Normative Portfolio Theory - Motivational … Portfolio Theory Yufen Fu · George Blazenko 1’st...
Transcript of Normative Portfolio Theory - Motivational … Portfolio Theory Yufen Fu · George Blazenko 1’st...
Normative Portfolio Theory
Yufen Fu · George Blazenko
1’st version, August 2014, latest version December 2016
Abstract
We correct the adverse impact of estimation risk on portfolio performance and weights with two
new equity allocation methods we implement with estimation-free and estimated ex-ante returns.
Portfolios with estimation-free ex-ante returns and systematic-to-unsystematic risk weights have
statistically higher Sharpe ratios than both similar portfolios with estimated ex-ante returns and
1/N’th portfolios. Optimal portfolio methods with well-behaved weights guide investors in a way
not hitherto possible (normative portfolio theory).
Keywords Equity allocation, estimation-free ex-ante returns, common share portfolio appeal
JEL Classification G14 · G15
______________________
G. Blazenko (✉)
Beedie School of Business,
Simon Fraser University,
500 Granville Street,
Vancouver, BC, Canada,
e-mail: [email protected]
Y. Fu (✉)
Department of Finance,
Tunghai University,
181, Sec.3, Taichung-kan Rd.,
Taichung, Taiwan,
e-mail: [email protected]
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1 Introduction
Despite the economic and methodologic elegance of modern portfolio theory first proposed by
Markowitz (1952), there is a good deal of evidence that estimation risk−substitution of parameters
presumed known with estimated equivalents−imposes application limits. A naïve equal-weight
portfolio (1/N’th investing) outperforms the sample tangency portfolio (Jobson and Korkie, 1981).
Mean-return and variance/covariance estimation encourages impractical portfolio weights that
vary wildly over time (Dickinson, 1974; Best and Grauer, 1991; Chopra and Ziemba, 1993;
Ziemba, 2003). Minor mean-return differences induce extreme short positions in some assets that
finance equally extreme long positions in others but a short-sale constraint encourages
undiversified portfolios with zero weights for most assets. Unrealistic weights offer investors little
guidance for their portfolio decisions.
There is a literature that tries to rehabilitate modern portfolio theory for application with
Bayesian and shrinkage mean-return estimators and methods that constrain portfolio weights (Kan
and Smith, 2008; Golosnoy and Okhrin, 2009; Best and Grauer, 1991, 1992). However, no version
of the mean-variance model consistently outperforms 1/N’th investing (Windcliff and Boyle, 2004;
DeMiguel, Garlappi, and Uppal, 2009). Because there exist exchange traded funds (ETFs) that
replicate equal weight portfolios of equities underlying major market indices (like the S&P 500),
1/N’th investing is a feasible strategy even for retail investors. An unfortunate lesson from these
results is that more than sixty years of portfolio research does not, as a rule, produce better
portfolios than naïve investing when confronted with estimation risk.
In Sharpe ratio analysis (Sharpe, 1964), we outperform 1/N’th investing with both Blazenko and
Fu’s (2013) estimation-free and Fama and French’s (2015) estimated ex-ante returns in two new
equity allocation methods that force well-behaved weights. Portfolios with estimation-free ex-ante
returns and systematic-to-unsystematic weights have higher out-of-sample Sharpe ratios than both
similar portfolios with estimated ex-ante returns and 1/N’th weights. We evaluate only portfolios
with well-behaved weights−including 1/N’th portfolios.
Extreme diversification in 1/N’th investing with the entire universe of common shares leads to
especially low portfolio return standard deviation. Methods we cite above that fail to outperform
this benchmark fail to identify returns high enough to offset high return standard-deviation from
less extreme diversification. This failure is, thus, the principal obstacle of estimation risk for
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portfolio construction. In security selection, with systematic-to-unsystematic risk weights and
estimation-free ex-ante returns, we find that portfolios with sixty-four common shares outperform
1/N’th investing in Sharpe-ratio analysis with higher average return and lower return standard-
deviation. Selected common shares have above average risk and, thus, diversification requires a
larger number of common shares than the existing literature often recommends.
The focus of asset-pricing in the finance literature is on returns, including Jensen’s (1968) alpha
and more modern alphas in multi-factor models (e.g., Fama and French, 1992, 2015). However,
only optimal portfolio weights that impound both return and risk (of all types) guide investors
completely. Weight misbehavior in the existing literature hinders this guidance. Superior
performance with well-behaved weights allows us to recommend portfolio construction
(normative portfolio theory) in a way not hitherto possible. We are the first to study the economic
determinants of optimal portfolios that outperform 1/N’th investing in a modern portfolio theory
application with well-behaved weights.
2 Equity allocation
In this section, we describe portfolio methods that force well-behaved weights and discuss the ex-
ante returns we use to implement them.
2.1 Systematic to Unsystematic Risk Weights
With equilibrium ex-ante returns from a linear model, Appendix A shows that there is a solution
to the Elton, Gruber, and Padberg (1976) equations that implicitly specify portfolio weights that
maximize the Sharpe ratio (mean portfolio return above a riskless rate divided by portfolio return
standard-deviation),
𝑥𝑖 = [
𝐸(�̃�𝑖)−𝑟𝑓
𝜎𝜀𝑖2 ]
∑ [𝐸(�̃�𝑘)−𝑟𝑓
𝜎𝜀𝑘2 ]𝑁
𝑘=1
, i=1,2,…,N (1)
where 𝑥𝑖 is the optimal weight for asset i in a universe of N risky assets, 𝐸(�̃�𝑖) is ex-ante return,
𝑟𝑓 is the riskless rate, and 𝜎𝜀𝑖
2 is unsystematic (unpriced) residual return risk. Eq. (1) is also the
tangency portfolio in mean-variance analysis (Jarrow and Rudd, 1983) when there is equivalence
between returns in the Arbitrage Pricing Theory (Ross, 1976) and the Capital Asset Pricing Model
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(Sharpe, 1964; Lintner, 1965; Mossin, 1966). We are the first to use these weights in applied
portfolio construction.
The numerator of Eq. (1) measures an asset’s portfolio appeal with own-return characteristics
solely. This result is especially useful to investors constructing non-equilibrium portfolios with
fewer than the universe of N assets to avoid the portfolio monitoring costs Merton (1987) identifies.
Investors can add or omit assets to or from a portfolio without changing the rank appeal of other
assets in or out of the portfolio. A closed-form weight expression and own return characteristics
facilitates equity screeners based on estimation-free ex-ante returns and the numerator of Eq. (1).
Equation (1) weights favour priced risk impounded in ex-ante excess return over unpriced risk
in residual return variance. The ratio of these measures ranks assets for portfolio appeal and
efficiently accumulates priced risk in a portfolio by eschewing unpriced risk. Only in the limit does
unpriced risk disappear as the number of assets N increases without bound and weights approach
zero leaving priced risk behind solely. If ex-ante excess returns are all positive, then weights are
all positive. In a general setting, the “all positive weight” condition is quite exacting and unlikely
satisfied in practice (Grauer and Best, 1992).
Weight misbehavior in the existing literature arises from incongruence between systematic risk
(priced) in ex-ante returns and estimated covariance between assets. By separating “good” priced
risk from “bad” unpriced risk, Eq. (1) weights negate the need to estimate covariance directly and,
thus, force congruence between covariance and priced risk in ex-ante returns. The combination of
own-return characteristics and inputs that don’t change dramatically in the short-term force well-
behaved weights in Eq. (1).
Equation (1) weights require no explicit factor representation other than for unsystematic risk.
Even with estimation-free ex-ante returns that we describe in section 2.3 and without covariance
estimation, systematic-to-unsystematic risk weights in Eq. (1) moderate but do not eliminate
estimation risk because we estimate unsystematic risk.1 For comparison purposes, we also estimate
ex-ante returns with the Fama and French (2015) five factor model.
1 Presuming that the beta of all companies in a market model is unity, one can calculate systematic-to-unsystematic
risk weights without estimation risk as estimation-free ex-ante excess return divided by residual volatility calculated
as implied volatility of a common share less implied volatility of a benchmark portfolio (both from the options market).
This methodology restricts security selection to common shares with exchange traded options. There is a literature on
historical volatility, implied volatility, and various econometric methods as forecasters of future volatility. Bartunek
and Chowdhury (1995) find no statistical difference.
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2.2 Ex-ante excess return weights
If we make an unrealistic presumption, we can construct portfolios without estimation risk. With
equal unsystematic risk for each asset, 𝜎𝜀𝑖
2 = 𝜎𝜀𝑗
2 , 𝑖 ≠ 𝑗, Eq. (1) simplifies,
𝑥𝑖 = [𝐸(�̃�𝑖)−𝑟𝑓]
∑ [𝐸(�̃�𝑘)−𝑟𝑓]𝑁𝑘=1
i=1,2,…,N (2)
We refer to Eq. (2) as ex-ante excess return weight portfolios because the weight for asset i is high
if ex-ante excess return is high.
