assume that the information is public three months after the firm’s
fiscal quarter. 1Pt
is one over market value of equity from the CRSP; Bt, xt, ∆NOAt, and
∆FINt, are book value of equity, earnings before extraordinary items, the change in net
operating assets, and the change in financial assets, from the Compustat quarterly files,
respectively. To avoid potential issues with outliers, we cross-sectionally standardize each
of the predictor variables following Hand and Green (2011) by converting each predictor
variable into a percentile rank, dividing by 99, and subtracting 0.5. This ensures each
predictor is mean zero with a stable distribution over time.
To avoid any look-ahead bias, every month before we construct portfolios, we col-
lect five years of historical data to estimate the coefficients A0, A1, A2, A3, A4, A5 by
regressing one-month ahead stock returns on the fundamentals within each of the ten
groups. Because we update our predictor variables quarterly (approximately every three
12
months) whereas returns are updated monthly, we include three lags in the model.6 We
use the estimated in-sample coefficients with the most recent fundamental variables to
generate expected returns and collect the in-sample residuals from these regressions and
estimate the covariance matrix using the non-linear shrinkage estimator of Ledoit and
Wolf (2017).7 We use these estimated expected returns and covariances to construct
portfolio weights at time t and examine out-of-sample portfolio returns from time t to
t+ 1.
While it is possible to expand the model to incorporate a larger set of variables
to potentially enhance out-of-sample predictability, maximizing predictability is not the
goal of our study. Nonetheless, if a fundamental-based strategy is to offer gains, it
should, at a minimum, be related to future stock returns. To test for predictability
of the model, we conduct (untabulated) out-of-sample predictability tests by regressing
future returns on current estimates of expected returns using the following specification:
Ri,t+1 = a + b × µi,t + εi,t+1. Our estimates of b using the Fama and MacBeth (1973)
approach is a highly significant 0.607 with a standard error of 0.0402, suggesting the
fundamentals model is a strong predictor of future stock returns.
3.3. Estimation of Performance Metrics
Our central research question is whether fundamental analysis when combined with
mean-variance portfolio optimization provides gains to investors. To answer this ques-
tion, we examine several different common measures used to evaluate the out-of-sample
performance of portfolios, such as average returns, return standard deviation, and Sharpe
6To see why we include lagged variables, suppose that a true model has the form yt+1 = xt + εt+1,but we only observe xt with delay (we wait for the accounting numbers to be reported in compustat)and update only every say three periods. In such a case if xt+1 = a + bxt + ξt+1, that is, the periodt + 1 predictor is related to the time t plus unpredictable noise, ξt+1. Thus for an arbitrary timet+ τ + 1 we have yt+τ+1 = a+ bxt+τ + εt+τ+1, but if we only have as a predictor xt, then yt+τ+1|xt =a b
τ−1b−1 + xtb
τ +∑τi=1 b
τ−iξt+1+i. The∑τi=1 b
τ−iξt+1+i term represents a moving average term, whichcan be interpreted as autocorrelation in stock returns, conditional on xt. Thus including lagged responsevariables when the predictor is updated slowly can help correct for this.
7We use the Ledoit and Wolf (2017) estimator because it represents the most recently developedcovariance estimator, dominates traditional estimators and was constructed for implementation in mean-variance portfolio optimization.
