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<reference>
Approaching Mean-Variance Efficiency for Large
Portfolios ∗
Mengmeng Ao†

Yingying Li‡

Xinghua Zheng§

First Draft: October 6, 2014
This Draft: November 11, 2017
Abstract

This paper studies the large dimensional Markowitz optimization problem. Given any
risk constraint level, we introduce a new approach for estimating the optimal portfolio,
which is developed through a novel unconstrained regression representation of the
mean-variance optimization problem, combined with high-dimensional sparse regression
methods. Our estimated portfolio, under a mild sparsity assumption, asymptotically
achieves mean-variance efficiency and meanwhile effectively controls the risk. To the
best of our knowledge, this is the first time that these two goals can be simultaneously
achieved for large portfolios. The superior properties of our approach are demonstrated
via comprehensive simulation and empirical studies.
Keywords: Markowitz optimization; Large portfolio selection; Unconstrained regression,
LASSO; Sharpe ratio

∗

Research partially supported by the RGC grants GRF16305315, GRF 16502014 and GRF 16518716 of

the HKSAR, and The Fundamental Research Funds for the Central Universities 20720171073.
†
Wang Yanan Institute for Studies in Economics & Department of Finance, School of Economics, Xiamen
University, China. mengmengao@xmu.edu.cn
‡
Hong Kong University of Science and Technology, HKSAR. yyli@ust.hk
§
Hong Kong University of Science and Technology, HKSAR. xhzheng@ust.hk

1

1
1.1

INTRODUCTION

Markowitz Optimization Enigma

The groundbreaking mean-variance portfolio theory proposed by Markowitz (1952) continues to play significant roles in research and practice. The optimal mean-variance portfolio has
a simple explicit expression1 that only depends on two population characteristics, the mean
and the covariance matrix of asset returns. Under the ideal situation when the underlying
mean and covariance matrix are known, mean-variance investors can easily compute the
optimal portfolio weights based on their preferred level of risk. In the real world, however,
the true parameters are unknown. Sample mean and sample covariance matrix are used as
proxies, and the resulting “plug-in” portfolio has been widely adopted. Such an approach
is justified by the classical statistics theory because the plug-in portfolio is an MLE of the
optimal portfolio. However, as is documented in Michaud (1989) and others, the out-of-sample
performance of the plug-in portfolio is poor. Moreover, the situation worsens as the number
of assets increases. (For additional details, see Best and Grauer (1991), Chopra and Ziemba
(1993), Britten-Jones (1999), Kan and Zhou (2007), and Basak et al. (2009) among others.)
Termed the “Markowitz Optimization Enigma” by Michaud (1989), the issues of constructing
the mean-variance optimal portfolio based on sample estimates limit the use of Markowitz’s
mean-variance framework.

1.2

Challenges for Large Portfolios

Modern portfolios often include a large number of assets. This makes the optimization
problem high-dimensional in nature, and induces numerous challenges. Take the plug-in
portfolio for example, as we will see below, the risk of the plug-in portfolio can be substantially
higher than the pre-specified risk level even when the portfolio weights are computed based
on simulated i.i.d. returns. On the other hand, such high risk is not well compensated by high
returns, resulting in significantly suboptimal Sharpe ratios. The key message is, even in the
ideal situation when all assumptions of Markowitz optimization are satisfied (i.e.,
no time-varying parameters, regime switching, etc.), there are intrinsic challenges towards
the estimation of the mean-variance efficient portfolio.
1

See details in Section 2.1.

2

Simulation Comparison of Sharpe Ratio

Sharpe ratio
1.0

0.05

0.5

0.03

0.04

risk

0.06

1.5

0.07

2.0

0.08

Simulation Comparison of Risk

0.02

True
Plug−in
0

20

40

60

80

100

True
Plug−in
0

replication

20

40

60

80

100

replication

Figure 1. Simulation comparisons of risks and Sharpe ratios of the plug-in portfolio versus the true optimal
portfolio. The portfolios are constructed based on data generated from i.i.d. multivariate normal distribution
with parameters specified in Section 3.2. The left panel plots the portfolio risks, and the right panel plots the
Sharpe ratios. The pool of assets includes 100 stocks and 3 factors, and the number of observations is 240.
The comparison is replicated 100 times.

Figure 1 shows the comparisons between the plug-in portfolio (black dashed lines) and the
theoretical optimal portfolio (black solid lines). The asset pool consists of 100 stocks and 3
(tradable) factors. The underlying mean and covariance matrix are calibrated from real data
(see Section 3.2 for details). We generate 20 years of monthly returns from i.i.d. multivariate
normal distribution, based on which we construct the plug-in portfolio and compare its risk
and Sharpe ratio with the theoretical optimal values. Such simulated returns satisfy all the
assumptions of Markowitz’s mean-variance framework. However, as we observe from Figure 1,
in all 100 replications, the plug-in portfolio carries a risk that is almost twice the specified
level. Meanwhile, as shown in the right panel of Figure 1, the Sharpe ratio of the plug-in
portfolio is only about 50% of the theoretical maximum Sharpe ratio.
The above phenomenon has been noted in Kan and Zhou (2007), and further investigated
in Bai et al. (2009) and El Karoui (2010). These papers document that the deviation of
the plug-in portfolio from the optimal portfolio is systematic, and the bias is due to the
dimension (number of assets) being not negligible compared with the sample size. For global
minimum variance portfolio (GMV), Basak et al. (2009) derive a result in a similar spirit,
which says that the plug-in GMV carries, on average, a risk that is a bigger than 1 multiple
3

of the true minimum risk, and the multiplier explicitly depends on the number of assets and
sample size.

1.3

Existing Alternative Methods

The plug-in portfolio is obtained by replacing the population mean and covariance matrix
in the formula for the optimal portfolio with their sample estimates. Alternatively, people
seek to improve portfolio performance by plugging in better estimates of the underlying
mean and covariance matrix. For estimation of covariance matrix, a widely used alternative
estimator is the linear shrinkage estimator proposed in Ledoit and Wolf (2003, 2004), which
estimates the covariance matrix by a suitable linear combination of the sample covariance
matrix and a target matrix (e.g., identity or single-index matrix). More recently, Ledoit
and Wolf (2017) propose a nonlinear shrinkage estimator of the covariance matrix and its
factor-model-adjusted version that are suitable for portfolio optimization. For estimation of
mean, among other works, Black and Litterman (1991) propose a quasi-Bayesian approach by
combining investors’ views with returns implied by CAPM. This quasi-Bayesian approach is
extended to a fully Bayesian approach by Lai et al. (2011), who consider the mean-variance
problem from a different angle and aim to maximize a certain utility function. Garlappi et al.
(2007) propose to adjust estimates of expected returns by a multi-prior approach, also with
an aim of maximizing a utility function. The aforementioned paper Bai et al. (2009), which
analyze the systematic bias in the plug-in portfolio, propose a “bootstrap-corrected” method
to estimate the optimal portfolio. However, as pointed out in a more recent working paper
(Bai et al. 2013), the bootstrap-corrected method fails to satisfy the risk constraint.
Another direction to improve portfolio performance is to modify the original framework
by imposing various constraints on portfolio weights. Most research in this direction focuses
on the GMV portfolio. Imposing constraints on the weights has been empirically shown to
be helpful; see, for example, Jagannathan and Ma (2003), DeMiguel et al. (2009), Brodie
et al. (2009) and Fastrich et al. (2012). Fan et al. (2012b) give theoretical justifications to
the empirical results in Jagannathan and Ma (2003), and also investigate the GMV portfolio
with gross-exposure constraints where some short positions are allowed. Fan et al. (2012a)
consider GMV portfolio with high-frequency data under gross-exposure constraints.
In addition to the approaches mentioned above, combinations of different portfolios have
been studied; see, for example, Kan and Zhou (2007) and Tu and Zhou (2011).

4

The aforementioned methods lead to improved portfolio performance. However, they are
still suboptimal. Take the latest development, the nonlinear shrinkage method in Ledoit and
Wolf (2017) as an example, we see in Figure 2 that although its risk is substantially lower
than that of the plug-in portfolio, it still violates the risk constraint, and is also significantly
suboptimal in terms of Sharpe ratio.
Simulation Comparison of Sharpe Ratio

1.0

0.03

0.04

risk

0.05

Sharpe ratio

0.06

1.5

0.07

2.0

0.08

Simulation Comparison of Risk

0

20

40

60

80

True
Plug−in
Nonlinear Shrinkage

0.5

0.02

True
Plug−in
Nonlinear Shrinkage
100

0

replication

20

40

60

80

100

replication

Figure 2. Simulation comparisons of the plug-in and nonlinear shrinkage portfolios. The portfolios are
constructed based on the same data used for Figure 1. The left and right panels plot the portfolio risks and
Sharpe ratios, respectively. There are 100 stocks and 3 factors in the asset pool, and the sample size is 240.
The comparison is replicated 100 times.

More comprehensive comparisons including several other benchmark strategies are made
in Sections 3 and 4, based on both simulated and empirical data. The comparisons reveal
similar conclusions.