The foundation of positive economic theory is that one judges an economic model by its
forecasts rather than underlying presumptions, which are less important than high Sharpe ratios
for investors, which we report in the current paper for Eq. (2) portfolios compared to 1/N’th
portfolios. In addition, it is instructive to compare Eq. (2) with Eq. (1) portfolios. Our results
illustrate that minimizing residual return risk is important for low portfolio return standard-
deviation and high Sharpe ratios for Eq. (1) compared to Eq. (2) portfolios. Eq. (2) portfolios
generate highest average returns but not highest Sharpe ratios.
2.3 Estimation-free ex-ante returns
Blazenko and Fu (2013) develop a closed-form estimation-free ex-ante return expression,
𝐸(�̃�) = 𝑅𝑂𝐸 + (1 − 𝑀/𝐵) ∗ 𝑑𝑦, (3)
where ROE is the forward rate of return on equity, M/B is the market to book ratio, and dy is the
forward dividend yield.2 Since Eq. (3) is dividend yield plus growth, forward corporate growth is,
𝐸(�̃�) = 𝑅𝑂𝐸 − 𝑀/𝐵 ∗ 𝑑𝑦 (4)
For non-dividend paying firms, both ex-ante return and forward growth is forward ROE because
shareholders receive a return entirely from capital gains.
Profitability, ROE, appears in Eq. (3) because growth capital-expenditures lever risk like fixed-
costs in operating-leverage (Blazenko and Pavlov, 2009). Profitability and growth relate positively
2 Eq. (3) arises from static equity valuation. In dynamic equity valuation, when managers suspend growth investment
upon stochastically poor profitability, ex-ante return is Eq. (3) plus a term that depends on profit volatility (Blazenko
and Pavlov, 2009).
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for two reasons. Businesses finance growth largely internally with profitability (Myers and Majluf,
1984) and profitability is a principal determinant of growth-option exercise (Blazenko and Pavlov,
2009). Eq. (3) is unique in the financial literature in that it specifies the ex-ante relation between
returns and profitability that leads to the ex-post relation documented empirically (Blazenko and
Fu, 2013; Novy-Marx, 2013; Hou, Xue, and Zhang, 2014; Fama and French, 2015; Fu and
Blazenko, 2015). Specifying this ex-ante relation is important because portfolio construction
requires ex-ante returns. Blazenko and Fu (2013) investigate the relation between ex-ante returns
in Eq. (3) and standard asset pricing models in the financial literature. They show that Eq. (3) ex-
ante returns relate positively with realized returns and generate positive alphas for high
profitability companies.
Other than the relation between profitability and growth, Eq. (3) identifies no economic
determinant of ex-ante returns, which, alternatively, is the focus of study in the asset pricing
literature. The riskless interest rate and risk appear only implicitly in Eq. (3) through share price
in M/B and dy. While valuable for investment understanding, modern portfolio theory requires
only ex-ante returns, like Eq. (3), with no necessary appreciation of risk determinants.
One way to calculate ROE with accounting constructs is with price/book divided by
price/forward-earnings (that is, (P/B)/(P/E)=E/B=ROE). Financial information providers report
both these measures for many public companies with consensus analyst earnings forecasts as
forward earnings. Rajan, Reichelstein, and Soliman (2007) review the literature on accounting
returns (like ROE) as economic returns. Since we find superior equity allocation with ex-ante
returns in Eq. (3), we conclude that forward ROE measures economic ROE meaningfully.
Many public companies have dividend policies that when divided by share-price is the forward
dividend-yield often reported by financial information providers. Alternatively, because we expect
dividends and ex-dividend share price to grow at the same rate, current dividend yield forecasts
future dividend yield. Market/book exceeds unity for most public companies, which, from Eq. (3),
means forward ROE exceeds ex-ante return and, thus, businesses make positive NPV investments
for shareholders as they grow.
Best and Grauer (1991), Chopra and Ziemba (1993) and Ziemba (2003) identify mean returns
as the principal source of weight misbehaviour in mean-variance models rather than variances or
covariances. Ziemba (2003) finds mean returns have 10 and 20 times the impact on portfolio
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weights, respectively. Observations like this lead Best and Grauer (1992) to use returns consistent
with market weights and estimated variances-covariances in their asset-allocation. Black and
Litterman (1992) use market weights as equilibrium aggregates from which investors apply
deviations for personal expectations. Because averaging does not eliminate return variability
completely, average historical returns produce especially acute anomalies with weights vigorously
following past returns. Because ex-ante return in Eq (3) is estimation-free and the residual return
variance we estimate for systematic-to-unsystematic risk weights in Eq. (1) is not a primary source
of weight misbehaviour our methodology is essentially without estimation risk. With estimation-
free ex-ante returns, Eq. (2) portfolios are entirely without estimation risk.
One might argue that ex-ante returns in Eq. (3) substitute specification error for estimation error
in portfolio construction. However, portfolios with Eq. (3) ex-ante returns and well-behaved
weights in Eq. (1) and Eq. (2) outperform 1/N’th portfolios and similar portfolios formed with
estimated ex-ante returns in Sharpe ratio analysis, which excuses specification error including
earnings forecast biases. We conclude that specification error is less problematic than estimation
error for portfolio construction.
2.4 Estimated ex-ante returns
To isolate the source of high Sharpe ratios we report, we compare portfolio construction with
estimation-free and estimated ex-ante returns. The Fama and French (2015) five factor model uses
book/market, size, investment, profitability, and a market factor to represent returns. The ex-post
version of the FF 5-factor model for realized returns is,
�̃�𝑖,𝑡 − 𝑅𝑓,𝑡 = 𝛼 + 𝛽𝑀(𝑅𝑀,𝑡 − 𝑅𝑓,𝑡) + 𝛽𝑆𝑀𝐵𝑆𝑀𝐵𝑡+𝛽𝐻𝑀𝐿𝐻𝑀𝐿𝑡 + 𝛽𝑅𝑀𝑊𝑅𝑀𝑊+𝛽𝐶𝑀𝐴𝐶𝑀𝐴𝑡 + 𝜖�̃�,𝑡, (5)
where �̃�𝑝,𝑡 is the return on asset i in month t, 𝑅𝑀,𝑡 is the return on the CRSP value weighted index
of common stocks in month t, f ,tR is the riskless interest rate at the beginning of month t, SMBt is
the return on a net-zero investment in small companies financed by a short position in large
companies, HMLt is the return on a net-zero investment in value stocks (high book to market)
financed by a short-position in growth stocks (low book to market), RMWt is the return on a net-
zero investment in robust stocks (high operating profitability) financed by a short-position in weak
stocks (low operating profitability), and CMAt is the return on a net-zero investment in
conservative stocks (low investment) financed by a short-position in aggressive stocks (high
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investment). We estimate factor betas with a time-series regression with common share returns in
the 36-month rolling window prior to portfolio rebalancing. Data is from Ken French’s website.3
The ex-ante version of the 5-factor model at the beginning of month t says that expected excess
return for asset i is,
𝐸[�̃�𝑖,𝑡] − 𝑅𝑓,𝑡 = �̂�𝑀(𝐸[𝑅𝑀,𝑡] − 𝑅𝑓,𝑡) + �̂�𝑆𝑀𝐵𝐸[𝑆𝑀𝐵𝑡]
+�̂�𝐻𝑀𝐿𝐸[𝐻𝑀𝐿𝑡] +�̂�𝑅𝑀𝑊𝐸[𝑅𝑀𝑊𝑡]+�̂�𝐶𝑀𝐴𝐸[𝐶𝑀𝐴𝑡], (6)
where �̂�𝑀, �̂�𝑆𝑀𝐵, �̂�𝐻𝑀𝐿 , �̂�𝑅𝑀𝑊, �̂�𝐶𝑀𝐴 are estimated factor betas and (𝐸[𝑅𝑀,𝑡] − 𝑅𝑓,𝑡), 𝐸[𝑆𝑀𝐵𝑡],
𝐸[𝐻𝑀𝐿𝑡], 𝐸[𝑅𝑀𝑊𝑡], 𝐸[𝐶𝑀𝐴𝑡] are factor risk premia.
Rather than average historical factor premia that Hu (2007) shows to be inefficient as forecasts,
we use his alternative methodology to estimate factor premia with several structural variables
known to predict future returns. These structural variables include lagged values for realizations
of the five factors, the difference in CRSP value weighted market returns with and without
dividends, the one-month Treasury bill rate, the growth rate of industrial production, the term
spread (measured as the difference between 10-year Treasury bond yields and the three-month
Treasury bill yield), and the credit spread (measured as the difference between Moody’s Bbb and
Aaa corporate bond yields). The regression of realized factor premia on structural variables in the
preceding 36-month rolling window is the mechanism for forecasting the following month’s factor
premia. Data for the growth rate of industrial production, the term spread, and the credit spread
are from the Federal Reserve Bank of St. Louis. Combining forecast factor premia and beta
estimates in Eq. (6) yields the estimated ex-ante excess return for asset i.