13
and Information ratios. Because we assume that the representative fundamental investor
trades off expected return with expected risk (variance), our main measures used for
performance evaluation are Sharpe (SR) and Information (IR) ratios, where the Sharpe
and Information ratios for portfolio return, are respectively given by:
SR = RP,t+1 − RF√σ2P,t+1 − σ2
RF
, (17)
IR = RP,t+1 − β × Rm,t+1√σ2P,t+1 − β2σ2
m,t+1. (18)
RP,t+1 and σP,t+1 are the out-of-sample mean and standard deviation of the portfolio
return respectively, RF and σRF are the mean and standard deviation of the risk free
rate respectively. Rm,t+1 and σm,t+1 are the mean and standard deviation of the market
portfolio respectively; β is the portfolio’s beta. To test for differences in SR and IR
between two portfolios, we estimate two equations simultaneously, one for each portfolio,
of the form RP,t+1 − RF = aSR + σSRεt+1for SR and Rp,t+1 = aIR + βRm,t+1 + σIRεt+1
for IR, and conduct a nonlinear test of the model parameters. Specifically, to test for
differences in the Sharpe (Information) ratios between portfolio x and y we test the
hypotheses, a(x)SR
σ(x)SR
− a(y)SR
σ(y)SR
= 0 ( a(x)IR
σI(x)SR
− a(y)IR
σ(y)IR
= 0), using a Wald test.8
4. Empirical Results
Our empirical tests are meant to examine the benefits, if any, of combining funda-
mental analysis with portfolio optimization. To do this, we first report key portfolio
performance metrics and characteristics for each of the portfolios (i.e, EW, MV, ER, and
MS portfolios) as well as the differences in Sharpe and Information ratios across port-
folios. Upon presenting these results, we then report the performance of portfolios for
different samples and time periods.
8We also verify the results of the Wald tests using both the likelihood-ratio test and the Lagrangemultiplier test.
14
4.1. Fundamental Portfolios Performance
Table 1 presents the key findings of our study as the table shows the portfolio perfor-
mance of the EW portfolios and the MV, ER, and MS portfolios using the fundamentals-
based model. For comparison to prior research, we also report the performance of portfo-
lios based on expected returns and covariances constructed from 60 months of historical
returns as opposed to a fundamentals-based model. The portfolios are constructed with
no constraints in terms of shorting stocks or portfolio allocation weights.
Table 1, Panel A reports the mean, volatility (Std.), Sharpe and Information ratios
for each portfolio using either the fundamentals model (FUND), calculated as in equa-
tion (9), or historical returns (HIST). We also report the portfolio weighted average of
characteristics of the variables used in the fundamentals model: firm size (Size), earnings-
to-price (EP), book-to-market (BM), change in net operating assets (∆NOA), and change
in financial assets (∆FIN). For example, for the MS portfolio, BM corresponds to the
time-series average of ∑Ni=1 ωi,t,MS
Bi,tPi,t
. Panels B and C report tests of differences in Sharpe
and Information ratios across portfolios for each model.
Column (1) of Table 1 Panel A reports the results for the EW portfolios, which
disregard cross-sectional differences in means and covariances and provide benchmark
summary statistics of cross-sectional average portfolios over our sample. Columns (2)
through (4) report results for the MV, ER, and MS portfolios based on the fundamentals
model, while columns (5) through (7) report results for the MV, ER, and MS portfolios
based on historical stock returns. Consistent with prior research (e.g., Engle et al.,
2017; Jagannathan and Ma, 2003; Jorion, 1985, 1986), we find that minimum variance
portfolios (MV) offer some performance improvements over naive EW portfolios. This
holds regardless of whether we use a covariance matrix constructed from the fundamentals
(column (2)) or historical returns (column (5)) model. Specifically, the MV portfolios
deliver the lowest volatility of all portfolios leading to higher Sharpe (SR) ratios and
significantly higher Information (IR) ratios.
15
Columns (3) and (6) of Table 1 Panel A show the impact on portfolio performance
of incorporating only information about expected returns (ER portfolios). The ER port-
folios based on the FUND model generate higher realized returns than either the EW
or MV portfolios; however, this comes at the cost of higher stock return volatility and
insignificantly lower Sharpe and Information ratios than the MV portfolios. The ER port-
folios based on historical returns perform very poorly, as they have lower stock returns
and higher volatility than either the EW or MV portfolios, leading to significantly lower
Sharpe and Information ratios. These results highlight that incorporating poor quality
expected returns in optimization can lead to portfolios that under-perform even naive
portfolios.
