Download Implied Expected Returns and the Choice of a Mean–Variance

M
A
T
R
N
Y
FO
Implied Expected Returns
and the Choice of a
Mean–Variance Efficient
Portfolio Proxy
A
TH
K RIS BOUDT
IT
IS
IL
LE
G
A
L
TO
R
EP
R
O
D
U
C
E
is associate professor of
finance and econometrics at Vrije Universiteit
Brussel and VU University
Amsterdam and research
partner at Finvex Group
in Brussels.
[email protected]
68
can be more accurately predicted than mean
returns.2
Second, if w is a mean–variance efficient portfolio with respect to a universe of
assets with known expected returns vector μ
and covariance matrix Σ, then there exists a
linear relation between the assets’ expected
return μ and their covariance with that efficient portfolio, i.e., Σw . Hence, if a mean–
variance efficient weight vector is known
and a covariance matrix has been accurately
estimated, the above-mentioned linear
relation can be exploited to create implied
expected returns which may have a higher
accuracy than the time series model based
return forecasts. This idea of reverse engineering for expected returns goes back to
Sharpe [1974], Fisher [1975] and Best and
Grauer [1985]. Its application in practice
was boosted by the research of Black and
Litterman [1991, 1992].
An elementary feature in the implementation of reverse engineering of expected
returns is the choice of proxy for the mean–
variance efficient portfolio. Traditionally the
standard choice has been to use the market
capitalization weighted portfolio. The proponents of this approach tend to emphasize
the result that under the capital asset pricing
model (CAPM) the market capitalization
portfolio is mean–variance efficient. However, for almost all portfolio managers, the
R
TI
growing amount of capital is
attracted by smart beta equity
solutions, such as the equalweighted, equal-risk-contribution, fundamental indexation, and maximum
diversification portfolio.1 The determination of portfolio weight does not require
any return forecast, but from the portfolio
weight and the covariance matrix, an implied
expected return can be computed. We
investigate the usefulness of this alternative
implied expected return, in comparison with
the standard approach of deriving implied
expected returns from the market capitalization weighted portfolio.
The implied expected return methodology is founded on two fundamental
insights. First, the availability of higher frequency data allows estimates of covariances
at a higher accuracy than expected returns. In
fact, as shown by Merton [1980], using higher
sampling frequencies improves significantly
the accuracy of the covariance estimator, but
not the estimation of the mean parameter.
As it is commonly done in practice, we use
weekly observations to estimate the covariance matrix over a monthly horizon. Another
reason is that the mean and covariance matrix
are time-varying parameters. Since the timevariation in the covariance matrix is much
more persistent than the time-variation in
the expected return, covariance matrices
IS
is assistant professor of
finance at Laval University
in Québéc City,
QC, Canada.
[email protected]
C
A
DAVID A RDIA
LE
IN
A
DAVID ARDIA AND KRIS BOUDT
I MPLIED E XPECTED R ETURNS AND THE CHOICE OF A M EAN –VARIANCE EFFICIENT PORTFOLIO P ROXY
SUMMER 2015
Copyright © 2015
JPM-ARDIA.indd 68
7/21/15 2:13:08 PM
investable universe does not correspond to all tradable
assets and hence, the CAPM conditions are by construction not verified. We therefore advocate that the
market capitalization weighted portfolio is only one possible proxy for a mean–variance efficient portfolio and
that other proxies may lead to more accurate expected
returns.
We test this hypothesis empirically for monthly
returns of the S&P 100 stocks over the 1987–2012
period. Importantly, the economic benefits of the choice
of proxy for computing implied expected returns are
examined not only in terms of forecast precision (as
measured by the hit ratio and root-mean-squared forecast error), but also in terms of stability, both crosssectionally (in order to avoid extreme investment bets,
as described for instance in Michaud [1989]) and from
a time series perspective (to reduce portfolio turnover).
In addition, we compare those implied expected returns
with the return forecasts obtained from various time
series models.
The analysis is done for a broad set of alternative
mean–variance portfolio proxies, namely the smoothed
market capitalization portfolio (Chen et al. [2007]),
the fundamentally weighted portfolio (Arnott et al.
[2005]), the equally weighted portfolio (DeMiguel
et al. [2009]), the inverse volatility weighted portfolio (Leote De Carvalho et al. [2012]), the equalrisk-contribution portfolio (Maillard et al. [2010]),
the maximum diversification portfolio (Choueifaty
and Coignard [2008]) and the risk-eff icient portfolio (Amenc et al. [2011]). We refer the reader to Lee
[2011] for a recent overview on smart beta portfolios
and Leote De Carvalho et al. [2012] and Clarke et al.
[2013] for empirical evidence on the risk factor exposures of those strategies.
We conclude that, for all proxies considered, the
implied expected returns outperform the time series
model based forecasts in terms of forecast precision and
stability. We find that the choice of a mean–variance
portfolio proxy matters and that the largest gains are
in terms of stability of the return forecasts. The use of
a maximum diversification or equal-risk-contribution
portfolio as proxy reduces significantly the cross-section
and time series dispersion in the implied expected returns
and leads to a small improvement in forecast precision,
compared with using a market capitalization, fundamental value or equal weighting scheme. This result
is shown to be robust with respect to the choice of the
SUMMER 2015
JPM-ARDIA.indd 69
level of risk aversion as well as the length of the estimation window.
REVERSE ENGINEERING EXPECTED
RETURNS FROM MEAN–VARIANCE
EFFICIENT PORTFOLIOS
A general description of the implied expected
return methodology starts by specifying the investment
universe, the first and second moments of the asset
returns, and the portfolio weights that will be assumed
to be mean–variance efficient. We consider a market
with N risky securities and denote a generic portfolio in
this market by the (N × 1) vector w. The expected arithmetic returns at the desired holding horizon are denoted
by the (
1) vector μ, and the corresponding (
)
covariance matrix of arithmetic returns is denoted by
Σ. We drop the time dimension for simplicity but consider the conditional moments as time-dependent. Let
ι be the (
1) vector
1) vector of ones and 0 the (
of zeros.
