It’s All in the Timing: Simple Active Portfolio
Strategies that Outperform Naı̈ve Diversification
Chris Kirby a , Barbara Ostdiek b
a John
E. Walker Department of Economics, Clemson University
b Jesse
H. Jones Graduate School of Business, Rice University
Abstract
DeMiguel et al. (2009) report that naı̈ve diversification dominates mean-variance optimization in
out-of-sample asset allocation tests. Our analysis suggests that this is largely due to their research
design, which focuses on mean-variance efficient portfolios that are subject to high estimation risk
and extreme turnover. We find that mean-variance optimization outperforms naı̈ve diversification
under many circumstances, but its advantage can easily be eroded by transactions costs. This
motivates us to propose two types of mean-variance timing strategies, both characterized by
low turnover. These strategies outperform naı̈ve diversification even in the presence of relatively
high transactions costs. In contrast to DeMiguel et al. (2009), therefore, we conclude that using
sample information to guide portfolio selection yields substantial benefits.
Key words: portfolio selection, mean-variance optimization, estimation risk, turnover, market
timing, volatility timing
JEL classification: G11; G12; C11
May 9, 2010
Initial draft: November 18, 2009
? Comments welcome. We are thankful for comments from Nick Bollen and from seminar participants at Rice University, Universidad Carlos III de Madrid, and University of North Carolina at Charlotte. Address correspondence to: Chris Kirby, John E. Walker Department of
Economics, Clemson University, P.O. Box 341309, Clemson, SC 29634-1309. e-mail addresses:
[email protected] (Chris Kirby) and [email protected] (Barbara Ostdiek).
1
Introduction
Mean-variance optimization is a cornerstone of modern portfolio theory. However, a recent
study by DeMiguel et al. (2009) questions the value of mean-variance optimization relative
to naı̈ve diversification, i.e., relative to a strategy that places a weight of 1/N on each of
the N assets under consideration. The authors of the study implement 14 variants of the
standard mean-variance model for a number of datasets and find that “there is no single
model that consistently delivers a Sharpe ratio or a CEQ return that is higher than that of
the 1/N portfolio.” This finding presents researchers with two clear challenges. The first is
to understand why the mean-variance approach to portfolio selection performs so poorly
in the DeMiguel et al. (2009) study. The second is to develop more effective procedures
for using sample information about means and variances in portfolio problems.
With respect to the first challenge, we show that the DeMiguel et al. (2009) research
design places the mean-variance model at an inherent disadvantage relative to naı̈ve diversification. Specifically, it delivers mean-variance efficient portfolios that tend to be
very aggressive, with target conditional expected excess returns that often exceed 100%
per year. Targeting conditional expected excess returns of this magnitude leads to poor
out-of-sample performance because it greatly magnifies both estimation risk and portfolio turnover. If the mean-variance model is implemented by targeting the conditional
expected return of the 1/N portfolio, the resulting mean-variance efficient strategies outperform naı̈ve diversification for most of the DeMiguel et al. (2009) datasets. However, it
is not clear that this finding is robust to transactions costs.
In response to the second challenge, we develop simple active portfolio strategies that
retain the most appealing features of the 1/N portfolio — no optimization, no covariance matrix inversion, and no short sales — while exploiting sample information about
the reward and risk characteristics of the assets under consideration. In particular, we
specify the portfolio weights in terms of conditional expected returns, conditional return
volatilities, and a tuning parameter that allows some control over portfolio turnover. The
empirical evidence shows that the proposed strategies outperform naı̈ve diversification by
statistically and economically significant margins. This is true even though we implement
the strategies using estimators of the conditional expected returns and conditional return volatilities that are likely to be relatively inefficient. Moreover, the advantage of the
proposed strategies persists even in the presence of relatively high transactions costs.
The strategies we develop in the paper are most naturally interpreted as mean-variance
timing rules, but they are rooted in an extensive literature on asset allocation in the presence of estimation error and constraints on portfolio holdings. There have been a number
of notable contributions to this literature in recent years. Pastor (2000) and Pastor and
Stambaugh (2000) use Bayesian methods to address the issue of parameter uncertainty.
Ledoit and Wolf (2003) develop an optimal shrinkage methodology for covariance matrix
estimation and find that it improves the out-of-sample performance of mean-variance optimization methods. Jagannathan and Ma (2003) consider ad hoc short-sale constraints
and position limits and show that these restrictions are a form of shrinkage that improves
1
portfolio performance by reducing the ex post effect of estimation error. Kan and Zhou
(2007) use an innovative approach to develop a three-fund asset allocation strategy that
optimally diversifies across both factor and estimation risk.
The recent contribution of Tu and Zhou (2008) is particularly relevant to our investigation. In this study, the authors develop a simple theory of portfolio choice in the presence
of estimation risk by assuming that asset returns are independently and identically distributed over time. Building on the idea that the 1/N portfolio constitutes a reasonable
shrinkage target, they propose a new strategy that optimally combines the 1/N portfolio
and the Kan and Zhou (2007) three-fund portfolio, with the degree of shrinkage towards
the 1/N portfolio determined by the level of estimation risk. Simulations demonstrate
that this “four-fund strategy” outperforms the 1/N portfolio under a range of assumptions about the data generating process for asset returns, providing convincing evidence
that optimization methods can be consistently useful for cases in which the means and
variances of returns are constant through time.
This paper addresses many of the issues studied by Tu and Zhou (2008), but does so from
the perspective of an investor who assumes that the conditional means and variances of
asset returns change through time. We start by proposing a new class of active portfolio
strategies that are designed to exploit sample information about volatility dynamics in
a way that mitigates the impact of estimation risk. Under our approach, which we refer
to as volatility timing, the portfolios are rebalanced monthly based solely on changes in
the estimated conditional volatilities of asset returns. We control the sensitivity of the
portfolio weights to these changes via a tuning parameter that can be interpreted as a
measure of timing aggressiveness. This allows us to keep the turnover of the proposed
strategies to a level competitive with that of naı̈ve diversification.
We also propose a more general class of timing strategies that incorporate sample information about the dynamics of conditional expected returns. Under this approach, which
we refer to as reward-to-risk timing, the portfolios are rebalanced monthly based solely on
changes in the estimated reward-to-risk ratios of the assets. We implement the proposed
strategies using two different estimators of conditional expected returns: a rolling estimator that imposes no parametric assumptions and an estimator that is designed to reduce
estimation risk by exploiting the predictions of asset pricing theory. Under the conditional
capital asset pricing model (CAPM), for example, we can express the timing weights in
terms of the conditional market betas and variances of the returns. We illustrate that
using estimates of the conditional betas in place of estimates of expected returns should
reduce the sampling variation in the weights if the conditional CAPM holds.
The empirical section of the paper compares the out-of-sample performance of the volatility and reward-to-risk timing strategies to that of the 1/N portfolio using datasets similar
to those used by DeMiguel et al. (2009). To provide additional perspective, we also report results for several strategies obtained from the standard mean-variance model. The
inputs needed to implement the strategies are estimated using the same rolling estimators
employed by DeMiguel et al. (2009). This allows us to focus sharply on the relative performance of the proposed timing strategies rather than on the impact of different estimation
2
techniques. We find that both types of timing strategies can significantly outperform the
1/N portfolio for a range of datasets even after accounting for the impact of transactions costs. Thus, in contrast to DeMiguel et al. (2009), we conclude that there can be
substantial value in using sample information to guide portfolio selection decisions.
For example, with proportional transactions costs of 50 basis points, the 1/N portfolio
has an estimated annualized Sharpe ratio of 0.46 for a dataset that contains 25 portfolios
formed on size and book-to-market characteristics. In comparison, the volatility-timing
strategies have estimated annualized Sharpe ratios that range from 0.47 to 0.49 for this
dataset. Even these small differences translate into economically and statistically significant performance gains. This level of precise inference is possible because the returns
for the volatility-timing and 1/N strategies are generally highly correlated, resulting in
small standard errors for the relative performance measures. We estimate that risk averse
investors would be willing to pay fees between 81 and 108 basis points per year to switch
from the 1/N strategy to our volatility-timing strategies. These estimated performance
fees are statistically different from zero at the 10% significance level.
The performance gains are more pronounced for the reward-to-risk timing strategies. If we
implement these strategies using a rolling estimator of conditional expected returns, then
the estimated annualized Sharpe ratios range from 0.52 to 0.54. Moreover, we estimate
that risk averse investors would be willing to pay fees between 127 and 167 basis points per
year to switch from the 1/N strategy to these strategies. Reward-to-risk timing appears
to be a particularly promising strategy when it is implemented using the Carhart (1997)
four-factor risk model to estimate the conditional factor betas. In this case, the estimated
annualized Sharpe ratios range from 0.52 to 0.57, the estimated performance fees range
from 118 to 220 basis points per year for a risk averse investor, and these gains are all
statistically significant at the 1% level.
The evidence also suggests that the performance gains become larger as we increase the
cross-sectional dispersion in the means and variances of returns. The dataset that poses
the biggest challenge to the timing strategies contains 10 industry portfolios. Sorting firms
into industries produces only a modest cross-sectional spread in the estimated means. In
contrast, some of the most compelling results are obtained using a dataset created by
sorting firms into 10 portfolios based on momentum, which produces a pronounced spread
in the estimated means. The 1/N portfolio has an estimated annualized Sharpe ratio of
0.28 for this dataset, while the reward-to-risk timing strategies have estimated annualized
Sharpe ratios that range from 0.43 to 0.47 when they are implemented using a rolling
estimator of conditional expected returns. Moreover, we observe similar performance gains
for the volatility timing strategies when we use a dataset that is constructed by sorting
firms into portfolios based on historical volatility.
The remainder of the paper is organized as follows. Section 2 considers the portfolio
choice problem of an investor with quadratic risk preferences and shows how the resulting
framework can be used to motivate volatility and reward-to-risk timing strategies. Section
3 describes our estimators of the conditional mean vector and conditional covariance
matrix of excess returns. Section 4 discusses our approach to performance evaluation
3
along with the methods we use to draw statistical inferences. Section 5 describes the data
and presents the empirical results. Section 6 provides concluding remarks.
2
Portfolio Strategies
This section describes the portfolio strategies we investigate. The first strategy — naı̈ve
diversification — requires little explanation. It consists of allocating an equal amount to
each asset in the portfolio and is unique in the sense that it implies time-invariant portfolio
weights. The remaining strategies fall into one of two general categories: strategies that
are ex ante optimal under quadratic loss, i.e., mean-variance efficient (MVE) strategies,
and strategies that do not entail formal optimization, but are designed to exploit sample
information about means and variances in a manner that mitigates estimation risk. We
assume throughout that there are N risky assets and a single risk-free asset. We refer to
naı̈ve diversification across the risky assets as the 1/N strategy.
2.1
Optimal strategies under quadratic loss
Let rt = Rt − ιRf,t where Rt is an N × 1 vector of risky-asset returns for period t, Rf,t
is the risk-free rate of interest for period t, and ι denotes an N × 1 vector of ones. Under
the standard approach to conditional mean-variance optimization, the investor’s objective
in period t is to choose the N × 1 vector of risky-asset weights ωp,t that maximizes the
quadratic objective function
γ 0
0
Q(ωp,t ) = ωp,t
µt − ωp,t
Σt ωp,t ,
2
(1)
where µt = Et (rt+1 ) is the conditional mean vector of the excess risky-asset returns,
0
Σt = Et (rt+1 rt+1
) − Et (rt+1 )Et (rt+1 )0 is the conditional covariance matrix of the excess
risky-asset returns, and γ denotes the investor’s coefficient of relative risk aversion. The
0
weight in the risk-free asset is determined implicitly by 1 − ωp,t
ι.
This problem has a straightforward and well known solution. The optimal choice of ωp,t
is given by
1
ωp,t = Σ−1
µt .
(2)
γ t
Equation (2) implies that, in general, the investor divides his wealth between the risk-free
asset and a “tangency portfolio” of risky assets with weights
ωT P,t =
Σ−1
t µt
.
0
ι Σ−1
t µt
(3)
That is, the investor holds a conditionally MVE portfolio. 1 The fraction of wealth allo0 −1
We assume for the discussion that ι0 Σ−1
t µt > 0. If ι Σt µt < 0 then the tangency portfolio is
conditionally inefficient. In this case, the investor will, in general, short the tangency portfolio
and invest everything — initial wealth and short sale proceeds — in the risk-free asset.
1
4
cated to the tangency portfolio is xT P,t = ι0 Σ−1
t µt /γ. As γ falls and the investor becomes
less risk averse, he invests more aggressively in the tangency portfolio to raise his condi0
tional expected excess return, µp,t = ωp,t
µt . Because there is a one-to-one correspondence
between γ and µp,t for each t, we can express equation (2) as
Σ−1
t µt
0 −1
µt Σ t µt
ωp,t = µp,t
!
(4)
and view the investor as choosing the period t portfolio by minimizing the conditional
risk of the portfolio for a specified value of µp,t .
We refer to the portfolio in equation (4) as the optimal unconstrained (OU) portfolio
because the sum of the risky-asset weights is unconstrained. DeMiguel et al. (2009) focus
on portfolios that constrain these weights to sum to one to ensure that performance
differences across portfolios are not simply the result of different allocations to the risk-free
and risky assets. They impose this constraint by rescaling the weights of the OU portfolio,
0
i.e., they divide each side of equation (2) by ωp,t
ι to obtain the tangency portfolio. 2
Because the tangency portfolio has the same conditional Sharpe ratio as the OU portfolio,
focusing on this portfolio may seem innocuous. However, the tangency portfolio differs
from the OU portfolio in two important respects: estimation risk and turnover. These
differences can have a substantial impact on the relative performance of the two portfolios.
2.1.1
Estimation risk, turnover, and the tangency portfolio
First consider the issue of estimation risk. Intuitively, estimation risk arises from uncertainty about the parameters of the data generating process. This uncertainty leads to
errors in estimating the portfolio weights, which drives up the risk of the portfolio. Suppose, for example, that µt = µ and Σt = Σ for all t. If the values of µ and Σ are known to
the investor, then the OU portfolio and the tangency portfolio have known, time-invariant
weights. In particular, the weights are given by ωp = Σ−1 µ/γ and ωT P = Σ−1 µ/ι0 Σ−1 µ.
Because the weights are proportional to Σ−1 µ in each case, the excess return on the OU
portfolio is perfectly correlated with the excess return on the tangency portfolio and the
two portfolios have the same unconditional Sharpe ratio.
This is not true, however, if we replace µ and Σ with the sample mean vector µ̂ and sample
covariance matrix Σ̂. The sampling variation in µ̂ and Σ̂, which translates into sampling
variation in the portfolio weights, inflates the variance of the portfolio returns and lowers
the unconditional Sharpe ratios. Although the sampling variation in µ̂ and Σ̂ affects both
portfolios, the tangency portfolio is likely to experience a more severe deterioration in
its Sharpe ratio. To see this, consider the estimated weights of the tangency portfolio,
ω̂T P = Σ̂−1 µ̂/ι0 Σ̂−1 µ̂. If the value of ι0 Σ̂−1 µ̂ is small in magnitude, then the tangency
0 −1
To be precise, DeMiguel et al. (2009) consider a strategy with ωT∗ P,t = Σ−1
t µt /|ι Σt µt |. This
strategy invests 100% in the tangency portfolio and 0% in the risk-free asset for cases in which
ι0 Σ−1
t µt > 0, and invests −100% in the tangency portfolio and 200% in the risk-free asset for
cases in which ι0 Σ−1
t µt < 0.
2
5
portfolio will typically be characterized by extreme weights. The problem with extreme
weights is that they tend to produce extreme returns that inflate the portfolio variance.
The OU portfolio does not suffer from this problem because the vector Σ̂−1 µ̂ is scaled by
1/γ rather than by 1/ι0 Σ̂−1 µ̂. The investor chooses γ, while ι0 Σ̂−1 µ̂ is a random variable