Columns (4) and (7) show the impact of simultaneously incorporating information
about expected returns and covariances (MS portfolios). The MS portfolios based on
the FUND model, which we refer to as optimal fundamental portfolios, fully exploit
information in expected returns and covariances, generating future stock returns of 3.85%,
the highest future stock returns of all the portfolios. This does come at a cost however,
as it has the highest volatility of 6.732 of all FUND portfolios. Nonetheless, this increase
in volatility does not overwhelm the increase in average returns, as the MS portfolios
generate the highest Sharpe ratio of 0.524 and Information ratio of 0.523. The differences
bertween MS portfolios and MV, ER, and EW portfolios are significant at the one percent
level. There is a general pattern with respect to characteristics of the portfolios such that
MV fundamental portfolios tend to put more weight on larger firms while MS fundamental
portfolios tend put more weight on firms with higher BM, EP and ∆FIN and lower ∆NOA.
The performance of MS portfolios based on historical returns highlights the fragility
of unconstrained mean-variance portfolio optimization when poor quality estimates of
expected returns are used - the portfolios have negative average returns and Sharpe and
Information ratios, and dramatically higher volatility than any other portfolios.
The results in Tables 1 are based on unconstrained optimization, which, by construc-
tion, allows for short positions in stocks. In practice short selling costs can be prohibitive
16
and even infeasible (Beneish et al., 2015). Table 2 presents the results of portfolio per-
formance after imposing a long-only constraint on each portfolio. The constraint has
the most pronounced effect on the fundamentals-based MS portfolios, which experience a
decrease in average returns and volatility. However, the inferences do not change as every
panel in Table 2 suggests that the fundamentals-based MS portfolios yield the highest
portfolio performance. The MS portfolios earn 1.92% future returns, and yield a Sharpe
ratio of 0.367 and Information ratio of 0.395, significantly higher than the MV, ER, and
EW portfolios. The pattern that MV fundamental portfolios tend to put more weight on
larger firms while MS fundamental portfolios tend put more weight on firms with higher
BM, EP and ∆FIN and lower ∆NOA holds in the long-only portfolios. In addition, con-
sistent with DeMiguel et al. (2009), historical returns-based portfolios offer little to no
benefit to investors. Because portfolios relying only on historical returns are known to
perform poorly and because we also find this result for our sample, we do not report the
results for historical-based portfolios in our remaining tables.
The portfolios in Tables 1 and 2 were either unconstrained or constrained on the
long-side, allowing for potentially very large allocations to a few stocks. To determine
the sensitivity of our results to allocation constraints, Table 3 presents the results for the
fundamentals-based portfolios when the magnitude of weights are constrained to 2.5%.
Long-short portfolios are constrained to between -2.5% and 2.5%, and long-only portfolios
are constrained to between 0% and 2.5%. Each panel in Table 3 suggests that imposing
these constraints tends to improve upon the performance of unconstrained portfolios,
suggesting that our initial results are not driven by extreme allocations.9
4.2. Portfolios Grouped by Size
Hou et al. (2018) find that most return-predictive signals documented in prior stud-
ies do not significantly predict returns when small firms are removed from the sample,
suggesting that most predictive signals are concentrated among smaller potentially illiq-
9We replicate this table using 1%, 3% and 5% constraints and find similar results.
17
uid firms. Given the importance of stock liquidity for portfolio construction, Table 4
presents the results for investment sets that progressively exclude a larger number of
smaller stocks, such that the portfolios are formed from the largest 1,000 stocks, 500
stocks, 200 stocks, and 100 stocks. While the portfolios in previous analyses exclude
penny stocks, excluding smaller stocks further removes stocks for which investors are
more likely to face liquidity issues and higher transactions costs. The table presents
the performance metrics for optimal fundamental portfolios which incorporate expected
returns from the fundamentals model with either ER, MV, or MS portfolio optimiza-
tion. Like Table 3, we report results here, and in remaining tables, based on portfolios
that are constrained to have weights with magnitude of no more than 2.5%. We present
the performance metrics for non-optimized (EW) fundamental portfolios as a benchmark
for comparison, and report the long-short portfolios as well as the long-only portfolios.