1.4

Our Contributions

In this paper, a new methodology for estimating the mean-variance efficient portfolio
is proposed, which we call the MAXimum-Sharpe-ratio Estimated & sparse Regression
(MAXSER) method. MAXSER is a general approach which can be applied to various
situations when the number of assets in portfolios is not negligible compared with sample
size. We show that, under mild assumptions, the MAXSER portfolio can asymptotically
(1) achieve mean-variance efficiency and (2) satisfy the risk constraint. To the best of our
5

knowledge, this is the first time that both objectives can be simultaneously achieved for large
portfolio optimization.
Our first main contribution is establishing an equivalent unconstrained regression representation of the Markowitz optimization problem. This special regression representation is a
novel finding made in this paper. There is existing literature on using regression to estimate
the optimal portfolio. Two most closely related approaches are Britten-Jones (1999), who use
1 as the response; and Brodie et al. (2009), who use the maximum expected return as the
response. The issues with the two approaches are that, the regression in Britten-Jones (1999)
results in a biased solution; while Brodie et al. (2009) deal with a constrained regression,
which is challenging and involves error and biases induced by sample estimates. Our regression representation, in contrast, is on the one hand, equivalent to the original optimization
problem, and on the other hand, unconstrained, so that it can be conveniently combined with
high-dimensional regression techniques. See more details in Section 2.1.
Our method further involves the following important aspects:
(1) Consistent estimation of the response in our regression representation, which depends on
consistent estimation of the maximum Sharpe ratio achieved by the tangency portfolio.
(2) Proper sparse regression and rigorous analysis under the framework of our regression
which possesses some unique features.
(3) The estimation in (1) and the sparse regression in (2) consists of the core of MAXSER.
Depending on whether factor structure is present in the asset pool or not, MAXSER
can be used directly on all assets or on idiosyncratic components.
• When there is no factor structure in returns, the combination of estimation of the
response and the sparse regression is directly applied to the assets. See Section 2.2.
• When factor structure does present, we develop a framework that decomposes the
portfolio estimation into the estimation of the optimal portfolio on factors and
that on idiosyncratic components. See Section 2.3.
Under both settings, we theoretically prove the convergences of the expected return
and risk of our estimated portfolio towards the theoretical maximum expected return
and risk constraint, respectively. These properties guarantee that our estimator can
asymptotically achieve mean-variance efficiency as both the number of assets and sample
size get large.
6

The theoretical properties of the MAXSER portfolio are supported by simulation and
empirical studies. We compare our method with a number of benchmark methods including
the plug-in portfolio, the equally weighted portfolio, the linear/nonlinear shrinkage portfolio of
Ledoit and Wolf (2004) and Ledoit and Wolf (2017), and several other variations of MV/GMV
portfolios with constraints on portfolio weights. The complete simulation results are given
in Section 3. Figure 3 below shows the comparison among the plug-in, nonlinear shrinkage
and MAXSER portfolios. The added blue dashed lines plot the results of our portfolio. We
see that our portfolio effectively controls the risk to satisfy the risk constraint. More
importantly, the comparisons of Sharpe ratios show that our MAXSER portfolio nearly
achieves the mean-variance efficiency and significantly outperforms others.
Simulation Comparison of Sharpe Ratio

Sharpe ratio
1.0

0.05
0.03

0.04

risk

0.06

1.5

0.07

2.0

0.08

Simulation Comparison of Risk

0

20

40

60

80

True
Plug−in
Nonlinear Shrinkage
MAXSER

0.5

0.02

True
Plug−in
Nonlinear Shrinkage
MAXSER
100

0

replication

20

40

60

80

100

replication

Figure 3. Comparison of our MAXSER portfolio with the plug-in and nonlinear shrinkage portfolios, based
on the same simulated data as for Figures 1 and 2. The added blue dashed lines represent the MAXSER
portfolio. The comparison is replicated 100 times.

Comprehensive empirical studies are presented in Section 4, in which we demonstrate the
favorable performance of our proposed strategy.

7

2
2.1

The MAXSER Methodology

An Unconstrained Regression Representation

Suppose that we have a pool of N risky assets. Denote their (random excess) returns
by r = (r1 , r2 , · · · , rN )0 , where for any vector v, v 0 stands for the transpose of v. Let µ
and Σ be the mean vector and covariance matrix of r, respectively, and let w be a vector of
portfolio weights on the risky assets. For a given level of risk constraint σ, the Markowitz
optimization problem is
arg max E(w0 r) = w0 µ subject to Var(w0 r) = w0 Σw ≤ σ 2 .

(2.1)

w

If we denote by θ = µ0 Σ−1 µ the square of the maximum Sharpe ratio of the optimal portfolio,
then the optimization problem (2.1) can be represented in its dual form with a return
√
constraint r∗ = σ θ:
arg min w0 Σw

subject to w0 µ = r∗ .

(2.2)

w

The optimal portfolio, w∗ , admits the following explicit expression:
σ
w∗ = √ Σ−1 µ.
θ

(2.3)

The optimal portfolio w∗ satisfies the following linear relationship:
r∗ = w0 r + ε,

ε ∼ (0, σ 2 ),

(2.4)

where ε = r∗ − w0 r is a random variable with mean 0 and variance σ 2 . Based on such an
equation, a natural way to solve for w∗ is the ordinary least squares (OLS):
w0 := arg min E(r∗ − w0 r)2 .

(2.5)

w

However, w0 is not the optimal portfolio. In fact, we have
w0 =

θ
w∗ .
1+θ

(2.6)

In particular, we see that the OLS solution yields a smaller expected return and carries a
lower risk.
The result above shows that the OLS (2.5) leads to biased solution. The bias is due to
a unique feature of the OLS (2.5): Unlike in the conventional regression where the noise is
independent of (or at least uncorrelated with) the predictor, in (2.5) the noise ε and the
8

predictor are correlated. To correct for the bias, if still r∗ is used as the response, then a
constraint must be added to least squares:
arg min E(r∗ − w0 r)2

subject to E(w0 r) = r∗ or Var(w0 r) = σ 2 .

w

The difficulty with such an approach is that the constraint has to be replaced with a sample
version as what is done in Brodie et al. (2009). In such a way, estimation error and even bias
are introduced. Our proposal, in contrast, is the following novel unconstrained regression.
Proposition 1. The unconstrained regression
arg min E(rc − w0 r)2 ,

where rc :=

w

1+θ ∗ 1+θ
r ≡ √ σ.
θ
θ

(2.7)

is equivalent to the Markowitz optimization (2.1) or (2.2).
The corrected response in our regression representation (2.7), rc , adjusts the bias of the
OLS solution w0 by rescaling the maximum expected return r∗ . In such a way, the Markowtiz
optimization problem is translated into an equivalent unconstrained regression.
Remark 1. In Britten-Jones (1999), the author connects the estimation of the tangency
portfolio with the regression coefficients in an OLS regression. There, the response is simply 1,
which results in a multiple of the plug-in tangency portfolio, hence is biased. To obtain the
tangency portfolio, one scales the weights so that the weights add up to one. After such
an adjustment, one gets the plug-in tangency portfolio, which, unfortunately, will not be
mean-variance efficient.
We emphasize that our regression representation (2.7) is intrinsically different from
existing regression representations in the literature for mean-variance portfolio estimation.
Our representation is unconstrained and equivalent to the Markowitz problem. The elimination
of constraint is particularly helpful for the estimation of the optimal portfolio of a large pool
of assets, for which important techniques like sparse regression becomes directly applicable.

2.2

When There Is No Factor Structure

Let us first consider the situation when asset returns do not possess a factor structure,
for example, when the asset pool consists of factor portfolios.

9

2.2.1

Estimating the Maximum Sharpe Ratio and the Regression Response

In our regression representation, the response rc is unknown2 and needs to be estimated.
The estimation of the response rc is closely related to the estimation of the maximum Sharpe
ratio, which has been considered in Kan and Zhou (2007). It is shown there that the square
of the plug-in Sharpe ratio follows a non-centralized F -distribution; see equation (49) in their
paper. Utilizing such a fact, we establish the following
Proposition 2. Define the following estimator of θ:
(T − N − 2)θbs − N
θb :=
,
T

(2.8)

b −1 µ
b0Σ
b is the sample estimate of θ. Under normality assumption on returns,
where θbs := µ
and assuming that N/T → ρ ∈ (0, 1), we have
P

|θb − θ| → 0.

(2.9)

1 + θb
rbc := p
θb

(2.10)

Consequently,

satisfies that
P

|rbc − rc | → 0.

(2.11)

We emphasize that our estimation of θ = µ0 Σ−1 µ is not via consistently estimating µ
and Σ, which would be impossible without imposing strong structural assumptions on
them. Instead, we estimate θ directly, and in such a way we overcome the challenge due to
high-dimensionality.
2.2.2

A LASSO-type Estimator

On the basis of our unconstrained regression representation (2.7), we aim to estimate the
optimal portfolio w∗ for a given risk constraint σ.
Suppose that Rt = (Rt1 , · · · , RtN )0 , t = 1, · · · , T , are T i.i.d. copies of the (excess) return.
Let R = (R1 , ..., Rn )0 be the T × N observation matrix. We focus on the case where
N and T are both large and the optimal portfolio has a bounded `1 -norm. In terms of
2

In the case when return constraint r∗ is given, the response is still unknown because the bias-correction

scalar depends on θ, the square of the underlying maximum Sharpe ratio.

10

the regression (2.7), this amounts to a high-dimensional regression problem with bounded
`1 -norm regression coefficients. Such a connection leads us to the widely used approach in
high-dimensional regression, LASSO (Tibshirani (1996)). More specifically, take a λ > 0 and

0
∗
c1∗ , ..., w
cN
c∗ = w
define a LASSO-type estimator w
as follows:
c∗ = arg min ||rbc − Rw||22
w

subject to ||w||1 ≤ λ,

(2.12)

w

where rbc = (rbc , ..., rbc )0 ∈ RT is the estimated response. Before we give the theoretical property
c∗ , we first list the assumptions that will be needed.
of w
Assumption:
A1 The (excess) return r ∼ N (µ, Σ);
A2 There exists an L < ∞ such that for all dimension N , Σ = (σjk )N ×N satisfies that
max |σjk | ≤ L;

1≤j,k≤N

A3 The eigenvalues of Σ are bounded from both above and below;
A4 There exists M < ∞ such that for all dimension N , ||µ||2 ≤ M ;
A5 ||w∗ ||1 ≤ λ for some constant λ;
A6 The number of assets N and the sample size T satisfy that ρT := N/T → ρ ∈ (0, 1).
Remark 2. About Assumption A1, in this paper we focus on the most fundamental form of
the mean-variance problem, so we assume normal distribution of returns. Numerically, we
find that our proposed method works well even when heavy-tailedness is present. Extensions
incorporating heteroscedasticity and heavy-tailedness will be studied in subsequent papers.
Remark 3. Under Assumption A3, Assumption A4 is equivalent to the common belief that
the theoretically optimal Sharpe ratio is bounded. Note that Assumption A4 also implies that
max |µj | ≤ M .