that can take on values close to zero if there is sufficient sampling variation in µ̂.
Estimation risk is particularly important when considering transactions costs. If transactions are greater than zero, then anything that increases portfolio turnover can cause the
after-transactions-costs performance of the portfolio to deteriorate. We define turnover as
the fraction of invested wealth traded in a given period to rebalance the portfolio. To see
how we compute turnover for the OU portfolio, note that if one dollar is invested in the
portfolio at time t − 1, there will be ωi,t−1 (1 + Ri,t ) dollars invested in the ith risky asset
at time t. Hence, the weight in asset i before the portfolio is rebalanced at time t is
ωi,t−1 (1 + Ri,t )
,
PN
i=1 ωi,t−1 (1 + Ri,t ) + (1 −
i=1 ωi,t−1 )(1 + Rf,t )
ω̃i,t = PN
(5)
and the turnover at time t is given by
τp,t =
N
X
i=1
N
X
|ωi,t − ω̃i,t | + (ωi,t
i=1
− ω̃i,t ) ,
(6)
where ωi,t is the desired weight in asset i at time t.
Since turnover increases with the variance of the portfolio weights, both the choice of γ
and the sampling variation in µ̂t and Σ̂t play a role in determining its value. Consider
our earlier example with µt = µ and Σt = Σ for all t. In this case, the estimated weights
of the OU portfolio are given by ω̂p = Σ̂−1 µ̂/γ. Changing γ has no effect on the beforetransactions-costs Sharpe ratio of the portfolio because both the mean and standard
deviation of the portfolio return are proportional to 1/γ. However, it can have a dramatic
impact on turnover because |ωi,t − ω̃i,t | is approximately proportional to 1/γ. 3 Reducing γ
causes the cost of rebalancing the portfolio to rise, which drives down the average return
of the portfolio and causes its after-transactions-costs Sharpe ratio to fall. Hence, the
choice of γ is an important consideration when developing a research design to evaluate
the effectiveness of mean-variance optimization.
The impact of turnover on transactions costs is an even greater concern for the tangency
portfolio because of the potential for extreme weights. If there is more than a small chance
that ι0 Σ̂−1 µ̂ is less than zero, then it is likely that realizations of this quantity that are
close to zero will push turnover to very high levels. Since the turnover of the tangency
portfolio is a function of the characteristics of the data generating process and the choice
of estimators for µ and Σ, there is little we can do to mitigate this problem. In contrast
to the OU portfolio, we cannot reduce turnover by specifying a higher value of γ. This
3
Note that ω̃i,t = ωi,t−1 (1 + Ri,t ) for the special case in which the portfolio has a zero return in
period t. Because ωi,t−1 and ωi,t are proportional to 1/γ, it follows that |ωi,t − ω̃i,t | is proportional
to 1/γ in this case. More generally, approximate proportionality holds.
6
is a major drawback in the presence of transactions costs. By focusing on the tangency
portfolio, DeMiguel et al. (2009) place the standard mean-variance model at an inherent
disadvantage with respect to the impact of turnover. Thus their results could produce a
overly pessimistic picture of the usefulness of mean-variance optimization.
2.1.2
Optimization over the risky assets only
If the objective is to investigate the performance of MVE portfolios that exclude the riskfree asset, an alternative to considering the tangency portfolio is to solve the investor’s
0
portfolio problem subject to the constraint ωp,t
ι = 1. The first-order condition for the
constrained problem is
µt − δt ι − γΣt ωp,t = 0,
(7)
where δt is the Lagrange multiplier associated with the constraint. Hence, the optimal
vector of constrained portfolio weights is
ωp,t =
δt
1 −1
Σt µt + Σ−1
ι.
γ
γ t
(8)
To interpret the solution, note that the first term on the right side of equation (8) is
0 −1
proportional to ωT P,t and the second is proportional to ωM V,t = Σ−1
t ι/ι Σt ι, which is
the vector of weights for the minimum-variance (MV) portfolio of risky assets, i.e., the
0
0
portfolio obtained by minimizing ωp,t
Σt ωp,t subject to ωp,t
ι = 1. Thus, the solution to the
constrained problem takes the same general form as the solution for the unconstrained
problem with the MV portfolio replacing the risk-free asset. 4
If we solve for δt and substitute the resulting expression into equation (8), we obtain
ωp,t = xT P,t
Σ−1
t µt
ι0 Σ−1
t µt
!
Σ−1 ι
+ (1 − xT P,t ) 0 t −1 ,
ι Σt ι
!
(9)
which implies that the conditional expected excess return on the portfolio is given by
µp,t = xT P,t µT P,t + (1 − xT P,t )µM V,t ,
(10)
where µT P,t and µM V,t denote the conditional expected excess returns on the tangency and
MV portfolios. Accordingly, we can express equation (9) as
ωp,t =
µp,t − µM V,t
µT P,t − µM V,t
!
Σ−1
t µt
ι0 Σ−1
t µt
!
4
µp,t − µM V,t
+ 1−
µT P,t − µM V,t
!
Σ−1
t ι
,
ι0 Σ−1
t ι
!
(11)
Kan and Zhou (2007) propose a “three-fund strategy” in which the investor is restricted to
hold a combination of the risk-free asset, the tangency portfolio, and the MV portfolio. One
could interpret the portfolio in equation (8) as a constrained version of their strategy. However,
the MV and tangency portfolios cease to play a special role in determining the optimal portfolio
once the weight in the risk-free is constrained to be zero: the conditional efficient frontier for the
risky assets is spanned by any two portfolios on the frontier.
7
and view the investor as choosing his portfolio for period t by minimizing the conditional
risk of the portfolio for a specified value of µp,t . We refer to the portfolio in equation (11)
as the optimal constrained (OC) portfolio.
Note that the OC portfolio is identical to the OU portfolio except that the weight in
the risk-free asset has been transfered to the MV portfolio. It follows, therefore, that the
increase in estimation risk from imposing the constraint is due solely to errors in estimating
ωM V,t . Two observations suggest that this increase should be considerably less than that
incurred by rescaling the OU portfolio weights (assuming that the OU portfolio is a convex
combination of the risk-free asset and tangency portfolio). First, ωM V,t does not depend
on µt . The variances and covariances of returns can typically be estimated with better
precision than the mean returns (see, e.g., Merton, 1980), so the errors in estimating ωM V,t
should be smaller than those in estimating the weights of other MVE portfolios. Second,
0 −1
the denominator in the expression ωM V,t = Σ−1
t ι/ι Σt ι is the reciprocal of the variance of
the MV portfolio. This suggests that the potential for generating extreme weights is much
lower than with the tangency portfolio because the value of ι0 Σ̂−1
t ι is, by construction,
both positive and as large as possible.
2.2
Monte Carlo evidence on estimation risk and turnover
To provide insights on the empirical relevance of the issues identified in Sections 2.1.1 and
2.1.2, we conduct a simple Monte Carlo experiment using five of the six datasets examined
by DeMiguel et al. (2009). 5 Each dataset consists of monthly returns on a collection of
broadly-based equity portfolios and the monthly risk-free rate of interest. Three of the
datasets are constructed by sorting U.S. firms into portfolios based on market value and
book-to-market value characteristics (the FF 1-Factor, FF 4-Factor, and Mkt/SMB/HML
datasets), one is constructed by sorting U.S. firms into industries using standard industrial
classification codes (the FF 10 Industry dataset), and one contains international equity
market indexes (the International dataset). The sample size is 497 observations except for
the International dataset which contains 379 observations.
The Monte Carlo experiment is designed to illustrate the impact of sampling variation
in the estimated portfolio weights on the performance of the OU, OC, and tangency
portfolios under conditions similar to those studied by DeMiguel et al. (2009). To highlight
the effect of estimation risk, we focus on a scenario with time invariant µt and Σt . Using
resampling methods, we simulate a time series of excess risky-asset returns that preserve
key characteristics of the actual data and then we compute the unconditional Sharpe
ratio and expected turnover of the OU, OC, and tangency portfolio via Monte Carlo
integration. We also report results for the 1/N and MV portfolios to provide additional
points of reference.
5
We thank Victor DeMiguel and Lorenzo Garlappi for sharing these data. The S&P sector
dataset is proprietary and thus not included in analysis.
8
2.2.1
The experiment
Suppose a given dataset contains T + h observations, where T is the number of months
used to evaluate the out-of-sample performance of the portfolios and h is the length of the
data window used to estimate the portfolio weights. The general design of the experiment
∗
is as follows. First, we generate a sequence {rt∗ }Tt=1+h of i.i.d. excess risky-asset returns
from a distribution with mean vector µ∗ and covariance matrix Σ∗ , where T ∗ denotes
the length of the out-of-sample period for the experiment. The data generating process
is such that µ∗ and Σ∗ match the sample mean vector and sample covariance matrix of
T +h
∗ T ∗ +h
{rt }t=h+1
. Next, we construct a sequence {rp,t
}t=h+1 of out-of-sample excess returns for
each portfolio using the weights implied by our estimates of µ∗ and Σ∗ . Finally, we use
∗ T ∗ +h
∗
∗
the sample moments of {rp,t
}t=h+1 to approximate E(rp,t
) and Var(rp,t
) for each portfolio.
The error in approximating these population moments goes to zero as T ∗ → ∞. We set
h = 120, which is the value used by DeMiguel et al. (2009), and T ∗ = 1,000,000.
∗
The sequence {rt∗ }Tt=1+h is generated by resampling the data. Specifically, we draw h times
with replacement from {rt }ht=1 to obtain the sample used to form the initial estimates of
T +h
the portfolio weights, and T ∗ times with replacement from {rt }t=h+1
to obtain the sample
∗
used for the Monte Carlo integration. Once we have the sequence {rt∗ }Tt=1+h for a given
dataset, we use a rolling-sample approach to construct the portfolios. First we compute
∗
∗
∗
ω̂p,h
for each portfolio by using {rt∗ }ht=1 to estimate µ∗ and Σ∗ , and multiply ω̂p,h
by rh+1
to
∗
∗
obtain rp,h+1 . Then we roll forward to the next period, compute ω̂p,h+1 for each portfolio
∗
∗
∗
∗
∗
by using {rt∗ }h+1
t=2 to estimate µ and Σ , and multiply ω̂p,h+1 by rh+2 to obtain rp,h+2 . We
∗
continue in this fashion until we reach period T + h.
To implement this procedure for the OU and OC portfolios, we have to specify a target
estimated conditional expected return for each t. Because turnover is sensitive to the target
selected, we set µ̂∗p,t equal to the estimated conditional expected excess return of the 1/N
6
portfolio, i.e., µ̂∗p,t = µ̂∗0
This should reduce the differences in turnover between the
t ι/N .
OU, OC, and 1/N portfolios, and allow them to compete on a more equal footing with
respect to transactions costs. We expect the 1/N portfolio to have very low turnover, so
we do not want to inflate the turnover of the OU and OC portfolios by targeting a µ̂∗p,t
that exceeds the value implicitly targeted by the 1/N portfolio.
2.2.2
The experimental results
Table 1 summarizes the results of the experiment. We report three quantities for each
portfolio: the population value of the annualized Sharpe ratio implied by the true weights,
λ∗ (ω), the population value of the annualized Sharpe ratio when the portfolio is rebalanced
monthly using the estimated weights, λ∗ (ω̂), and the population value of expected monthly
turnover when the portfolio is rebalanced monthly using the estimated weights, τ ∗ (ω̂). The
6
Occasionally, targeting the estimated expected excess return of the 1/N portfolio delivers a
conditionally inefficient portfolio. In these cases, we replace µ̂∗p,t − µ̂∗M V,t in the sample analog of
equation (11) with |µ̂∗p,t − µ̂∗M V,t |. This delivers a conditionally efficient portfolio with the same
conditional volatility as the identified inefficient portfolio.
9
reported values of λ∗ (ω) are exact because they are computed directly from the values of
µ∗ and Σ∗ , e.g., λ∗ (ω) = (µ∗0 Σ∗−1 µ∗ )1/2 for the OU portfolio and λ∗ (ω) = µ∗0 ι/(ι0 Σ∗ ι)1/2
for the 1/N portfolio. Since ω̂ = ω for the 1/N portfolio, the difference between λ∗ (ω̂) and
λ∗ (ω) for this portfolio is due solely to the Monte Carlo error in computing population
expectations. The difference is negligible for every dataset.
The OU and tangency portfolios have the maximum possible value of λ∗ (ω) by construction. It ranges from 0.74 for the International dataset to 1.92 for the FF 4-Factor dataset.
The difference between the values of λ∗ (ω) for the OU portfolio and for the OC portfolio
provides a direct measure of the cost of constraining the weight in the risk-free asset to
be zero. This cost varies a good deal across the datasets. It is small for the FF 4-Factor
dataset (a difference of 0.04) and large for the FF 1-Factor dataset (a difference of 0.71).
However, the cost of naı̈ve diversification is typically much larger. The difference between
the value of λ∗ (ω) for the OU portfolio and that for the 1/N portfolio ranges from 0.21
for the Mkt/SMB/HML dataset to 1.31 for the FF 4-Factor dataset. Thus the gains to
optimization are substantial in the absence of estimation risk.
A very different picture emerges when we take estimation risk into account. The most
striking change is a dramatic deterioration in the performance of the tangency portfolio. The value of λ∗ (ω̂) ranges from zero for the International dataset to 0.05 for the
Mkt/SMB/HML dataset. This finding is consistent with our earlier discussion regarding
the impact of rescaling the OU portfolio weights. The evidence points to a non-negligible
probability of observing values of ι0 Σ̂∗−1 µ̂∗ that are close to zero. This is an undesirable characteristic that greatly magnifies estimation risk. As a consequence, the tangency
portfolio does not perform nearly as well as the other MVE portfolios.
To investigate further, we find the minimum value of ι0 Σ̂−1 µ̂ that is observed when we
compute the estimated tangency portfolio weights for months h+1 to T +h using the actual
data. Then we use the subset of simulated excess returns for which the value of ι0 Σ̂∗−1 µ̂∗
exceeds this cutoff to generate a second set of results for the tangency portfolio. These
results, which are reported in the last row of the table, show how the tangency portfolio
performs conditional on a degree of rescaling that is no more extreme than that observed
in the actual sample analyzed by DeMiguel et al. (2009). The value of λ∗ (ω̂) increases for
every dataset, but in general the tangency portfolio still performs markedly worse than the
other portfolios. The one exception is for the Mkt/SMB/HML dataset, which is the only
dataset that has predominately negative estimated return correlations. 7 With negative
correlations, which imply large benefits to diversification, the tangency portfolio is less
likely to be characterized by extreme long and short positions.
The results for the other portfolios yield additional insights. The value of λ∗ (ω̂) is substantially lower than the value of λ∗ (ω) for every portfolio except the 1/N portfolio. This
reflects the impact of estimation risk. The OC portfolio, however, has a value of λ∗ (ω̂)
that is greater than that of the 1/N portfolio for every dataset, and greater than or equal
to that of the OU portfolio for all but one of the datasets (the FF 1-Factor dataset). This
7
The estimates of corr(rM kt,t , rSM B,t ) and corr(rM kt,t , rHM L,t ) are −0.29 and −0.47.
10
indicates that the OC portfolio can outperform naı̈ve diversification and shows that in
the presence of estimation risk constraining the position in the risk-free asset to be zero
does not necessarily entail a positive cost.
We can see why this occurs by comparing the values of λ∗ (ω) to the values of λ∗ (ω̂) for
the OC and OU portfolios. With the exception of the FF 4-Factor dataset, the difference
between λ∗ (ω) and λ∗ (ω̂) for the OC portfolio is relatively small, ranging from 0.06 to
0.07, while the difference for the OU portfolio for the same datasets ranges from 0.13 to
0.68. Because the risk-free asset has a Sharpe ratio of zero, this is not surprising. If an
estimation error results in an overallocation to the risk-free asset, we would expect this
to have a bigger impact on λ∗ (ω̂) than if the error results in an overallocation to the
MV portfolio. The MV portfolio has a relatively high value of λ∗ (ω̂) except for the FF
4-Factor dataset, which is the same dataset for which the difference between λ∗ (ω) and
λ∗ (ω̂) for the OC portfolio is relatively large. In general, the MV portfolio is subject to
low estimation risk (the maximum difference between λ∗ (ω̂) and λ∗ (ω) is 0.26), so the OC
portfolio performs relatively well.
The expected turnover numbers provide further evidence that the performance of the tangency portfolio is not representative of the performance of MVE portfolios more generally.
The value of τ ∗ (ω̂) for the OU and OC portfolios is typically in the 25% to 60% range.
This is considerably higher than the value of τ ∗ (ω̂) for the 1/N portfolio, which is less
than 3% in each case. However, the turnover for OU and OC portfolios is tiny compared to
the value of τ ∗ (ω̂) for the tangency portfolio, which exceeds 10,000% for three of the five
datasets. It is under 1000% only for the Mkt/SMB/HML dataset and, even in this case,
it is 3 to 9 times larger than the expected monthly turnover of the other MVE portfolios.
Clearly, considering the results for the tangency portfolio in isolation provides a distorted
picture of the after-transactions-cost performance of MVE portfolios.
2.3
The DeMiguel et al. (2009) results revisited
The evidence from our Monte Carlo experiment suggests that it would interesting to revisit the DeMiguel et al. (2009) results for the tangency, MV, and 1/N portfolios, and to
compare them to the results obtained for the OC portfolio. We report the results of this
comparison in Table 2. Panel A reports the annualized mean, annualized standard deviation, and annualized Sharpe ratio for the time series of monthly excess returns generated
by each of these portfolios using the actual DeMiguel et al. (2009) datasets. We use rolling
estimators with a 120 month window length and assume that transactions costs are zero
when computing these statistics. Panel B reports the minimum, median, and maximum
value of the estimated conditional expected return for each portfolio over the 377 months