Consistent with the results in our prior tables, we find that the MS fundamental-based
portfolios yield the highest Sharpe ratios, Information ratios, and returns for each size.
Consistent with the “size effect,” returns, Sharpe ratios, and Information ratios decline
as the investment set is further constrained to fewer larger stocks. Nonetheless, the MS
portfolios dominate within each size group even within the most constrained set of 100
largest stocks.
4.3. Fundamental Portfolios Performance Grouped by Expected Returns
One of the most common academic tests of return predictability is to form groups
based on a characteristic of interest and then form portfolios based on these groups and
compare average returns across portfolios. Table 5 presents the performance of portfolios
after progressively shrinking the available set of stocks to the most extreme group of stocks
based on the expected returns from the fundamentals-based model. For the long-short
portfolios, we constrain firms in the top group to have non-negative weights that sum to
one and firms in the bottom groups to have non-positive weights that sum to negative one
such that the resulting long-short portfolio represents a “zero cost” portfolio. Long-only
18
portfolios are portfolios constructed from those stocks with the highest expected returns.
Consistent with our prior results, Table 5 shows that the Sharpe and Information
ratios for the fundamental portfolios are higher with MS optimization relative to the
other fundamental portfolios within each expected return grouping. However, a curious
result that emerges in Table 5 is that the gains of MS over MV portfolios are smaller when
we exclude stocks based on expected returns. This might seem surprising, but as was
outlined in subsection 2.2.1, when cross-sectional differences in expected returns shrink,
the MS portfolios should come closer to the MV portfolios. Excluding stocks based on
expected returns essentially reduces cross-sectional differences in expected returns, and
the MS and MV portfolios should perform more similarly.
4.4. Fundamental Portfolios Performance Over Time
Our analysis thus far has focused on summary performance measures over the period
from 1981 to 2017. In Table 6, we report the portfolio performance metrics for non-
overlapping 10-year periods (except for the 8 year period 1981-1987) to provide insight
into the performance of portfolios over time. For each portfolio, there is a negative time
trend in average returns, Sharpe ratios, and Information ratios. However, within each
time period, optimized portfolios outperform the naive EW portfolios and MS portfo-
lios consistently outperform all other portfolios. Figures (1) and (2) provide graphical
analyses of differences in conditional Sharpe ratios over time.10 Both plots are broadly
consistent with Table 6 in that, outside of a few months, the Sharpe ratio of the MS port-
folios dominate alternative portfolios, suggesting the relative gains of the MS portfolios
are not driven by a particular time period.
10The conditional Sharpe Ratio of a portfolio at time t is calculated as the ratio of the conditional meanto the conditional standard deviation of the portfolio. The portfolio excess return (defined as portfolioraw return less the risk free rate) series is assumed to follow an AR (1) process with the variance ofthe error term following GARCH(1,1) process. The time series behavior of differences in conditionalInformation ratios are similar and available upon request.
19
5. Factor Portfolios and Alternative Optimization
In this section, we first examine whether common factor portfolios can explain the
returns of fundamental portfolios. We then examine portfolios constructed using an
alternative optimization approach developed by Brandt et al. (2009).
5.1. Factor Alphas
The number of return predictors documented in the literature has grown to exceed 300
(see for example, Bartram and Grinblatt, 2018; Green et al., 2013, 2017; Harvey et al.,
2016) and, as the number of predictors have grown over time, so has the number of factors
used to explain returns: from one, the market factor, to at least five, the market factor
plus four factors related to firm characteristics (Fama and French, 2015). The Fama and
French (2015) factor models represent returns of portfolios which are constructed from
characteristics that are similar to some of the variables we use in our fundamentals-based
returns model and hence use to form portfolios. There are two natural concerns with this.