A mean–variance efficient portfolio is defined as a
portfolio that for some value of the risk-aversion parameter γ , 0 < γ < ∞ , maximizes the mean–variance utility
function under a full investment constraint:
w ∗ ≡ argmax
w ∈CFFI
{
}
(1)
where C FI ≡ {w
{w R N | w ′ 1} is the set of portfolio
weights satisfying the full-investment constraint.
While this problem is easy to solve explicitly, it
still remains that accurate estimates of the expected
return vector μ are difficult to obtain. At the same
time, investors often have a good proxy available for a
mean–variance efficient portfolio. A common approach
in practice is therefore to turn the problem around and
instead of optimizing the portfolio weights with a set of
expected returns and covariance matrix, extract from
a given portfolio structure the set of implied expected
returns for which the portfolio is mean–variance
efficient.
To derive these implied expected returns formally,
we need the first order conditions of mean–variance
efficiency. The Lagrangian corresponding to (1) is
L( w l )
w ′μ
γ
w ′ w l( w ′
2
1)
THE JOURNAL OF PORTFOLIO M ANAGEMENT
69
7/21/15 2:13:09 PM
with l ∈R . The corresponding first order conditions
are
μ − γΣw lι
0
w ′ι 1
(2)
(3)
From (2)–(3), it follows that the generic solution to
the mean–variance optimization problem in (1) is
w=
1 −1
Σ (μ l )
γ
(4)
It also follows from the first order condition in (2)
that there exists a linear relationship between μ and Σw:
μ = lι + γ Σw
(5)
Note that as γ is finite, (5) excludes the minimum
variance portfolio.
The linear relationship in (5) is well known, and for
given values of l, γ, Σ and w, it can be used to compute
the implied expected returns. The intercept parameter
l is a consequence of imposing (3), i.e., the budget constraint, in the calculation of implied expected returns,
as recommended by Herold [2005]. He shows that this
is necessary to obtain meaningful implied expected
returns. As in the original publication of Black and
Litterman [1991], we set l to the risk-free rate and γ =
2.4. In the sensitivity study, we also present results for
γ = 1 and γ = 5.
OVERVIEW OF PROXIES FOR MEAN–
VARIANCE EFFICIENT PORTFOLIOS
Our main analysis pertains to the choice of proxy
for the mean–variance efficient portfolio weights w. The
most popular one is the market capitalization weighted
portfolio, which we denote by w mkt . It has the mean–
variance efficiency property under the CAPM assumptions, but in most practical cases, these assumptions are
unrealistic (Goltz and Le Sourd [2011]). This claim is supported by an increasing body of literature that criticizes
the mean–variance efficiency of the market capitalization
weighted portfolio and proposes alternatives that (under
different assumptions) are mean–variance efficient.
The first strand of research that we consider is the
fundamental indexing literature (Arnott et al. [2005],
70
JPM-ARDIA.indd 70
Treynor [2005]). The main motivation for this approach
is that in noisy markets, overpriced stocks receive a
higher weight in the market capitalization weighted
portfolio than in the mean–variance efficient one, and
vice versa for underpriced stocks. If pricing errors are
corrected, this leads to a drag on returns for market-capitalization-based portfolios. Perold [2007] and Graham
[2012] emphasize, however, that this result is sensitive
to the assumption that stock prices will systematically
revert to fair value, which is not directly testable since
the fair value is not observable.
Chen et al. [2007] argue that by smoothing the
market capitalization weights, pricing errors are reduced
and hence a more efficient estimate of the fundamental
market capitalization weights are obtained. We take the
same setup as in Chen et al. [2007] and consider the
median capitalization weights over a five-week period.
We denote this portfolio by w smkt .
An alternative is to proxy the firm’s fair value using
fundamental metrics of company size such as book value
of common equity, dividends, net operating cash f low
and sales. This leads to the fundamentally weighted portfolios popularized by Arnott et al. [2005]. We follow
their framework and compute the fundamental portfolio
weights as the average value of the normalized version
of the fundamental size proxies, that is,
w fund ≡
4
1
4
⎛
∑ ⎜⎝ ∑
k =1
max{xk ,1, 0}
N
j =1
max{{xk j , 0}}
, …,
⎞′
∑ j 1 max{xk j, 0} ⎟⎠
max{xk ,N , 0}
N
with x1,i the most recently observed end of fiscal year
value of the firm i ’s book value of common equity and
x 2,i , x 2,i , x 4,i five-year trailing averages of the yearly
value of dividends, net operating cash f low, and sales,
respectively. In our application, the yearly fundamental
data are retrieved from the Compustat database and are
lagged by one quarter to avoid any possible look-ahead
bias. Net operating cash f low is taken as the difference
between the operating income before depreciation and
total accruals (Kothari et al. [2005]).
A second strand of research relates to the general
objective of achieving diversified portfolios, whereby the
focus of diversification implicitly leads to a portfolio that
is mean–variance efficient under some assumptions. The
most popular mean–variance efficient portfolio proxies
that achieve efficiency by focusing on diversification
are the following.
I MPLIED E XPECTED R ETURNS AND THE CHOICE OF A M EAN –VARIANCE EFFICIENT PORTFOLIO P ROXY
SUMMER 2015
7/21/15 2:13:10 PM
Equally weighted portfolio. The equally
weighted portfolio, which we denote by w ew , spreads
its capital investment equally over all N assets, i.e.,
w ew (1
(1
)ι. DeMiguel et al. [2009] show that,
for various popular datasets, the 1/N allocation rule
outperforms several optimized portfolios. Plyakha
et al. [2014] benchmark the equal weighting approach
against value and price weighted strategies. They
find an outperformance in approximately 65% of the
investigated periods. Part of the outperformance is
explained by higher exposure to the market, size, value
and momentum factors.
From (4) it follows that this portfolio is mean–
variance efficient when the expected excess returns of
the stocks, i.e., (μ − lι), are proportional to the sum total
of their covariances, i.e., Σι.