in the out-of-sample period. All of the estimated Sharpe ratios reported in Panel A for
the tangency, MV, and 1/N portfolios match those √
reported by DeMiguel et al. (2009) in
their Table 3 (after multiplying their estimates by 12 to obtain annualized statistics).
Panel A reveals that the tangency portfolio has markedly different reward and risk characteristics than the other portfolios. The estimated mean and estimated standard deviation
of its excess return are greater than 100% per year for two of the datasets. This contrasts
11
sharply with the results for the OC portfolio which has reward and risk characteristics
similar to those of the 1/N and MV portfolios. Moreover, the tangency portfolio typically has an estimated expected monthly turnover that is orders of magnitude higher
than that of the OC portfolio. The only exception to the general pattern is provided by
the Mkt/SMB/HML dataset. In this case, the tangency portfolio has the same estimated
expected turnover as the OC portfolio.
Panel B of the table points to why the Mkt/SMB/HML dataset produces atypical results. It appears that the negative estimated return correlations have a substantial impact on the aggressiveness of the tangency portfolio. The median value of µ̂T P,t for the
Mkt/SMB/HML dataset is 6.7% per year. This is relatively low compared to a median
value that ranges from 30.9%to 60.8% per year for the remaining datasets. More importantly, the maximum value of µ̂T P,t is only 15.5% for Mkt/SMB/HML, but ranges from
2,486% to 12,216% per year for the remaining datasets. It is hardly surprising to find that
target estimated expected excess returns of this magnitude produce extreme turnover.
The weights that deliver these targets could not be implemented in practice.
In comparison, the maximum value of µ̂p,t for the OC portfolio ranges from 8.6% to 37.8%
per year, a much more reasonable range. If we discount the results for the tangency portfolio, then the picture that emerges from Table 2 is far more supportive of mean-variance
optimization than that suggested by DeMiguel et al. (2009). The estimated Sharpe ratio
of the OC portfolio exceeds that of the 1/N portfolio for four of the datasets. Indeed,
for the FF 1-Factor and FF 4-Factor datasets, the estimated Sharpe ratio for the OC
portfolio is greater than one and about twice that of the 1/N portfolio. Once again the
Mkt/SMB/HML dataset provides the exception to the general pattern, with an estimated
Sharpe ratio of 0.76 for the OC portfolio versus 0.78 for the 1/N portfolio.
These results, along with the evidence from the Monte Carlo experiment, demonstrate
that changing the target conditional expected excess return can have a substantial impact
on the performance of MVE portfolios. In contrast to our approach, the DeMiguel et al.
(2009) research design does not attempt to match the reward and risk characteristics of
the MVE portfolios under consideration to those of the naive diversification benchmark.
As a result, their analysis is skewed in favor of naı̈ve diversification, especially with respect
to turnover and after-transactions-costs portfolio performance.
For example, DeMiguel et al. (2009) implement a version of the “three-fund” strategy
proposed by Kan and Zhou (2007), but it contains only two funds — the tangency and
MV portfolios — because they rescale the weights for the risky assets to sum to one. The
estimated expected turnover for the resulting portfolio exceeds that for the 1/N portfolio
by a factor of more than a thousand for several of the datasets. However, we find that the
OC portfolio, which is a combination of the same two funds, has a vastly lower turnover
when we target the conditional expected excess return of the 1/N portfolio. We should not
interpret the turnover and other performance figures reported by DeMiguel et al. (2009)
as representative of the three-fund strategy.
12
2.4
Mean-variance timing strategies
Although the OC portfolio performs considerably better than the tangency portfolio, it
is not clear that the OC portfolio would consistently outperform the 1/N portfolio under
plausible transactions costs assumptions. For instance, if we assume that establishing
or liquidating a portfolio position costs 50 basis points, then the estimates of expected
turnover for the OC portfolio reported in Table 2 would entail transactions costs of between
0.4% and 5.7% per year. In view of the potential impact of transactions costs, we regard
turnover as the primary barrier to capitalizing on the gains promised by mean-variance
optimization.
It might be possible to reduce turnover for the OC portfolio by using various techniques
proposed in the literature to improve the performance of mean-variance optimization. 8
However, our interest lies in a different direction. Instead of focusing strictly on portfolio
optimization, we expand the scope of the investigation to include alternative methods of
exploiting sample information about the means and variances of returns. Our objective
is to develop methods of portfolio selection that retain many of the features that make
naı̈ve diversification appealing — nonnegative weights, low turnover, wide applicability —
while improving on its performance. We begin by developing the idea of volatility timing
within the context of the standard portfolio problem of Section 2.1.
2.4.1
Volatility timing
Fleming et al. (2001, 2003) study a class of active portfolio strategies in which the portfolio
weights are rebalanced based on changes in the estimated conditional covariance matrix
of returns. They find that these “volatility-timing” strategies outperform unconditionally
mean-variance efficient portfolios by statistically significant margins. This points to the
potential for a volatility-timing approach to outperform naı̈ve diversification. The question
is how to implement volatility timing in the present setting. Unlike Fleming et al. (2001,
2003), who use futures contracts for their analysis, we want to avoid short sales and
keep turnover as low as possible. Accordingly, we propose a new class of volatility-timing
strategies characterized by four notable features: they do not require optimization, they
do not require covariance matrix inversion, they do not generate negative weights, and
they allow the sensitivity of the weights to volatility changes to be adjusted via a tuning
parameter. The last feature facilitates control over turnover and transactions costs.
To motivate our approach, consider a scenario in which all of the estimated pair-wise
correlations between the excess risky-asset returns are zero, i.e., Σ̂t is a diagonal matrix.
In this case, the weights for the sample MV portfolio are given by
(1/σ̂it2 )
ω̂it = PN
,
2
i=1 (1/σ̂it )
8
i = 1, 2, . . . , N,
(12)
Some recent examples of work in this area include Pastor (2000), Pastor and Stambaugh
(2000), MacKinlay and Pastor (2000), Jagannathan and Ma (2003), Ledoit and Wolf (2004),
Garlappi et al. (2007), and Kan and Zhou (2007).
13
where σ̂it is the estimated conditional volatility of the excess return on the ith risky asset.
Thus, if Σ̂t is restricted to be diagonal for all t, the investor will follow a very simple
volatility timing strategy, i.e., he will rebalance his portfolio based solely on changes in
the relative volatilities of the risky assets.
Obviously we do not expect Σ̂t to actually be diagonal. However, the sample MV portfolio
obtained by setting the off-diagonal elements of Σ̂t to zero might perform better than that
obtained using the usual estimator of Σt . To see why, note that weights in equation (12)
are strictly nonnegative, while the weights obtained using a non-diagonal estimator of
Σt will typically involve short positions in one or more assets. In general, strategies that
permit short sales are more likely to be characterized by extreme weights. We view setting
the off-diagonal elements of Σ̂t to zero as an aggressive form of shrinkage along the lines of
that proposed in Ledoit and Wolf (2003, 2004). Although it may seem unusual to ignore
the return correlations, this results in N (N − 1)/2 fewer parameters to estimate from the
data. Thus the reduction in estimation risk could outweigh the loss of information.
There is also a possibility that we can reduce the impact of the information loss by
modifying the way in which the portfolio weights respond to volatility changes. To see
how, consider the N = 2 case. The estimated weights of the MV portfolio are in general
given by
2
σ̂2t
− σ̂1t σ̂2t ρ̂t
ω̂1t = 2
(13)
2
σ̂1t + σ̂2t
− 2σ̂1t σ̂2t ρ̂t
and ω̂2t = 1 − ω̂1t , where ρ̂t is the estimated conditional correlation between the excess
returns on the two risky assets. Now suppose that σ̂1t = σ̂2t so that ω̂p,t = (1/2, 1/2)0 .
If asset one’s estimated conditional volatility doubles in period t + 1, then we adjust the
portfolio weights to ω̂t+1 = (0, 1)0 for ρ̂t+1 = 1/2, and to ω̂t+1 = (1/5, 4/5)0 for ρ̂t+1 = 0.
Thus the weights are more responsive to volatility changes when the estimated correlation
between the returns is positive.
Although the strategy in equation (12) provides no flexibility in determining how the portfolio weights respond to volatility changes, it belongs to a more general class of volatilitytiming strategies with weights of the form
(1/σ̂it2 )η
,
ω̂it = PN
2 η
i=1 (1/σ̂it )
i = 1, 2, . . . , N,
(14)
where η ≥ 0. The idea behind this generalization is straightforward. The tuning parameter
η is a measure of timing aggressiveness, i.e., it determines how aggressively the investor
adjusts the portfolio weights in response to volatility changes. Setting η > 1 should compensate to some extent for the information loss caused by ignoring the return correlations.
We refer to the portfolio in equation (14) as the VT(η) strategy.
The choice of η also controls the average level of diversification achieved by the VT
strategies. As η approaches zero the cross-sectional variation in the portfolio weights
goes to zero. In the limiting case, η = 0, we recover the naı̈ve diversification strategy.
In contrast, as η approaches infinity, the weight for the asset with the lowest estimated
volatility approaches one and all the other weights approach zero. We expect that, in
14
general, the best choice of η will depend on the level of estimation risk, the level of
transactions costs, and the number of assets, N .
If estimation risk is high, then much of the variation in the parameter estimates may simply
reflect estimation errors. Reducing the value of η would reduce the impact of these errors
on portfolio performance. If transactions costs are high, then reducing portfolio turnover
becomes more important. Reducing the value of η should also reduce turnover. The number
of assets is likely to be important because, for a given cross-sectional dispersion in return
volatilities, the maximum portfolio weight tends to zero as N → ∞. If we keep η fixed
and increase N , we would expect both the cross-sectional and time-series variation in the
portfolio weights to decrease. Allowing η to vary with N counteracts this effect.
2.4.2
Reward-to-risk timing
The volatility-timing strategies of Section 2.4.1 ignore information about conditional expected returns. It is natural to ask, therefore, whether we can improve upon their performance by incorporating such information. Suppose we again consider a scenario in which
all of the estimated pair-wise correlations between the excess risky-asset returns are zero.
The weights for the sample tangency portfolio in this case are given by
(µ̂it /σ̂it2 )
ω̂it = PN
2
i=1 (µ̂it /σ̂it )
i = 1, 2, . . . , N,
(15)
where µ̂it is the estimated conditional mean of the excess return on the ith risky asset.
Thus, if Σ̂t is restricted to be diagonal for all t, the investor will follow a simple rewardto-risk timing strategy.
Because expected returns are typically estimated with less precision than variances, the
strategy in equation (15) is likely to entail significantly higher levels of estimation risk than
the volatility timing strategies. Setting the off-diagonal elements of Σ̂t to zero reduces the
tendency for the sample tangency portfolio to be characterized by extreme long and short
weights, but we could still see extreme weights if µ̂it is negative for some assets because
this could cause the denominator of the fraction on the right side of equation (15) to be
close to zero. We address this possibility by assuming that the investor has a strong prior
belief that µit ≥ 0 for all i and therefore constructs the reward-to-risk timing weights as
(µ̂+ /σ̂ 2 )
ω̂it = PN it + it 2
i=1 (µ̂it /σ̂it )
i = 1, 2, . . . , N,
(16)
where µ̂+
it = max (µ̂it , 0). This is equivalent to assuming that the investor eliminates any
asset with µ̂it ≤ 0 from consideration in period t.
Using the same approach as in Section 2.4.1, we can view equation (16) as one example
of a more general class of reward-to-risk timing strategies that have weights of the form
(µ̂+ /σ̂ 2 )η
ω̂it = PN it + it 2 η
i=1 (µ̂it /σ̂it )
15
i = 1, 2, . . . , N,
(17)
where η ≥ 0. These strategies approach naı̈ve diversification across the assets with positive
estimated expected excess returns as η → 0 and put a weight that approaches one on the
asset with the maximum estimated reward-to-risk ratio as η → ∞. We refer to the portfolio
in equation (17) as the RRT(µ+
t , η) strategy.
It would be easy to allow for more flexibility with respect to the tuning parameters. For
instance, we could consider strategies with weights of the form
(µ̂+ )ηµ /(σ̂ 2 )ησ
ω̂it = PN it + η it 2 η
µ
σ
i=1 (µ̂it ) /(σ̂it )
i = 1, 2, . . . , N,
(18)
where ηµ , ησ ≥ 0. This would allow an investor to specify the sensitivity of the weights to
changes in the means independently from the sensitivity to changes in the volatilities. It
might be advantageous, for example, to reduce ηµ and/or increase ησ for cases in which
estimation risk for expected returns is much higher than that for return volatilities. An
analysis of this generalized approach to reward-to-risk timing should prove interesting.
We focus, however, on the ηµ = ησ case because the resulting timing strategies are based
on the reward-to-risk ratio that is central to mean-variance optimization.
3
Estimating the conditional moments of monthly excess returns
To construct sample analogs of the MVE portfolios discussed in Section 2, we must estimate µt and Σt for each portfolio rebalancing date t. For our baseline analysis we use
fixed-window rolling estimators of µt and Σt . This allows us to directly compare our results
to those of DeMiguel et al. (2009). In the case of the RRT strategies, we also consider
an alternative estimator of µt that is designed to reduce estimation risk by exploiting the
predictions of asset pricing theory. We provide the details below.
3.1
Rolling estimators
Using a rolling data window to estimate µt and Σt is designed to balance the tradeoff between the efficiency gains from using more observations and the loss in forecast precision
from including less timely observations that are less likely to reflect current market conP
ditions. To implement this approach, we define our estimators to be µ̂t = (1/L) L−1
i=0 rt−i
PL−1
and Σ̂t = (1/L) i=0 (rt−i − µ̂t )(rt−i − µ̂t )0 for some window length L. Common choices of
the window length for monthly data are L = 60 and L = 120, i.e., five-year and ten-year
rolling windows. We follow DeMiguel et al. (2009) and set L = 120.
Although rolling estimators of conditional expected excess returns have the advantage of
simplicity, using these estimators in portfolio optimization is likely to entail a high level of
estimation risk. It is well known that we need a long time series of returns to estimate µt
accurately (Merton, 1980). This is true even if µt is time invariant. In general, we would
not expect the estimates of µt obtained using a five- or ten-year window to display good
precision. We therefore consider an alternative estimator of µt for implementing the RRT
strategies that should reduce estimation risk under the circumstances described below.
16
3.1.1
Alternative estimator of conditional expected returns
Many asset pricing models imply a direct relation between the first and second moments
of excess returns. To see how we might be able to exploit this relation in the context
of reward-to-risk timing, suppose that a conditional version of the capital asset pricing
model (CAPM) holds. The conditional CAPM implies that the cross-sectional variation
in conditional expected excess returns is due to cross-sectional variation in conditional
betas. Since the market risk premium is just a scaling factor that multiplies each of the
conditional betas, we can express the weights for the RRT strategy as
(β + /σ 2 )η
ωit = PN it + it 2 η ,
i=1 (βit /σit )
i = 1, 2, . . . , N,
(19)
where βit+ = max(βit , 0), and βit is the period t conditional market beta of asset i.
If we estimate the conditional betas using the same L used for the rolling estimator of
µt , then imposing the pricing restriction implied by the conditional CAPM can lower the
sampling variation of the weights. Consider, for illustration purposes, a scenario in which
rt+1 ∼ i.i.d. N (µ, Σ). Upon further simplification, equation (19) reduces to
(ρ+
/σi )η
ωit = PN i +
,
η
i=1 (ρi /σi )
i = 1, 2, . . . , N,
(20)
where ρ+
i = max(ρi , 0) and ρi is the correlation between the excess return on asset i and
the excess return on the market. Hence, we have replaced µ̂i with σ̂i ρ̂i in the sample
version of the strategy. The asymptotic variance of σ̂i ρ̂i is given by (σi2 /L)(1 − ρ2i /2). 9
In comparison, the asymptotic variance of µ̂i is given by σi2 /L. Although a scenario in
which the expected returns, variances, and covariances are time varying is more complex,
we would expect a similar reduction in estimation risk to obtain.
Now suppose in the alternative that the conditional CAPM does not hold, but conditional
betas capture at least some of the cross-sectional variation in conditional expected returns.
In this case, replacing µ̂it with σ̂it ρ̂it will introduce bias. The substitution may still prove
beneficial, however, if we are replacing an unbiased but high variance estimator with a
biased but lower variance estimator. Consider an estimator of the form µ̂t = µ0 ι where
µ0 > 0 is a scaler. This estimator is undoubtedly biased. Nonetheless, an investor who uses
it, and imposes the constraint ωt0 ι = 1, holds the sample analog of the MV portfolio. This
9
2 and σ
To see this, let σm
im denote the variance of the excess market return and the covariance
between the excess return on asset i and the excess market return. It is easy to show that
√