First, resultant portfolios from our model may simply mimic some combination of factors
used in the Fama and French models. Second, because more return predictors have been
found with the passage of time, it is possible that the fundamental portfolio returns we
document could be driven by large returns from one or several predictors prior to the
publication date of the predictor. We test this formally by running standard asset pricing
portfolio tests over different time periods following the publication date of fundamentals
and characteristics related to those in our model. Our empirical tests use the following
regression:
Rp,t+1 = α +m∑j=1
βjfj,t+1 + εt+1,
where α is the average return of portfolio that is not explained by the factor(s), fj,t+1,
i.e., the factor alpha. If the factors do not explain the average return of our portfolios,
we expect α to be non-zero.
20
While factor models may have some conceptual similarities to the fundamentals model
we use, we do not anticipate that they will fully explain our portfolio returns. To see
why, ex ante, it is unclear whether the Fama and French factors can explain our portfolio
returns, recall that the vector of weights of our portfolios are of the form
ω∗ = Σ−1µ
eTΣ−1µ,
where the ith expected return is µi = Ai,0 + Ai,11Pi
+ Ai,2BiPi
+ Ai,3xiPi
+ Ai,4∆NOAi
Pi+
Ai,5∆FINiPi
. For a factor model to explain the average return of the optimized portfolios,
the weights of the factor portfolios would need to roughly align with our portfolio weights
with respect to the underlying fundamental characteristic. However, the Fama and French
factor models represent “zero cost” portfolios with respect to each characteristic. For
example, the book-to-market factor is long high book-to-market stocks and short low
book-to-market stocks and the weights of the book-to-market (or HML) factor portfolio
have a functional form similar to
ωi,hml =
ωi,H ,Bi,tMi,t≥ TH
−ωi,L, Bi,tMi,t≤ TL
0, TL <Bi,tMi,t
< TH
where ωi,H represents a market value weighting for stocks that have a book-to-market
ratio that is above the threshold TH and ωi,L represents a market value weighting for stocks
that have a book-to-market below the threshold TL. Because the Fama and French factor
portfolio weights have a different functional form than our portfolio weights, there is no
ex ante reason that our fundamental portfolios would be explained by standard factor
pricing models.
The results of this analysis is reported in Table 7 where we report α’s for each portfolio
for four different factor models: the CAPM, the Fama and French (1993, 2015) three,
21
four, and five factor models. We report the α’s over four different time periods. The first
time period represents our full out-of-sample time period from 1981-2017. Our second
time period is 1993-2017, which follows the publication date of Fama and French (1992)
(related to size and book-to-market). Our third is 1996-2017 and represents a period
after publication of Sloan (1996) (related to accruals) and our final is 2003-2017 which is
after publication of Fairfield et al. (2003) (related to growth in operating assets).
As Table 7 shows, there is very little evidence to suggest that any of the portfolios we
test can be replicated by any of the factor models. In fact almost all optimized funda-
mental portfolios generate significant factor alphas, even over the most recent 2003-2017
time period. Moreover, similar to our prior tables, the MS portfolios which incorporate
information about expected returns and covariances have the highest factor alphas in
virtually every specification.
5.2. BSV Portfolios Performance
As an attempt to overcome the difficulty in estimating expected returns and other
return moments within mean-variance optimization, Brandt et al. (2009) propose the
combination of a power utility function with firm characteristics to solve for portfolio
weights via non-linear estimation. Specifically, Brandt et al. (2009) propose a method to
incorporate firm-level characteristics by utilizing the following estimation approach:
maxθ
1t
t−1∑j=0
Rp,j1−γ
1− γ , (19)
s.t. ωi,j = ωi,j + 1Nj
θci,j,
where ωi,j is the weight of a benchmark portfolio at time j, Nj is the number of firms in
the portfolio at time j, θ is a vector of parameters to be estimated, and ci,j a vector of firm
characteristics. Portfolio weights at time t are then given by ωi,t = ωi,t + 1Nθci,t, subject
to ∑Ni=1 ωi,t = 1. Consistent with our prior analyses, we use the equal weighted portfolio
22
as the benchmark portfolio, and we use two different sets of characteristics in portfolio
construction. We first use our estimate of expected returns as a single characteristic, we
then include all of the characteristics 1Pi,t, Bi,tPi,t
, xi,tPi,t, goi,tPi,t
, gfi,tPi,t as individual variables in
the optimization program. Like Brandt et al. (2009) and Hand and Green (2011) we set
γ = 5 and estimate the parameter θ using equation (19) with an expanding window, with
a minimum of 10 years historical data. The minimum 10 year window changes our out-of-
sample time period to 1991-2017 in this analysis, so we report the portfolio performance
for the EW, MV, ER, MS portfolios as well as the Brandt et al. (2009) (BSV, henceforth)
portfolios for this time period. The results are reported in Table 8. For brevity, we report
the results of long-only portfolios because these portfolios are more representative of what
an investor could actually earn as opposed to long-short portfolios.