Inverse volatility weighted portfolio. The
inverse volatility weighted portfolio, called the equalrisk-budget portfolio by Leote De Carvalho et al. [2012],
allocates to the N stocks with volatility σ 1 ,…,σ N the
weight
⎛
⎞′
1/σ
w iv ≡ ⎜ 1N/σ1 ,…, N N ⎟
∑ j 11/σ j ⎠
⎝ ∑ j =111/σ j
As noted by Leoto De Carvalho et al., the inverse
volatility weighted portfolio is mean–variance efficient
with the highest Sharpe ratio if the Sharpe ratio for
each stock is the same and all pairwise correlations are
equal.
Equal-risk-contribution portfolio. The equalrisk-contribution portfolio is the portfolio for which
all assets contribute equally to the overall portfolio
volatility. Or equivalently, it is the portfolio for which
the percentage volatility risk contribution of all N
assets equals 1/N, where percentage volatility risk contribution of the ith asset is given by
%RC i ≡
w i [ Σw ]i
w ′Σ
′ w
We denote this portfolio by w erc. Numerically,
it is computed by solving the following optimization
problem 3 :
⎧N
⎫
w erc ≡ argmin ⎨∑ (%RC i − N1 )2 ⎬
w ∈CFFI ⎩ i =1
⎭
SUMMER 2015
JPM-ARDIA.indd 71
with CFI≡{ w N w 1}, the set of weights guaranteeing that
the portfolio is fully invested.
Maillard et al. [2010] and Roncalli [2013] derive
the theoretical properties of the equal-risk-contribution
portfolio and show that its volatility is located between
those of the minimum variance and the equally
weighted portfolio. Bruder and Roncalli [2012] show
that the equal-risk-contribution portfolio is mean–
variance efficient when all assets contribute equally to
the portfolio excess return w ′μ − l .
Maximum diversif ication portfolio. Let
σ ≡ diag( Σ ) be the (
1) vector of standard deviations
of arithmetic returns of the N assets in the universe. An
important property of the portfolio standard deviation is
its sub-additivity, meaning that, because of diversification
effects, the portfolio’s standard deviation is always less or
equal to the weighted average volatility:
w ′′Σw ≤ w ′σ
or equivalently that the ratio between the weighted
average volatility and the portfolio volatility exceeds
one:
DR( w ) ≡
w ′σ
≥1
w ′Σ
′ w
(6)
Choueifaty and Coignard [2008] call (6) the
portfolio’s diversification ratio and define the maximum
diversification portfolio as the portfolio that has the
highest diversification ratio. From (6) it is clear that
the maximum diversification portfolio coincides with
the maximum Sharpe ratio portfolio in the special case
where the stocks’ expected returns (in excess of the riskfree rate) are proportional to their volatility. This special
case is also discussed by Martellini [2008]. We denote
this portfolio by w md .
Risk-eff icient portfolio. Amenc et al. [2011]
recommend to construct a maximum Sharpe portfolio
under the assumption that the stock’s expected return
is a deterministic function of its semi-deviation and
the cross-sectional distribution of semi-deviations, and
where portfolio weights are constrained to be between
1/(2N ) and 1/N. More precisely, they sort stocks by
their semi-deviation, form decile portfolios, and then
compute the median semi-deviation of stocks in each
decile portfolio, which we denote by ξ j ( j = 1 …,10).
The risk-efficient portfolio is then given by
THE JOURNAL OF PORTFOLIO M ANAGEMENT
71
7/21/15 2:13:11 PM
w ref ≡ argmax
w ∈C
FI∗
{
}
where J is a (
10) matrix of zeros whose
(i,j )th element is 1 if the semi-deviation of
stock i belongs to decile j, ξ ≡ ( ξ1 , ξ10 )′ and
CFFI∗ ≡ {w
{w R N | w ′ 1,(1 / (2N ))ι ≤ w ≤ (2
( / N ) } is the
full-investment constraint with bound restrictions on
the portfolio weights.
Results
The differences in allocation between the alternative mean–variance efficient portfolio proxies and the
market capitalization weighted portfolio can be interpreted as active allocation bets, which lead to bets in
the implied returns. The translation of these allocation
bets into implied returns bets interacts with the correlation and variance properties of the stocks in the mean–
variance efficient portfolio. This effect can be seen in
more detail from the definition of the implied return for
stock i that follows from the general expression in (5):
⎛
⎜
N
⎞
⎟
⎝
j =1
⎠
μ i = λ + γ ( Σw )i = λ + γ cov ⎜⎜ ri , ∑ w j r j ⎟⎟
=λ+γΣ
⎛
⎜
1/2 ⎜
⎜
ii ⎜
⎜
⎜⎝
N
Σ w i + ∑ ρij Σ w
1/2
ii
j=1
j =i
/
1/2
jj
⎞
⎟
⎟
⎟
j⎟
⎟
⎟⎠
(7)
(8)
with ρij the correlation between the returns of stock i and
j. We thus see that the implied expected return for stock
i is proportional to the covariance between the stock’s
return and the return on the mean–variance efficient
portfolio. Ceteris paribus, this covariance will tend to be
higher for stocks that have: (i) a relatively larger weight
in the efficient portfolio proxy, (ii) a high volatility, and
(iii) a high correlation with the stocks in the efficient
portfolio. The propagation of the weights bets to implied
returns bets thus interacts with the variance and correlation properties of the individual stock returns.
The aim of this section is to provide a better
understanding of the sensitivity of the stability and forecast
precision of the monthly implied expected returns to the
choice of a mean–variance efficient portfolio proxy, and
compare those with common time series model based
72
JPM-ARDIA.indd 72
forecasts of expected returns. The empirical analysis
provides a better insight in the conclusion of Lee [2011]
that, if risk-based allocation approaches outperform the
market portfolio, it must be that the implied expected
returns are a better set of return forecasts than the market’s implied expected returns. The analysis is done for a
sample of large U.S. stocks over the period January 1984
to December 2012. The expected return forecast horizon
that we consider is one month and the prediction is made
at the end of the month. The baseline results assume a
rolling estimation window of three years, such that the
out-of-sample evaluation period ranges from January
1987 to December 2012 (312 monthly observations).