1≤j≤p

Remark 4. Assumption A5 is our sparsity requirement on the optimal portfolio when there
is no factor structure. Assumptions A3 and A4 imply that ||w∗ ||2 is bounded, so the only
difference is between the `2 norm and the `1 norm. Note that Assumption A5 does not require
most weights to be zero. For example, it does not rule out value-weighted portfolios. We will
see in Section 2.3 that when there is factor structure, the requirement will be only imposed
upon the weights on idiosyncratic components.
More generally, if indeed ||w∗ ||1 is unbounded, then one can truncate w∗ at the level
√
of ε/ N for some ε > 0 to reduce nonzero but small elements and possibly largely reduce the
11

`1 norm. Note that the difference between w∗ and its truncated version has a difference of `2
norm at most ε. Using Assumptions A3 and A4, one can show that the differences between
the variances and expected returns of the two portfolios are both O(ε). In other words, one
sacrifices little in terms of both return and risk and yet can possibly largely reduce the number
of nonzero but small positions.
Remark 5. Assumption A6 says that we are in a high-dimensional setting where the number
of assets N and the sample size T proportionally grow up to infinity. We require the sample
size T to be larger than the dimension N due to that we need to take inverse of the sample
covariance matrix in estimating the maximum Sharpe ratio; see Proposition 2.
2.2.3

Main Result I: MAXSER Without Factor Structure

We now state our first main result, which establishes the near-optimality of our MAXSER
portfolio when returns do not admit factor structures.
c∗ defined in (2.12) with
Theorem 1. Under Assumptions A1∼A6, the MAXSER portfolio w
rbc given by (2.10) satisfies that, as N → ∞,

and

P
c∗ ||2 →
||w∗ − w
0,

(2.13)

P
c∗ | →
|r∗ − µ0 w
0,

(2.14)

q

P
c∗ 0 Σw
c∗ − σ →
w
0.

(2.15)

c∗ asymptotically (1) achieves the
Theorem 1 guarantees that our MAXSER portfolio w
maximum expected return and (2) satisfies the risk constraint. An immediate implication is
c∗ also approaches the mean-variance efficiency.
that w

2.3

When Factor Structure Presents

2.3.1

The Optimal Portfolio: A Factor-Idiosyncratic Components Separation

Motivated by the large literature on factor models for stock returns, we next propose an
alternative version of MAXSER under the following approximate factor model:
ri = αi +

K
X
j=1

βij fj + ei :=

K
X

βij fj + ui ,

j=1

12

i = 1, · · · , N,

(2.16)

where the βij ’s represent individual stock sensitivities to the factors, and ei ’s are idiosyncratic
disturbances independent of the factor returns (fj ). The factors can be any well-recognized
factors like Fama-French three factors or other factors identified in the large literature of asset
pricing (see, e.g., Jegadeesh and Titman (2001) and Korajczyk and Sadka (2008)). They
can also be statistical factors identified via principal component analysis (see, e.g., Connor
et al. (2010)). As to the idiosyncratic disturbances (ei )’s, we emphasize that they can still be
cross-sectionally dependent. The terms ui = αi + ei will be referred to as the idiosyncratic
returns. Model (2.16) can be written in a compact form as
r = βf + u,

(2.17)

where β = (βij )N ×K , f = (f1 , · · · , fK )0 , and u = (u1 , · · · , uN )0 . Let µf be the mean of the
factor returns, and α = (α1 , · · · , αN )0 which is the mean of idiosyncratic return u. Let Σf
and Σu be the covariance matrix of factor and idiosyncratic returns, respectively. Then the
return vector r has the following mean µ and covariance matrix Σ:
µ = βµf + α,

Σ = βΣf β 0 + Σu .

(2.18)

Denote the mean and covariance matrix of the returns on the full pool of factors and assets
by µall and Σall , respectively.
Such a model has been widely adopted in the literature. In particular, in light of Section 5
of Chamberlain and Rothschild (1983), we make the following
Assumption:
B1 The eigenvalues of Σu := Cov(u) is bounded from both above and below, and
σij := Σu (i, j) satisfies that max |σi,j | ≤ L < ∞.
1≤i,j≤N
qP
N
2
B2 ||α||2 :=
j=1 αj is bounded. In particular, there exists M < ∞ such that
max |αj | ≤ M .

1≤j≤N

B3 The number of factors K is bounded.
Given the factor structure, when we build portfolios, we shall invest in not only the N
assets but also the K factors. Such a strategy is straightforward when the factors are
taken to be investable factors like the Fama-French factors and many others (see, e.g., the
supplementary file of Feng et al. (2017)), and is also feasible when the factors are statistical
factors thanks to that both the number of factors K and the factors can be consistently
estimated (see, for example, Bai and Ng (2002)).
13

f
We aim to find an optimal weight (w1f , . . . , wK
; w1 , . . . , wN ) := (wf , w), where wf and w

represent the weight vectors put on the K factors and N assets, respectively. The following
result shows that finding such an optimal weight can be decomposed into three steps:
(i) Find the optimal portfolio on the factors with 1 unit of risk (denoted by wf∗ ),
(ii) Find the optimal portfolio on the idiosyncratic components with 1 unit of risk (denoted
by wu∗ ), and
(iii) Suitably combine these two portfolios.
Proposition 3. For any given risk constraint level σ, the optimal portfolio wall := (wf , w)
is given by
r
σ

θf ∗
w −σ
θall f

r

θu 0 ∗
β wu ,
θall

r
σ

!
θu ∗
w ,
θall u

−1
0 −1
0
where θf = µ0f Σ−1
f µf , θu = α Σu α, and θall = µall Σall µall are the squared maximum Sharpe

ratios of portfolios on the factors, the idiosyncratic components, and the full set of factors and
individual assets, respectively. Moreover, wf∗ and wu∗ admit the following explicit expressions:
1
wf∗ = p Σ−1
f µf ,
θf

1
α.
wu∗ = √ Σ−1
θu u

(2.19)

Remark 6. In the case when αi ≡ 0, we have θu = 0 and hence the optimal weight is given by

σwf∗ , 0 . In other words, the optimal portfolio will be fully invested in factors. However, this
is only the case when all the underlying factors are identified and included, which is unlikely
in practice especially when one wants to only include a small number of strong factors.
According to Proposition 3, in order to estimate the optimal portfolio (wf , w), we need to
estimate θf , θu , wf∗ and wu∗ . We will deal with them one by one, starting with the estimation
of the maximum Sharpe ratios.
2.3.2

Estimating the Maximum Sharpe ratios

Estimation of the maximum Sharpe ratio is still essential under the current setting with
factor structure. There are three Sharpe ratios that are of interest in this setting: θf , θu
and θall , the squared maximum Sharpe ratios on factors, idiosyncratic components and all
assets, respectively. For estimating θf and θall , parallel to Proposition 2 we have the following

14

Proposition 4. Define the following estimators of θf and θall :
(T − K − 2)θbsf − K
θbf :=
,
T

(2.20)

(T − N − K − 2)θbsa − N − K
,
(2.21)
θball :=
T
b
b −1 µ
b −1 µ
b 0f Σ
b 0all Σ
where θbsf := µ
f b f and θsa := µ
all b all are the sample estimates of θf and θall ,
respectively. Under normality assumption on factor and idiosyncratic returns, assuming that
N/T → ρ ∈ (0, 1), we have
P

|θbf − θf | → 0,

P

|θball − θall | → 0.

and

There is one more Sharpe ratio to be estimated,

√
θu , the maximum Sharpe ratio on

the idiosyncratic components. This quantity is a bit trickier to deal with, because the
b , the estimated
idiosyncratic return U is not observable. A natural idea is to work with U
idiosyncratic return. However, it can be shown that an estimator similar to (2.20) and (2.21)
b will be biased.
applied to U
The solution to the aforementioned difficulty lies in the relationship among θf , θu and θall .
Based on the model (2.17), one can show that
θall = θf + θu .
By Proposition 4, both θf and θu can be consistently estimated, so we get the following
Proposition 5. Define
θbu := θball − θbf .
Under the assumptions of Proposition 4, we have
P

|θbu − θu | → 0.
√
Therefore for rc := (1 + θu )/ θu , if we define
1 + θbu
rbc = q ,
θbu
then
P

|rbc − rc | → 0.
15

(2.22)

2.3.3

Estimating the Optimal Portfolio on Idiosyncratic Components

The optimal portfolio on the idiosyncratic components, wu∗ , solves the following Markowitz
optimization problem:
arg max α0 w

subject to w0 Σu w ≤ 1.