3 Data and Summary Statistics
3.1 Data
Our sample firms have data from COMPUSTAT (Standard & Poor’s), the Center for Research in
Security Prices at the University of Chicago (CRSP), and the Institutional Brokers’ Estimate
System (I/B/E/S) with book equity per share of at least one US dollar at portfolio rebalancing. We
3 http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/Data_Library
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exclude closed-end funds (CEF) and exchange traded funds (ETF). These are mostly US
corporations, although there are some inter-listed foreign firms (mainly Canadian). Corporate
financial data is from COMPUSTAT, stock market data is from CRSP, and consensus earnings per
share (EPS) forecasts for the next unreported fiscal year from a portfolio rebalancing date are from
I/B/E/S. The time-period of our study is January 1976 (earliest date for I/B/E/S forecasts) to July
2014. The number of firms in our sample, N, ranges from a low of 565 (January, 1976) to a high
of 3,893 (July, 1998).
We report summary statistics by industry with Fama and French (1997) industry definitions that
use 4-digit SIC sub-sectors listed in Table 1. Our only modification to this industry classification
is to move REITs (SIC 6798) from “Trading” to “Real Estate.” We use COMPUSTAT SIC codes
to sort firms into industries and CRSP SIC codes when the former are unavailable.
3.2. Realized industry returns, estimated beta, return volatility, market weights
For forty-nine industry portfolios, Table 1 reports annualized value-weighted average monthly
returns, estimated beta, annualized monthly return standard-deviation, average monthly industry
value-weights, and average number of firms in an industry. One has to be careful interpreting these
results because we average both realized industry returns and market weights over the time series
so that high returns generate high weights. This ex post relation need not exist in ex-ante analysis.
Tobacco Products has the highest average annualized realized monthly return (20.9%) and
Fabricated Metal Products has the lowest (4.4%). Entertainment has the highest estimated beta
(1.479) and Utilities has the lowest (0.483). Coal has the highest monthly return standard-deviation
(13.4%) and Utilities has the lowest (2.2%). Petroleum & Natural Gas has the largest average
market weight (10.5%) and four industries jointly have the lowest (0.1% each): Agriculture,
Shipbuilding & Railroad Equipment, Coal, and Real Estate. Banking has the most number of firms
on average (284) and Tobacco Products has the fewest (4).
3.3. Forward ROE, forward growth, estimation-free ex-ante return, unsystematic risk, weights
We calculate estimation-free annualized ex-ante industry returns with Eq. (3). ROE is the sum of
forward earnings/annum (forward eps times outstanding shares) over firms in an industry divided
by the sum of book equity over firms. Forward earnings is the consensus forecast of analysts for
the next unreported fiscal year-end. M/B is the total value of outstanding shares at the time of
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portfolio formation divided by the sum of book equity over firms in an industry. Forward dividend-
yield is trailing twelve month (TTM) dividends times the number of outstanding shares summed
over firms times one plus Eq. (4) divided by industry market-value at the time of portfolio
formation. Book equity is from the most recently reported annual financial statement prior to
portfolio formation. Unsystematic risk is residual return variance in a market model regression of
excess industry returns versus excess market returns for thirty-six months prior to portfolio
formation. The “market” is the value-weighted portfolio of all common-shares in our sample
(month by month). We use US Treasury bills for the “riskless rate.”
For the purpose of the forward dividend-yield in Eq. (3), we do not use share repurchases as a
dividend substitute. Grullon and Michaely (2002) and Grullon, Paye, Underwood and Weston
(2011) report that most share repurchasing firms also pay dividends but not conversely. Lee and
Rui (2007) report that permanent earnings determine dividends whereas temporary earnings
determine share repurchases. In addition, it is difficult to identify if firms actually repurchase
shares despite announcements because they often leave them incomplete or un-started (Chung,
Dusan, and Perignon 2007).
For the 49 industry portfolios, Table 2 presents average annual forward industry ROE, forward
industry growth, estimation-free ex-ante industry return, annualized unsystematic risk, and
systematic-to-unsystematic risk weights from Eq. (1) (with estimation-free industry ex-ante
returns and unsystematic risk estimated from the market model for thirty-six months prior to
portfolio formation at the end of calendar months) over all 49 industries. Other than unsystematic
risk, which is over the entire time series, we update each of these variates monthly and average
from January 1976 to July 2014.
Tobacco Products has the highest forward ROE (30.0% per annum) and Recreation has the
lowest (7.2% per annum). Defense has the greatest forward growth (14.9% per annum) and
Utilities has the smallest (3.7% per annum). Defense has the highest ex-ante return (18.1% per
annum) and Precious Metals has the lowest (6.8% per annum). Precious Metals has the highest
unsystematic risk (131%2 per annum) and Business Services has the lowest (7.2%2 per annum).
Wholesale has the largest systematic-to-unsystematic risk weight (3.7%) and Precious Metals has
the lowest (0.2%).
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4 Security selection
In our portfolio performance assessment, we study “non-equilibrium” portfolios. Investors follow
naïve 1/N’th investing, they make unrealistic presumptions that lead to ex-ante excess return
weight portfolios in Eq. (2), and/or they own less than the universe of N common shares in security
selection for either Eq. (1) or Eq. (2) portfolios. The monitoring costs Merton (1987) identifies
prevent active investors from owning all common shares and, thus, our security selection relates
to one of the earliest financial research questions: how many common shares diversify a portfolio?
Since both are important to investors, only the optimal trade-off between risk and return in
portfolios with well-behaved weights adequately answers the optimal diversification question.
Early evidence suggests that as few as ten common shares diversify a portfolio (Evans and
Archer, 1968; Lorie, 1975). Statman (1987) argues that diversification requires up to thirty
common shares. These authors plot average portfolio return standard-deviations against common-
share numbers in randomly constructed portfolios to identify when the plot “levels out” so that
there are no further diversification benefits. Elton and Gruber (1977) develop an analytic
expression for variance of portfolio return variance with respect to common-share numbers so that
investors can choose the volatility they accept from incomplete diversification. Bernstein (2000)
argues that the only way to correctly diversify a portfolio is to own the entire market. The superior
performance of 1/N’th investing in Jobson and Korkie (1981), Windcliff and Boyle (2004) and
DeMiguel, Garlappi, and Uppal (2009) also suggests extreme diversification. In our optimal
security selection, we weight high-return and high-risk common shares more highly than average
common shares, and thus we require more common shares than early evidence recommends but
markedly less than 1/N’th investing for diversification.
4.1 Methodology
We construct both systematic-to-unsystematic risk weight portfolios in Eq. (1) and ex-ante excess
return weight portfolios in Eq. (2) with both estimation-free and estimated ex-ante returns in Eq.
(3) and Eq. (6), respectively. Along with 1/N’th investing (that uses all available common shares
that meet our sample selection criteria at portfolio rebalancing), we compare five portfolios in
Sharpe ratio analysis. Since financial ratios have their vagaries when applied to individual common
shares, we limit the impact of extreme observations, by capping the weight of any common share
for portfolios of five or more common shares at twenty-five percent or less.
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4.2 The Realized Portfolio Out-Of-Sample Sharpe Ratio
Because weights in Eq. (1) maximize the ex-ante portfolio Sharpe ratio (see Appendix A),
consistency requires we measure portfolio performance with a realized Sharpe ratio. Portfolio
rebalancing is at the beginning of a calendar month (end of the prior calendar month) and we
measure the realized Sharpe ratio out-of-sample for the month following rebalancing. The one-
month realized Sharpe ratio (RSR) is the realized monthly portfolio return (with systematic-to-
unsystematic risk weights, ex-ante excess return weights, or 1/N’th weights as appropriate) less
the riskless interest rate for a one-month holding period divided by the portfolio return standard-
deviation (total rather than residual) for thirty-six months prior to portfolio rebalancing. Thus,
𝑅𝑆𝑅 = (�̃�𝑝,𝑡−𝑟𝑓,𝑡
�̃�𝑝,𝑡) (7)
Because of non-normality (confirmed with Kolmogorov-Smirnov tests), we use the temporal-
median as a central-tendency measure for RSR and, in addition, statistical tests are non-parametric,
𝑀𝑒𝑑𝑖𝑎𝑛 𝑅𝑆𝑅 =𝑚𝑒𝑑𝑖𝑎𝑛𝑡 = 1, 𝑇
(�̃�𝑝,𝑡−𝑟𝑓,𝑡
�̃�𝑝,𝑡) (8)
4.3 Security selection with estimation-free ex-ante returns
For estimation-free ex-ante returns in Eq. (3), forward earnings is the consensus forecast of
analysts for common share i for the next unreported fiscal year-end. M/B is share-price divided by
book equity. Dividend yield is trailing twelve month dividend per share divided by share-price (all
at portfolio rebalancing date t). Book equity is from the most recently reported quarterly or annual
financial statement prior to a portfolio rebalancing date t.