Data and Estimation Methods
For each month between January 1984 and
December 2012, the universe is restricted to the 100
end-of-month largest S&P 500 equities for which there
are no missing observations in the estimation sample
used to forecast the covariance matrix and expected
returns. For convenience, we call this henceforth the
S&P 100 universe. Results are presented for the overall
universe and for each of the ten GICS sectors separately.
The adjusted price data, market capitalization, and firm
fundamentals have been retrieved from COMPUSTAT.
The accounting data used to compute the fundamental
weights have a yearly frequency. We follow Arnott et al.
[2005] by using five-year trailing averages of the yearly
value of dividends, net operating cash f low, and sales
to avoid excessive volatility in the index weights due
to year-to-year variability in the accounting data. We
obtained the Fama and French [1993] or Carhart [1997]
factors computed for the U.S. market from Kenneth
French’s website.4 The risk-free rate is the three-month
U.S. Treasury bill. All figures are in U.S. dollars.
In total, we are comparing twelve return-forecast
methods, eight of which are obtained using the implied
expected return methodology. The others are classified as
time series model based forecasts because they explicitly
model the dynamics in the stock returns. We consider the
constant mean model, the autoregressive model of order
one (AR(1)), as well as the Fama and French [1993] and
Carhart [1997] three- and four-factor models, whereby
the factors also follow an AR(1) process. All models are
estimated on a rolling estimation sample of three years
(156 weeks), which is common practice in the financial
I MPLIED E XPECTED R ETURNS AND THE CHOICE OF A M EAN –VARIANCE EFFICIENT PORTFOLIO P ROXY
SUMMER 2015
7/21/15 6:43:07 PM
industry. As shown by Merton [1980], using higher sampling frequencies improves significantly the accuracy of
the covariance estimator, but not the estimation of the
mean parameter. As it is commonly done in practice,
we use monthly returns to estimate the constant mean,
AR(1), and factors models by ordinary least squares. For
the covariance matrix, we use weekly log-returns and
the exponentially weighted moving average (EWMA)
estimator with a decay parameter of 0.94 as advocated
by RiskMetrics Group [1996]. The covariance matrix
is initialized at the Ledoit and Wolf [2003] shrinkage
estimate of the unconditional covariance matrix over
the estimation window. The covariance matrix is then
projected into a monthly arithmetic returns covariance
matrix estimator as in Meucci [2004]. In the sensitivity
analysis, we also consider rolling samples of two and
four years.
Performance and Sector Allocation
of Mean–Variance Efficient Portfolio Proxies
tribution portfolio and the lowest for the fundamental
value portfolio (0.35). The performance of the market
capitalization weighted portfolio is in between those two
extremes, with a volatility of 15.6%, a VaR95 of −7%,
and a Sharpe ratio of 0.41. We thus find that the allocation
style has only a minor effect on the out-of-sample average
return but matters for the portfolio risk: risk-optimized
portfolios are characterized by a significantly lower risk
and a similar return as the market capitalization weighted
portfolio.
The observed difference in risk-adjusted performance of the various mean–variance efficient portfolio
proxies can be partly attributed to underlying sector bets,
which become visible when aggregating the portfolio
weights at the sector level. Panel B of Exhibit 1 reports
the average end-of-month sector allocation for each
of the mean–variance efficient portfolio proxies. Note
that the sector bets are intuitive and can be explained by
the cardinality per sector and the average risk (correlation
and volatility) of the stocks per sector, as reported in Columns 2–4 of Exhibit 2. We see for instance in Exhibit 2
that there are proportionally fewer stocks in the IT sector,
explaining why the equally weighted portfolio has a relatively lower weight for that sector compared with the
The intrinsic motivation to compute implied
expected returns from alternative proxies for a mean–
variance efficient portfolio than the market capitalization
weighted portfolio is that other portfolios
show a better out-of-sample performance
in term of risk-adjusted returns. Exhibit 1
verifies that this prediction, observed by E X H I B I T 1
various authors on different datasets, also Out-of-Sample Performance of the Various Mean–Variance Efficient
holds for the sample under investigation. Portfolio Proxies and Their Average End-of-Month Sector Allocation
The mean–variance efficient portfolio
proxies that we consider are rebalanced at
the end of the month, when the return
forecast is made.5 In Panel A of Exhibit 1,
we see that the annualized average return of
all proxies is around 10% for all portfolios.
The portfolio risk is thus the distinctive
factor. The out-of-sample annualized portfolio volatility and monthly value-at-risk
(at the 95% risk level, denoted by VaR95)
are the lowest for the risk-based portfolios wiv, werc, wmd, and wref (around 14%
and −6%, respectively) and the highest for
the fundamental value portfolio (around
16% and −8%, respectively). These differences in volatility are statistically signifi- Notes: Mean: annualized mean (in percent). Vol: annualized volatility (in percent). Sharpe:
annualized Sharpe ratio. VaR95: empirical monthly value-at-risk at the 95% risk level (in
cant at the 1% level.6 The Sharpe ratio is percent). The evaluation period ranges from January 1987 to December 2012 for a total of 312
the highest (0.50) for the equal-risk-con- monthly observations.
SUMMER 2015
JPM-ARDIA.indd 73
THE JOURNAL OF PORTFOLIO M ANAGEMENT
73
7/21/15 2:13:14 PM
EXHIBIT 2
Average Value of Stock Characteristics per Sector
Notes: N: (rounded) average number of stocks per sector. Vol: average annualized volatility (in percent). ρ: average correlation (in percent). The last eight
columns report the average value per sector of the stock returns’ covariance with the various mean–variance efficient portfolio proxies (in permil). Row All
refers to the overall S&P 100 universe. All numbers reported are averages of stock characteristics per sector and then averaged over the 312 months in the
out-of-sample evaluation period.
market capitalization weighted portfolio. The largest
sector bets compared with the market capitalization
weighted portfolio are made by the most diversified portw md ), which is under-invested especially in Finanfolio (w
cial and Industrial stocks (5.7% and 6.6% compared with
13.4% and 9.8%) and over-invested in Consumer Goods,
Energy, and Healthcare stocks. The under-investment in
Financial and Industrial stocks is for our sample due to
the relatively lower diversification potential of Financial
and Industrial stocks (their correlation with other stocks
is on average 35% compared with 29% for the Healthcare
sector for instance). The other risk-based portfolios
(wiv and wref ) make similar but less extreme sector bets.