(2.23)

w

By (2.19), the optimal portfolio yields an expected return of
ru∗ =

p
θu ,

(2.24)

which is the maximum Sharpe ratio of portfolios on (ui ). Following our regression representation in Section 2.1, we will estimate wu∗ based on the following:
2

arg min E (rc − u0 w) ,

where rc :=

w

1 + θu ∗ 1 + θu
r = √ .
θu u
θu

(2.25)

One major difference here is that the idiosyncratic returns are not observable.
Suppose that Rt = (Rt1 , · · · , RtN )0 and Ft = (Ft1 , · · · , FtK )0 , t = 1, · · · , T , are T
i.i.d. copies of the (excess) return r and the factor (excess) return f , respectively. Let
R = (R1 , ..., RT )0 and F = (F1 , · · · , FT )0 . We estimate the coefficient β in (2.17) by
b =
regressing R on F . Denote by βb the estimated beta matrix. Correspondingly, let U
b N )0 = R − F βb be the estimator of U = R − F β. Our estimation of wu∗ will be
(b
u1 , · · · , u
b.
based on the T × N matrix U
Similarly to Section 2.2.2, our estimator of wu∗ is the following LASSO-type estimator:
b w||2
cu∗ = arg min ||rbc − U
w
2
w

subject to

N
X

|wi | ≤ λ,

(2.26)

i=1

where rbc = (rbc , · · · , rbc )0 is the estimator of the response defined in (2.22), and λ is a positive
cu∗ , the following assumptions will be needed.
constant. To give the theoretical properties of w
Assumption:
C1 f ∼ N (µf , Σf ), u ∼ N (α, Σu );
C2 ||wu∗ ||1 ≤ λ for some constant λ;
C3 The number of assets N and the sample size T satisfy that ρT := N/T → ρ ∈ (0, 1).
Remark 7. Assumption C2 is our sparsity requirement under factor models. We emphasize
that the sparsity assumption is put on the weights on the idiosyncratic components. As
16

discussed in Remark 6, wu∗ ≡ 0 if αi ≡ 0, and the assumption is certainly satisfied. More
generally, ||α||2 is bounded by Assumption B2. If there are only finitely many none zero of
P
αi , then Assumption C2 would be satisfied when maxi j=1,...,N |Ωu (i, j)| is bounded, where
Ωu := Σ−1
u . Recall that under the Gaussian assumption C1, Ωu (i, j) = 0 if and only if ui
and uj are conditionally independent given {uk : k 6= i, j}, and more generally, Ωu (i, j) is
related to the partial correlation between ui and uj . Because (ui ) are idiosyncratic returns, it
is possibly reasonable to expect that for each i, ui is only correlated with finitely many other
(uj )’s, in which case the assumption naturally holds.
cu∗ . Note that
We are now ready to give the asymptotic properties of the portfolio w
b contains estimation errors, Theorem 1 does not readily apply to this case.
because U
Proposition 6. Under Assumptions B1 ∼ B3 and C1 ∼ C3, we have as N → ∞,
P
cu∗ ||2 →
||wu∗ − w
0,

(2.27)

P

cu∗ | → 0,
|α0 wu∗ − α0 w

(2.28)

q
P
cu∗ 0 Σu w
cu∗ − 1 →
0.
w

(2.29)

and

cu∗ asymptotically attains the maximum expected
Proposition 6 states that the portfolio w
return and carries a risk that is close to the given risk constraint, in this case, 1.
2.3.4

Main Result II: MAXSER Under Factor Models

By far we have achieved consistency in estimating wu∗ , θu and θf . There is one more
item to be estimated, wf∗ , the optimal weight on factors (with risk equal to 1). This is easy
because the number of factors is fixed, and the simple “plug-in” estimator works. Combining
these results with Proposition 3, we obtain the following main result for our estimator of the
d
optimal full portfolio w
all .
∗
b −1 µ
c∗ := √1 Σ
Theorem 2. Let w
f
f b f be the estimator of wf . Our estimator of the optimal
b
θf

full portfolio wall is
 s
d
cf , b
w
w) = σ
all := (w

θbf c∗
wf − σ
θball

s

θbu b0 ∗
cu , σ
βw
θball

s


b
θu ∗ 
cu .
w
θball

(2.30)

Under Assumptions B1 ∼ B3 and C1 ∼ C3, as N → ∞, we have
0

P

∗
d
|w
all µall − r | → 0,

and
17

0

P

2
d
d
|w
all Σall w
all − σ | → 0.

(2.31)

Theorem 2 guarantees that our MAXSER portfolio under the factor model setting can
again asymptotically achieve the maximum expected return and meanwhile satisfy the risk
constraint, and consequently, achieve mean-variance efficiency.

2.4

Practical Implementation of MAXSER

2.4.1

Choosing λ in (2.12) and (2.26)

In practice, it is important to choose λ in (2.12) and (2.26). Because one of our goals is
to meet the risk constraint, we naturally want to choose a λ such that the estimated portfolio
possesses a risk that is close to the given risk constraint. In practice, we do not know the
underlying covariance matrix Σ/Σall . To circumvent this difficulty, we use a cross-validation
method to choose the λ. Specifically, for a 10-fold cross-validation procedure, we randomly
split the sample into 10 groups to form 10 validation sets. For each validation set, the training
set is taken to be the rest of the observations. Next, for each such training set i, we let λ
vary from 0 to the maximum value (the `1 -norm of the least squares solution) to obtain the


c∗ (λ) ((w
d
whole solution path w
all (λ))λ under factor model setting), and find the value
λ

of λ such that the estimated portfolio minimizes the difference between the “out-of-sample”
risk computed using the validation set and the given risk constraint. Denote such a value by
b is then taken to be the average of (λ(i), i = 1, · · · , 10).
λ(i). The ultimate λ
To our knowledge, the above cross-validation method for determining the constraint
parameter in norm-constrained mean-variance portfolio optimization is new and constitutes
another contribution of ours to the literature. By choosing the parameter selection criterion
to be the risk, our cross-validation method effectively helps control out-of-sample risk. The
procedure can be easily applied to other portfolio optimizations where norm-constraints
are imposed. In our numerical studies, we apply such a cross-validation method to other
norm-constrained portfolios and compare their performance with ours.
2.4.2

b θbf , θbu and θball
Adjustment of θ,

Kan and Zhou (2007) notice that the unbiased estimator of the square of maximum
Sharpe ratio θb often takes negative values, and they propose the adjusted estimator that
improves over the unbiased one:
(T − N − 2)θbs − N
2(θbs )N/2 (1 + θbs )−(T −2)/2
θba =
+
,
T
T Bθbs /(1+θbs ) (N/2, (T − N )/2)
18

(2.32)

where, recall that, θbs is the plug-in estimators of θ, and
Z x
Bx (a, b) =
y a−1 (1 − y)b−1 dy.
0

Under the factor model setting, we adopt the following adjustments of θbf and θball :
(T − K − 2)θbsf − K
2(θbsf )K/2 (1 + θbsf )−(T −2)/2
θbf a =
+
,
T
T Bθbsf /(1+θbsf ) (K/2, (T − K)/2)

(2.33)

(T − N − K − 2)θbsa − N − K
2(θbsa )(N +K)/2 (1 + θbsa )−(T −2)/2
θball,a =
+
, (2.34)
T
T Bθbsa /(1+θbsa ) ((N + K)/2, (T − N − K)/2)
c
b
where, recall that, θc
sf and θsa are the plug-in estimators of θf and θall . The adjusted θu is
θbua := θball,a − θbf a .
2.4.3

Implementation Steps

When there is no factor structure
To sum up, our method consists of the following steps:
Step 1 Compute the estimates of the square of the maximum Sharpe ratios θb by (2.32),
and compute the response rbc ;
Step 2 Choose λ by cross-validation according to the procedure described in Section 2.4.1.
b
Denote the chosen value by λ;
b and solve for w
c∗ , the MAXimum-Sharpe-ratio Estimated
Step 3 Set λ in (2.12) to be λ
sparse Regression (MAXSER) portfolio.
When factor structure presents
Under the factor model setting, MAXSER is implemented as follows:
Step 1 Perform OLS regressions of observed asset returns X on observed factor returns F
b;
to obtain βb and U
Step 2 Compute the estimates of the square of the maximum Sharpe ratios θbf , θball and θbu ,
and compute the response rbc ;
Step 3 Choose λ by cross-validation according to the procedure described in Section 2.4.1
0
b
d
d
(based on risk w
all Σall w
all ). Denote the chosen value by λ;
19

b and solve for w
cu∗ ;
Step 4 Set λ in (2.26) to be λ
c∗ and combine the estimates from the previous steps to obtain the
Step 5 Compute w
f
d
MAXSER portfolio weight w
all .

3
3.1

SIMULATION STUDIES

Methods to be Compared with

In addition to the plug-in and nonlinear shrinkage portfolios that we discussed in the
Introduction, we include several other strategies in our simulation comparisons. The complete
list is given in Table 1.
Table 1
List of portfolios under comparison and their abbreviations. “MV” stands for mean-variance portfolio, and
“GMV” stands for global minimum variance portfolio.

Portfolio

Abbreviation

Plug-in MV on factors

Factor

Three-fund portfolio by Kan and Zhou (2007)

KZ

MV/GMV with estimated covariance matrix inputs
MV with sample cov

MV-P

MV with linear shrinkage cov

MV-LS

MV with nonlinear shrinkage cov

MV-NLS

Nonlinear shrinkage adjusted for factor models by Ledoit and Wolf (2017)
GMV with linear shrinkage cov

NLSF
GMV-LS

GMV with nonlinear shrinkage cov

GMV-NLS

MV with no-short-sale constraint
MV with sample cov & no-short-sale constraint

MV-P-NSS

MV with linear shrinkage cov & no-short-sale constraint

MV-LS-NSS

MV with nonlinear shrinkage cov & no-short-sale constraint

MV-NLS-NSS

MV with short-sale constraint & cross-validation
MV with sample cov & short-sale-CV

MV-P-SSCV

MV with linear shrinkage cov & short-sale-CV

MV-LS-SSCV

MV with nonlinear shrinkage cov & short-sale-CV
20

MV-NLS-SSCV

Among the portfolios under comparison, a special one is the portfolio Factor, which is
cf are the sample mean
cf and Σ
the Markowitz portfolio on factors. Specifically, suppose µ
and sample covariance matrix computed from the observed factor returns, then the Factor
portfolio has the following explicit form:
σ
cf −1 µ
bF ac := q
cf .
w
Σ
0 c −1
cf Σf µ
cf
µ

(3.1)

This portfolio is special in the sense that it only involves a small number of assets (3 in
our case). Consequently, the plug-in formula (3.1) indeed gives a nearly optimal portfolio.
Including such a portfolio in the comparison would reveal whether there is benefit to invest
in idiosyncratic components.
On the other hand, the MV/GMV portfolios are constructed by replacing covariance
matrix with the sample/linear shrinkage/nonlinear shrinkage (adjusted for factor model)
estimators in the formulas of MV/GMV portfolio weights. Details about portfolios “KZ”3
and “NLSF” can be found in Kan and Zhou (2007) and Ledoit and Wolf (2017), respectively.
In addition, we construct portfolios with either no-short-sale or short-sale constraints
on portfolio weights. The “MV-P-NSS”, “MV-LS-NSS” and “MV-NLS-NSS” portfolios are
with no-short-sale constraints, and are using the sample/linear shrinkage/nonlinear shrinkage
covariance matrix, respectively. More generally, the MV portfolios with short-sale constraints4 ,
“MV-P-SSCV”, “MV-LS-SSCV” and “MV-NLS-SSCV”, are having short position thresholds
determined by the cross-validation method that we proposed in Section 2.4.1. In such a way,
these portfolios enjoy the same benefit as our MAXSER portfolio in terms of risk control.
We include these portfolios to demonstrate the effectiveness of our cross-validation procedure,
and that the advantage of MAXSER is not only due to the `1 -norm constraints, but rather,
more fundamentally, due to its methodology.
3
4

Following Kan and Zhou (2007), the risk aversion is set to be 3.
b all is an estimate
b all is the sample mean of returns on stocks and factors, and Σ
Specifically, suppose that µ

of the covariance matrix, which can be the sample/linear shrinkage/nonlinear shrinkage covariance matrix.
Then the MV portfolio with short-sale constraint is solved by
b all ,
wSSCV = arg max w0 µ
w

b all w ≤ σ 2 and wi > −λSS for all i.
subject to w0 Σ

Here λSS > 0 is the short position threshold determined via a 10-fold cross-validation as follow: split the
sample into 10 groups of validation sets, with the rest of the observations being the corresponding training
set. For each training set, we solve the optimization for a sequence of λSS to get a solution path, and find the
value of λSS such that the difference between the risk on validation sets and the given constraint is minimized.