Fig. 2 plots temporal median realized monthly Sharpe ratios in Eq. (8) for the 1/N’th common
share strategy (that allocates a 1/N’th weight to each common share at the beginning of a month),
and portfolios with N*=1 to N*=200 highest ranked common-shares using two ranking methods:
first, by the ratio of estimation-free ex-ante excess return to unsystematic risk and, second, by
estimation-free ex-ante excess return (that is, Eq. (1) and Eq (2) portfolios). Weights for these
portfolios use the normalized value of each ranking variate. There are 200 portfolios for each
ranking variate. Unsystematic risk is from a market model regression for thirty-six months prior
Y. Fu, G. Blazenko
12
to portfolio formation. The “market” is the value-weighted portfolio of all common-shares in our
sample. We eliminate common shares from our sample at a month if they do not have at least 20
months for this regression. The “riskless rate” is from US Treasury bills.
So, how many common shares diversify a portfolio? The conclusion we draw from Fig. 2 is
that N*=64 highest ranked systematic-to-unsystematic risk common shares optimally diversify a
portfolio. The Sharpe ratio plot increases with common shares, N*, increases from N*=1 to N*=64
where it levels and then declines. Above N*=64, portfolios gives up return faster than standard-
deviation decreases so the Sharpe ratio decreases.
At the risk of over-interpreting, our results suggest a market inefficiency. The advice we might
give investors from our results is that they can focus their portfolios the sixty-four common- shares
with highest systematic-to-unsystematic risk for highest portfolio Sharpe ratios. If investors had,
in fact, followed this advice in the past, then prices would differ from those in our analysis so that
results alternatively be that investors hold all N common-shares in the common-share universe for
highest realized Sharpe ratios. That we did not find this result suggests investors make errors in
portfolio construction.
The 1/N’th common share portfolio has extreme but indiscriminate diversification. On the other
hand, in Fig. 2 and Panel A of Table 3, a portfolio of only N*=3 highest ranked systematic-to-
unsystematic risk common shares has a Sharpe ratio above 1/N’th investing (although, of course
not statistically significant) with higher average returns and a slightly lower return standard
deviation (see Table 3 for details). The combination of high realized returns we identify with
estimation-free ex-ante returns in Eq. (3) and low portfolio return standard deviation we identify
with low residual return variance in the weights of Eq. (1) produces a portfolio with extraordinary
Sharpe ratio performance given the small number of common shares.
For portfolios of more common shares (N*=54 to N*=74 in Table 3), this focused
diversification yields portfolio return standard deviations lower than 1/N’th investing. While the
focus of Eq. (1) weights is a maximum Sharpe ratio rather than high returns or low portfolio return
standard deviation specifically, in application, portfolios with these weights have higher average
returns, lower return standard deviations, and statistically greater Sharpe ratios than the 1/N’th
portfolio (at less than the 1% level). These results illustrate that minimizing residual return risk in
Y. Fu, G. Blazenko
13
systematic-to-unsystematic risk portfolios in Eq. (1) is important for low portfolio return standard-
deviation and high Sharpe ratios.
In Panel B, because the focus of estimation-free ex-ante excess return weight portfolios is high
return with no risk consideration, they have the highest realized average returns in Table 3 but also
the highest return standard-deviations. Nonetheless, high returns dominate and ex-ante excess
return weight portfolios with N*=137 to N*=157 highest ranked ex-ante excess returns have
Sharpe ratios that statistically exceed the 1/N’th common share portfolio. Diversifying the high
risk of high return common shares returns requires a large number of common shares. A portfolio
with N*=147 ex-ante excess return selected common shares has the highest Sharpe ratio above
1/N’th investing. A portfolio with as few as N*=28 common shares has a Sharpe ratio about the
same as 1/N’th investing because of high realized returns we identify with estimation-free ex-ante
returns. This portfolio has the highest average realized monthly excess return and the highest return
standard deviation in Table 3 (also the highest return in either Table 3 or Table 4).
Fig. 2 Median Sharpe ratios for portfolios with estimation-free ex-ante returns
-0.01
0.04
0.09
0.14
0.19
0.24
0.29
0.34
0.39
1 51 101 151
Median
Sharpe
Ratio
Number of Common Shares in Portfolio, N*
1/N'th Common Share Strategy
Ranked by Estimation-Free Ex-Ante Excess Return
Ranked by Estimation-Free Ex-Ante Excess Return/Unsystematic Risk
Y. Fu, G. Blazenko
14
4.4 Security selection with estimated ex-ante returns
We estimate ex-ante returns with Eq. (6) at a monthly portfolio rebalancing date. Portfolio
rebalancing is at the beginning of a calendar month and we measure the realized Sharpe ratio for
the month following rebalancing. For systematic-to-unsystematic risk weights in Eq. (1),
unsystematic risk is from the regression in Eq. (5) that estimates factor betas for thirty-six months4
prior to portfolio formation at month t. We eliminate common shares from our sample at a month
if they do not have at least 20 months for this regression. The “riskless rate” is from US Treasury
bills.
Fig. 3 plots temporal median realized monthly Sharpe ratios in Eq. (8) for the 1/N’th common
share strategy (that allocates a 1/N’th weight to each common share at the beginning of a month),
and portfolios with N*=1 to N*=200 highest ranked common-shares using two ranking methods:
first, by the ratio of estimated ex-ante excess return to unsystematic risk and, second, by estimated
ex-ante excess return (that is, Eq. (1) and Eq. (2) portfolios). Weights for these portfolios use the
normalized value of each ranking variate. There are 200 portfolios for each ranking variate.
Details of highest Sharpe ratio portfolios in Fig. 3 are reported in Panel A of Table 4 for
portfolios with N*=46 to N*=66 common shares. These portfolios have higher average returns,
lower return standard-deviation, and statistically greater Sharpe ratios (at about the 10% level)
than the 1/N’th portfolio. The modest statistical significance of Sharpe ratio differences illustrates
the difficulty of outperforming the 1/N’th portfolio. The highest Sharpe ratio portfolio in Fig. 3
has N*=56 common shares.
In Panel B of Table 4, because the focus of ex-ante excess return weight portfolios is high return
with no risk consideration, they have the highest realized average returns in Table 4 but also the
highest return standard-deviations. However, there is no ex-ante excess return weight portfolio
with estimated ex-ante returns with a higher Sharpe ratio than the 1/N’th portfolio.
4 We use 36 months to be as temporally current as possible and to be consistent with realized Sharpe ratios in Eq. (7)
and Eq. (8). We get similar results when using a 60 month estimation window for ex-ante returns.
Y. Fu, G. Blazenko
15
Fig. 3 Median Sharpe ratios for portfolios formed with estimated ex-ante returns
4.5 Estimation-free versus estimated ex-ante returns
The answer to the question of the number of common shares to diversify a portfolio is almost
identical in Fig. 2 with estimation-free ex-ante returns and in Fig. 3 with estimated FF 5-factor ex-
ante returns. In Fig. 2, N*=64 highest ranked systematic-to-unsystematic risk common-shares
diversify a portfolio optimally, whereas, in Fig. 3, N*=56 highest ranked systematic-to-
unsystematic risk common-shares optimally diversify a portfolio. Table 5 compares these
portfolios against one another. The Sharpe ratio of the estimation-free ex-ante return portfolio in
Fig. 2 with N*=64 common shares statistically exceeds that of the estimated ex-ante return
portfolio in Fig. 3 with N*=56 common shares (at a significance level less than one percent). Better
Sharpe ratio performance for systematic-to-unsystematic risk portfolios in Fig 2 with estimation-
free ex-ante returns compared to Fig 3 with estimated FF 5-factor ex-ante returns is primarily from
lower return standard deviation (see Table 3 versus Table 4).
A comparison of Panel B in Table 3 versus Panel B in Table 4 confirms that estimation-free ex-
ante returns identify higher realized returns than estimated ex-ante returns (in ex-ante excess return
weight portfolios). This difference is consistent with estimation-free ex-ante returns in Eq. (3)
being more timely and better able to capture temporal risk changes.