They under-invest in the relatively higher volatility sectors (such as Communications, Financials, and IT) and
over-invest in the low-risk sectors, such as Consumer
Staples, Healthcare, and Utilities.
Descriptive Analysis of Expected Return
Forecasts
Let us now study how the differences in the mean–
variance efficient portfolio allocations translate into differences in implied expected returns. Exhibit 3 reports
the average annualized implied expected return (in percent) per sector for each benchmark portfolio (Panel A),
and compares them with the average return prediction
under the constant mean, AR(1), Fama–French threefactor, and Carhart four-factor models (Panel B). We see
74
JPM-ARDIA.indd 74
that the implied expected return predictions are, overall,
more pessimistic than the time series model predictions
in the sense of predicting a lower return.
The implied expected return and time series model
forecasts also disagree in terms of ranking sectors by
expected returns. The time series model based return
forecasts predict the highest return performance for
the IT sector, while the implied expected return based
forecasts predict the highest return for the Financial
sector.
A further interesting result in Exhibit 3 is that
the most diversified portfolio (w
w md ) implies the least
pronounced bets of all mean–variance efficiency portfolio proxies. This can be attributed to the fact that the
implied expected return for stock i is proportional to
the covariance between the returns of that stock with
the returns of the mean–variance efficient portfolio
proxy; see (7). Since the maximum diversif ication
portfolio explicitly seeks portfolio diversification, it
comes as no surprise that the stocks have a relatively
lower covariance with this portfolio than with the other
mean–variance efficient portfolios. This is further confirmed by Columns 5–12 of Exhibit 2, which report
the average covariance of the stock returns with the
various mean–variance efficient portfolio proxies; we
see that the lowest covariance is systematically observed
for the maximum diversification portfolio. The most
pronounced bets in terms of implied expected returns
are observed for the market capitalization weighted
I MPLIED E XPECTED R ETURNS AND THE CHOICE OF A M EAN –VARIANCE EFFICIENT PORTFOLIO P ROXY
SUMMER 2015
7/21/15 2:13:15 PM
EXHIBIT 3
Averages of Annualized Monthly Expected Return Predictions (in percent) per Sector for Implied Expected
Returns and Time Series Model Based Approaches
Notes: ier-: implied expected returns forecasts. Constant: forecasts under constant mean model. AR(1): autoregressive model forecasts. Fama–French and
Carhart: three- and four-factor model forecasts. Each entry is the (equally weighted) mean end-of-month stock return forecast per sector and then averaged
over all months in the sample. For convenience, results are reported in percentages and annualized. Column All refers to the overall S&P 100 universe.
portfolio, the fundamental weighting portfolio, and the
equally–weighted portfolio. Since implied expected
returns are proportional to the stock’s covariance with
the efficient portfolio proxy, we also see that implied
expected returns tend to be higher for sectors with
stocks that have: (i) a large weight in the eff icient
portfolio, (ii) a high volatility and (iii) a high correlation with other stocks. This is for instance the case for
the Financial sector, as can be seen from Exhibits 1
and 2.
Overall, we find substantial (but not extreme)
differences in implied expected return predictions
depending on the mean–variance portfolio proxy used.
In general, it seems that using the (smoothed) market
capitalization weighted or equally weighted portfolio,
more optimistic return predictions are obtained than
using the risk-based portfolios.
The fact that the differences in implied expected
returns at the sector level are not extreme suggests that,
while Lee [2011] is right to point out that smart beta
strategies can be seen as views on expected returns, these
views can be seen as being reasonable from a practical
perspective.
The analysis in Exhibit 3 is static and hides the
strong time variation in the predicted return. This is
shown in Exhibit 4, which displays the monthly time
SUMMER 2015
JPM-ARDIA.indd 75
series of the average annualized predicted return (across
all stocks). The solid lines show the minimum, median,
and maximum value across all implied expected return
methods, while the dashed lines correspond to the time
series of the minimum, median, and maximum forecasts obtained by the various time series model forecast
approaches. It is interesting to see that the two families
disagree especially in periods of high volatility. The
implied expected return methods impose the risk–return
tradeoff in the prediction such that higher volatility is
associated with a higher expected return. The time series
model forecasts extrapolate past returns into the future,
which, in bear markets, can lead to the (unrealistic)
prediction of negative returns.
Forecast Precision Comparison of Expected
Return Forecasts
The ultimate question is which expected return
prediction is most useful? We have essentially portfolio
applications in mind, where expected returns need to
be not only precise but also stable and not excessively
dispersed. The results of this analysis are reported in
Exhibit 5 where we zoom in on the statistical properties
of the implied expected returns (panel A) compared with
the time series forecasts (panel B).
THE JOURNAL OF PORTFOLIO M ANAGEMENT
75
7/21/15 2:13:15 PM
E
X HIBI T
4
Evolution of the Annualized Monthly Average Expected Return Forecasts across All S&P 100 Constituents
Solid lines are for implied expected returns approaches and dashed lines for time series model based return forecasts. The three lines per
family of estimation methods correspond to, for each time point, the minimum, median, and maximum value (across methods) of the average
forecasts (across stocks).
The precision of the forecasts is analyzed in Columns 2–3 of Exhibit 5. We consider two precision measures, namely the hit ratio (HR), which reports the
percentage of times the sign of a stock return is correctly
predicted by the corresponding forecast method, and the
root-mean-squared forecast error (RMSFE) of the onemonth-ahead predicted return compared with the actual
return. Both measures are computed for each month
across the 100 predicted returns for the S&P 100 constituents and then averaged over time. We see that the hit
ratio is 55.5% for all implied expected return methods
and slightly higher than the hit ratios obtained using
the time series models (between 53.3% and 53.8%).