21

3.2

Parameter Setting

We simulate data from a three-factor model with the parameters calibrated from real
data. Specifically, out of the stocks that stayed in S&P 500 index during the period of 2007
– 2016, we randomly pick 100 of them. We then regress the monthly excess returns on the
100 stocks over the Fama-French three factor (FF3) returns, and set the resulting slopes to
be the βi ’s; the αi ’s in (2.16) are obtained by hard thresholding the estimated intercepts
with a threshold of 2 standard errors. The covariance matrix of idiosyncratic returns, Σu ,
is obtained by applying the soft-thresholding method proposed in Rothman (2012)5 to the
sample covariance matrix of the residuals in the regression above. For the parameters of
factors, the mean and covariance matrix are taken to be the sample mean and the sample
covariance matrix of the Fama-French three factor returns, respectively.

3.3

Simulation Comparisons

3.3.1

When returns are normally distributed

We first show simulation results for data generated under multivariate normal distribution.
Returns of 100 stocks and 3 factors are generated using the parameters described in Section 3.2.
The level of risk constraint is fixed to be σ = 0.04.
We run 1,000 replications to evaluate the performance of the portfolios in terms of the
risk and (annualized) Sharpe ratio. The comparison results for sample sizes T = 120 and 240
are summarized in Tables 2 and 3.
5

The soft-thresholding method can be implemented in R by the package “PDSCE”. In this case the penalty

parameter “lam” is set to be 0.5.

22

Table 2
Simulation comparison of risks and Sharpe ratios of the portfolios under comparison based on 1,000 replications
where returns are generated from multivariate normal distribution. “True” stands for the theoretical optimal
portfolio. The risk constraint is set to be 0.04. Both average value and standard deviation (in brackets) of
each performance measure are reported.

Normal Distribution

σ = 0.04

T = 120

Portfolio

Risk

Sharpe Ratio

True

0.040

1.882

Factor

0.041 (0.003)

0.401 (0.169)

KZ

0.052 (0.040)

0.329 (0.184)

MAXSER

0.041 (0.005)

1.183 (0.276)

MV-P

0.296 (0.072)

0.367 (0.168)

MV-LS

0.082 (0.006)

0.697 (0.160)

MV-NLS

0.054 (0.017)

0.945 (0.183)

NLSF

0.044 (0.002)

0.837 (0.139)

GMV-LS

0.013 (0.001)

0.438 (0.132)

GMV-NLS

0.015 (0.003)

0.553 (0.148)

MV-P-NSS

0.044 (0.003)

0.399 (0.040)

MV-LS-NSS

0.044 (0.003)

0.409 (0.036)

MV-NLS-NSS

0.043 (0.003)

0.416 (0.035)

MV-P-SSCV

0.044 (0.003)

0.399 (0.040)

MV-LS-SSCV

0.044 (0.003)

0.409 (0.036)

MV-NLS-SSCV

0.044 (0.004)

0.501 (0.169)

MV/GMV with estimated covariance matrix inputs

MV with no-short-sale constraint

MV with short-sale constraint & cross-validation

23

Table 3
Simulation comparison of risks and Sharpe ratios of the portfolios under comparison based on 1,000 replications
where returns are generated from multivariate normal distribution. “True” stands for the theoretical optimal
portfolio. The risk constraint is set to be 0.04. Both average value and standard deviation (in brackets) of
each performance measure are reported.

Normal Distribution

σ = 0.04

T = 240

Portfolio

Risk

Sharpe Ratio

True

0.040

1.882

Factor

0.041 (0.002)

0.467 (0.108)

KZ

0.091 (0.031)

0.909 (0.130)

MAXSER

0.040 (0.003)

1.508 (0.154)

MV-P

0.070 (0.005)

0.911 (0.123)

MV-LS

0.061 (0.004)

0.943 (0.117)

MV-NLS

0.049 (0.004)

1.199 (0.117)

NLSF

0.042 (0.001)

1.068 (0.104)

GMV-LS

0.009 (0.000)

0.450 (0.102)

GMV-NLS

0.009 (0.001)

0.539 (0.167)

MV-P-NSS

0.042 (0.002)

0.415 (0.032)

MV-LS-NSS

0.042 (0.002)

0.420 (0.031)

MV-NLS-NSS

0.041 (0.002)

0.427 (0.030)

MV-P-SSCV

0.042 (0.002)

0.415 (0.032)

MV-LS-SSCV

0.042 (0.002)

0.420 (0.031)

MV-NLS-SSCV

0.042 (0.003)

0.468 (0.137)

MV/GMV with estimated covariance matrix inputs

MV with no-short-sale constraint

MV with short-sale constraint & cross-validation

From Tables 2 and 3, we observe that
• In terms of risk control,
– The risk of our MAXSER portfolio is close to the given constraint, whereas
MV portfolios with covariance matrix estimated by sample, linear shrinkage and
nonlinear shrinkage estimators violate the risk constraint by about 640%, 105%
24

and 35% when T = 120, respectively.
– The risks of Factor, NLSF, and MV portfolios with (no-)short-sale constraints are
similar to the risk of MAXSER.
• In terms of Sharpe ratio,
– MAXSER achieves the highest Sharpe ratio among all portfolios under comparison.
In the T = 240 case, MAXSER achieves about 80% of the theoretical maximum
Sharpe ratio on average, whereas the Sharpe ratio of the MV-NLS portfolio, the
second highest among all portfolios, is about 64% of the theoretical maximum
value on average.
– For the T = 240 case, the 95% confidence interval of the mean Sharpe ratio of
MAXSER is [1.498, 1.518]. In terms of comparison, the 95% confidence interval
for the difference between the mean Sharpe ratios of MAXSER and MV-NLS is
[0.301, 0.317]. In particular, we conclude that the higher Sharpe ratio of MAXSER
is statistically significant.
– Compared with MV-P-SSCV, MV-LS-SSCV and MV-NLS-SSCV portfolios, the
Sharpe ratio of MAXSER is substantially higher, indicating that the outstanding
performance of MAXSER is fundamental and does not rely on cross-validation
only.
• In summary,
– Our MAXSER portfolio effectively controls risk, and is significantly more meanvariance efficient than the other portfolios.
– The comparison with Factor portfolio suggests that investing in idiosyncratic
components, or equivalently, individual stocks, helps improve the mean-variance
efficiency.
3.3.2

When returns are heavy-tailed

Given the empirical evidence that financial returns tend to be heavy-tailed, in the following
we conduct a simulation study for data with heavy-tails. More specifically, we shall let the
factor and idiosyncratic returns be all Student-t distributed with 6 degrees of freedom. The
mean and covariance matrix parameters are taken to be the same as in Section 3.2.
25

Table 4
Simulation comparison of risks and Sharpe ratios of the portfolios under comparison based on 1,000 replications
where returns are generated from t-distribution with 6 degrees of freedom. The underlying mean and covariance
matrix are the same as in Section 3.3.1. “True” stands for the theoretical optimal portfolio. The risk constraint
is set to be 0.04. Both average value and standard deviation (in brackets) of each performance measure are
reported.

t(6) Distribution

σ = 0.04

T = 120

Portfolio

Risk

Sharpe Ratio

True

0.040

1.882

Factor

0.034 (0.003)

0.350 (0.202)

KZ

0.039 (0.031)

0.288 (0.191)

MAXSER

0.033 (0.005)

1.035 (0.281)

MV-P

0.246 (0.060)

0.321 (0.174)

MV-LS

0.062 (0.005)

0.635 (0.169)

MV-NLS

0.042 (0.009)

0.845 (0.179)

NLSF

0.036 (0.002)

0.716 (0.150)

GMV-LS

0.013 (0.001)

0.459 (0.130)

GMV-NLS

0.014 (0.003)

0.572 (0.125)

MV-P-NSS

0.036 (0.003)

0.394 (0.039)

MV-LS-NSS

0.036 (0.003)

0.406 (0.035)

MV-NLS-NSS

0.035 (0.003)

0.411 (0.036)

MV-P-SSCV

0.036 (0.003)

0.394 (0.039)

MV-LS-SSCV

0.036 (0.003)

0.406 (0.035)

MV-NLS-SSCV

0.037 (0.004)

0.531 (0.196)

MV/GMV with estimated covariance matrix inputs

MV with no-short-sale constraint

MV with short-sale constraint & cross-validation

26

Table 5
Simulation comparison of risks and Sharpe ratios of the portfolios under comparison based on 1,000 replications
where returns are generated from t-distribution with 6 degrees of freedom. The underlying mean and covariance
matrix are the same as in Section 3.3.1. “True” stands for the theoretical optimal portfolio. The risk constraint
is set to be 0.04. Both average value and standard deviation (in brackets) of each performance measure are
reported.

