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 (m-lyle@kellogg.northwestern.edu) is an Associate Professor at the Kellogg School of Man-agement, and Yohn (teri.yohn@kellogg.northwestern.edu) 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

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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.

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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.

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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:

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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

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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:

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peR

atio

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nsF

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disp

lays

the

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peR

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gain

sfo

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ulat

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ket

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erpa

nel)

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ance

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ror

term

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RC

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)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:

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ityG

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Fig

ure

2di

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ins

for

the

max

imum

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port

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gth

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ates

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sk-a

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1996

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rian

ceof

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AR

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(1)

proc

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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

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re3:

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paris

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asa

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tion

ofIn

vest

able

Ass

ets

Fig

ure

3di

spla

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arpe

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afu

ncti

onof

the

num

ber

ofas

sets

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able

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for

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esti

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usin

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for

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tim

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riod

1996

-20

15.

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inve

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top

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onth

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pres

ents

the

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the

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whi

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less

the

risk

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rate

divi

ded

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epo

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lays

the

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per

unit

num

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inth

ein

vest

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univ

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(N)

and

the

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per

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num

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ocks

inve

sted

inth

epo

rtfo

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In

v)

for

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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

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0

Num

ber

of

Investa

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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

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kre

turn

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expe

cted

retu

rnpr

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s,µ

t,u

sing

two

diffe

rent

esti

mat

ion

mod

els

from

1996−

2015

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colu

mns

(1)

and

(3),µ

tis

HIS

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hich

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oric

alm

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rn(o

ver

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ths)

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mns

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and

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tis

FU

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chis

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tals

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nce

leve

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Tabl

e3:

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folio

sw

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tO

ptim

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Tabl

e3

pres

ents

port

folio

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rics

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38

Tabl

e3:

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folio

sw

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ptim

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tinue

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(b)

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959

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024

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bmt

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344

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0)1.

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4517

4517

417

417

417

4

39

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e4:

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imiz

edPo

rtfo

lios

with

out

Fund

amen

tals-

Base

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pect

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etur

nsTa

ble

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llSt

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Pan

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Info

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ess

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turn

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Sign

ifica

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1%,5

%,a

nd10

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dby

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ecti

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for

over

lap.

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Pane

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Met

rics

Stoc

ksIn

clud

ed:

All

Stoc

ksA

llSt

ocks

Top

Dec

ileEx

pect

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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*

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89-0

.522

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82-0

.529

FF3

0.55

3*-0

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19-0

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FF4

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7-0

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17-0

.411

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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

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top

deci

leba

sed

onth

eex

pect

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turn

esti

mat

eF

UN

D.E

Wre

pres

ents

aneq

ually

wei

ghte

dpo

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lioof

the

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ksin

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top

deci

leba

sed

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pect

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turn

esti

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MS

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arpe

Rat

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rist

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port

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rics

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elB

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the

char

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rist

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char

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ele

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and

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ifth

era

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ofth

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gnifi

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2)co

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Met

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pect

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DFU

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Port

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stru

ctio

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MV

TM

SB

SVM

VT

MS

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chm

ark

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Shar

peR

atio

0.42

60.

447

0.45

6**

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50.

473*

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473*

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279

Info

rmat

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io0.

451

0.47

60.

495*

0.38

50.

520*

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522*

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176

λ∗

3.01

92.

556

9.12

00.

926

0.69

6∞

Alpha’s

CA

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023*

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803*

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888*

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161*

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192*

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220*

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397*

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32.

520*

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302*

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384*

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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

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ileEx

pect

edR

etur

n:FU

ND

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DFU

ND

Port

folio

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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

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40-3

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87-3

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gf t

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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

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pect

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turn

esti

mat

esca

lcul

ated

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gth

efu

ndam

enta

lsm

odel

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UN

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hen

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inve

stab

leun

iver

seof

stoc

ksis

rest

rict

edto

top

80%

stoc

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sed

onm

arke

tca

pita

lizat

ion

atti

met.

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hold

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em

onth

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nequ

arte

r).

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Stoc

ksre

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port

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atin

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ple

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*,an

d*,

resp

ecti

vely

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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

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rger

than

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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

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rick

(199

2)co

rrec

tion

toac

coun

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rov

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p.

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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

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ortf

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Cha

ract

erist

ics

Stoc

ksIn

clud

ed:

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ileA

llSt

ocks

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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

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98-3

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91-3

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14-4

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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

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heho

ldin

gpe

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ree

mon

ths

(one

quar

ter)

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anel

Are

port

spo

rtfo

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etri

csov

erro

lling

ten

year

peri

ods

and

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elB

repo

rts

port

folio

met

rics

usin

gno

nov

erla

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eye

arpe

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s.T

heSh

arpe

Rat

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ple

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npo

rtfo

liore

turn

less

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risk

free

rate

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ded

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est

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ion

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lio.

The

Info

rmat

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Rat

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Rat

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Port

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Met

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MS

Opt

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ndam

enta

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10-y

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Rol

ling

Win

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(1)

(2)

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’96-

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’06

’98-

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’99-

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’05-

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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*

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540*

**1.

676*

**1.

801*

**FF

43.

017*

**2.

765*

**2.

904*

**3.

143*

**3.

492*

**3.

194*

**3.

152*

**2.

596*

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632*

**1.

777*

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892*

**FF

52.

901*

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842*

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905*

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674*

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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*

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853*

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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*

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656*

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998*

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524*

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42.

697*

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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

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base

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ulat

edus

ing

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tals

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elov

erdi

ffere

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heho

ldin

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riod

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ree

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ter)

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Wre

pres

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ually

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lios.

MS

repr

esen

tsth

em

axim

umSh

arpe

Rat

iopo

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lious

ing

mea

n-va

rian

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ptim

izat

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The

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peR

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retu

rnle

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ket

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ifica

nce

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lsof

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Sign

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(1)

(2)

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ecile

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arpe

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366

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MS-

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Info

rmat

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Rat

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004

0.15

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0.15

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0.13

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7***

0.11

2***

0.09

6***

0.10

3***

0.07

1***

48