Results for RMSFE indicate a better performance for
implied expected returns methods compared with the
time series models. For instance, the forecasts implied
by the market capitalization weighted portfolios lead
to a significantly lower RMSFE than each of the time
76
JPM-ARDIA.indd 76
series model based forecasts (at the 1% level). Also, we
find that within the family of implied expected returns
approaches, the most precise predictions are obtained
using the risk-based portfolios (wiv, werc, wmd, and wref ).
The RMSFE accuracy gains of switching from the use
of the market capitalization weighted portfolios to the
inverse volatility weighted portfolios are statistically significant at the 5% level. They are significant at the 10%
level for the risk-efficient portfolio. For other risk-based
portfolios, the difference is not statistically significantly
different.
Next, we study in Column 4 of Exhibit 5 the magnitude of the cross-sectional bets as measured by the
time series average of the monthly standard deviation
of return forecasts across the S&P 100 constituents (σ i ).
The higher this dispersion is the more extreme views are
in terms of differences in future performance of stocks.
We see that the standard deviation of the time series
I MPLIED E XPECTED R ETURNS AND THE CHOICE OF A M EAN –VARIANCE EFFICIENT PORTFOLIO P ROXY
SUMMER 2015
7/21/15 2:13:16 PM
EXHIBIT 5
Precision, Dispersion, and Instability of Monthly Return Forecasts for Implied Expected Returns and Time
Series Approaches
Notes: Precision measures: HR (hit ratio, in percent) and RMSFE (root–mean–squared forecast error, in permil). Dispersion measure: σi (cross-sectional
dispersion, in percent). Instability measures: σi (monthly time series volatility, in percent) and ρi ( first-order autocorrelation, in percent). Precision and
dispersion measures are computed at each time point for the one hundred stocks in the portfolio, and then averaged over time. The instability measures are
computed first for the individual predicted return series, for which at least twelve consecutive observations are available, and then averaged over the stocks in
the sample. ier-: implied expected returns forecasts. Constant: forecasts under constant mean model. AR(1): autoregressive model forecasts. Fama-French
and Carhart: three- and four-factor model forecasts.
model based return predictions is more than four times
the one of the implied expected return prediction. These
differences are economically and statistically significant
(at the 1% level). When used in portfolios, this implies
that the time series model based forecasts of return have
a higher risk to act like an error maximizer, as described
by Michaud [1989], and to exhibit a much larger portfolio turnover. The lowest dispersion is observed for the
maximum diversification portfolio.
Finally, we analyze in Columns 5–6 of Exhibit 5
the time series instability of the return forecasts, by considering the volatility (σ t ) and the first-order autocorrelation (ρ1 ) of the forecasts, computed for the union
of all stocks included in the universe over the backtest
period (and provided at least twelve consecutive observations are available). These quantities are then averaged
over stocks. We see that the standard deviation of the
expected return predictions is more than two times the
one of the implied expected return prediction. These
differences are economically and statistically significant
(at the 1% level). The first-order autocorrelation shows
that the implied expected return based forecast are much
more persistent. This is expected, since the time series
based return forecasts are a linear combination of the
SUMMER 2015
JPM-ARDIA.indd 77
past stock and factor returns, while the implied expected
returns are a linear combination of the risk-free rate and
the assets’ covariances with the portfolio proxy ( Σw ).
Because stock returns tend to be more volatile and more
dispersed than the assets’ covariances with the portfolio
proxy, we expect to find a higher stability in the implied
expected return forecasts compared with the time series
model based forecasts. This is confirmed by our results.
Note also that the implied expected returns based on
pure risk-optimized portfolios are more stable than
those using a market capitalization, fundamental value,
or equal weighting scheme.
Robustness of Results
The results hitherto obtained are for a particular
choice of estimation window (three years) and a specific
value of the the risk-aversion parameter γ (set to 2.4, as in
the original publication of Black and Litterman [1991]).
In Exhibit 6, we present the results of our robustness
analysis with respect to those two choices. Our analysis
confirms the robustness of the higher forecast accuracy
of implied expected returns compared with the time
series based model forecasts considered and the gains of
THE JOURNAL OF PORTFOLIO M ANAGEMENT
77
7/21/15 2:13:16 PM
E
X HIBI T
6
Sensitivity of Forecast Precision (RMSFE, in permil) of Implied Expected Returns and Time Series Model Based
Return Forecasts to the Estimation Window and the Risk-Aversion Parameter γ
Notes: RMSFE (in permil) for various value of risk-aversion parameter γ and estimation window (two, three, and four years). ier-: implied expected
returns forecasts. Constant: forecasts under constant mean model. AR(1): autoregressive model forecasts; Fama–French and Carhart: three- and four-factor
model forecasts. The time series model based approaches are insensitive to the choice of the risk-aversion parameter.
using a risk-based portfolio proxy compared with the
capitalization weighted portfolio. We find that for the
time series model based forecasts, the use of a longer time
span (four instead of three years) improves the RMSFE,
but it remains substantially higher than the one obtained
using the implied expected return estimates. The length
of the estimation sample has almost no impact on the
precision of the implied expected return estimates and
the observed differences in precision are statistically
insignificant. Moreover, increasing or decreasing the
value of γ to 1 or 5 does not lead to a statistically significant difference in RMSFE (at the 1% level). As such,
our results are robust to the choice of γ. It is however,
interesting, to note that for the risk-based approaches,
the accuracy is the highest for γ = 2.4, while for the
(smoothed) market capitalization weighted portfolio, the
fundamental value portfolio, and the equally weighted
portfolio, the accuracy is improved by reducing the value
of γ, and thus giving less weight to the portfolio weights
in the implied return calculation.
Forecast Combination
We conclude our empirical analysis by studying
the gains of forecast combination for predicting expected
returns. From the literature on forecast combination (see,
78
JPM-ARDIA.indd 78
e.g., Tu and Zhou [2011]), it can be expected that combinations of the different forecasts improve further the
stability of the forecasts. We study the impact of four types
of forecast combinations in Exhibit 7 and consider combinations over: (i) all time series model based forecasts, (ii) all
implied expected return based forecasts, and (iii) all pure
risk-based forecasts. Panel A corresponds to the results
obtained by simple averaging of the return forecasts.