  
4
0
2σm
d
→
L
N   , 
2 σ
σ̂im − σim
0
2σm
im
2 − σ2
σ̂m
m

2 σ
2σm
im
2 σ 2 (1 + ρ2 )
σm
i
i

 .
See, e.g., Hamilton (1994), page 301. The asymptotic variance of σ̂i ρ̂i = σ̂im /σ̂m follows immediately via the delta method.
17
portfolio often performs better than other sample efficient portfolios because its weights
do not depend on µ̂t , which reduces estimation risk (see, e.g., Jagannathan and Ma, 2003).
This methodology can be extended to allow for multiple risk factors. Consider a K-factor
model and let βijt denote the period t conditional beta of the ith asset with respect to
the jth factor. With a single factor the portfolio weights do not depend on the factor risk
premium provided that ωt0 ι = 1. This is not the case for K > 1. Estimating the factor risk
premiums would introduce additional errors that could easily overwhelm any benefits from
employing the model. Instead, we mimic the approach used to obtain the MV portfolio,
i.e., we assume for the purpose of computing the weights that the factors have identical
risk premiums. The resulting weights for the RRT strategy are
(β̄it+ /σit2 )η
ωit = PN
+
2 η
i=1 (β̄it /σit )
,
i = 1, 2, . . . , N,
(21)
where β̄it+ = max(β̄it , 0) and β̄it = (1/K) K
j=1 βijt is the average conditional beta of asset
i with respect to the K factors. We refer to this portfolio as the RRT(β̄t+ , η) strategy.
P
Whether the gains from using a multi-factor model outweigh the additional bias introduced
by ignoring differences in risk premiums across factors is an empirical question. For the
empirical analysis, we implement the RRT(β̄t+ , η) strategy using the Carhart (1997) fourfactor extension of the Fama and French (1993) three-factor model. We use this model
rather than the conditional CAPM because the literature finds little evidence of a relation
between market betas and average returns after accounting for size and book-to-market
characteristics (Fama and French, 1993). This finding suggests that the conditional CAPM
would perform poorly for a least two of the datasets that we examine.. To estimate the
conditional betas, we use a rolling estimator of the joint conditional covariance matrix of
the excess risky asset and factor returns with the same L that is used to estimate µt .
4
Evaluating Portfolio Performance
In this section we describe the methods used for performance assessment and statistical
inference. As in the Monte Carlo experiment, we use a rolling-sample approach to obtain
+h
a sequence {rpt }Tt=h+1
of out-of-sample excess returns for each portfolio considered. Our
assessment of the out-of-sample performance of the portfolios is based on two criteria.
The first is the Sharpe ratio for the portfolio, i.e., λp = µp /σp where µp = E(rpt ) and σp2 =
Var(rpt ). We estimate this ratio using the sample mean and variance of the excess returns
P +h
P +h
for the out-of-sample period: µ̂p = (1/T ) Tt=h+1
rpt and σ̂p2 = (1/T ) Tt=h+1
(rpt − µ̂p )2 .
The difference in the estimated Sharpe ratios for any two portfolios is one measure of
their relative performance in the out-of-sample tests. We base the reported values of λ̂p
on the annualized values of µ̂p and σ̂p for each portfolio strategy.
Following Fleming et al. (2001, 2003), we also report a measure of relative performance
that is based on quadratic utility. The idea is that quadratic utility can be viewed as a
second-order approximation to the investor’s true utility function. Under this approxima18
tion, the investor’s realized utility in period t + 1 can be expressed as
1
U (Rp,t+1 ) = Wt (1 + Rp,t+1 ) − αWt2 (1 + Rp,t+1 )2 ,
2
(22)
0
where Rp,t+1 = Rf,t+1 + ωp,t
rt+1 is the portfolio return in period t + 1, Wt is wealth in
period t, and α is the coefficient of absolute risk aversion. To facilitate comparisons across
portfolios, we hold αWt constant. This is equivalent to setting the investor’s coefficient of
relative risk, γt = αWt /(1 − αWt ), equal to some fixed value γ. Our performance measure
is the fee (expressed as a fraction of wealth invested) that would equate the expected
utilities generated by two alternative strategies.
Suppose, for example, that with a fee of ∆γ per period imposed on strategy j, the strategies
i and j yield the same expected utility, i.e., E[U (Rpi ,t )] = E[U (Rpj ,t − ∆γ )]. Because the
investor would be indifferent between these two alternatives, we can interpret ∆γ as the
maximum per period fee the investor would be willing to pay to switch from strategy i to
strategy j. Recognizing that E[U (Rpi t )] = E[U (Rpj t )] ⇐⇒ ∆γ = 0, it follows from the
quadratic formula that
∆γ = −γ −1 (1 − γE[Rpj t ]) + γ −1 ((1 − γE[Rpj t ])2 − 2γE[U (Rpi t ) − U (Rpj t )])1/2 .
(23)
We consider two levels of relative risk aversion (γ = 1 and γ = 5), specify naı̈ve diversifiˆ γ , the sample analog of ∆γ , as an annualized basis point
cation as strategy i, and report ∆
value for each active strategy j considered.
4.1
Portfolio turnover and adjustments for transactions costs
Since naı̈ve diversification generates low turnover, active strategies that generate high
turnover are disproportionately affected by the imposition of transactions costs. Consequently, estimating expected turnover is an important aspect of assessing the performance
of active trading strategies. We document the impact of turnover by reporting a second
set of estimated Sharpe ratios and performance fees using returns measured net of transactions costs. The cost of rebalancing to the desired period t+1 weights is subtracted from
the excess portfolio return for period t. Our analysis assumes that the level of transactions
costs is constant across assets and over the sample period.
To illustrate, let R̃p,t denote the portfolio return net of transactions costs for period t.
Under our assumptions, this return is given by
R̃p,t = (1 + Rp,t )
N X
1−c
ω̂i,t
i=1
!
ω̂i,t−1 (1 + Ri,t ) −
− 1,
P
1+ N
i=1 ω̂i,t−1 Ri,t
(24)
where c is the level of proportional costs per transaction. 10 To a close approximation,
the impact of imposing transactions costs can be deduced by subtracting τ̂ c from sample
10
Note that we omit the risk-free asset term from the turnover calculation because the empirical
0 ι = 1 for all t.
analysis is confined to portfolios that impose the restriction ωp,t
19
mean of Rpt . Because we report τ̂ for each strategy, the choice of c used to compute the
net returns is not critical. DeMiguel et al. (2009) follow Balduzzi and Lynch (1999) and
set c equal to 50 basis points. We do the same to facilitate comparisons with their results.
4.2
Statistical inference
We conduct inferences about the relative performance of different strategies using largesample t-statistics. These statistics are constructed using the generalized method of moments (GMM). The analysis is as follows. Suppose we have a set of moment restrictions
of the form E(e(Yt , θ)) = 0, where e(Yt , θ) is a J × 1 vector of disturbances, Yt is a vector
T +h
of random variables, and θ is a J × 1 vector of parameters. If we observe {Yt }t=h+1
, then
PT +h
the GMM estimator θ̂ is the value of θ for which (1/T ) t=h+1 e(Yt , θ) = 0. Hansen (1982)
shows that, subject to regularity conditions, the limiting distribution of θ̂ is
√
d
T (θ̂ − θ) → N (0, V ),
(25)
0
where V = D−1 SD−10 , D = E(∂e(Yt , θ)/∂θ0 ) and S = ∞
j=−∞ E(e(Yt , θ)e(Yt−j , θ) ). We use
equation (25) to derive the asymptotic standard errors of the differences in performance
criteria across strategies and then draw inferences based on the corresponding t-statistics.
P
For instance, let Yt = (rpi t , rpj t ) and consider the Sharpe ratios for strategies i and j. To
obtain the asymptotic standard error of λ̂pj − λ̂pi , we specify the disturbance vector as

e(Yt , θ) =
rpi t − θ1 θ3







rpj t − θ2 θ4



.
 (r − θ θ )2 − θ 2 
 pi t
1 3
3 


(26)
(rpj t − θ2 θ4 )2 − θ42
With this specification of e(Yt , θ), the GMM estimators of θ1 and θ2 coincide with the usual
estimators of λpi and λpj , e.g., θ̂1 = µ̂pi /σ̂pi where µ̂pi and σ̂p2i are the sample analogs of
µpi and σp2i . It follows, then, from equation (25) that
√
d
T ((λ̂pj − λ̂pi ) − (λpj − λpi )) → N (0, dλ Vλ d0λ ),
(27)
where dλ = (−1, 1, 0, 0) and Vλ is the asymptotic covariance matrix implied by equation
(26). If the two strategies have the same Sharpe ratio, we have
√


λ̂pj − λ̂pi  a
T
∼ N (0, 1),
(dλ V̂λ d0λ )1/2
(28)
where V̂λ denotes a consistent estimator of Vλ . 11 To identify strategies that outperform
naı̈ve diversification, we specify naı̈ve diversification as strategy i and report p-values for
11
This estimator is constructed using the sample analog of D together with a heteroscedasticity
20
H0: λpj − λpi ≤ 0 based on the t-statistic in equation (28).
We follow a similar approach to assess the significance of the estimated performance fees.
Recall that ∆γ is a function of 1 − γE(Rpj t ) and E[U (Rpi t ) − U (Rpj t )]. We can therefore
ˆ γ . To do so, we let
use the delta method to derive the asymptotic standard error of ∆
Yt = (Rpi t , Rpj t ) and specify the disturbance vector as

e(Yt , θ) = 

1 − γRpj t − θ1

Rpi t − Rpj t −
(γ/2)(Rp2i t
−
Rp2j t )
− θ2 ,

,
(29)
where the value of γ is prespecified. With this specification of e(Yt , θ), it follows from
equation (25) and the definition of ∆γ that
√
a
ˆ γ − ∆γ ) ∼
T (∆
N (0, d∆ V∆ d0∆ ),
(30)
where d∆ = ∂∆γ /∂θ0 and V∆ is the asymptotic covariance matrix implied by equation
(29). If the two strategies deliver the same expected utility, we have
√
T
ˆγ
∆
(dˆ∆ V̂∆ dˆ0∆ )1/2
!
a
∼ N (0, 1)
(31)
where dˆ∆ and V̂∆ denote consistent estimators of d∆ and V∆ . 12 To identify strategies that
outperform naı̈ve diversification, we specify naı̈ve diversification as strategy i and report
p-values for H0: ∆γ ≤ 0 based on the t-statistic in equation (31).
4.2.1
Bootstrapping the p-values
Since we have no evidence on the quality of the approximation provided by equation
(25) in our setting, we employ a block bootstrap approach to obtain p-values. Let Y =
(Yh+1 , Yh+2 , . . . , YT +h ) denote the sample under consideration. Each bootstrap trial con∗
∗
sists of two steps. First, we construct a resample Y ∗ = (Yh+1
, Yh+2
, . . . , YT∗+h ) using the
stationary bootstrap of Politis and Romano (1994). The resample is such that, in general,
∗
∗
if Yi∗ = Yt , then Yi+1
= Yt+1 with probability π and Yi+1
is drawn randomly from Y with
probability 1−π. This delivers an expected block length of 1/(1−π). Second, we calculate
and autocorrelation consistent estimator of S given by
Ŝ = Γ̂0 +
m
X
(1 − l/(m + 1))(Γ̂l + Γ̂0l ),
l=1
P +h
where Γ̂l = (1/T ) Tt=h+l+1
e(Yt , θ̂)e(Yt−l , θ̂)0 For the empirical analysis we set m = 5.
12 In this case, straightforward calculations show that d = (0, (γE(R ) − 1)−1 ).
pj t
∆
21
√

ϑ̂∗1 = T 
(λ̂∗pj − λ̂∗pi ) − (λ̂pj − λ̂pi )
(dλ V̂λ∗ d0λ )1/2


,
(32)