-0.01
0.04
0.09
0.14
0.19
0.24
0.29
0.34
0.39
1 51 101 151
Median
Sharpe
Ratio
Number of Common Shares in Portfolio, N*
1/N'th Industry Strategy
Ranked by FF 5Factor Estimated Ex-Ante Excess Return
Ranked by FF 5 Factor Ex-Ante Return / Unsystematic Risk
Y. Fu, G. Blazenko
16
An interesting question is why the lower return standard deviation in Panel A of Table 3 versus
Panel A of Table 4 (both for systematic-to-unsystematic risk weight portfolios)? Common shares
with high systematic-to-unsystematic risk weights in Eq. (1) minimize unsystematic risk but only
for a given ex-ante return. Thus, when we identify high returns with estimation-free ex-ante returns
(see Panel B of Table 3 compared to Panel B of Table 4), we are also better able to identify high
returns in combination with low unsystematic risk leading to lower portfolio return standard-
deviation. This is also the reason that systematic-to-unsystematic risk portfolios with both
estimation-free and FF 5-factor estimated ex-ante returns in Panel A of Table 3 and Panel A of
Table 4 have lower return standard deviation that the 1/N’th portfolio. Maximizing return relative
to risk is at the expense of some return. Average realized returns in Panel A are lower than in Panel
B in both Table 3 and Table 4. This “expense” is somewhat greater with estimation-free ex-ante
returns compared to estimated ex-ante returns with the result that both average returns and return
standard-deviations are lesser in Panel A of Table 3 compared to Panel A of Table 4.
6 Summary, conclusion, and suggestions for future research
In this paper, we jointly control portfolio weights and outperform 1/N’th investing in Sharpe ratio
analysis. We correct the weight misbehaviour of mean-variance analysis with systematic-to-
unsystematic risk weights that separate “good” priced risk in ex-ante returns from “bad” unpriced
risk in residual return variance. This separation obviates the need to estimate covariance directly.
Consistent with the theoretical work of Best and Grauer (1991), Chopra and Ziemba (1993), and
Ziemba (2003), we empirically find that mean returns are indeed paramount for portfolio
performance. In security selection, estimation-free ex-ante returns in systematic-to-unsystematic
risk weight portfolios have the highest Sharpe ratio.
Theoretical portfolio construction in Eq. (1) weights guides our analysis. But, we have been
crude in our application of this theory to assure readers that our results do not arise from data
“fishing.” Outperforming 1/N’th investing with well-behaved weights is the objective of our study.
Having achieved this objective, future research can attempt to better this performance with
alternative methods.
In general terms, the study of optimal portfolio weights should guide investor behaviour
(normative portfolio analysis). Unfortunately, weight misbehaviour in the existing mean-variance
literature impairs the ability of modern portfolio theory to offer this guidance. The well-behaved
Y. Fu, G. Blazenko
17
weights we propose have this ability. We find highest Sharpe ratios with sixty-four common shares
in security selection. These results challenge the advice of Bernstein (2000) and 1/N’th investing
to own the entire market for diversification. Beyond this result, we leave a detailed study of optimal
portfolio weights to future research, which we now merely anticipate.
An interesting investment question is whether there are performance “costs” to carbon or sin-
free “ethical” portfolios (carbon is oil & gas and coal and the sin industries are defense, adult
entertainment, tobacco, gambling, and alcohol). In the current paper, because we outperform the
1/N’th portfolio with sixty-four common shares, perhaps we will find that carbon is seldom a
component of high Sharpe ratio portfolios. Perhaps there are Sharpe ratio “costs” to investment
portfolios that are carbon or sin free. Perhaps sin common shares exceed their market weights.
We investigate equity allocation in a domestic context. An obvious extension is global equity
allocation. Our results suggest that global investing differs for investors in large and small
countries. We find domestically that sixty-four common shares in security selection maximize the
Sharpe ratio. Thus, despite theoretical equilibrium analysis to the contrary, for individually optimal
portfolios, investors need not own all common shares in security selection. Internationally, large
countries have a large number of common shares. Large numbers make investors more likely to
find high systematic-to-unsystematic risk common shares domestically and they can, thus, avoid
the exchange rate risk of global investing. Opposite results are true for investors in small countries.
We investigate a particular mean-variance investor who maximizes the Sharpe ratio. We can
undertake a similar analysis for an investor with general mean-variance utility. A principal
question is whether a low risk-averse investor favours ex-ante excess return weight portfolios and
a high risk-averse investor favours systematic-to-unsystematic risk portfolios. If different risk-
averse investors choose different portfolios, then we uncover a violation of the hypothesis of two-
fund separation.
To maximize Sharpe ratios, our results suggest that investors need not own all common shares
in security selection. However, we investigate only portfolios with long-positions. Alternatively,
common shares with least portfolio appeal measured by Eq. (1) may be candidates for short-
positions in non-equilibrium long-short portfolios for enhanced Sharpe ratios. Transactions costs
for short-positions may exceed those of long-positions.
Of course, any conjecture we make above requires confirmation in future research.
Y. Fu, G. Blazenko
18
6 Appendix A
In this appendix, with equilibrium returns from a linear return model, we derive the portfolio
weights in Eq. (1) that maximize the Sharpe ratio. Realized return, �̃�𝑖, for asset i depends on the
return on a benchmark portfolio, �̃�𝐺 , and a zero-mean perturbation that is independent over time,
independent over assets, and independent of the return on the benchmark portfolio,
�̃�𝑖 = 𝛼𝑖 + 𝛽𝑖 ∙ �̃�𝐺 + 𝜀�̃� i=1,2,…,N (A1)
The variance of the perturbation, 𝜎𝜀𝑖
2 , is unsystematic risk.
A single factor in Eq. (A1) is without loss of generality. Alternatively, if multiple factors
determine returns, we show in appendix B that an asset’s excess return is, nonetheless, a beta times
the excess return of a benchmark portfolio. Bali and Cakici (2010) find no difference in global
asset-pricing tests with single versus multiple factors. In domestic asset-pricing tests, Fama and
French (1992) find multiple return factors but a beta estimate over a multi-year estimation window
does not capture temporal risk changes. Thus, within the confines of estimation error,
supplementary factors figure prominently in testing because they are more current than a beta
estimate and act as updates for the unobserved true beta. For example, the likelihood that a high-
risk asset has high average past and future returns is greater than for a low risk asset and, thus, a
“momentum” factor (high average returns in the recent past) helps explain realized returns beyond
a dated single-factor beta estimate.
Let 𝐸(�̃�𝑖) and 𝜎𝑖2 be ex-ante return and return variance for asset i. Similar measures for a
portfolio are 𝐸(�̃�𝑝) and 𝜎𝑝2. The investor’s portfolio problem is,
𝑚𝑎𝑥
𝑥1, 𝑥2, … , 𝑥𝑁[
𝐸(�̃�𝑝)−𝑟𝑓
𝜎𝑝] subject to
1
1N
k
k
x
(A2)
where 𝑟𝑓 is the riskless rate and 𝑥𝑖 is the portfolio weight for asset i from among N risky assets.
Since the Sharpe ratio in Eq. (A2) is homogeneous of degree zero, we can ignore the constraint if
we subsequently normalize weights to sum to unity.
With returns from (A1), the portfolio Sharpe ratio is,
Y. Fu, G. Blazenko
19
∑ 𝑥𝑘(𝐸(�̃�𝑘)−𝑟𝑓)𝑁
𝑘=1
[(∑ 𝑥𝑘𝛽𝑘𝑁𝑘=1 )
2𝜎𝐺
2+∑ 𝑥𝑘2𝑁
𝑘=1 𝜎𝜀𝑘2 ]
1/2 (A3)
where 𝜎𝐺2 is the variance of the rate of return for the common factor. First order conditions are,
[𝐸(�̃�𝑖) − 𝑟𝑓]𝜎𝑝2 − [(∑ 𝑥𝑘𝛽𝑘
𝑁𝑘=1 )𝜎𝐺
2𝛽𝑖 + 𝑥𝑖𝜎𝜀𝑖
2 ] ∙ [∑ 𝑥𝑘(𝐸(�̃�𝑘) − 𝑟𝑓)𝑁𝑘=1 ]=0, i=1,2,…,N (A4)
which are the Elton, Gruber, Padberg (1976) equations that implicitly specify optimal portfolio
weights when a linear return model generates returns.