Consistent with Brandt et al. (2009) and Hand and Green (2011), the BSV ap-
proach offers benefits to an investor over naive portfolio construction. Using either the
fundamentals-based expected return (BSV1) or all six characteristics (BSV2) results in
portfolios that outperform EW and MV portfolios. However, the ER and MS portfolios
offer performance gains over the BSV approach and, consistent with our prior findings,
the MS portfolios dominate on virtually all dimensions of portfolio performance.
6. Conclusion
Constructing an investment portfolio generally consists of two activities: forming
beliefs about future stock returns and optimally allocating wealth in a portfolio based on
those beliefs. We formalize how fundamental analysis can be mapped directly into beliefs
through expected returns and covariances to construct optimal fundamental portfolios
and provide initial large sample evidence of gains to investors of combining fundamental
analysis and portfolio optimization.
Long-only optimizal fundamental portfolios, which incorporate fundamentals-based
expected returns and covariances and MS optimization, produce CAPM alphas of over
1% per month and 5-factor alphas of over 0.8% per month, with high Sharpe and In-
23
formation ratios. Gains to investors over naive equal weighted strategies remain 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 provides important contributions to both practice and research on funda-
mental analysis and portfolio optimization. Fundamental analysis is aimed at identifying
stocks that are likely to experience higher or lower future returns, but provides little
insight with respect to creating portfolios. Our study provides an implementable method
of developing portfolios that improve the performance of fundamental analysis. Similarly,
portfolio optimization provides theoretical arguments for optimizing portfolios, but there
is little empirical evidence to date suggesting that it results in portfolio performance pre-
dicted by theory. Our findings suggest that portfolio optimization, when combined with
fundamental analysis, can help investors realize substantial portfolio gains.
24
References
Bartram, S. M. and M. Grinblatt (2018). Agnostic fundamental analysis works. Journal
of Financial Economics 128 (1), 125–147.
Beneish, M. D., C. M. Lee, and D. C. Nichols (2015). In short supply: Short-sellers and
stock returns. Journal of accounting and economics 60 (2-3), 33–57.
Benjamin, G. and D. L. Dodd (1934). Security analysis. Me Graw Hill Ine, New York.
Bradshaw, M. T., S. A. Richardson, and R. G. Sloan (2006). The relation between corpo-
rate financing activities, analysts’ forecasts and stock returns. Journal of Accounting
and Economics 42 (1-2), 53–85.
Brandt, M. W., P. Santa-clara, and R. Valkanov (2009). Parametric portfolio policies:
Exploiting characteristics. The Review of Financial Studies 22 (9), 3411–3447.
Chattopadhyay, A., M. R. Lyle, and C. C. Wang (2018). Accounting data, market values,
and the cross section of expected returns worldwide. Working Paper .
Christensen, P. O. and G. Feltham (2012). Economics of Accounting: Information in
markets, Volume 1. Springer.
Cooper, M. J., H. Gulen, and M. J. Schill (2008). Asset growth and the cross-section of
stock returns. The Journal of Finance 63 (4), 1609–1651.
DeMiguel, V., L. Garlappi, and R. Uppal (2009, May). Optimal Versus Naive Diver-
sification: How Inefficient is the 1/N Portfolio Strategy? The Review of Financial
Studies 22 (5), 1915–1953.