ˆ∗ − ∆
ˆγ
∆
,
ϑ̂∗2 = T  ∗ γ ∗ ∗0
(dˆ V̂ dˆ )1/2
√
(33)
∆ ∆ ∆
ˆ ∗γ , V̂λ∗ , V̂∆∗ , and dˆ∗∆ denote the estimates for the resample. After carrying
where λ̂∗pi , λ̂∗pj , ∆
out M bootstrap trials in total, we compute the p-values for the t-statistics in equations
(28) and (31) using the observed percentiles of ϑ̂∗1 and ϑ̂∗2 . We set M = 10,000 and π = 0.9,
which delivers an expected block length of 10.
5
Data and Empirical Results
The data for the empirical analysis consist of monthly excess returns on broadly-based
U.S. equity portfolios. The sample period is July 1963 to December 2008 (T + h = 546
monthly observations with h = 120). We consider four datasets in total. Three are drawn
from the data library maintained by Ken French, and one is constructed using the Center
for Research in Security Prices (CRSP) daily stock file. The data library is also the
source of the Treasury bill rate and the factor returns that are used to estimate the
beta coefficients for the four-factor risk model. 13 The risk factors are the excess return
on the market index and the returns on a set of three zero-investment portfolios. These
portfolios are designed to mimic the unobserved factors that lead to systematic differences
in expected excess returns between small and large capitalization stocks (SMB), low and
high book-to-market equity stocks (HML), and low and high momentum stocks (UMD).
We begin the analysis with a dataset formed by using standard industrial classification
(SIC) codes to sort firms into 10 Industry portfolios (10 Industry). We expect this dataset
to pose a substantial challenge to our timing strategies. DeMiguel et al. (2009) find that
with similar data, none of the 14 portfolio selection methods considered performs better
than naı̈ve diversification by a statistically significant margin. Next, we consider a dataset
formed by using market capitalization and book-to-market value to sort firms into 25
portfolios (25 Size/BTM). Sorting firms on these criteria is known to produce a large crosssectional dispersion in average returns. Thus the 25 Size/BTM dataset could potentially
afford better reward-to-risk timing opportunities than the 10 Industry dataset.
In addition to the Size/BTM portfolios, we consider two other datasets that are chosen
based on their potential to provide improved timing opportunities. The first is obtained
by using a momentum measure to sort firms into 10 portfolios (10 Momentum). This is
another sorting scheme that is known to spread average returns. The second is obtained
by using estimated return standard deviations to sort firms into 10 portfolios (10 Volatility). 14 These portfolios are formed in January of each year and rebalanced monthly to
13
14
See http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data library.html.
We are grateful to Richard Price for making these data available to us. The volatility
22
equal weights. The standard deviation breakpoints for the portfolios are estimated using
daily data for the one-year period prior to portfolio formation. The monthly portfolio
returns used for the analysis are obtained by compounding the daily portfolio returns.
5.1
Settings for the empirical analysis
Most of the settings for the empirical analysis have been described earlier. The one remaining task is to specify η for the timing strategies. Setting η = 1 is a natural choice for
the baseline analysis because it delivers the volatility and reward-to-risk timing strategies
that are implied by mean-variance optimization with a diagonal covariance matrix. We
also consider two other values of the timing aggressiveness parameter: η = 2 and η = 4.
These choices are motivated by our analysis of the effect of ignoring the estimated return
correlations, which suggests setting η > 1 to mitigate the information loss.
To document how the timing strategies perform relative to mean-variance optimization, we
report results for four MVE strategies: the MV, OC, and tangency portfolio (TP) strategies, plus a version of the OC strategy that prohibits short sales (OC+ ). We implement
the OC and OC+ strategies by targeting the estimated conditional expected excess return
of the 1/N strategy for each date t in the sample. 15 Targeting the estimated expected
excess return of the 1/N strategy should produce reasonable performance benchmarks for
the timing strategies because the conditional expected excess returns generated by these
strategies must lie between the maximum and minimum conditional expected excess returns of the individual assets.
5.2
Results for the 10 Industry dataset
Table 3 documents the out-of-sample performance of the 1/N , timing, and MVE strategies
for the 10 Industry dataset. The results for the 1/N strategy are reported in the leading
row of the table. In the absence of transactions costs, the values of µ̂p and σ̂p are 5.80%
and 15.04%, which translates into a λ̂p of 0.386. 16 As expected, the estimated expected
turnover is quite low: 2.3% per month. Thus, imposing transactions costs of 50 basis points
has a minor impact on performance. Specifically, λ̂p falls from 0.386 to 0.376.
The results for the VT strategies are shown in Panel A. In the absence of transactions
costs, λ̂p is 0.426 for η = 1, 0.454 for η = 2, and 0.463 for η = 4. These values exceed the
estimated Sharpe ratio for the 1/N strategy and the differences for η = 1 and η = 2 are
portfolios are constructed using the methodology developed in Crawford et al. (2009). See
http://richardp.rice.edu/research/ for details.
15 As discussed in Section 2, we occasionally find that targeting the estimated conditional expected excess return of the 1/N strategy delivers a conditionally inefficient portfolio. In these
cases, we target the estimated conditional expected excess return of the conditionally efficient
portfolio with the same conditional volatility as the identified inefficient portfolio.
16 Note that the results reported in Table 3 for this dataset differ from those reported in Table
2 because the sample period is different and because the SIC codes included in each industry
were changed in 2004. See the data library webpage for details.
23
statistically significant at the 5% level. Moreover, the evidence suggests that risk averse
investors could reap substantial benefits from volatility timing. The value of σ̂p for the
VT strategies is about 0.8 to 2.2 percentage points lower than for the 1/N strategy. As
ˆ γ ranges from 89 to 172 basis points for γ = 5, and all of the estimates
a consequence, ∆
are statistically significant at the 10% level. Of course the gains to volatility timing are
ˆ γ range from 37 to 54 basis points
smaller for less risk averse investors. The values of ∆
for γ = 1 and none of the estimates is statistically significant at the 10% level.
Importantly, all of the VT strategies have low estimated expected turnover. It ranges from
2.4% with η = 1 to 3.6% with η = 4. These values are close to the estimated expected
turnover for the 1/N strategy, so imposing transactions costs has little impact on relative
performance. With transactions costs of 50 basis points, λ̂p falls to 0.416 for η = 1, 0.442
ˆ γ becomes 90 to 166 basis points
for η = 2, and 0.446 for η = 4, and the range of ∆
for γ = 5 and 36 to 51 basis points for γ = 1. Thus turnover is not a concern for the
VT strategies. The difference in the estimated Sharpe ratios of the VT and 1/N strategies
remains statistically significant at the 5% level for η = 1 and η = 2 and all of the estimated
performance fees for γ = 5 remain statistically significant at the 10% level.
The results for the RRT strategies are shown in Panel B. First consider the case in which
we implement the strategies using the standard rolling estimator of µt . The results are
clearly less favorable than for the VT strategies: λ̂p is 0.328 for η = 1, 0.320 for η = 2,
and 0.310 for η = 4 and all of the estimated performance fees are negative, ranging from
−181 to −82 basis points. Moreover, the estimated expected turnover is higher than for
the VT strategies. It ranges from 7.7% per month for η = 1 to 11.9% per month for η = 4.
Hence, the performance of the RRT strategies deteriorates when we impose transactions
costs. The value of λ̂p falls to 0.298 for η = 1, 0.285 for η = 2, and 0.266 for η = 4 and
ˆ γ range from −241 to −114 basis points.
the values of ∆
Although we find no support for reward-to-risk timing in these results, this may be because
the 10 Industry dataset poses an especially difficult challenge for the RRT strategies. Even
if sorting firms according to SIC codes is a reasonable way to identify different industries,
there is no guarantee that the industries will display significant cross-sectional variation
in conditional expected excess returns. If the variation in conditional expected excess
returns across assets is relatively low, then the estimates of µt may convey little useful
information. Our analysis suggests that this plays a role in the unimpressive performance
of the RRT strategies for the 10 Industry dataset.
Consider, for instance, the evidence in panel A of Figure 1. The plot on the left side of the
figure shows the cross-sectional dispersion in the annualized sample mean of the excess
industry portfolio returns. The sample means lie in a relatively narrow range: 8.7% to 14%.
Moreover, seven of the values are between 10% and 13%. We would expect to find greater
variation in these values if there were substantial cross-sectional dispersion in conditional
expected excess returns. In contrast, the plot on the right side of the figure reveals that
the dispersion in the sample volatilities is considerably larger than the dispersion in the
sample means. The range is from 14.6% to 24.5%. This is consistent with finding that the
volatility timing outperforms reward-to-risk timing.
24
To investigate further, we turn to the case in which we implement the RRT strategies
using an estimator of µt derived from our four-factor risk model. Since this estimator
should display less sampling variation than the standard rolling estimator of µt , we anticipate better performance from the RRT strategies. The improvement in performance
is substantial. In the absence of transactions costs, λ̂p is 0.418 for η = 1, 0.430 for η = 2,
ˆ γ range from 38 to 51 basis points for γ = 1, and
and 0.437 for η = 4. The values of ∆
from 64 to 106 basis points for γ = 5. These results are comparable to those for the VT
strategies. Although the differences in the estimated Sharpe ratios of the RRT and 1/N
strategies are not statistically significant, the estimated performance fees for γ = 5 are
statistically significant at the 10% level.
These findings point to a sizeable reduction in estimation risk from using the four-factor
risk model. Additional evidence of this reduction can be found in the estimates of expected
turnover, which are roughly half as large as those generated by using the standard rolling
estimator of µt . The range is from 3% per month for η = 1 to 6.2% per month for η = 4.
As a consequence, imposing transactions costs has a minor impact on performance. The
value of λ̂p falls to 0.406 for η = 1, 0.413 for η = 2, and 0.410 for η = 4 and the range of
ˆ γ becomes 27 to 39 basis points for γ = 1 and 61 to 83 basis points for γ = 5. Two of the
∆
ˆ γ for γ = 5 remain statistically significant at the 10% level. Hence, we
three values of ∆
find that the RRT strategies are capable of outperforming naı̈ve diversification provided
that they are implemented using our alternative estimator of µt .
The evidence in panel C, which documents the performance of the MVE strategies, lends
additional context to these findings. In the absence of transactions costs, the MVE strategies deliver mixed results. The TP strategy displays the worst performance by far, with
a negative µ̂p and a value of σ̂p that exceeds the estimated volatility of every other MVE
strategy by a factor of four. Its estimated Sharpe ratio is exceedingly low: −0.252. In
comparison, the MV and OC strategies have estimated Sharpe ratios of 0.506 and 0.447.
Both outperform the 1/N strategy, although the differences in the estimated Sharpe ratios
are not statistically significant. Prohibiting short sales causes the performance of the OC
strategy to deteriorate. Its estimated Sharpe ratio falls to 0.414.
It may seem odd to find that the performance gains for the MV and OC strategies, which
are larger than those for the VT strategies, are statistically insignificant. This is explained
by the correlation between the excess returns for the different strategies. For example, the
estimated correlation between the excess returns for the 1/N and VT(1) strategies is 0.99,
while that for the 1/N and OC strategies is 0.74. Since equation (27) implies that the
standard error of λ̂pj − λ̂pi is a decreasing function of this correlation, it takes a larger
difference in the estimated Sharpe ratios to conclude that the OC strategy outperforms
the 1/N strategy. This is also the case for the estimated performance fees. The values of
ˆ γ for the MV and OC strategies range from 40 to 228 basis points, but only one of the
∆
estimates is statistically significant at the 10% level.
High turnover is the most troublesome aspect of the performance of the MVE strategies.
The TP strategy is clearly an outlier in this respect, with an estimated expected turnover
of 2,729% per month. This value exceeds the estimated expected turnover for the other
25
MVE strategies by two orders of magnitude. Indeed, there are a number of months in which
the turnover of the TP strategy exceeds 20,000%. Since this level of turnover implies that
rebalancing costs would consume all of the investor’s wealth, it is impossible to compute
meaningful after-transactions-costs values of the performance measures. This is indicated
by a “–” entry in panel C.
The remaining estimates of expected turnover are much more reasonable: 16.1% for the
MV strategy, 28.5% for the OC strategy, and 10.9% for the OC+ strategy. Nonetheless, the
values are such that there is substantial deterioration in the performance of the strategies
in the presence of transactions costs. The estimated Sharpe ratios for the MV and OC
strategies fall to 0.431 and 0.317, and the estimated performance fees are −118 and 16
basis points for γ = 1 and −20 and 147 basis points with γ = 5. None of the performance
gains is statistically significant at the 10% level. Note also that once we account for
transactions costs the OC+ strategy has a higher Sharpe ratio than the OC strategy
because prohibiting short sales leads to a substantial reduction in turnover.
5.3
Results for the 25 Size/BTM dataset
The results for the 10 Industry dataset foreshadow the crucial role of turnover in our
investigation. With the exception of the TP strategy, all of the MVE strategies perform
better than naı̈ve diversification in the absence of transactions costs. Although the analysis is inconclusive because most of the performance gains are statistically insignificant,
the picture that emerges from the initial evidence is generally supportive of mean-variance
optimization. This is no longer true in the presence of transactions costs because the advantage of the MVE strategies is eroded by high turnover. The results indicate, therefore,
that controlling turnover is key to improving mean-variance methods of portfolio selection.
The timing strategies are largely successful in this regard for the 10 Industry dataset, but
additional evidence is needed to draw firm conclusions.