Define,
𝑧𝑖 = [∑ 𝑥𝑘(𝐸(�̃�𝑘)−𝑟𝑓)𝑁
𝑘=1
𝜎𝑝2 ] 𝑥𝑖, i=1,2,…,N (A5)
Substitute (A5) into Eq. (A4) and solve for 𝑧𝑖,
𝑧𝑖 =𝐸(�̃�𝑖)−𝑟𝑓
𝜎𝜀𝑖2 − [∑ 𝑧𝑘𝛽𝑘
𝑁𝑘=1 ] [
𝛽𝑖𝜎𝐺2
𝜎𝜀𝑖2 ] (A6)
Multiply Eq. (A6) by 𝛽𝑖, sum, and solve for ∑ 𝑧𝑘𝛽𝑘𝑁𝑘=1 ,
∑ 𝑧𝑘𝛽𝑘𝑁𝑘=1 =
∑ [𝐸(�̃�𝑘)−𝑟𝑓]𝛽𝑘/𝜎𝜀𝑘2𝑁
𝑘=1
1+∑ 𝛽𝑘2𝜎𝐺
2/𝜎𝜀𝑘2𝑁
𝑘=1
(A7)
Define 𝐴 = (∑ 𝛽𝑘2/𝜎𝜀𝑘
2𝑁𝑘=1 )𝜎𝐺
2 (A8)
Substitute Eqs. (A7) and (A8) into Eq. (A6),
𝑧𝑖 =𝐸(�̃�𝑖)−𝑟𝑓
𝜎𝜀𝑖2 − [
∑ [𝐸(�̃�𝑘)−𝑟𝑓]𝛽𝑘/𝜎𝜀𝑘2𝑁
𝑘=1
1+𝐴] [
𝛽𝑖𝜎𝐺2
𝜎𝜀𝑖2 ] (A9)
Since the excess return of asset i is 𝛽𝑖 times that of portfolio G,
𝐸(�̃�𝑖) − 𝑟𝑓 = 𝛽𝑖[𝐸(�̃�𝐺) − 𝑟𝑓] i=1,2,…,N (A10)
where GE R is the expected return on the benchmark portfolio. Substitute Eq. (A10) into (A9),
𝑧𝑖 =𝛽𝑖[𝐸(�̃�𝐺)−𝑟𝑓]
𝜎𝜀𝑖2 (1+𝐴)
(A11)
Y. Fu, G. Blazenko
20
Because the numerator of (A11) is the risk premium for security i from Eq. (A10),
𝑧𝑖 =𝐸(�̃�𝑖)−𝑟𝑓
𝜎𝜀𝑖2 (1+𝐴)
(A12)
Normalize to eliminate the constant of proportionality to produce the weights in Eq. (1).
7 Appendix B
In this appendix, we show that if multiple factors determine returns, then, nonetheless, excess
return for an asset is the asset-beta multiplied by excess return of a benchmark portfolio.
Consider two portfolios, 1 and 2. Two factors (A and B) determine returns (the extension to
more than two factors is transparent). First,
�̃�2 = 𝐸(�̃�2) + 𝑔𝐴 ∙ 𝑓𝐴 + 𝑔𝐵 ∙ 𝑓𝐵 + 𝜉+ (B1)
where, 𝑓𝐴 = �̃�𝐴 − 𝐸(�̃�𝐴) and 𝑓𝐵 = �̃�𝐵 − 𝐸(�̃�𝐵) are the unexpected parts of factor A and B,
respectively. The excess return of portfolio 2 is
𝐸(�̃�2) − 𝑟𝑓 = 𝑔𝐴 ∙ [𝐸(�̃�𝐴) − 𝑟𝑓] + 𝑔𝐵 ∙ [𝐸(�̃�𝐵) − 𝑟𝑓] (B2)
The return of portfolio 2 determines the return of portfolio 1 plus an error,
�̃�1 = 𝛼 + 𝛽 ∙ �̃�2 + 𝜀̃ (B3)
Substitute (B1) into (B3) and,
�̃�1 = 𝛼 + 𝛽 ∙ [𝐸(�̃�2) + 𝑔𝐴 ∙ 𝑓𝐴 + 𝑔𝐵 ∙ 𝑓𝐵 + 𝜉] + 𝜀̃
= 𝛼 + 𝛽 ∙ 𝐸(�̃�2) + 𝛽 ∙ 𝑔𝐴 ∙ 𝑓𝐴 + 𝛽 ∙ 𝑔𝐵 ∙ 𝑓𝐵 + 𝛽 ∙ 𝜉 + 𝜀̃
= [𝛼 + 𝛽 ∙ 𝐸(�̃�2)] + (𝛽 ∙ 𝑔𝐴) ∙ 𝑓𝐴 + (𝛽 ∙ 𝑔𝐵) ∙ 𝑓𝐵 + [𝛽 ∙ 𝜉 + 𝜀̃] (B4)
Take the expectation of (B3),
𝐸(�̃�1) = 𝛼 + 𝛽 ∙ 𝐸(�̃�2) (B5)
Replace the first term of (B4) with (B5),
�̃�1 = 𝐸(�̃�1) + (𝛽𝑔𝐴) ∙ 𝑓𝐴 + (𝛽𝑔𝐵) ∙ 𝑓𝐵 + [𝛽 ∙ 𝜉 + 𝜀̃] (B6)
Eq. (B6) shows that, like portfolio 2, factors A and B determine the return of portfolio 1 but with
factor sensitivities Ag and Bg . The excess return of portfolio 1 is,
Y. Fu, G. Blazenko
21
𝐸(�̃�1) − 𝑟𝑓 = (𝛽𝑔𝐴) ∙ [𝐸(�̃�𝐴) − 𝑟𝑓] + (𝛽𝑔𝐵) ∙ [𝐸(�̃�𝐵) − 𝑟𝑓]
=𝛽 ∙ {𝑔𝐴 ∙ [𝐸(�̃�𝐴) − 𝑟𝑓] + 𝑔𝐵 ∙ [𝐸(�̃�𝐵) − 𝑟𝑓]} = 𝛽 ∙ [𝐸(�̃�2) − 𝑟𝑓]
Thus, the excess return of portfolio 1 is 𝛽 times that of portfolio 2.
Y. Fu, G. Blazenko
22
Table 1 Realized industry portfolio returns
No. Industry
Annualized
Average Monthly
Realized Return
Estimated
Beta
Average Monthly
Return Standard-
Deviation
Average Weight
in Market
Portfolio
Average #
of Firms in
Portfolio
1 Agriculture 0.151 0.952 0.079 0.001 6 2 Food Products 0.145 0.582 0.024 0.022 47 3 Candy & Soda 0.150 0.709 0.044 0.011 7 4 Beer & Liquor 0.146 0.616 0.033 0.011 9 5 Tobacco Products 0.209 0.572 0.060 0.010 4 6 Recreation 0.134 1.097 0.053 0.004 19 7 Entertainment 0.177 1.479 0.098 0.007 35 8 Printing & Publishing 0.119 1.013 0.041 0.007 22 9 Consumer Goods 0.109 0.687 0.029 0.024 50
10 Apparel 0.168 1.200 0.062 0.005 36 11 Healthcare 0.156 1.016 0.078 0.004 40 12 Medical Equipment 0.137 0.976 0.041 0.011 75 13 Pharmaceutical Products 0.140 0.773 0.031 0.075 121 14 Chemicals 0.130 1.086 0.046 0.030 59 15 Rubber & Plastic Products 0.121 1.018 0.054 0.002 20 16 Textiles 0.144 1.258 0.078 0.002 20 17 Construction Materials 0.125 1.178 0.047 0.011 55 18 Construction 0.124 1.479 0.076 0.003 34 19 Primary Metal Products 0.117 1.428 0.083 0.012 48 20 Fabricated Metal Products 0.044 1.354 0.113 0.000 7 21 Machinery 0.139 1.226 0.048 0.039 100 22 Electrical Equipment 0.127 1.122 0.047 0.008 37 23 Automobiles & Trucks 0.123 1.179 0.063 0.022 46 24 Aircraft 0.166 1.091 0.054 0.012 15 25 Shipbuilding, Railroad
Eq.
0.197 1.182 0.096 0.001 6 26 Defense 0.169 0.811 0.062 0.002 6 27 Precious Metals 0.121 0.720 0.134 0.005 16 28 Mining (non-coal) 0.144 1.262 0.068 0.003 11 29 Coal 0.137 1.152 0.116 0.001 6 30 Petroleum & Nat. Gas 0.143 0.830 0.040 0.105 118 31 Utilities 0.129 0.483 0.022 0.056 130 32 Telecommunications 0.121 0.794 0.031 0.067 72 33 Personal Services 0.131 1.046 0.054 0.004 29 34 Business Services 0.119 1.138 0.040 0.017 126 35 Computer Hardware 0.143 1.383 0.093 0.031 71 36 Computer Software 0.118 1.137 0.058 0.060 188 37 Electronic Equipment 0.137 1.446 0.081 0.040 162 38 Measuring & Control Eq. 0.139 1.274 0.059 0.009 59 39 Business Supplies 0.113 0.947 0.039 0.020 44 40 Containers & Boxes 0.140 0.969 0.039 0.002 10 41 Transportation 0.143 1.087 0.042 0.020 80 42 Wholesale 0.122 0.993 0.033 0.011 85 43 Retail 0.139 0.989 0.042 0.055 152 44 Restaurants, Hotels,
Motels
0.136 0.910 0.045 0.008 45 45 Banking 0.135 1.083 0.048 0.080 284 46 Insurance 0.141 1.021 0.038 0.044 110 47 Real Estate 0.164 1.153 0.076 0.001 13 48 Trading 0.155 1.438 0.063 0.021 104 49 Plumbing, HVAC 0.151 1.029 0.073 0.005 18
Annualized return is twelve-times the value weighted average monthly return for firms in an industry from January 1976 to July 2014. The beta uses the value-weighted return of all firms in our sample as the “market” portfolio. Monthly return standard-deviation is for the value weighted
average return for firms in an industry from January 1976 to July 2014. Industry value weights are for all firms in an industry in our sample
measured monthly and averaged over the entire time-period.