Panel B takes a more outlier-robust approach by setting
the combination forecast equal to the median value of the
individual forecasts. Following Stock and Watson [2004]
and Genre et al. [2013], we also report performance-based
combination forecasts in which the predicted return is a
weighted average of the individual forecasts, with weights
proportional to the one-month lagged hit ratio (Panel C)
or inversely proportional to a 12-months average of the
mean squared forecast error (Panel D).7
Compared with the individual component forecasts, we find that the combination-based forecast tends
to be the second best and has the advantage of not
requiring the hindsight information of the ex post best
forecast method. The forecast combination method
(simple averaging, median aggregation, or performancebased aggregation) has only a minor impact on the
performance.
I MPLIED E XPECTED R ETURNS AND THE CHOICE OF A M EAN –VARIANCE EFFICIENT PORTFOLIO P ROXY
SUMMER 2015
7/21/15 2:13:17 PM
EXHIBIT 7
Precision, Dispersion, and Instability of Monthly Return Forecasts for
Combination-Based Approaches
Notes: ier-all: equally–weighted implied expected return based forecasts; ier-risk-based: equally–
weighted risk-based implied expected return based forecasts. See Exhibit 5 for more details.
CONCLUSION
We study the performance of implied expected
return predictions obtained using various mean–variance efficient portfolio proxies and compare them with
time series model based forecasts.
Our motivation is that, even though traditionally
the market capitalization weighted portfolio dominates in terms of reference portfolio from which the
implied expected returns are derived, there is growing
empirical evidence that risk-based portfolios may be a
better proxy for a mean–variance efficient portfolio.
In fact, in his important review on risk-based asset
allocation, Lee [2011] concludes that if risk-based allocation approaches outperform the market portfolio,
it must be that the implied expected returns are a
better set of return forecasts than the market’s implied
expected returns. We investigate empirically this prediction for the monthly returns of the S&P 100 stocks
over a period ranging from January 1987 to December
2012.
While Lee [2011] is right to point out that smart
beta strategies can be seen as views on expected returns,
we additionally show that these implied returns are rea-
SUMMER 2015
JPM-ARDIA.indd 79
sonable and potentially useful from a
practical perspective. We find that
there is only a small forecast accuracy gain in using implied expected
returns from risk-based portfolios
compared with the market capitalization weighted portfolio but conclude
that those forecasts are indeed better in
the sense of being substantially more
stable and less dispersed. When used
in dynamic portfolio optimization, we
expect them to yield optimized portfolio weights that are more stable and
less extreme, implying lower turnover costs and a reduced sensitivity to
the sampling variability in the return
data. Our current work on the use of
implied expected returns from riskbased portfolios in a mean–variance
portfolio optimization framework
seems to confirm this intuition.
ENDNOTES
The authors thank the Editor, an anonymous referee,
Marie-Hélène Gagnon, Georges Hübner, Thierry Roncalli,
Enrico Schumann, Piet Sercu, Marjan Wauters and participants at the IAF seminar (University of Neuchâtel 2013), the
R/Finance 2013, the SSES 2013 and the AFFI 2014 conference for useful suggestions. Financial support from the
Dutch science foundation, the IFM2 Montréal and the Hercules Foundation (Project No. AKUL/11/02) is gratefully
acknowledged.
1
According to Flood [2013], global assets held in
advanced beta strategies grew from USD 58 billion at the
end of 2010 to USD 142 billion at the end of March 2013.
2
We restrict our attention to time series methods that
extrapolate observed return dynamics into the future. The
predictability of asset returns using state variables such as
dividend-price or dividend-earnings ratios is a matter of controversy in the literature and beyond the scope of this article.
See, e.g., Campbell and Thompson [2007].
3
If a numerical solution is not found, we follow the recommendation in Maillard et al. [2010] and slightly modify the
problem and optimize over the N-dimensional vector u such
that w ≡ u / ( u ′ ) , under the constraint that u ≥ 0 and u ′ι 0.
This new optimization problem is easier to solve numerically as an inequality constraint is less restrictive than the full
investment equality constraint.
THE JOURNAL OF PORTFOLIO M ANAGEMENT
79
7/21/15 2:13:17 PM
4
http://mba.tuck.dartmouth.edu/pages/faculty/ken.
french/.
5
As noted by a referee, the actual implementation of
smart beta portfolios typically uses a quarterly or annual
rebalancing to reduce the high costs associated with portfolio turnover. Our focus is on the implied expected return
prediction, which requires the use of portfolios for which the
rebalancing is aligned with the return prediction horizon.
6
Throughout the article, statistical significance of differences in performance are computed using a t-test with
heteroskedasticity and autocorrelation consistent (HAC)
standard errors.
7
For more details, see Equation (1) in Genre et al.
[2013], which we implemented with a rolling window length
h = 12 and by initializing the weights for the first 11 months
as equal weights.
REFERENCES
Amenc, N., F. Goltz, L. Martellini, and P. Retkowsky. “Efficient Indexation: An Alternative to Cap-Weighted Indices.”
Journal of Investment Management, Vol. 9, No. 4 (2011),
pp. 1-23.
Arnott, R., J. Hsu, and P. Moore. “Fundamental Indexation.”
Financial Analysts Journal, Vol. 61, No. 2 (2005), pp. 83-99.
Best, M., and R. Grauer. “Capital Asset Pricing Compatible
with Observed Market Value Weights.” Journal of Finance,
Vol. 40, No. 1 (1985), pp. 85-103.
Black, F., and R. Litterman. “Asset Allocation: Combining
Investor Views with Market Equilibrium.” The Journal of Fixed
Income, Vol. 1, No. 2 (1991), pp. 7-18.
Chen, C., C. Rong, and G. Bassett. “Fundamental Indexation
via Smoothed Cap Weights.” Journal of Banking and Finance,
Vol. 31, No. 11 (2007), pp. 3486-3502.
Choueifaty, Y., and Y. Coignard. “Toward Maximum Diversification.” The Journal of Portfolio Management, Vol. 35, No. 1
(2008), pp. 40-51.