To develop this evidence, we turn to the 25 Size/BTM dataset. If our hypothesis regarding the relation between the cross-sectional dispersion in conditional expected returns and
the performance of the RRT strategies is correct, then we should find stronger support
for reward-to-risk timing with this dataset. As panel B of Figure 1 shows, the annualized
sample mean of the excess returns for the size and book-to-market portfolios ranges from
7.2% to 18.8%, which is more than twice the range for the industry portfolios. The estimates of µt should therefore have more value than in the case of the 10 Industry dataset.
We anticipate little change in the performance of VT strategies since the range of the
annualized sample volatilities is 15.3% to 28.4%, which is only a few percentage points
larger than that for the industry portfolios.
Table 4 documents the out-of-sample performance of the 1/N , timing, and MVE strategies
for the 25 Size/BTM dataset. The layout of the table is identical to that of Table 3. In the
absence of transactions costs, the values of µ̂p and σ̂p for the 1/N strategy are 8.25% and
17.64%, which translates into a λ̂p of 0.468. Once again this strategy has low estimated
expected turnover — 1.7% per month — so the impact of imposing transactions costs is
26
quite small. The value of λ̂p falls to 0.462.
The results for VT strategies are similar to those for the 10 Industry dataset. In the
absence of transactions costs, volatility timing delivers higher values of λ̂p than naı̈ve
diversification: 0.492 for η = 1, 0.496 for η = 2, and 0.484 for η = 4. The difference in
the estimated Sharpe ratios is statistically significant at the 10% level for η = 1. The
estimated performance fees range from −39 to 15 basis points for γ = 1, and from 81 to
111 basis points for γ = 5. In the latter case, all of the fees are statistically significant at
the 10% level. Because the VT strategies have low turnover (1.8% to 3.1% per month),
imposing transactions costs does not alter the nature of these findings.
As anticipated, the evidence in favor of reward-to-risk timing is more compelling for the
10 Momentum dataset than for the 10 Industry dataset. In the absence of transactions
costs, using the standard rolling estimator of µt produces a λ̂p of 0.535 for η = 1, 0.555 for
η = 2, and 0.569 for η = 4. The increase in the estimated Sharpe ratio relative to naı̈ve
diversification is statistically significant at the 5% level in each case. Moreover, all of the
estimated performance fees, which range from 101 to 147 basis points for γ = 1 and from
137 to 206 basis points for γ = 5, are statistically significant at the 5% level. These gains
are achieved with only a modest increase in estimated expected turnover relative to the
1/N strategy: 3.6% for η = 1, 5.0% for η = 2, and 8.0% for η = 4. Hence, the gains remain
statistically significant in the presence of transactions costs, with estimated performance
fees that range from 90 to 109 basis points for γ = 1 and from 127 to 167 basis points for
γ = 5.
Interestingly, using the estimator of µt derived from our four-factor risk model produces
similar estimated Sharpe ratios for the RRT strategies. In the absence of transactions
costs, the value of λ̂p is 0.524 for η = 1, 0.555 for η = 2, and 0.579 for η = 4. Because
these values are nearly identical to those obtained using the rolling estimator of µt , it
may appear that there is no benefit from employing the risk model. This is not the case,
however. The estimated expected turnover is 1.8% for η = 1, 2.0% for η = 2, and 2.9%
for η = 4. These values are two to three times lower than those obtained using the rolling
estimator of µt . Thus the risk model achieves substantial reductions in turnover while
delivering comparable performance gains.
With such low levels of turnover, imposing transactions costs has little impact on the
performance of the RRT strategies. The value of λ̂p falls to 0.518 for η = 1, 0.547 for
η = 2, and 0.568 for η = 4. In each case, the increase in the estimated Sharpe ratio relative
to naı̈ve diversification is statistically significant at the 1% level. All of the estimated
performance fees, which range from 83 to 152 basis points for γ = 1 and from 118 to 220
basis points for γ = 5, are statistically significant at the 5% level, and all but one are
statistically significant at the 1% level.
The evidence for the MVE strategies provides additional perspective on these results. The
performance of the TP strategy is again exceedingly poor. In the absence of transactions
costs, it has an estimated annualized mean return of −58.8% and an estimated annualized
volatility of 777.8%. This translates into an estimated Sharpe ratio of −0.076. Note that
27
we report a “–” in the table for the estimated performance fees. This is because the results
for the TP strategy are so poor that the quadratic equation used to find the performance
fees has no real roots. Thus there is no fixed fee that equates the estimated expected
utilities generated by the TP and 1/N strategies. The TP strategy is also characterized
by extreme estimated expected turnover: 23,380% per month.
In contrast, the MV and OC strategies perform well in the absence of transactions costs.
Their estimated Sharpe ratios — 0.853 and 1.006 — are much higher than the estimated
Sharpe ratio of the 1/N strategy, and the differences are statistically significant at the 5%
level. Moreover, the estimated performance fees for the MV and OC strategies, which are
386 and 651 basis points for γ = 1, and 664 and 892 basis points for γ = 5, are statistically
significant at the 10% level. Thus the evidence suggests that mean-variance optimization
is superior to naı̈ve diversification if transactions costs are zero.
Notice, however, that imposing transactions costs leads to a marked reduction in the
performance of the MVE strategies. The estimated expected turnover for the MV and
OC strategies is 79% and 100% per month. As a consequence, λ̂p falls to 0.496 for the
MV strategy and to 0.569 for the OC strategy in the presence of transactions costs.
These values are still higher than the estimated Sharpe ratio for the 1/N strategy, but
the differences are no longer statistically significant. This is also true for the estimated
performance fees. Unlike the timing strategies, the MVE strategies simply generate too
much turnover to be competitive with naı̈ve diversification.
5.4
Results for the 10 Momentum dataset
In view of the results for the 25 Size/BTM portfolios, it seems clear that performance
of the timing strategies is influenced by dataset characteristics. This is not a surprise.
Intuitively, we would expect the effectiveness of volatility and reward-to risk timing to
depend on both the cross-sectional and time series variation in the conditional means and
volatilities. These characteristics are undoubtedly affected by the scheme used to sort firms
into portfolios. In the case of the 25 Size/BTM dataset, the sorting scheme is explicitly
designed to increase the cross-sectional dispersion in conditional expected returns. To the
extent that it does so, we should see an improvement in the signal-to-noise ratio of our
rolling estimator of µt , which should benefit both the timing and MVE strategies. The
evidence is consistent with the presence of such benefits, but the performance gains for
the MVE strategies are eroded by transactions costs.
To see if these findings hold more generally, we consider a dataset that is obtained by
sorting firms into portfolios using a momentum measure. This is another case in which
the sorting scheme is explicitly designed to spread conditional expected returns (see,
e.g., Jegadeesh and Titman, 1993). The evidence in panel C of Figure 1 suggests that it
succeeds in this respect. The annualized sample mean for the momentum portfolios ranges
from zero to 17.9%. Since this is larger than the range for the size and book-to-market
portfolios, we again anticipate that the RRT strategies will perform well relative to naı̈ve
diversification. The range of the annualized sample volatilities — 15.4% to 27.0% — is
comparable to that for both the 25 Size/BTM and 10 Industry datasets.
28
Table 5 documents the out-of-sample performance of the 1/N , timing, and MVE strategies
for the 10 Momentum dataset. In the absence of transactions costs, the values of µ̂p , σ̂p ,
and λ̂p for the 1/N strategy are 4.70%, 16.68%, and 0.282. Imposing transactions costs
reduces the value of λ̂p to 0.276. In comparison, the value of λ̂p for the VT strategies
ranges from 0.330 to 0.375 with no transactions costs and the increase in the estimated
Sharpe ratio relative to naı̈ve diversification is statistically significant at the 1% level in
each case. These gains translate into estimated performance fees of 68 to 128 basis points
for γ = 1, and 117 to 215 basis points for γ = 5, all of which are statistically significant at
the 5% level. Because the estimated expected turnover is only 1.7% to 2.6% per month,
imposing transactions costs has little effect on the results.
The performance of the RRT strategies is even more compelling. In the absence of transactions costs, the standard rolling estimator of µt produces a λ̂p of 0.455 for η = 1, 0.476 for
η = 2 and 0.497 for η = 4. The differences relative to naı̈ve diversification are statistically
significant at the 1% level. This is also the case for the estimated performance fees, which
range from 285 to 381 basis points. The estimates of expected turnover — 5.8%, 6.1%,
and 6.8% — are higher than for the 1/N strategy, but the differences are not dramatic.
Hence, imposing transactions does not alter our inferences.
The results using the estimator of µt derived from our four-factor risk model are similar.
In the absence of transactions costs, the value of λ̂p is 0.398 for η = 1, 0.447 for η = 2,
and 0.478 for η = 4 and the estimated performance fees range from 179 to 335 basis
points. All of the gains are statistically significant at the 1% level. As was the case for the
other datasets, the estimates of expected turnover are two to three times lower than those
produced by the rolling estimator of µt . And, again, the impact of imposing transactions
costs is minimal.
We also find that the performance of the MVE strategies is more competitive for this
dataset. The TP strategy clearly delivers extreme results: an estimated annualized mean
return of 8,321%, and estimated annualized volatility of 5,680%, and an estimated expected monthly turnover of 18,490%. However, the MV and OC strategies have estimated
Sharpe ratios of 0.496 and 0.589 in the absence of transactions costs and the estimated
performance fees range from 302 to 536 basis points. All of the gains are statistically
significant at the 10% level. Turnover is still an issue, but the estimates are considerably
lower than for the 25 Size/BTM dataset. Thus the majority of the performance gains
remain statistically significant in the presence of transactions costs.
5.5
Results for 10 Volatility dataset
The evidence for the 25 Size/BTM and 10 Momentum datasets suggests that the performance of RRT strategies is related to the cross-sectional dispersion in conditional expected
excess returns. Accordingly, we posit that the performance of the VT strategies is related
to the cross-sectional dispersion in conditional return volatilities. To test this hypothesis,
we consider a final dataset that is obtained by sorting firms into portfolios based on estimates of historical volatility. Panel D of Figure 1 shows that the resulting cross-sectional
29
dispersion in the annualized sample volatilities, which range from 10.9% to 34.9%, is approximately twice that for the other three datasets. The range for the annualized sample
means, on the other hand, is relatively narrow at 12.0% to 16.5%.
Table 6 documents the out-of-sample performance of the 1/N , timing, and MVE strategies
for the 10 Volatility dataset. As anticipated, the VT strategies perform particularly well.
In the absence of transactions costs, the 1/N strategy has an estimated Sharpe ratio
of 0.477 and an estimated expected turnover of 1.7%. The VT strategies have estimated
Sharpe ratios that range from 0.576 to 0.712, and estimated expected turnovers that range
from 1.5% to 1.6%. The reductions in σ̂p relative to naı̈ve diversification, which are in 3
to 7 percentage point range, would be quite valuable to risk averse investors, even in the
presence of transactions costs. The estimated performance fees range from 297 to 498 basis
points with γ = 5 and all of the estimates are statistically significant at the 1% level.
The RRT strategies also perform well for this dataset. In the absence of transactions costs,
the value of λ̂p ranges from 0.547 to 0.647 using the standard rolling estimator of µt and
from 0.520 to 0.623 using the estimator of µt derived from our four-factor risk model.
In each case, however, the RRT strategy for a given value of η performs worse than the
corresponding VT strategy. Apparently the estimates of µt convey little useful information
for the volatility portfolios, which is not surprising given the low cross-sectional dispersion
in the sample means. The estimates of expected turnover are similar to those obtained
with the other datasets: 2.9% to 6.6% using the standard rolling estimator and 1.7% to
2.7% using the four-factor risk model.
The results for the MVE strategies follow the same general pattern as for previous datasets.
If we ignore the TP strategy, then the evidence indicates that turnover is the primary
barrier to outperforming naı̈ve diversification. In the absence of transactions costs, the
MV and OC strategies have estimated Sharpe ratios of 0.766 and 0.930 and the differences
relative to naı̈ve diversification are statistically significant at the 10% level. The estimated
performance fees range from 34 to 751 basis points. However, the estimates of expected
turnover are 35% and 54.7% per month, so the performance deteriorates sharply in the
presence of transactions costs. The estimated Sharpe ratios fall to 0.566 and 0.641, and
the differences relative to naı̈ve diversification are no longer statistically significant.
5.6
Sensitivity analysis
We have established that volatility timing and reward-to-risk timing are capable of outperforming naı̈ve diversification, even in the presence of relatively high transactions costs.
However, the results for the RRT strategies are somewhat sensitive to the choice of estimator for µt , suggesting that estimator efficiency is an important consideration. Two
aspects of the efficiency issue are worth noting. First, we use what is probably an inefficient estimator of Σt to facilitate comparisons with DeMiguel et al. (2009). There are
almost certainly better ways to model the dynamics of conditional variances and covariances that could be implemented in our framework. This suggests an avenue for future
research. Second, we set L = 120 for the rolling estimators, again to facilitate comparisons with DeMiguel et al. (2009). Although we expect our conclusions to be robust to
30
the choice of L, this bears further investigation.
To address the issue, we repeat the entire analysis using L = 60, i.e., a window length
of five years. Cutting the window length in half could result in a substantial increase in
estimation risk. For example, it doubles the asymptotic variance of the rolling estimators
under circumstances in which µt and Σt are time invariant. If this is the dominant effect,