t(6) Distribution

σ = 0.04

T = 240

Portfolio

Risk

Sharpe Ratio

True

0.040

1.882

Factor

0.033 (0.002)

0.427 (0.141)

KZ

0.059 (0.023)

0.802 (0.154)

MAXSER

0.033 (0.003)

1.374 (0.203)

MV-P

0.058 (0.004)

0.807 (0.140)

MV-LS

0.048 (0.003)

0.847 (0.133)

MV-NLS

0.040 (0.004)

1.071 (0.138)

NLSF

0.034 (0.001)

0.931 (0.117)

GMV-LS

0.010 (0.000)

0.469 (0.107)

GMV-NLS

0.010 (0.001)

0.538 (0.182)

MV-P-NSS

0.034 (0.002)

0.406 (0.035)

MV-LS-NSS

0.034 (0.002)

0.412 (0.033)

MV-NLS-NSS

0.034 (0.002)

0.418 (0.032)

MV-P-SSCV

0.034 (0.002)

0.406 (0.035)

MV-LS-SSCV

0.034 (0.002)

0.414 (0.046)

MV-NLS-SSCV

0.035 (0.003)

0.529 (0.221)

MV/GMV with estimated covariance matrix inputs

MV with no-short-sale constraint

MV with short-sale constraint & cross-validation

Tables 4 and 5 show that MAXSER portfolio continues to clearly outperform other
portfolios. Another observation is that, if we compare Tables 4 and 5 with Tables 2 and 3 for
the normal case, we see that heavy-tailedness does to some extent hurt all the strategies in
terms of Sharpe ratios.

27

4

EMPIRICAL STUDIES

We investigate the performance of our strategy through two types of empirical studies:
• Practical evaluation:
The asset pool containing the constituents of the DJIA 30 index is considered. Under
a rolling-window scheme to be specified below, at each rebalancing time point, only
the constituents at that time are considered to invest in. Portfolio performances are
compared on the basis of both raw returns and returns net of transaction costs; see
Section 4.2 for the details.
• General statistical evaluation:
We compare the performance of the compared strategies using 100 random datasets, in
which the stocks are randomly picked historical constituents of the S&P 500 index. See
Section 4.3 for the details.

4.1

Portfolios Under Comparison in Empirical Study

In addition to the strategies we compared in simulation studies, we further investigate the
performance of five more portfolios in our empirical study, including the index, the equally
weighted portfolio (“1/N” rule), and three `1 -norm constrained mean-variance portfolios.
Based on different covariance matrices estimated by the sample estimator, the linear shrinkage
estimator (Ledoit and Wolf (2004)), and the nonlinear shrinkage estimator (Ledoit and Wolf
(2017)), we construct the portfolios “MV-P-l1CV”, “MV-LS-l1CV” and “MV-NLS-l1CV” by
imposing the `1 -norm constraint6 for which the tuning parameter λ is determined by the
cross-validation method we proposed in Section 2.4.1. We include these `1 -norm constrained
portfolios to examine the effect of imposing `1 -norm constraint, and, more importantly,
to demonstrate that the advantage of MAXSER is not only due to the constraint and
cross-validation, but rather more due to its methodology.
6

Due to the time cost of solving `1 -norm constrained mean-variance optimizations, the latter three `1 -norm

constrained methods are only applied to the DJIA data with around 30 stocks.

28

4.2

Practical Evaluation

4.2.1

Data & Investment Rolling–Window Scheme

We first evaluate our proposed portfolio, MAXSER, based on the stock universe of DJIA
30 index constituents. We obtain the lists of DJIA 30 index constituents from COMPUSTAT
and CRSP. Fama-French three factors are also included in our asset pools. We evaluate the
portfolios based on a practical rolling-window scheme. More specifically, at the beginning
of each month, one asset pool is formed by including the current constituents of DJIA 30
index and the Fama-French three factors. The portfolios are constructed using the monthly
excess returns during the past T months, where T is the sample size to be specified. If a
stock has missing data in the T -month training period, it is excluded from the asset pool. As
a consequence, the number of stocks would vary over time and can be slightly smaller than
the total number of constituents. The risk constraint is fixed to be the standard deviation of
the DJIA 30 index returns during the first training period. The portfolios are held for one
month, and the corresponding returns are recorded. We then evaluate the performance of
the portfolios under comparison based on the out-of-sample monthly portfolio returns.
4.2.2

Performance Summary

We evaluate the performance of MAXSER and other competing portfolios in terms of
risk, annualized Sharpe ratio7 and statistical test of Sharpe ratio. We also investigate the
effect of transaction costs, and demonstrate the comparisons based on portfolio returns net
of transaction costs.
Without transaction costs
For the DJIA data set with around 30 stocks in each investment pool, we use a sample
size T = 60, which means that each training set contains 5 years returns. The testing period
is February 1967 – December 2016, which results in, for each strategy, 599 out-of-sample
monthly returns. In addition to comparing out-of-sample risks and Sharpe ratios, to show
the significance of the advantage of our portfolio MAXSER, we conduct hypothesis tests
about the Sharpe ratio. More specifically, we test
H0 : SRM AXSER ⩽ SR0

vs Ha : SRM AXSER > SR0 ,

(4.1)

where SRM AXSER denotes the Sharpe ratio of MAXSER portfolio, and SR0 denotes the Sharpe
7

For computing Sharpe ratios, we obtain the risk-free rate rf from Fama/French Data Library.

29

ratio of the portfolio under comparison. To conduct such a test, we adopt Memmel (2003)’s
corrected version of Jobson and Korkie (1981)’s test. The summary without considering
transaction costs is reported in Table 6, which shows the risk, Sharpe ratio, and the p-value
of test (4.1) for each competing portfolio.

30

Table 6
Summary of risk, Sharpe ratio and p-value of Sharpe ratio test of the portfolios under comparison on DJIA 30
index constituents and Fama-French three factors. The testing period is February 1967 – December 2016. The
risk constraint σ = 0.0366 is the standard deviation of the excess returns on DJIA 30 index during February
1962 – January 1967.

DJIA 30 & FF3

σ=0.0366

T = 60

Portfolio

Risk

Sharpe Ratio

p-value

Index

0.044

0.102

0.000

Equally weighted

0.043

0.064

0.000

Factor

0.040

0.425

0.000

KZ

0.124

0.535

0.000

MAXSER

0.044

0.701

–

MV-P

0.089

0.595

0.008

MV-LS

0.053

0.303

0.000

MV-NLS

0.053

0.457

0.000

NLSF

0.051

0.524

0.000

GMV-LS

0.017

0.397

0.000

GMV-NLS

0.017

0.337

0.000

MV-P-NSS

0.040

0.463

0.000

MV-LS-NSS

0.035

0.431

0.000

MV-NLS-NSS

0.035

0.399

0.000

MV-P-SSCV

0.042

0.543

0.000

MV-LS-SSCV

0.047

0.371

0.000

MV-NLS-SSCV

0.041

0.294

0.000

MV-P-l1CV

0.038

0.357

0.000

MV-LS-l1CV

0.045

0.440

0.000

MV-NLS-l1CV

0.046

0.386

0.000

MV/GMV with estimated covariance matrix inputs

MV with no-short-sale constraint

MV with short-sale constraint & cross-validation

MV with `1 -norm constraint & cross-validation

From Table 6, one can observe the following:
31

• In terms of risk control,
– The risk of MAXSER is close to those of the index, the equally weighted, the
Factor, the short-sale constrained and `1 -norm constrained portfolios.
– The risk of the plug-in (“MV-P”) portfolio is more than twice of that of our
MAXSER portfolio, which is far beyond the risk constraint level and hardly
bearable for investors.
• In terms of Sharpe ratio,
– Our MAXSER portfolio yields the highest Sharpe ratio.
– Compared with portfolios with similar risks to ours, the Sharpe ratio of MAXSER
is 29% higher than that of MV-P-SSCV portfolio, which performs better than
other portfolios under comparison at the same risk level.
– The NLSF portfolio, which also takes factor structure into account, yields a 16%
higher risk and a 25% lower Sharpe ratio compared with our MAXSER portfolio.
– The three `1 -norm constrained MV portfolios (MV-P-l1CV, MV-LS-l1CV and MVNLS-l1CV) also possess the sparsity property that only a small portion of stocks
are invested in. However, they yield much lower Sharpe ratio than MAXSER.
Such a comparison shows that the outstanding performance of MAXSER is largely
due to its methodology rather than solely imposing `1 -norm constraint.
– The small p-values of Sharpe ratio tests against all the other portfolios demonstrate
the statistical significance of the advantage of MAXSER.
– The comparison between the Sharpe ratios of MAXSER (0.7) and Factor (0.4)
portfolios indicates that investing on individual stocks in addition to factors using
our strategy MAXSER can substantially improve the performance.
In summary, our MAXSER strategy effectively controls out-of-sample portfolio risk, and
dominates the competing portfolios in terms of mean-variance efficiency.
With transaction costs
Next, we take the transaction cost into account and compute the returns of each portfolio
net of transaction costs. Here we adopt a simple and widely-used formula of transaction

32

costs, which is closely related to the portfolio turnover. The turnover is defined as
Turnover(t) :=

N
X

|wj (t + 1) − wj (t+)|,

(4.2)

j=1

where wj (t + 1) is the weight on asset j at the beginning of period t + 1, and wj (t+) is the
weight of the same asset at the end of period t. The transaction cost of the portfolio at time t
is proportional to Turnover(t) and a cost level c0 , which measures transaction cost per dollar
traded. It can be derived that the portfolio return net of transaction cost in period t, rnet (t),
has the following relation with the total portfolio return r(t):
rnet (t) = (1 − c0 Turnover(t)) (1 + r(t)) − 1.

(4.3)

In Engle et al. (2012), it is found that the average cost level for NYSE stocks is around
0.088%. In the following analysis we adopt c0 = 0.1%. Table 7 shows the risk and Sharpe
ratio net of transaction costs.