Y. Fu, G. Blazenko
23
Table 2 Estimation-free ex-ante industry returns
No. Industry Average Annual
Forward ROE
Average
Annual
Forward Growth
Average
Estimation-
Free Ex-Ante Return
Annualized
Unsystematic
Risk
Average
Systematic/Un
-systematic Risk Weight
1 Agriculture 0.140 0.097 0.117 0.645 0.012 2 Food Products 0.150 0.090 0.120 0.132 0.037 3 Candy & Soda 0.205 0.109 0.138 0.309 0.015 4 Beer & Liquor 0.188 0.117 0.146 0.202 0.030 5 Tobacco Products 0.300 0.132 0.180 0.554 0.021 6 Recreation 0.072 0.057 0.075 0.321 0.008 7 Entertainment 0.087 0.065 0.079 0.358 0.007 8 Printing & Publishing 0.137 0.081 0.109 0.149 0.026 9 Consumer Goods 0.185 0.108 0.138 0.155 0.037
10 Apparel 0.153 0.120 0.141 0.249 0.021 11 Healthcare 0.141 0.117 0.128 0.408 0.014 12 Medical Equipment 0.152 0.125 0.138 0.177 0.030 13 Pharmaceutical Products 0.220 0.132 0.159 0.162 0.037 14 Chemicals 0.146 0.088 0.122 0.158 0.027 15 Rubber & Plastic Products 0.129 0.097 0.118 0.210 0.018 16 Textiles 0.110 0.087 0.109 0.355 0.010 17 Construction Materials 0.118 0.072 0.100 0.133 0.023 18 Construction 0.105 0.084 0.102 0.334 0.009 19 Primary Metal Products 0.090 0.067 0.093 0.345 0.007 20 Fabricated Metal Products 0.102 0.068 0.093 0.711 0.004 21 Machinery 0.147 0.091 0.119 0.108 0.035 22 Electrical Equipment 0.106 0.070 0.097 0.120 0.021 23 Automobiles & Trucks 0.117 0.084 0.122 0.295 0.020 24 Aircraft 0.160 0.112 0.138 0.274 0.018 25 Shipbuilding, Railroad
Eq.
0.127 0.106 0.119 0.487 0.012 26 Defense 0.233 0.149 0.181 0.360 0.021 27 Precious Metals 0.077 0.053 0.068 1.310 0.002 28 Mining (non-coal) 0.123 0.093 0.122 0.399 0.011 29 Coal 0.111 0.077 0.103 1.019 0.005 30 Petroleum & Nat. Gas 0.123 0.077 0.113 0.260 0.015 31 Utilities 0.091 0.037 0.095 0.146 0.019 32 Telecommunications 0.100 0.055 0.096 0.114 0.022 33 Personal Services 0.127 0.097 0.113 0.241 0.017 34 Business Services 0.135 0.092 0.113 0.072 0.051 35 Computer Hardware 0.151 0.135 0.142 0.367 0.015 36 Computer Software 0.184 0.131 0.152 0.228 0.030 37 Electronic Equipment 0.135 0.108 0.122 0.257 0.018 38 Measuring & Control Eq. 0.131 0.104 0.120 0.160 0.026 39 Business Supplies 0.123 0.072 0.103 0.130 0.025 40 Containers & Boxes 0.133 0.094 0.121 0.226 0.019 41 Transportation 0.090 0.062 0.086 0.130 0.016 42 Wholesale 0.118 0.084 0.106 0.095 0.037 43 Retail 0.152 0.112 0.132 0.174 0.030 44 Restaurants, Hotels,
Motels
0.162 0.123 0.137 0.219 0.023 45 Banking 0.134 0.085 0.124 0.188 0.034 46 Insurance 0.109 0.080 0.107 0.150 0.025 47 Real Estate 0.089 0.058 0.080 0.365 0.009 48 Trading 0.116 0.067 0.103 0.170 0.020 49 Plumbing, HVAC 0.130 0.096 0.113 0.403 0.012
We calculate ex-ante annualized industry returns with Eq. (3). ROE is the sum of forward/annum earnings (forward eps times outstanding shares) for firms in an industry divided by the sum of book equity over firms. Forward earnings is the consensus forecast of analysts for the next unreported
fiscal year-end. M/B is the total value of outstanding shares at the time of portfolio formation summed over firms divided by the sum of book-
equity. Forward dividend yield is trailing twelve month dividends per share times the number of outstanding shares summed over firms divided by the sum of market value over all firms in an industry at the time of portfolio formation. Book equity is from the most recently reported annual
financial statement prior to portfolio formation. We use US Treasury bills for the “riskless rate.” Ex-ante annualized growth is from Eq. (4). For
systematic-to-unsystematic risk weights of Eq. (1), unsystematic risk is residual variance in the market model regression of annualized monthly excess industry returns (value weighted) versus annualized monthly excess market returns for the thirty-six months prior to portfolio formation.
The “market” is all firms in our sample. “Annualized unsystematic risk” is residual variance over the entire time series. Otherwise, we average
monthly measures over the time-period January 1976 to July 2014.
Y. Fu, G. Blazenko
24
Table 3 Realized Sharpe ratios for portfolios formed by ranking individual common shares by
systematic-to-unsystematic risk (Panel A) and by ex-ante excess return (Panel B) using, in each
case, estimation-free ex-ante returns
N* Ranked
Firms in Portfolio
Median
Realized Sharpe Ratio
Average Monthly
Excess-return
Average Monthly
Return Standard Deviation
Median Sharpe Ratio Difference
with 1/N’th
portfolio
Wilcoxon Signed
Rank test for Sharpe ratio
difference between
N* and 1/N’th portfolio (p value)
Kolmogorov-Smirnov
Normality test
(p value)
Panel A: Portfolios Ranked by Estimation Free Ex-ante Excess Return/Unsystematic Risk
N*=3 0.238 0.00952 0.0539 -0.026 0.5727 0.06
N*=54 0.348 0.00954 0.0384 0.098 0.0000 0.01
N*=56 0.366 0.00956 0.0383 0.108 0.0000 0.01
N*=58 0.377 0.00968 0.0383 0.122 0.0000 0.01
N*=60 0.375 0.00981 0.0383 0.114 0.0000 0.01
N*=62 0.398 0.00993 0.0382 0.109 0.0000 0.01
N*=64 0.401 0.00982 0.0382 0.113 0.0000 0.01
N*=66 0.393 0.00982 0.0381 0.114 0.0000 0.01
N*=68 0.379 0.00989 0.0381 0.125 0.0000 0.01
N*=70 0.380 0.00992 0.0381 0.128 0.0000 0.01
N*=72 0.377 0.00998 0.0381 0.131 0.0000 0.01
N*=74 0.388 0.00989 0.0381 0.123 0.0000 0.01
Panel B: Portfolios Ranked by Estimation Free Ex-ante Excess Return
N*=28 0.214 0.01534 0.0765 0.045 0.0298 0.01
N*=137 0.261 0.01334 0.0676 0.052 0.0004 0.01
N*=139 0.263 0.01355 0.0675 0.053 0.0002 0.01
N*=141 0.263 0.01327 0.0675 0.052 0.0002 0.01
N*=143 0.265 0.01344 0.0675 0.054 0.0001 0.01
N*=145 0.269 0.01350 0.0674 0.051 0.0001 0.01
N*=147 0.272 0.01354 0.0674 0.051 0.0001 0.01
N*=149 0.261 0.01345 0.0674 0.051 0.0001 0.01
N*=151 0.266 0.01344 0.0673 0.046 0.0001 0.01
N*=153 0.265 0.01329 0.0673 0.047 0.0001 0.01
N*=155 0.263 0.01341 0.0672 0.043 0.0002 0.01
N*=157 0.263 0.01319 0.0672 0.043 0.0002 0.01
1/N’th
Common Share Portfolio
0.235 0.0083 0.0563
We rank common shares for portfolio inclusion in two ways: with the ratio of systematic-to-unsystematic risk in Eq. (1) and with ex-ante excess
returns in Eq. (2). In both cases, ex-ante return is from equation (3). Portfolio weights use normalized values for these two measures, respectively.