then the performance of the timing strategies should deteriorate. On the other hand, reducing the window length could also result in an improved ability to track the dynamics
of µt and Σt . If this is the dominant effect, then the performance of the timing strategies should improve. In general, therefore, we expect dataset characteristics to play an
nontrivial role in determining the sensitivity of the results to the choice of L.
Table 7 summarizes the results of the investigation. In the interest of space, we omit the
findings for the 10 Industry dataset and report only a few key statistics for the other
three datasets. Specifically, we report the estimated Sharpe ratio for each strategy in the
absence and presence of transactions costs, the p-values of the differences in estimated
Sharpe ratios using the 1/N strategy as our benchmark, and the estimated expected
turnover. A supplemental appendix that contains the full set of results is available from
the authors on request.
Using a shorter window length has little impact on the performance of the VT strategies.
In the absence of transactions costs, λ̂p ranges from 0.49 to 0.50 for the 25 Size/BTM
portfolios, 0.32 to 0.36 for the 10 Momentum portfolios and from 0.57 to 0.70 for the 10
Volatility portfolios. The estimates of expected turnover are somewhat higher than with
L = 120 but the increase is not dramatic. Hence, the performance of the strategies in the
presence of transactions costs is similar to that observed with the longer window length.
The values of λ̂p for the 25 Size/BTM and 10 Momentum portfolios are higher than λ̂p
for the 1/N strategy and the differences relative to the 1/N portfolios are statistically
significant at the 10% level in four of the six cases.
The shorter window length also has little impact on the performance of the RRT strategies.
The results obtained using the rolling estimator of µt show some erosion in estimated
Sharpe ratios for the 10 Momentum portfolios and some improvement for the 25 Size/BTM
portfolios and the 10 Volatility portfolios. Again, however, the changes are not dramatic.
The same is true for the changes that occur using the estimator of µt derived from the four
factor risk model. Although the estimates of expected turnover are somewhat higher than
with L = 120, the increase is not large enough to have much impact on transactions costs.
Thus, it appears that our conclusions regarding the performance of the timing strategies
are not overly sensitive to the choice of L.
6
Closing Remarks
In a recent article, DeMiguel et al. (2009) raise serious questions about the value of meanvariance optimization as a method of portfolio selection. Using a range of datasets, the
authors investigate the out-of-sample performance of the standard mean-variance model
along with a large number of variants developed in the literature to mitigate the impact
31
of estimation risk. They find that “there is no single model that consistently delivers a
Sharpe ratio or a CEQ return that is higher than that of the 1/N portfolio” and conclude
that “there are still many ‘miles to go’ before the gains promised by optimal portfolio
choice can actually be realized out of sample.”
We provide a different perspective on the performance of the mean-variance model vis-a-vis
the 1/N strategy. Our analysis indicates that DeMiguel et al. (2009) casts mean-variance
optimization in such an unfavorable light largely because their research design implicitly
targets a conditional expected return that greatly exceeds the conditional expected return of the 1/N strategy. This magnifies estimation risk and leads to excessive portfolio
turnover. If the mean-variance model is instead implemented by targeting the conditional
expected return of the 1/N strategy, it generally performs better than this strategy in the
absence of transactions costs. It is only after we consider the differences in transactions
costs across strategies that the mean-variance model has difficulty outperforming the 1/N
strategy by statistically significant margins.
Motivated by our findings for the mean-variance model, we propose two alternative methods of mean-variance portfolio selection — volatility timing and reward-to-risk timing —
that are designed to exploit sample information in a way that mitigates the impact of
estimation risk. Importantly, each of these methods allows an investor to exercise some
control over turnover, and hence transactions costs, via a tuning parameter that can be
interpreted as a measure of timing aggressiveness. We find that both types of timing
strategies have the ability to outperform naı̈ve diversification for a range of datasets. This
is true even after we incorporate relatively high transactions costs. Reward-to-risk timing
appears to be a particularly promising strategy when it is implemented using estimates
of conditional expected returns obtained from a four-factor risk model.
A convenient feature of timing strategies is that they require only univariate models for the
conditional first and second moments of excess returns. We use simple rolling estimators
of these moments in the empirical analysis above. It would be interesting to investigate
whether more sophisticated estimation techniques lead to significant benefits. For example,
we might model the daily excess return on each asset as a GARCH(1,1) process and use
the resulting forecasts of daily excess return variances to construct forecasts of monthly
excess return variances. Switching to daily data might significantly improve the precision
of our variance estimates. We leave this and other extensions to future research.
32
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34
0.47
0.75
0.56
0.47
0.75
0.75
1/N
OU
OC
MV
TP
TP†
λ∗ (ω)
0.02
0.10
0.41
0.49
0.45
0.47
λ∗ (ω̂)
179.8
138.5
0.349
0.322
0.234
0.022
τ ∗ (ω̂)
FF 10 Industry
0.74
0.74
0.74
0.64
0.57
0.44
λ∗ (ω)
0.00
0.13
0.41
0.57
0.55
0.45
λ∗ (ω̂)
146.6
105.1
0.284
0.238
0.163
0.029
τ ∗ (ω̂)
International
0.99
0.99
0.99
0.97
0.94
0.78
λ∗ (ω)
0.05
0.87
0.86
0.90
0.93
0.78
λ∗ (ω̂)
0.215
0.041
0.064
0.043
0.025
0.024
τ ∗ (ω̂)
Mkt/SMB/HML
1.82
1.82
1.82
1.11
1.01
0.56
λ∗ (ω)
0.03
0.33
1.14
1.05
0.90
0.56
λ∗ (ω̂)
31.38
22.43
0.507
0.859
0.725
0.016
τ ∗ (ω̂)
FF 1-Factor
1.92
1.92
1.92
1.88
0.15
0.61
λ∗ (ω)
0.01
0.32
1.06
1.06
-0.11
0.61
λ∗ (ω̂)
167.2
92.5
0.604
0.502
0.146
0.020
τ ∗ (ω̂)
FF 4-Factor
The table documents the impact of sampling variation in the portfolio weights on the out-of-sample performance of the 1/N , OU, OC, MV, and
tangency portfolio (TP) strategies. A resampling approach is employed to generate an i.i.d. sequence of length T ∗ + h from the empirical distribution
of excess risky-asset returns for each of the following DeMiguel et al. (2009) datasets: the ten Fama-French industry portfolios plus the market
portfolio (“FF 10 Industry”), the eight MSCI developed market portfolios plus the MSCI world market portfolio (“International”), the Fama-French
market, size, and value factor portfolios (“Mkt/SMB/HML”), the 20 Fama-French size and book-to-market portfolios plus the market portfolio (“FF
1-Factor”), and the 20 Fama-French size and book-to-market portfolios plus the market, size, value, and momentum factor portfolios (“FF 4-Factor”).
This sequence is used to compute population values — via Monte Carlo integration — of the following performance criteria for the each strategy: the
annualized Sharpe ratio implied by the true, time-invariant weights (λ∗ (ω)), the annualized Sharpe ratio when the portfolio is rebalanced monthly
using the estimated weights (λ∗ (ω̂)), and the expected monthly turnover (as a fraction of wealth invested) when the portfolio is rebalanced monthly
using the estimated weights (τ ∗ (ω̂)). The OU, OC, MV, and TP strategies are implemented using a 120-month rolling estimator of the conditional
mean vector and conditional covariance matrix of excess returns, and the OU and OC strategies target the estimated conditional expected excess
return of the 1/N strategy. In each case, the data are generated by drawing h = 120 times with replacement from the first 120 observations of the
dataset to obtain the sample used to initialize the rolling estimators, and T ∗ = 1, 000, 000 times with replacement from the remaining observations to
obtain the sample used for the Monte Carlo integration. The results reported in the last line of the table are obtained by using a subset of the TP
strategy returns for the Monte Carlo integration. See the text for additional details.
Table 1
Monte Carlo Evidence on Estimation Risk and Turnover
473
τ̂p
0.47 0.02
0.54 0.46
0.49 0.65
0.24
λ̂p
λ̂p
22.7
-3.2
-1.7
-0.3
TP
1/N
MV
OC
Max.
7.9
4.2
8.4
14.1
12.2
15.8
47.4 12,216
Min. Med.
0.44
0.52
0.48
-2.4
-3.4
-2.4
7.6
6.4
8.9
12.6
12.0
16.2
30.9 2,486
Min. Med. Max.
17.2
τ̂p
0.03
0.21
0.29
170 -0.11 1077
σ̂p
6.66 15.07
7.60 14.72
7.10 14.75
-19.5
µ̂p
International
Panel B: Estimated Conditional Expected Returns
7.14 15.23
7.07 13.13
6.55 13.30
1/N
MV
OC
457
108
σ̂p
TP
µ̂p
Panel A: Summary Statistics
FF 10 Industry
6.36
5.56
6.26
7.47
σ̂p
1.3
1.8
2.6
3.4
3.7
4.3
5.0
6.7
τ̂p
0.78 0.02
0.86 0.02
0.76 0.06
0.76 0.06
λ̂p
8.5
8.6
8.6
15.5
Min. Med. Max.
4.93
4.80
4.77
5.66
µ̂p
Mkt/SMB/HML
812
σ̂p
-0.5
-3.1
-0.5
38.1
16.5
22.5
36.7
60.8 9,191
8.6
14.2
16.8
212
τ̂p
0.56 0.02
0.96 0.74
1.15 0.95
0.04
λ̂p
Min. Med. Max.
10.47 18.62
12.86 13.37
16.32 14.18
35.92
µ̂p
FF 1-Factor
219
σ̂p
τ̂p
0.64 69.5
λ̂p
0.4
-0.7
0.4
3.6
8.0
-0.1
8.0
15.5
0.8
15.5
34.6 7,635
Min. Med. Max.
9.93 16.35 0.61 0.02
-0.20 3.09 -0.06 0.13
5.69 4.94 1.15 0.46
139
µ̂p
FF 4-Factor
The table documents key sample characteristics of the 1/N , MV, OC, and tangency portfolio (TP) strategies for five of the six DeMiguel et al.
(2009) datasets: the ten Fama-French industry portfolios plus the market portfolio (“FF 10 Industry”), the eight MSCI developed market portfolios
plus the MSCI world market portfolio (“International”), the Fama-French market, size, and value factor portfolios (“Mkt/SMB/HML”), the 20
Fama-French size and book-to-market portfolios plus the market portfolio (“FF 1-Factor”), and the 20 Fama-French size and book-to-market
portfolios plus the market, size, value, and momentum factor portfolios (“FF 4-Factor”). Panel A reports the annualized mean excess return
(µ̂p ), annualized excess return standard deviation (σ̂p ), annualized Sharpe ratio (λ̂p ), and average monthly turnover expressed as a fraction
of wealth invested (τ̂ ) for each strategy under the assumption that transactions costs are zero. Panel B reports the minimum, median, and
maximum of the annualized time-series estimates of the conditional expected excess return for each strategy. The MV, OC, and TP strategies
are implemented using a 120-month rolling estimator of the conditional mean vector and conditional covariance matrix of excess returns, and
the OC strategy targets the estimated conditional expected excess return of the 1/N strategy. The sample period is July 1963 to November
2004 for the Fama-French datasets and January 1970 to July 2001 for the international dataset (497 and 379 monthly observations, respectively).
In each case, the first 120 observations are held out to initialize the rolling estimators. See the text for a detailed description of each strategy.
Table 2
Characteristics of the 1/N and MVE Strategies for the DeMiguel et al. (2009) Datasets
5.80
15.04
σ̂p
0.386
0.418
0.430
0.437
Panel C: Mean-Variance Efficient
MV
6.48 12.81 0.506
OC
5.96 13.35 0.447
5.39 13.04 0.414
OC+
TP
-14.86 59.02 -0.252
14.63
14.41
14.13
RRT(β̄t+ ,1)
RRT(β̄t+ ,2)
RRT(β̄t+ ,4)
6.12
6.20
6.17
Timing
0.328
0.320
0.310
Panel B: Reward-to-Risk
5.01 15.30
RRT(µ+
t ,1)
RRT(µ+
,2)
5.00 15.65
t
,4)
5.06 16.35
RRT(µ+
t
37
54
44
38
49
51
Strategies
0.129
99
0.290
40
0.364
-13
0.995
-3,717
0.115
0.130
0.181
Strategies
0.908
-82
0.865
-89
0.830
-94
Panel A: Volatility Timing Strategies
VT(1)
6.04 14.18 0.426
0.015
VT(2)
6.12 13.48 0.454
0.040
VT(4)
5.93 12.81 0.463
0.163
1/N
µ̂p
0.269
0.407
0.539
0.985
0.169
0.190
0.260
0.890
0.823
0.767
0.119
0.196
0.355
228
139
104
-13,423
64
88
106
-99
-129
-181
89
147
172
No Transactions Costs
vs. 1/N
vs. 1/N
ˆ
ˆ5
p-val
∆1 p-val
∆
λ̂p
0.084
0.211
0.212
0.754
0.064
0.065
0.096
0.883
0.880
0.902
0.004
0.012
0.083
p-val
0.161
0.285
0.109
27.59
0.030
0.040
0.062
0.077
0.089
0.119
0.024
0.028
0.036
0.023
τ̂
0.431
0.317
0.363
–
0.406
0.413
0.410
0.298
0.285
0.266
0.416
0.442
0.446
0.376
λ̂p
0.303
0.683
0.561
–
0.140
0.174
0.272
0.960
0.929
0.914
0.016
0.046
0.187
16
-118
-64
–
34
39
27
-114
-128
-151
36
51
36
0.461
0.758
0.698
–
0.198
0.247
0.366
0.949
0.909
0.877
0.124
0.209
0.378
147
-20
52
–
61
78
83
-131
-169
-241
90
145
166
0.189
0.542
0.347
–
0.076
0.091
0.154
0.934
0.932
0.957
0.004
0.013
0.090
Transactions Costs = 50bp
vs. 1/N
vs. 1/N
ˆ
ˆ 5 p-val
p-val
∆1 p-val
∆
The table summarizes the out-of-sample performance of the 1/N strategy (row one), three volatility timing strategies (Panel A), six reward-to-risk
timing strategies (Panel B), and four MVE strategies (Panel C) for the 10 Industry portfolios. It reports the following sample statistics for the
time series of monthly excess returns generated by each strategy: the annualized mean (µ̂p ), the annualized standard deviation (σ̂p ), the annualized
Sharpe ratio (λ̂p ), the average monthly turnover expressed as a fraction of wealth invested (τ̂ ), the annualized basis point fee that an investor with
quadratic utility and constant relative risk aversion of γ = 1 and γ = 5 would be willing to pay to switch from the 1/N strategy to the timing
ˆ γ ), the p-value for the difference between the annualized Sharpe ratio produced by the timing or MVE strategy and the 1/N
or MVE strategy (∆
strategy (“vs. 1/N p-val”), and the p-values (“p-val”) of the basis point fees. The timing and MVE strategies are implemented using a 120-month
rolling estimator of the conditional mean vector and conditional covariance matrix of the excess returns, and the OC and OC+ strategies target
the estimated conditional expected excess return of the 1/N strategy. The performance measures are reported assuming no transactions costs and
assuming proportional transactions costs of 50 basis points, and the p-values are determined from 10,000 trials of a stationary block bootstrap with
expected block length of 10. The values of µ̂p and σ̂p are not reported for the results with transactions costs. An entry of “–” for the TP strategy
indicates that the corresponding sample statistic cannot be computed. This occurs if there is no real value of the performance fee that makes the
investor indifferent between the TP strategy and 1/N strategy, or if the turnover for the TP strategy exceeds 20,000% in one or more months, which
drives wealth to zero under the assumed level of transactions costs. The sample period is July 1963 to December 2008 (546 monthly observations).
The first 120 observations are held out to initialize the rolling estimators. See the text for a detailed description of each strategy.
Table 3
Results for the 10 Industry Dataset
8.25
17.64
σ̂p
0.068
0.018
0.752
–
0.001
0.001
0.002
Panel C: Mean-Variance Efficient Strategies
MV
11.45 13.43
0.853
0.021
386
OC
14.19 14.11
1.006
0.006
651
7.18 15.12
0.475
0.455
-65
OC+
TP
-58.80 777.8 -0.076
0.982
–
0.524
0.555
0.579
0.004
0.004
0.008
17.17
16.93
16.72
83
126
158
9.00
9.39
9.68
0.002
0.004
0.013
0.312
0.498
0.698
RRT(β̄t+ ,1)
RRT(β̄t+ ,2)
RRT(β̄t+ ,4)
15
2
-39
101
128
147
Strategies
0.492
0.074
0.496
0.178
0.484
0.382
0.468