33

Table 7
The summary of risk, Sharpe ratio and p-value of Sharpe ratio test based on returns net of transaction costs
of the portfolios on DJIA 30 constituents and Fama-French three factors. The out-of-sample testing period is
February 1967 – December 2016.

DJIA & FF3

c0 = 0.1%

σ=0.0366

Portfolio

Risk

Sharpe Ratio

p-value

Equally weighted

0.043

0.059

0.000

Factor

0.040

0.403

0.000

KZ

0.123

0.440

0.000

MAXSER

0.044

0.630

–

MV-P

0.089

0.516

0.004

MV-LS

0.053

0.251

0.000

MV-NLS

0.053

0.379

0.000

NLSF

0.051

0.466

0.000

GMV-LS

0.017

0.368

0.000

GMV-NLS

0.017

0.266

0.000

MV-P-NSS

0.040

0.437

0.000

MV-LS-NSS

0.035

0.409

0.000

MV-NLS-NSS

0.035

0.373

0.000

MV-P-SSCV

0.042

0.449

0.000

MV-LS-SSCV

0.047

0.282

0.000

MV-NLS-SSCV

0.041

0.150

0.000

MV-P-l1CV

0.038

0.269

0.000

MV-LS-l1CV

0.045

0.339

0.000

MV-NLS-l1CV

0.045

0.273

0.000

MV/GMV with estimated covariance matrix inputs

MV with no-short-sale constraint

MV with short-sale constraint & cross-validation

MV with `1 -norm constraint & cross-validation

Table 7 again shows the clear advantage of our MAXSER portfolio. With transaction
costs deducted, all methods have a lower Sharpe ratio, and MAXSER still yields a Sharpe

34

ratio that is significantly higher than the other strategies.

4.3

General Statistical Evaluation

The comparisons in Section 4.2.2 are from a practical viewpoint, where for each period the
stock pools are updated to include all index constituents. In this section, we further evaluate
the portfolio performances from a more statistical point of view, and base our investigation on
100 random stock pools formed by historical constituents of S&P 500 index. Specifically, each
stock pool consists of 100 stocks randomly chosen form the stock universe, which contains
369 stocks that have been included into S&P 500 index from 1964 to 2016, and also have
complete price data during the period of January 1992 – December 2016, the whole study
period. The stock pool is then kept fixed throughout the study period. In such a way, we
reduce the effect of inclusion/exclusion of stocks. We will make overall evaluations based on
the 100 randomizations.
4.3.1

An Overall Comparison

The following results are again based on the same rolling-window scheme as described
in Section 4.2.1. The sample size is T = 120, and we still include the Fama-French factors
into our investment pools. The means and standard deviations of portfolio risks and Sharpe
ratios are shown in Table 8.

35

Table 8
Summary of risks and Sharpe ratios of the portfolios under comparison for 100 random datasets, each
containing 100 stocks randomly selected from S&P 500 index historical constituents. The testing period is
January 2002 – December 2016. The risk constraint is taken to be the standard deviation of the index excess
returns during January 1992 – December 2001, and is fixed over time and for all asset pools. Both average
value and standard deviation (in brackets) of each performance measure are reported.

S&P 500 & FF3

σ = 0.041

T = 120

Portfolio

Risk

Sharpe Ratio

Index

0.042

0.223

Factor

0.041

0.320

Equally weighted

0.050 (0.002)

0.261 (0.041)

KZ

0.072 (0.019)

0.311 (0.234)

MAXSER

0.043 (0.003)

0.532 (0.185)

MV-P

0.334 (0.031)

0.325 (0.230)

MV-LS

0.065 (0.004)

0.194 (0.180)

MV-NLS

0.061 (0.004)

0.188 (0.179)

NLSF

0.057 (0.003)

0.353 (0.161)

GMV-LS

0.025 (0.001)

0.420 (0.127)

GMV-NLS

0.025 (0.001)

0.414 (0.123)

MV-P-NSS

0.047 (0.002)

0.345 (0.120)

MV-LS-NSS

0.043 (0.002)

0.288 (0.138)

MV-NLS-NSS

0.042 (0.002)

0.285 (0.146)

MV-P-SSCV

0.047 (0.002)

0.345 (0.120)

MV-LS-SSCV

0.043 (0.002)

0.286 (0.139)

MV-NLS-SSCV

0.043 (0.002)

0.286 (0.153)

MV/GMV with estimated covariance matrix inputs

MV with no-short-sale constraint

MV with short-sale constraint & cross-validation

Table 8 shows that
• The plug-in portfolio in general carries a risk much higher than other portfolios. The
KZ portfolio, on average carries a risk 67% higher than the average risk of our MAXSER
portfolio, which is close to the risk constraint.
36

• Among the portfolios with reasonable risk levels, MAXSER achieves the highest Sharpe
ratio on average, which is 27% higher than the average Sharpe ratio of GMV-LS
portfolio, the second highest among the portfolios under comparison.
Next, we again take transaction costs into account. The returns net of transaction costs
are computed by formula (4.3). The transaction cost level is again taken to be 0.1%. The
comparisons are summarized in Table 9.
Table 9
Comparison of risks and Sharpe ratios based on portfolio returns net of transaction costs, for 100 random
asset pools of size 103 formed by S&P 500 constituents and Fama-French three factors. Both average value
and standard deviation (in brackets) of each performance measure net of transaction costs are reported. The
length of training period is T = 120.

S&P 500 & FF3

c0 = 0.1%

σ = 0.0247

Portfolio

Net Risk

Net Sharpe Ratio

Factor

0.041

0.303

Equally weighted

0.050 (0.002)

0.256 (0.041)

Kan

0.072 (0.018)

0.140 (0.234)

MAXSER

0.043 (0.003)

0.467 (0.185)

MV-P

0.328 (0.029)

0.101 (0.231)

MV-LS

0.065 (0.004)

0.139 (0.182)

MV-NLS

0.061 (0.004)

0.123 (0.181)

NLSF

0.057 (0.003)

0.300 (0.159)

GMV-LS

0.025 (0.001)

0.388 (0.126)

GMV-NLS

0.025 (0.001)

0.365 (0.122)

MV-P-NSS

0.047 (0.002)

0.326 (0.121)

MV-LS-NSS

0.043 (0.002)

0.272 (0.139)

MV-NLS-NSS

0.042 (0.002)

0.269 (0.147)

MV-P-SSCV

0.047 (0.002)

0.326 (0.121)

MV-LS-SSCV

0.043 (0.002)

0.269 (0.140)

MV-NLS-SSCV

0.043 (0.002)

0.265 (0.154)

MV/GMV with estimated covariance matrix inputs

MV with no-short-sale constraint

MV with short-sale constraint & cross-validation

37

Table 9 and its comparison with Table 8 reveal the following:
• Transaction costs cause more harm to the portfolios with relatively high risks such as the
plug-in and KZ portfolios, whose Sharpe ratios drop from 0.311/0.325 to 0.140/0.101,
after the deduction of transaction costs.
• MAXSER portfolio maintains its advantage over other portfolios. Its Sharpe ratio is
more than 20% higher than that of GMV-LS, the second best portfolio in terms of
Sharpe ratio among the portfolios under comparison.
In summary, both without or with taking transaction costs into account, our MAXSER
portfolio generally outperforms other portfolios. Moreover, as we will see in the next section,
the comparisons that we observe in terms of Sharpe ratio are statistically significant.
4.3.2

Statistical Tests About Sharpe Ratio

In order to show the statistical significance of the advantage of our MAXSER portfolio
in terms of Sharpe ratio, for the 100 random asset pools formed by S&P 500 stocks and
Fama-French three factors, we also conduct the Sharpe ratio tests (4.1) based on raw returns
and returns net of transaction costs. The histograms of the p-values based on the 100 random
investment pools are given in Figures 4∼5.

38

MAXSER VS Equally weighted

MAXSER VS Factor

MAXSER VS KZ

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

Density

0.2
0.1
0.0
0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

p−value

p−value

p−value

MAXSER VS NLSF

MAXSER VS MV−P

MAXSER VS MV−LS

MAXSER VS MV−NLS

0.8

1.0

0.8

1.0

0.8

1.0

0.8

1.0

0.6
Density

0.4

Density

0.3

Density

0.2

0.4

0.4

0.6

0.5

0.8

0.8

0.6
0.5
0.4
0.3

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0
0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

p−value

p−value

p−value

p−value

MAXSER VS GMV−LS

MAXSER VS GMV−NLS

MAXSER VS MV−P−NSS

MAXSER VS MV−LS−NSS

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Density

0.3
0.2
0.1
0.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

p−value

p−value

p−value

p−value

MAXSER VS MV−NLS−NSS

MAXSER VS MV−P−SSCV

MAXSER VS MV−LS−SSCV

MAXSER VS MV−NLS−SSCV

0.2

0.4

0.6
p−value

0.8

1.0

0.6
0.5

0.6
0.0

0.2

0.4

0.6

0.8

1.0

0.4
Density
0.0

p−value

0.3
0.0

0.1

0.2

Density

0.2
0.0

0.1

0.1
0.0

0.1
0.0
0.0

0.3

0.3

Density

0.2

0.3
0.2

Density

0.4

0.4

0.4

0.5

0.5

0.5

0.6

0.6

0.0

0.0

0.0

0.0

0.1

0.1

0.1

0.2

0.3

Density

Density

0.2

0.2

Density

0.4

0.4

0.3

0.3

0.5

0.5

0.6

0.4

0.4

0.6

0.0

0.0

0.0

0.0

0.1

0.1

0.2

0.2

0.2

Density

0.3

0.4

0.5
Density

0.1
0.0

p−value

0.6

0.2

0.0

0.1
0.0

0.0
0.0

0.3
0.2

Density

0.3
0.2

0.4
0.2

Density

0.4

0.4

0.5

0.6

0.5

0.6

0.6

0.6

0.8

0.7

MAXSER VS Index

0.2

0.4

0.6
p−value

0.8

1.0

0.0

0.2

0.4

0.6
p−value

Figure 4. Histograms of p-values for the Sharpe ratio test (4.1) against the portfolios under comparison,
based on 100 random investment pools. In this figure transaction costs are not taken into account.