N* is the number of common shares in a portfolio selected with the highest ranked measure for all N common shares available at portfolio rebalancing (N*<N). The 1/N’th common share strategy has a 1/N weight in each common share. Portfolio formation is at the beginning of a month
and realized returns are for the month following rebalancing. The one-month realized Sharpe ratio (RSR) is the realized monthly return for a portfolio
less the riskless interest rate for a one-month holding period divided by the realized one-month portfolio return standard-deviation (total rather than residual) for the 36 months prior to portfolio rebalancing. We report the temporal median of realized Sharpe ratios over the 456 months from
January 1976 to July 2014. The Wilcoxon signed-rank test detects a Sharpe ratio difference between N* portfolios and 1/N’th common share
investing when the Kolmogorov-Smirnov test identifies the underlying population as non-normal. In comparing results in Tables 3, 4, and 5, the reader should recognize that the difference in median Sharpe ratios is not the same as the median Sharpe ratio difference. Statistical tests use the
monthly realized Sharpe ratios difference as the primary unit of observation.
Y. Fu, G. Blazenko
25
Table 4 Realized Sharpe ratios for portfolios formed by ranking individual common shares by
systematic-to-unsystematic risk (Panel A) and ex-ante excess return (Panel B) using, in each
case, estimated FF 5-Factor ex-ante returns
N* Ranked
Firms in Portfolio
Median
Realized Sharpe Ratio
Average Monthly
Excess-return
Average Monthly
Return Standard Deviation
Median Sharpe Ratio Difference
with 1/N’th
portfolio
Wilcoxon Signed
Rank test for Sharpe ratio
difference between
N* and 1/N’th portfolio (p value)
Kolmogorov-Smirnov
Normality test
(p value)
Panel A: Portfolios Ranked by Estimation Free Ex-ante Excess Return/Unsystematic Risk
N*=3 0.157 0.00782 0.0567 0.034 0.8728 0.01
N*=46 0.285 0.01069 0.0518 0.022 0.0558 0.01
N*=48 0.280 0.01061 0.0518 0.021 0.0599 0.01
N*=50 0.266 0.01053 0.0518 0.006 0.0797 0.01
N*=52 0.264 0.01051 0.0519 0.015 0.0793 0.01
N*=54 0.274 0.01038 0.0519 0.018 0.0781 0.01
N*=56 0.294 0.01036 0.0519 0.021 0.0800 0.01
N*=58 0.289 0.01036 0.0519 0.022 0.0769 0.01
N*=60 0.280 0.01038 0.0519 0.017 0.0780 0.01
N*=62 0.277 0.01029 0.0519 0.010 0.0878 0.01
N*=64 0.270 0.01027 0.0519 0.011 0.1045 0.01
N*=66 0.270 0.01023 0.0519 0.011 0.1091 0.01
Panel B: Portfolios Ranked by Estimation Free Ex-ante Excess Return
N*=28 0.082 0.01132 0.1570 -0.083 0.0034 0.01
N*=137 0.123 0.01135 0.1126 -0.067 0.0011 0.01
N*=139 0.121 0.01118 0.1121 -0.068 0.0009 0.01
N*=141 0.120 0.01118 0.1119 -0.076 0.0010 0.01
N*=143 0.128 0.01116 0.1116 -0.074 0.0009 0.01
N*=145 0.125 0.01111 0.1113 -0.063 0.0009 0.01
N*=147 0.131 0.01116 0.1111 -0.062 0.0010 0.01
N*=149 0.124 0.01115 0.1108 -0.063 0.0010 0.01
N*=151 0.124 0.01113 0.1105 -0.064 0.0010 0.01
N*=153 0.123 0.01112 0.1103 -0.061 0.0011 0.01
N*=155 0.122 0.01114 0.1100 -0.062 0.0012 0.01
N*=157 0.123 0.01115 0.1097 -0.064 0.0011 0.01
1/N’th
Common Share
Portfolio
0.235 0.0083 0.0563
We rank common shares for portfolio inclusion in two ways: with the ratio of systematic-to-unsystematic risk in Eq. (1) and with ex-ante excess
returns in Eq. (2). In both cases, we estimate ex-ante return with the FF 5-factor model. Portfolio weights use normalized values for these two
measures, respectively. N* is the number of common shares in a portfolio selected with the highest ranked measure for all N common shares available at portfolio rebalancing (N*<N). The 1/N’th common share strategy has a 1/N weight in each common share. Portfolio formation is at the
beginning of a month and realized returns are for the following month. The one-month realized Sharpe ratio (RSR) is the realized monthly return
for a portfolio less the riskless interest rate for a one-month holding period divided by the realized one-month portfolio return standard-deviation (total rather than residual) for the 36 months prior to portfolio rebalancing. We report the temporal median of realized Sharpe ratios over the 456
months from January 1976 to July 2014. The Wilcoxon signed-rank test detects a Sharpe ratio difference between N* portfolios and 1/N’th common
share investing when the Kolmogorov-Smirnov test identifies the underlying population as non-normal. In comparing results in Tables 3, 4, and 5, the reader should recognize that the difference in median Sharpe ratios is not the same as the median of Sharpe ratio differences. Statistical tests
use the monthly realized Sharpe ratios difference as the primary unit of observation.
Y. Fu, G. Blazenko
26
Table 5 Median difference in realized monthly Sharpe ratios for systematic-to-unsystematic risk portfolios using
estimation free ex-ante returns versus FF 5-factor estimated ex-ante returns
N* Ranked Firms in Portfolio Using
Estimation-Free
Ex-ante Returns for Systematic-to-
Unsystematic
Risk Weights
Median Difference in
Monthly
Sharpe Ratios
Wilcoxon Signed Rank test for Sharpe Ratio Difference
between systemtatic to
unsystematic risk portfolios with estimation-free versus
estimated ex-ante returns
(p value)
Kolmogorov-Smirnov
Normality test
(p-value)
N* Ranked Firms in Portfolio
Using Estimated
Ex-ante Returns for Systematic-
to-Unsystematic
Risk Weights
N*=54 0.072 0.0280 0.01 N*=46
N*=56 0.078 0.0190 0.01 N*=48
N*=58 0.072 0.0121 0.01 N*=50
N*=60 0.074 0.0098 0.01 N*=52
N*=62 0.074 0.0072 0.01 N*=54
N*=64 0.070 0.0056 0.01 N*=56
N*=66 0.0697 0.0049 0.01 N*=58
N*=68 0.069 0.0042 0.01 N*=60
N*=70 0.094 0.0029 0.01 N*=62
N*=72 0.097 0.0020 0.01 N*=64
N*=74 0.087 0.0017 0.01 N*=66
We rank common shares for portfolio inclusion with the ratio of systematic-to-unsystematic risk in Eq. (1) and with estimation-free ex-ante excess
returns in Eq. (3) and FF 5-factor estimated ex-ante returns in Eq. (6). Portfolio weights use normalized values for systematic-to-unsystematic risk.
N* is the number of common shares in a portfolio selected with the highest ranked measure for all N common shares available at portfolio rebalancing (N*<N). Portfolio formation is at the beginning of a month and realized returns are for the month following rebalancing. The one-
month realized Sharpe ratio (RSR) is the realized monthly return for a portfolio less the riskless interest rate for a one-month holding period divided
by the realized one-month portfolio return standard-deviation (total rather than residual) for the 36 months prior to portfolio rebalancing. We report the temporal median of realized Sharpe ratios over the 456 months from January 1976 to July 2014. The Wilcoxon signed-rank test detects a Sharpe
ratio difference between systematic-to-unsystematic risk portfolios with estimation-free versus estimated ex-ante returns when the Kolmogorov-
Smirnov test identifies the underlying population as non-normal. In studying the results in Tables 3, 4, and 5, the reader should recognize that the difference in median Sharpe ratios is not the same as the median of Sharpe ratio differences. Statistical tests use the monthly realized Sharpe ratios
difference as the primary unit of observation.
Y. Fu, G. Blazenko
27
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Yufen Fu is Assistant Professor of Finance, Department of Finance, Tunghai University, Taichung,
Taiwan. Her interests are in the areas of financial markets, international finance, and asset-pricing,
She has published numerous articles in the academic finance literature.
George Blazenko is Professor of Finance at the Beedie School of Business, Simon Fraser
University, Vancouver, BC, Canada. His research interests are in the areas of corporate finance,
real-options, real-options for innovative firms, real-estate, the economics of insurance, investments,
financial markets, asset-pricing, portfolio theory, and international finance. Among other outlets,
he has published his research in the Journal of Finance, the American Economic Review,
Management Science, Managerial Finance, and Financial Management.