Panel B: Reward-to-Risk Timing Strategies
9.17 17.15
0.535
0.001
RRT(µ+
t ,1)
RRT(µ+
,2)
9.41
16.96
0.555
0.001
t
,4)
9.58
16.84
0.569
0.003
RRT(µ+
t
Panel A: Volatility Timing
VT(1)
8.24 16.74
VT(2)
8.01 16.15
VT(4)
7.51 15.52
1/N
µ̂p
664
892
110
–
118
179
226
137
178
206
81
109
111
No Transactions Costs
vs. 1/N
vs. 1/N
ˆ
ˆ5
p-val
∆1
p-val
∆
λ̂p
0.003
0.001
0.143
–
0.000
0.000
0.000
0.000
0.000
0.001
0.002
0.012
0.081
p-val
0.789
0.996
0.164
233.8
0.018
0.020
0.029
0.036
0.050
0.080
0.018
0.021
0.031
0.017
τ̂
0.496
0.569
0.409
–
0.518
0.547
0.568
0.522
0.537
0.540
0.486
0.488
0.472
0.462
λ̂p
0.443
0.300
0.789
–
0.001
0.001
0.004
0.001
0.004
0.019
0.078
0.192
0.430
-81
56
-154
–
83
124
152
90
108
109
14
-1
-47
0.641
0.435
0.941
–
0.004
0.005
0.011
0.004
0.015
0.050
0.317
0.518
0.731
193
283
20
–
118
178
220
127
159
167
81
108
104
Transactions Costs = 50bp
vs. 1/N
vs. 1/N
ˆ
ˆ5
p-val
∆1
p-val
∆
0.224
0.175
0.432
–
0.000
0.000
0.000
0.000
0.001
0.004
0.002
0.013
0.093
p-val
The table summarizes the out-of-sample performance of the 1/N strategy (row one), three volatility timing strategies (Panel A), six reward-to-risk
timing strategies (Panel B), and four MVE strategies (Panel C) for the 25 Size/BTM portfolios. It reports the following sample statistics for the
time series of monthly excess returns generated by each strategy: the annualized mean (µ̂p ), the annualized standard deviation (σ̂p ), the annualized
Sharpe ratio (λ̂p ), the average monthly turnover expressed as a fraction of wealth invested (τ̂ ), the annualized basis point fee that an investor with
quadratic utility and constant relative risk aversion of γ = 1 and γ = 5 would be willing to pay to switch from the 1/N strategy to the timing
ˆ γ ), the p-value for the difference between the annualized Sharpe ratio produced by the timing or MVE strategy and the 1/N
or MVE strategy (∆
strategy (“vs. 1/N p-val”), and the p-values (“p-val”) of the basis point fees. The timing and MVE strategies are implemented using a 120-month
rolling estimator of the conditional mean vector and conditional covariance matrix of the excess returns, and the OC and OC+ strategies target
the estimated conditional expected excess return of the 1/N strategy. The performance measures are reported assuming no transactions costs and
assuming proportional transactions costs of 50 basis points, and the p-values are determined from 10,000 trials of a stationary block bootstrap with
expected block length of 10. The values of µ̂p and σ̂p are not reported for the results with transactions costs. An entry of “–” for the TP strategy
indicates that the corresponding sample statistic cannot be computed. This occurs if there is no real value of the performance fee that makes the
investor indifferent between the TP strategy and 1/N strategy, or if the turnover for the TP strategy exceeds 20,000% in one or more months, which
drives wealth to zero under the assumed level of transactions costs. The sample period is July 1963 to December 2008 (546 monthly observations).
The first 120 observations are held out to initialize the rolling estimators. See the text for a detailed description of each strategy.
Table 4
Results for the 25 Size/BTM Dataset
4.70
16.68
σ̂p
0.000
0.000
0.000
0.068
0.018
0.014
–
Panel C: Mean-Variance Efficient Strategies
MV
7.47 15.05 0.496
0.051
302
OC
9.18 15.59 0.589
0.015
465
6.49 15.71 0.413
0.007
195
OC+
TP
8,321 5,680 0.146
0.680
–
0.000
0.000
0.000
179
261
320
0.398
0.447
0.478
RRT(β̄t+ ,1)
RRT(β̄t+ ,2)
RRT(β̄t+ ,4)
16.08
16.19
16.47
0.000
0.000
0.000
Panel B: Reward-to-Risk Timing Strategies
7.62 16.75 0.455
0.000
290
RRT(µ+
t ,1)
RRT(µ+
,2)
8.14
17.09
0.476
0.000
336
t
,4)
8.64
17.41
0.497
0.000
381
RRT(µ+
t
6.40
7.23
7.87
0.017
0.018
0.022
Strategies
0.330
0.004
0.355
0.004
0.375
0.005
0.282
68
102
128
Panel A: Volatility Timing
VT(1)
5.27 15.96
VT(2)
5.55 15.63
VT(4)
5.77 15.38
1/N
µ̂p
409
536
261
–
220
295
335
285
306
328
117
173
215
No Transactions Costs
vs. 1/N
vs. 1/N
ˆ
ˆ5
p-val
∆1
p-val
∆
λ̂p
0.036
0.013
0.004
–
0.000
0.000
0.001
0.000
0.001
0.002
0.001
0.001
0.001
p-val
0.281
0.348
0.129
184.9
0.019
0.024
0.034
0.058
0.061
0.068
0.017
0.018
0.026
0.018
τ̂
0.385
0.454
0.363
–
0.391
0.438
0.466
0.434
0.454
0.473
0.324
0.349
0.365
0.276
λ̂p
0.191
0.096
0.044
–
0.000
0.000
0.000
0.000
0.000
0.001
0.004
0.004
0.006
144
266
127
–
178
257
310
266
310
351
69
102
123
0.242
0.112
0.072
–
0.000
0.000
0.000
0.000
0.000
0.001
0.017
0.019
0.026
253
337
193
–
220
292
325
260
279
298
119
174
211
Transactions Costs = 50bp
vs. 1/N
vs. 1/N
ˆ
ˆ5
p-val
∆1
p-val
∆
0.132
0.081
0.031
–
0.000
0.000
0.001
0.000
0.002
0.005
0.001
0.001
0.002
p-val
The table summarizes the out-of-sample performance of the 1/N strategy (row one), three volatility timing strategies (Panel A), six reward-to-risk
timing strategies (Panel B), and four MVE strategies (Panel C) for the 10 Momentum portfolios. It reports the following sample statistics for the
time series of monthly excess returns generated by each strategy: the annualized mean (µ̂p ), the annualized standard deviation (σ̂p ), the annualized
Sharpe ratio (λ̂p ), the average monthly turnover expressed as a fraction of wealth invested (τ̂ ), the annualized basis point fee that an investor with
quadratic utility and constant relative risk aversion of γ = 1 and γ = 5 would be willing to pay to switch from the 1/N strategy to the timing
ˆ γ ), the p-value for the difference between the annualized Sharpe ratio produced by the timing or MVE strategy and the 1/N
or MVE strategy (∆
strategy (“vs. 1/N p-val”), and the p-values (“p-val”) of the basis point fees. The timing and MVE strategies are implemented using a 120-month
rolling estimator of the conditional mean vector and conditional covariance matrix of the excess returns, and the OC and OC+ strategies target
the estimated conditional expected excess return of the 1/N strategy. The performance measures are reported assuming no transactions costs and
assuming proportional transactions costs of 50 basis points, and the p-values are determined from 10,000 trials of a stationary block bootstrap with
expected block length of 10. The values of µ̂p and σ̂p are not reported for the results with transactions costs. An entry of “–” for the TP strategy
indicates that the corresponding sample statistic cannot be computed. This occurs if there is no real value of the performance fee that makes the
investor indifferent between the TP strategy and 1/N strategy, or if the turnover for the TP strategy exceeds 20,000% in one or more months, which
drives wealth to zero under the assumed level of transactions costs. The sample period is July 1963 to December 2008 (546 monthly observations).
The first 120 observations are held out to initialize the rolling estimators. See the text for a detailed description of each strategy.
Table 5
Results for the 10 Momentum Dataset
0.329
0.333
0.380
0.462
0.146
0.448
0.904
Panel C: Mean-Variance Efficient Strategies
MV
8.09 10.56 0.766
0.078
34
OC
10.84 11.66 0.930
0.008
296
8.31 13.52 0.615
0.032
21
OC+
TP
138.5 242.6 0.571
0.338
-21,691
0.360
0.340
0.364
0.298
0.357
0.438
22
35
38
0.017
0.011
0.012
0.520
0.559
0.623
17.07
15.71
13.67
RRT(β̄t+ ,1)
RRT(β̄t+ ,2)
RRT(β̄t+ ,4)
8.88
8.78
8.51
32
51
54
Timing Strategies
0.547
0.045
0.598
0.032
0.647
0.031
0.477
Panel B: Reward-to-Risk
8.81 16.12
RRT(µ+
t ,1)
RRT(µ+
,2)
8.79 14.70
t
,4)
8.62 13.32
RRT(µ+
t
18.79
46
52
32
8.96
σ̂p
Panel A: Volatility Timing Strategies
VT(1)
8.83 15.34 0.576
0.006
VT(2)
8.58 13.17 0.652
0.010
VT(4)
8.17 11.47 0.712
0.023
1/N
µ̂p
541
751
381
–
154
262
390
230
341
426
296
432
498
No Transactions Costs
vs. 1/N
vs. 1/N
ˆ
ˆ5
p-val
∆1 p-val
∆
λ̂p
0.034
0.006
0.006
–
0.001
0.001
0.002
0.004
0.004
0.005
0.000
0.002
0.007
p-val
0.350
0.547
0.237
191.1
0.017
0.019
0.027
0.029
0.042
0.066
0.015
0.015
0.016
0.017
τ̂
0.566
0.641
0.508
–
0.514
0.552
0.611
0.536
0.581
0.617
0.570
0.645
0.703
0.472
λ̂p
0.324
0.176
0.322
–
0.017
0.011
0.015
0.063
0.050
0.061
0.006
0.010
0.024
-168
-25
-113
–
22
34
32
24
36
24
47
53
32
0.725
0.538
0.786
–
0.326
0.338
0.399
0.395
0.389
0.437
0.293
0.354
0.437
340
425
247
–
154
260
384
224
327
395
297
433
498
Transactions Costs = 50bp
vs. 1/N
vs. 1/N
ˆ
ˆ5
p-val
∆1 p-val
∆
0.129
0.072
0.054
–
0.001
0.001
0.002
0.005
0.005
0.009
0.000
0.001
0.007
p-val
The table summarizes the out-of-sample performance of the 1/N strategy (row one), three volatility timing strategies (Panel A), six reward-to-risk
timing strategies (Panel B), and four MVE strategies (Panel C) for the 10 Volatility portfolios. It reports the following sample statistics for the
time series of monthly excess returns generated by each strategy: the annualized mean (µ̂p ), the annualized standard deviation (σ̂p ), the annualized
Sharpe ratio (λ̂p ), the average monthly turnover expressed as a fraction of wealth invested (τ̂ ), the annualized basis point fee that an investor with
quadratic utility and constant relative risk aversion of γ = 1 and γ = 5 would be willing to pay to switch from the 1/N strategy to the timing
ˆ γ ), the p-value for the difference between the annualized Sharpe ratio produced by the timing or MVE strategy and the 1/N
or MVE strategy (∆
strategy (“vs. 1/N p-val”), and the p-values (“p-val”) of the basis point fees. The timing and MVE strategies are implemented using a 120-month
rolling estimator of the conditional mean vector and conditional covariance matrix of the excess returns, and the OC and OC+ strategies target
the estimated conditional expected excess return of the 1/N strategy. The performance measures are reported assuming no transactions costs and
assuming proportional transactions costs of 50 basis points, and the p-values are determined from 10,000 trials of a stationary block bootstrap with
expected block length of 10. The values of µ̂p and σ̂p are not reported for the results with transactions costs. An entry of “–” for the TP strategy
indicates that the corresponding sample statistic cannot be computed. This occurs if there is no real value of the performance fee that makes the
investor indifferent between the TP strategy and 1/N strategy, or if the turnover for the TP strategy exceeds 20,000% in one or more months, which
drives wealth to zero under the assumed level of transactions costs. The sample period is July 1963 to December 2008 (546 monthly observations).
The first 120 observations are held out to initialize the rolling estimators. See the text for a detailed description of each strategy.
Table 6
Results for the 10 Volatility Dataset
0.39
0.44
0.46
0.50
0.67
0.37
0.16
Panel C: Mean-Variance Efficient Strategies
MV
0.74
0.05
1.88 -0.02
1.00
OC
0.86
0.01
2.39
0.02
0.99
0.43
0.73
0.31
0.31
0.99
OC+
TP
-0.14
1.00
114
–
–
0.52
0.54
0.55
0.00
0.00
0.02
0.02
0.04
0.06
RRT(β̄t+ ,1)
RRT(β̄t+ ,2)
RRT(β̄t+ ,4)
0.00
0.00
0.01
0.41
0.42
0.44
Panel B: Reward-to-Risk Timing Strategies
0.50
0.27
0.08
0.47
0.44
RRT(µ+
t ,1)
RRT(µ+
0.52
0.22
0.11
0.48
0.42
t ,2)
0.52
0.22
0.15
0.46
0.50
RRT(µ+
t ,4)
0.53
0.56
0.57
0.32
0.35
0.36
0.07
0.19
0.45
Panel A: Volatility Timing Strategies
VT(1)
0.50
0.05
0.02
0.49
VT(2)
0.50
0.14
0.03
0.49
VT(4)
0.49
0.32
0.06
0.47
0.02
0.28
0.47
τ̂
0.06
0.01
0.06
0.72
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.01
0.02
0.62
0.80
0.22
258
0.03
0.05
0.08
0.12
0.13
0.16
0.02
0.03
0.06
0.02
0.27
0.39
0.29
–
0.38
0.42
0.43
0.37
0.38
0.38
0.32
0.34
0.34
0.28
0.51
0.24
0.40
–
0.00
0.00
0.00
0.01
0.01
0.02
0.01
0.02
0.04
TC = 50bp
vs 1/N
λ̂p
p-val
10 Momentum
No TC
vs 1/N
λ̂p
p-val
0.46
1/N
τ̂
TC = 50bp
vs 1/N
λ̂p
p-val
25 Size/BTM
No TC
vs 1/N
λ̂p
p-val
0.77
0.93
0.62
0.24
0.52
0.56
0.65
0.58
0.63
0.67
0.57
0.64
0.70
0.48
τ̂
0.08
0.01
0.04
0.82
0.03
0.01
0.01
0.02
0.01
0.02
0.01
0.01
0.03
0.71
1.17
0.31
98.15
0.02
0.03
0.06
0.06
0.08
0.11
0.02
0.03
0.04
0.02
0.38
0.37
0.49
–
0.51
0.55
0.63
0.55
0.59
0.61
0.56
0.63
0.68
0.47
0.67
0.71
0.42
–
0.04
0.02
0.02
0.04
0.04
0.05
0.01
0.02
0.03
TC = 50bp
vs 1/N
λ̂p
p-val
10 Volatility
No TC
vs 1/N
λ̂p
p-val
The table summarizes the out-of-sample performance of the 1/N strategy (row one), three volatility timing strategies (Panel A), six reward-to-risk
timing strategies (Panel B), and four MVE strategies (Panel C) for the 25 size/book-to-market portfolios (“25 Size/BTM”), 10 momentum portfolios
(“10 Momentum”), and 10 volatility portfolios (“10 Volatility”). It reports the following sample statistics for the time series of monthly excess returns
generated by each strategy: the annualized Sharpe ratio (λ̂p ), the p-value for the difference between the annualized Sharpe ratio produced by the
timing or MVE strategy and the 1/N strategy (“vs. 1/N p-val”) assuming no transactions costs (“No TC”) and assuming proportional transactions
costs of 50 basis points (“TC = 50bp”), and the average monthly turnover expressed as a fraction of wealth invested (τ̂ ). The timing and MVE
strategies are implemented using a 60-month rolling estimator of the conditional mean vector and conditional covariance matrix of the excess returns,
and the OC and OC+ strategies target the estimated conditional expected excess return of the 1/N strategy. The p-values are determined from 10,000
trials of a stationary block bootstrap with expected block length of 10. An entry of “–” for the TP strategy indicates that the corresponding sample
statistic cannot be computed. This occurs if there is no real value of the performance fee that makes the investor indifferent between the TP strategy
and 1/N strategy, or if the turnover for the TP strategy exceeds 20,000% in one or more months, which drives wealth to zero under the assumed level
of transactions costs. The sample period is July 1963 to December 2008 (546 monthly observations). As in Tables 3 to 6, the first 120 observations
are held out, and observations 61 to 120 are used to initialize the rolling estimators. See the text for a detailed description of each strategy.
Table 7
Results Using a 60-Month Window for the Rolling Estimators
Figure 1
Reward and Risk Characteristics of the Datasets
The figure summarizes the sample reward and return characteristics for the 10 Industry dataset (Panel A), 25
Size/BTM dataset (Panel B), 10 Momentum dataset (Panel C), and 10 Volatility dataset (Panel D). The first
graph in each panel shows the cross-section of annualized mean returns and the second graph shows the crosssection of annualized return standard deviations. The sample period is July 1963 to December 2008 (546
monthly observations). However, the reported statistics correspond to the subperiod used to evaluate the outof-sample performance of the portfolio strategies, i.e, observations 121 to 546.
Mean Return
Panel A. 10 Industry Portfolios
Volatility
20%
35%
30%
15%
25%
20%
10%
15%
10%
5%
5%
0%
0%
1
2
3
4
5
6
7
8
9
1
10
2
3
4
5
6
7
8
9
10
Panel B. 25 Size/BTM Portfolios
20%
35%
30%
15%
25%
20%
10%
15%
10%
5%
5%
Small ‐
all ‐ 1
Small ‐
all ‐ 2
Small ‐
all ‐ 3
Small ‐
all ‐ 4
Small ‐
all ‐ 5
2 ‐ 1
2 ‐ 2
2 ‐ 3
2 ‐ 4
2 ‐ 5
3 ‐ 1
3 ‐ 2
3 ‐ 3
3 ‐ 4
3 ‐ 5
4 ‐ 1
4 ‐ 2
4 ‐ 3
4 ‐ 4
4 ‐ 5
Big ‐ 1
Big ‐ 2
Big ‐ 3
Big ‐ 4
Big ‐ 5
Small ‐
all ‐ 1
Small ‐
all ‐ 2
Small ‐
all ‐ 3
Small ‐
all ‐ 4
Small ‐
all ‐ 5
2 ‐ 1
2 ‐ 2
2 ‐ 3
2 ‐ 4
2 ‐ 5
3 ‐ 1
3 ‐ 2
3 ‐ 3
3 ‐ 4
3 ‐ 5
4 ‐ 1
4 ‐ 2
4 ‐ 3
4 ‐ 4
4 ‐ 5
Big ‐ 1
Big ‐ 2
Big ‐ 3
Big ‐ 4
Big ‐ 5
0%
0%
Panel C. 10 Momentum Portfolios
20%
35%
30%
15%
25%
20%
10%
15%
10%
5%
5%
0%
0%
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Panel D. 10 Volatility Portfolios
20%
35%
30%
15%
25%
20%
10%
15%
10%
5%
5%
0%
0%
1
2
3
4
5
6
7
8
9
10