39

MAXSER VS Factor

MAXSER VS KZ

0.4

Density

0.3

Density

0.2

0.3

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

p−value

p−value

p−value

MAXSER VS NLSF

MAXSER VS MV−P

MAXSER VS MV−LS

0.8

1.0

0.8

1.0

0.8

1.0

0.8

1.0

0.8

1.0

0.4

Density

0.4

Density

0.3

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

p−value

p−value

p−value

MAXSER VS MV−NLS

MAXSER VS GMV−LS

MAXSER VS GMV−NLS

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

p−value

p−value

p−value

MAXSER VS MV−P−NSS

MAXSER VS MV−LS−NSS

MAXSER VS MV−NLS−NSS

0.4
0.3

Density

0.3

Density
0.4

0.6

0.8

1.0

0.1
0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

p−value

p−value

p−value

MAXSER VS MV−P−SSCV

MAXSER VS MV−LS−SSCV

MAXSER VS MV−NLS−SSCV

0.4

0.5

0.6

0.3

Density

0.3

Density
0.0

0.2

0.4

0.6
p−value

0.8

1.0

0.2
0.1
0.0

0.0

0.0

0.1

0.1

0.2

0.2

Density

0.3

0.4

0.4

0.5

0.6

0.2

0.5

0.0

0.0

0.0

0.0

0.1

0.1

0.2

0.2

0.2

Density

0.3

0.4

0.4

0.5

0.5

0.6

0.6

0.2

0.5

0.0

0.0

0.0

0.00

0.05

0.2

0.1

0.2

Density

0.20

Density

0.15
0.10

0.4

Density

0.6

0.3

0.25

0.8

0.30

0.4

0.35

0.0

0.0

0.0

0.0

0.1

0.2

0.2

0.2

Density

0.4

0.6

0.6

0.5

0.8

0.6

0.8

0.0

0.0

0.0

0.0

0.1

0.1

0.2

0.2

Density

0.4

0.4

0.6

0.5

0.5

0.6

MAXSER VS Equally weighted

0.0

0.2

0.4

0.6
p−value

0.8

1.0

0.0

0.2

0.4

0.6
p−value

Figure 5. Histograms of p-values for the Sharpe ratio test (4.1) against the portfolios under comparison,
based on 100 random investment pools. In this figure transaction costs are deducted from portfolio returns.

40

Note that if the null hypothesis H0 in (4.1) holds, then the p-values would be roughly
uniformly distributed. This is obviously not the case here. Furthermore, we observe from
Figures 4 and 5 that in all comparisons, the p-values are mostly small. The histograms
indicate that, among all portfolios under comparison, MAXSER does in general yield the
highest Sharpe ratio.

5

CONCLUSION

In this paper, we propose a novel approach to construct the mean-variance efficient
portfolio when the number of assets in the investment pool is not small compared with sample
size. We prove that, under a mild sparsity assumption on the optimal portfolio, the MAXSER
portfolio asymptotically achieves the mean-variance efficiency and meanwhile controls the
risk effectively. To the best of our knowledge, this is the first time that these two objectives
can be simultaneously achieved for large portfolios.
In addition to the sound statistical properties, the MAXSER portfolio possesses an
attractive feature for practical implementation. Being a sparse portfolio strategy, MAXSER
performs favorably by investing in only a small portion of assets. Moreover, our strategy
involves relatively low transaction costs as demonstrated in the empirical studies.

References
Bai, J. and Ng, S. “Determining the number of factors in approximate factor models.”
Econometrica, 70(1):191–221 (2002).
Bai, Z., Li, H., and Wong, W.-K. “The best estimation for high-dimensional Markowitz
mean-variance optimization.” (2013).
Bai, Z., Liu, H., and Wong, W.-K. “Enhancement of the applicability of Markowitz’s portfolio
optimization by utilizing random matrix theory.” Mathematical Finance, 19(4):639–667
(2009).
URL http://dx.doi.org/10.1111/j.1467-9965.2009.00383.x
Basak, G. K., Jagannathan, R., and Ma, T. “Jackknife estimator for tracking error variance
of optimal portfolios.” Management Science, 55(6):990–1002 (2009).

41

Best, M. J. and Grauer, R. R. “On the sensitivity of mean-variance-efficient portfolios to
changes in asset means: some analytical and computational results.” Review of Financial
Studies, 4(2):315–342 (1991).
Black, F. and Litterman, R. B. “Asset allocation: combining investor views with market
equilibrium.” The Journal of Fixed Income, 1(2):7–18 (1991).
Britten-Jones, M. “The Sampling Error in Estimates of Mean-Variance Efficient Portfolio
Weights.” The Journal of Finance, 54(2):655–671 (1999).
Brodie, J., Daubechies, I., De Mol, C., Giannone, D., and Loris, I. “Sparse and stable
Markowitz portfolios.” Proceedings of the National Academy of Sciences, 106(30):12267–
12272 (2009).
Chamberlain, G. and Rothschild, M. “Arbitrage, factor structure, and mean-variance analysis
on large asset markets.” Econometrica, 51(5):1281–1304 (1983).
URL http://dx.doi.org/10.2307/1912275
Chopra, V. K. and Ziemba, W. T. “The effect of errors in means, variances, and covariances
on optimal portfolio choice.” The Journal of Portfolio Management, 19(2):6–11 (1993).
Connor, G., Goldberg, L. R., and Korajczyk, R. A. Portfolio risk analysis. Princeton
University Press (2010).
DeMiguel, V., Garlappi, L., Nogales, F. J., and Uppal, R. “A generalized approach to portfolio
optimization: Improving performance by constraining portfolio norms.” Management
Science, 55(5):798–812 (2009).
El Karoui, N. “High-dimensionality effects in the Markowitz problem and other quadratic programs with linear constraints: Risk underestimation.” The Annals of Statistics, 38(6):3487–
3566 (2010).
URL http://dx.doi.org/10.1214/10-AOS795
Engle, R., Ferstenberg, R., and Russell, J. “Measuring and Modeling Execution Cost and
Risk.” The Journal of Portfolio Management, 38(2):14–28 (2012).
Fan, J., Li, Y., and Yu, K. “Vast volatility matrix estimation using high-frequency data for
portfolio selection.” Journal of the American Statistical Association, 107(497):412–428
42

(2012a).
URL http://dx.doi.org/10.1080/01621459.2012.656041
Fan, J., Zhang, J., and Yu, K. “Vast portfolio selection with gross-exposure constraints.”
Journal of the American Statistical Association, 107(498):592–606 (2012b).
URL http://dx.doi.org/10.1080/01621459.2012.682825
Fastrich, B., Paterlini, S., and Winker, P. “Constructing Optimal Sparse Portfolios Using
Regularization Methods.” Available at SSRN 2169062 (2012).
Feng, G., Giglio, S., and Xiu, D. “Taming the Factor Zoo.” (2017).
Garlappi, L., Uppal, R., and Wang, T. “Portfolio selection with parameter and model
uncertainty: A multi-prior approach.” Review of Financial Studies, 20(1):41–81 (2007).
Jagannathan, R. and Ma, T. “Risk reduction in large portfolios: Why imposing the wrong
constraints helps.” The Journal of Finance, 58(4):1651–1684 (2003).
Jegadeesh, N. and Titman, S. “Profitability of momentum strategies: An evaluation of
alternative explanations.” The Journal of Finance, 56(2):699–720 (2001).
Jobson, J. D. and Korkie, B. M. “Performance hypothesis testing with the Sharpe and
Treynor measures.” The Journal of Finance, 36(4):889–908 (1981).
Kan, R. and Zhou, G. “Optimal portfolio choice with parameter uncertainty.” Journal of
Financial and Quantitative Analysis, 42(3):621 (2007).
Korajczyk, R. A. and Sadka, R. “Pricing the commonality across alternative measures of
liquidity.” Journal of Financial Economics, 87(1):45–72 (2008).
Lai, T. L., Xing, H., and Chen, Z. “Mean–variance portfolio optimization when means and
covariances are unknown.” The Annals of Applied Statistics, 5(2A):798–823 (2011).
URL http://dx.doi.org/10.1214/10-AOAS422
Ledoit, O. and Wolf, M. “Improved estimation of the covariance matrix of stock returns with
an application to portfolio selection.” Journal of Empirical Finance, 10(5):603–621 (2003).
—. “A well-conditioned estimator for large-dimensional covariance matrices.” Journal of
Multivariate Analysis, 88(2):365–411 (2004).
URL http://dx.doi.org/10.1016/S0047-259X(03)00096-4
43

—. “Nonlinear shrinkage of the covariance matrix for portfolio selection: Markowitz meets
goldilocks.” The Review of Financial Studies (2017).
Markowitz, H. “Portfolio selection.” The Journal of Finance, 7(1):77–91 (1952).
Memmel, C. “Performance hypothesis testing with the Sharpe ratio.” Finance Letters, 1(1)
(2003).
Michaud, R. O. “The Markowitz optimization enigma: is “optimized” optimal?” Financial
Analysts Journal , 31–42 (1989).
Rothman, A. J. “Positive definite estimators of large covariance matrices.” Biometrika,
99(3):733–740 (2012).
Tibshirani, R. “Regression shrinkage and selection via the lasso.” Journal of the Royal
Statistical Society. Series B. Methodological , 58(1):267–288 (1996).
Tu, J. and Zhou, G. “Markowitz meets Talmud: A combination of sophisticated and naive
diversification strategies.” Journal of Financial Economics, 99(1):204–215 (2011).

44
</reference>

<statements>
1. The MV optimizer is notoriously sensitive to \(\mu\): small estimation errors in expected returns can produce extreme, unstable portfolios, especially in high dimensions.
2. Mean–Variance: Main limitations: Highly sensitive to return estimates; ignores non‑Gaussian tails; static, single‑period.
</statements>

Begin the assessment now. Output only the JSON list, without any conversational text or explanations.