Clarke, R., H. de Silva, and S. Thorley. “Risk Parity, Maximum Diversification, and Minimum Variance: An Analytic
Perspective.” The Journal of Portfolio Management, Vol. 39,
No. 3 (2013), pp. 39-53.
DeMiguel, V., L. Garlappi, and R. Uppal. “Optimal versus
Naive Diversification: How Inefficient Is the 1/N Portfolio
Strategy.” Review of Financial Studies, Vol. 22, No. 5 (2009),
pp. 1915-1953.
Fama, E., and K. French. “Common Risk Factors in the
Returns of Stocks and Bonds.” Journal of Financial Economics,
Vol. 33, No. 1 (1993), pp. 3-56.
Fisher, L. “Using Modern Portfolio Theory to Maintain an
Efficiently Diversified Portfolio.” Financial Analysts Journal,
Vol. 31, No. 3 (1975), pp. 73-85.
Flood, C. “Large Spikes in ‘Smart Beta’ Investments.” Financial Times, June 16, 2013.
Genre, V., G. Kenny, A. Meyler, and A. Timmermann.
“Combining Expert Forecasts: Can Anything Beat the Simple
Average?” International Journal of Forecasting, Vol. 29, No. 1
(2013), pp. 108-121.
——. “Global Portfolio Optimization.” Financial Analysts
Journal, Vol. 48, No. 5 (1992), pp. 28-43.
Goltz, F., and V. Le Sourd. “Does Finance Theory Make the
Case for Capitalisation-Weighted Indexing?” The Journal of
Index Investing, Vol. 2, No. 2 (2011), pp. 59-75.
Bruder, B., and T. Roncalli. “Managing Risk Exposure
using the Risk Budgeting Approach.” Working paper, Lyxor
Research, 2012.
Graham, J. “Comment on the Theoretical and Empirical Evidence of Fundamental Indexing.” Journal of Investment Management, Vol. 10, No. 1 (2012), p. 90.
Campbell, J., and S. Thompson. “Predicting Excess Stock
Returns Out of Sample: Can Anything Beat the Historical
Average?” Review of Financial Studies, Vol. 21, No. 4 (2007),
pp. 1509-1531.
Herold, U. “Computing Implied Returns in a Meaningful
Way.” Journal of Asset Management, 6 (2005), pp. 53-64.
Carhart, M. “On Persistence in Mutual Fund Performance.”
Journal of Finance, Vol. 52, No. 1 (1997), pp. 57-82.
80
JPM-ARDIA.indd 80
Kothari, S., A. Leone, and C. Wasley. “Performance Matched
Accrual Measures.” Journal of Accounting and Economics, Vol. 39,
No. 1 (2005), pp. 163-197.
I MPLIED E XPECTED R ETURNS AND THE CHOICE OF A M EAN –VARIANCE EFFICIENT PORTFOLIO P ROXY
SUMMER 2015
7/21/15 2:13:18 PM
Ledoit, O., and M. Wolf. “Improved Estimation of the
Covariance Matrix of Stock Returns with an Application
to Portfolio Selection.” Journal of Empirical Finance, Vol. 10,
No. 5 (2003), pp. 603-621.
Lee, W. “Risk-Based Asset Allocation: A New Answer to an
Old Question?” The Journal of Portfolio Management, Vol. 37,
No. 4 (2011), pp. 11-28.
Plyakha, Y., R. Uppal, and G. Vilkov. “Equal or Value
Weighting? Implications for Asset-Pricing Tests.” Working
paper, SSRN, 2014.
RiskMetrics Group. RiskMetrics Technical Document. J. P.
Morgan/Reuters, 1996.
Roncalli, T. Introduction to Risk Parity and Budgeting. Chapman
and Hall/CRC, 2013.
Leote De Carvalho, R., X. Lu, and P. Moulin. “Demystifiying Equity Risk-Based Strategies: A Simple Alpha plus
Beta Description.” The Journal of Portfolio Management, Vol. 38,
No. 3 (2012), pp. 56-70.
Sharpe, W. “Imputing Expected Returns from Portfolio
Composition.” Journal of Financial and Quantitative Analysis,
Vol. 9, No. 3 (1974), pp. 463-472.
Maillard, S., T. Roncalli, and J. Teiletche. “The Properties of
Equally Weighted Risk Contribution Portfolios.” The Journal
of Portfolio Management, Vol. 36, No. 4 (2010), pp. 60-70.
Stock, J., and M. Watson. “Combination Forecasts of Output
Growth in a Seven-Country Data Set.” Journal of Forecasting,
Vol. 23, No. 6 (2004), pp. 405-430.
Martellini, L. “Toward the Design of Better Equity Benchmarks Rehabilitating the Tangency Portfolio from Modern
Portfolio Theory.” The Journal of Portfolio Management, Vol. 34,
No. 4 (2008), pp. 34-41.
Treynor, J. “Why Market-Valuation-Indifferent Indexing
Works.” Financial Analysts Journal, Vol. 61, No. 5 (2005),
pp. 65-69.
Merton, R. “On Estimating the Expected Return on the
Market.” Journal of Financial Economics, Vol. 8, No. 4 (1980),
pp. 323-361.
Meucci, A. “Broadening Horizons.” Risk, Vol. 17, No. 12
(2004), pp. 98-101.
Tu, J., and G. Zhou. “Markowitz Meets Talmud: A Combination of Sophisticated and Naive Diversification Strategies.” Journal of Financial Economics, Vol. 99, No. 1 (2011),
pp. 204-215.
To order reprints of this article, please contact Dewey Palmieri
at dpalmieri@ iijournals.com or 212-224-3675.
Michaud, R. “The Markowitz Optimization Enigma: Is
Optimized Optimal?” Financial Analysts Journal, Vol. 45,
No. 1 (1989), pp. 31-42.
Perold, A. “Fundamental Flawed Indexing.” Financial Analysts
Journal, Vol. 63, No. 6 (2007), pp. 31-37.
SUMMER 2015
JPM-ARDIA.indd 81
THE JOURNAL OF PORTFOLIO M ANAGEMENT
81
7/21/15 2:13:18 PM