Engle, R. F., O. Ledoit, and M. Wolf (2017). Large dynamic covariance matrices. Journal
of Business & Economic Statistics, 1–13.
Fairfield, P. M., S. Ramnath, and T. L. Yohn (2009). Do industry-level analyses improve
forecasts of financial performance? Journal of Accounting Research 47 (1), 147–178.
25
Fairfield, P. M., J. S. Whisenant, and T. L. Yohn (2003). Accrued earnings and
growth: Implications for future profitability and market mispricing. The accounting
review 78 (1), 353–371.
Fama, E. F. and K. R. French (1992). The cross-section of expected stock returns. the
Journal of Finance 47 (2), 427–465.
Fama, E. F. and K. R. French (1993). Common risk factors in the returns on stocks and
bonds. Journal of financial economics 33 (1), 3–56.
Fama, E. F. and K. R. French (2015). A five-factor asset pricing model. Journal of
Financial Economics 116 (1), 1 – 22.
Fama, E. F. and J. D. MacBeth (1973). Risk, return, and equilibrium: Empirical tests.
Journal of political economy 81 (3), 607–636.
Frankel, R. and C. M. Lee (1998). Accounting valuation, market expectation, and cross-
sectional stock returns. Journal of Accounting and economics 25 (3), 283–319.
Gebhardt, W. R., C. M. Lee, and B. Swaminathan (2001). Toward an implied cost of
capital. Journal of accounting research 39 (1), 135–176.
Green, J., J. R. Hand, and X. F. Zhang (2013). The supraview of return predictive
signals. Review of Accounting Studies 18 (3), 692–730.
Green, J., J. R. Hand, and X. F. Zhang (2017). The characteristics that provide inde-
pendent information about average us monthly stock returns. The Review of Financial
Studies 30 (12), 4389–4436.
Hand, J. R. and J. Green (2011). The importance of accounting information in portfolio
optimization. Journal of Accounting, Auditing & Finance 26 (1), 1–34.
Harvey, C. R., Y. Liu, and H. Zhu (2016). ... and the cross-section of expected returns.
The Review of Financial Studies 29 (1), 5–68.
26
Hou, K., C. Xue, and L. Zhang (2018). Replicating Anomalies. The Review of Financial
Studies.
Jagannathan, R. and T. Ma (2003). Risk reduction in large portfolios: Why imposing
the wrong constraints helps. The Journal of Finance 58 (4), 1651–1683.
Johannesson, E. and J. A. Ohlson (2019). Explaining returns through valuation. Columbia
Business School Working Paper (17-38).
Jorion, P. (1985). International portfolio diversification with estimation risk. Journal of
Business, 259–278.
Jorion, P. (1986). Bayes-stein estimation for portfolio analysis. Journal of Financial and
Quantitative Analysis 21 (3), 279–292.
Ledoit, O. and M. Wolf (2017). Nonlinear shrinkage of the covariance matrix for portfolio
selection: Markowitz meets goldilocks. The Review of Financial Studies 30 (12), 4349–
4388.
Lee, C. M., J. Myers, and B. Swaminathan (1999). What is the intrinsic value of the
dow? The Journal of Finance 54 (5), 1693–1741.
Lewellen, J. (2015). The cross-section of expected stock returns. Critical Finance Re-
view 4 (1), 1–44.
Lyle, M. R., J. L. Callen, and R. J. Elliott (2013). Dynamic risk, accounting-based
valuation and firm fundamentals. Review of Accounting Studies 18 (4), 899–929.
Lyle, M. R. and C. C. Wang (2015). The cross section of expected holding period returns
and their dynamics: A present value approach. Journal of Financial Economics 116 (3),
505–525.
Markowitz, H. (1952). Portfolio selection. Journal of Finance 7 (1), pp. 77–91.
27
Michaud, R. O. (1989). The markowitz optimization enigma: Is ’optimized’ optimal?
Financial Analysts Journal 45 (1), 31–42.
Sloan, R. G. (1996). Do stock prices fully reflect information in accruals and cash flows
about future earnings? Accounting review, 289–315.
