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ARTICLE TEMPLATE
Linear programming models for portfolio optimization using a
benchmark
ARTICLE HISTORY
Compiled September 27, 2018
ABSTRACT
We consider the problem of constructing a perturbed portfolio by utilizing a bench-
mark portfolio. We propose two computationally efficient portfolio optimization
models, the mean-absolute deviation risk and the Dantzig-type, which can be solved
using linear programming. These portfolio models push the existing benchmark to-
ward the efficient frontier through sparse and stable asset selection. We implement
these models on two benchmarks, a market index and the equally-weighted port-
folio. We carry out an extensive out-of-sample analysis with 11 empirical datasets
and simulated data. The proposed portfolios outperform the benchmark portfolio in
various performance measures, including the mean return and Sharpe ratio.
Keywords: mean-absolute deviation risk; Dantzig; portfolio optimization;
perturbation; sparsity; linear programming
JEL Classification Code: G0
1. Introduction
Markowitz’s (1952) mean-variance portfolio model has been extensively developed to
improve its performance. In the mean-variance portfolio model, expected return and
risk constitute the most important factors. The mean-variance portfolio is based on the
idea that investors seek high returns on investment and minimize risk. As the target
is to find portfolios with lower risks for the fixed expected return or higher expected
returns for the given risk, many risk minimization or return maximization approaches
have been considered in constructing such portfolios (Alexander and Baptista 2004;
Benati and Rizzi 2007; Mansini et al. 2007). For optimization problems, one popular
risk measure is the variance of a portfolio return (Jobson and Korkie 1980; Best and
Grauer 1991; Broadie 1993; Britten-Jones 1999; DeMiguel et al. 2009). Although the
variance has a simple quadratic form and can easily be applied to portfolio optimization
problems, it is not easily interpreted and is sensitive to outliers. Additionally, solving
quadratic optimization problems is time-consuming, especially when the data contain
a large number of assets.
With advances in finance, the variance was replaced by more sophisticated risk
measures using the following models: lower semi-standard deviation (Markowitz 1959);
worst conditional expectation (Benati 2003); minimum return (Young 1998); below
target risk (Fishburn 1977); maximum (Young 1998); minimax (Cai et al. 2000; Deng
et al. 2005); L1risk or mean-absolute deviation (MAD) (Konno and Yamazaki 1991);
MAD with the downside risk (Kamil and Ibrahim 2007); and conditional value-at-
risk (CVaR) (Rockafellar and Uryasev 2000; Mansini et al. 2007; Alexander 2009;
Sawik 2009; Durand et al. 2011; Sawik 2012; Lwin et al. 2017; Bernard et al. 2017).
Portfolio optimization problems with some of these risk measures can be solved using
linear programming (LP), which is less time-consuming than solving optimization
problems involving quadratic terms (e.g., variance). Mansini et al. (2003a,b) provided a
systematic overview of the LP solvable models with a wide discussion of their properties
and the first complete computational comparison of the discussed models on real-life
data.
LP allows us to solve large optimization problems with hundreds of thousands of
assets and is numerically stable (Gondzio and Terlaky 1996; Ill´es and Terlaky 2002).
It is also computationally fast because it reduces the number of possible solutions that
need to be evaluated. In addition, LP can accommodate integer variables in the portfo-
lio optimization problem (Sawik 2012), making it easier to use more complex decision
models, including practical constraints such as fixed transaction costs and Boolean-
type constraints on allocations (Young 1998). Analysis of portfolio selection models
in the LP framework by considering appropriate risk measures has been conducted
2
in the literature (Levary 1983; Konno and Yamazaki 1991; Ogryczak 2000; Mansini
et al. 2007; Sawik 2016). Among these, the optimization problem with MAD risk can
be easily extended into frictional cases (Konno and Wijayanayake 1999, 2001), while
the mean-variance model may be more difficult to implement in such cases. The MAD
risk model could constitute an alternative to the widely used mean-variance model,
because MAD is more robust to underlying distributions. The MAD model is also
easily extended to account for downside risk aversion of an investor, and at the same
time preserves the simplicity and linearity of the original MAD model (Michalowski
and Ogryczak 2001a,b). Since the MAD model possesses numerous advantages, its
extension is worth considering.
This paper assumes that the investor is characterized by mean-variance utility com-
bined with some regularization functions related to sparse and stable asset selection
when utilizing a particular portfolio strategy. We propose two perturbation models
allowing the investor to utilize an existing benchmark motivated by the work by Lee
and Park (2016). Instead of using the quadratic optimization (i.e. using the variance)
as in the original work (Lee and Park 2016), we propose two computationally effi-
cient portfolio optimizations utilizing MAD risk and Dantzig optimization (Dantzig),
respectively. The Dantzig optimization basically uses the variance as portfolio risk,
but it can be efficiently solved via LP. Note that solving quadratic optimization prob-
lems is usually time-consuming, especially when the data contain a large number of
assets. The proposed LP models which replace the existing quadratic risk term with
appropriate linear terms are computationally efficient than the existing work. More
importantly, the proposed perturbation methods enjoy several interesting properties
that cannot be found in original methods. We include details of the advantages of the
methods in Section 1.1.
Suppose there exists a benchmark that is stable and close to an efficient frontier.
An investor may utilize this benchmark to construct a better portfolio by allowing a
few additional assets. Specifically, we utilize the benchmark by advancing it to the effi-
cient frontier to construct a superior portfolio. For a reasonable 0 < α ≤1, we suggest
investing 1 −αin a few assets selected by the proposed algorithm and investing αin
3
the benchmark. We propose two methods, West perturbation and North perturbation,
which aim to push a benchmark to the west and north on the risk-return plane, respec-
tively, using a few stable assets from the market. The West perturbation constitutes a
risk-based portfolio, while the North perturbation is return and risk-based. The opti-
mal portfolio depends on the investors’ risk and return preferences. If one is interested
in reducing risk, he or she can use the West perturbation method. North perturbation
considers the situation in which the investor is interested in an active portfolio. The
fundamental aim of the proposed portfolio is to assist investors in improving the qual-
ity of their decisions, along with effectively implementing their investment strategies.
To the best of our knowledge, this paper is the first attempt to construct a portfolio
via LP by holding a few stable and sparse assets with a benchmark.
The motivation of our work is that we hold a well-diversified benchmark with an
additional few sparse and stable stocks, so that the investor could maintain the ad-
vantages of a well-diversified portfolio with an expectation that the investor will also
benefit from holding a few risky assets. As an example, utilizing an appropriate bench-
mark, such as Standard&Poor’s 500 index, can be an effective approach for construct-
ing a portfolio. Note that index funds have become popular, and they offer attractive
risk-return profiles at low costs (O.Strub and P.Baumann 2018). Investment decisions
are usually delegated to professional fund managers. Since managers are often eval-
uated against some benchmarks, utilizing a proper benchmark has become the norm
(Costa and Soares 2004).
As Gaivoronski et al. (2005) argued, decisions involved in constructing the portfolio
using a benchmark would comprise the following elements: (1) the choice of a bench-
mark; (2) the choice of risk and performances measures relative to the benchmark; and
(3) the update strategy to adapt the portfolio to market changes. Our paper focuses
on the update strategy to construct a portfolio. We consider two different benchmarks,
the market index and the equally-weighted portfolio formed from assets in the market.
Note that this naive equally-weighted portfolio is a popular investment strategy for
private investors (Benartzi and Thaler 2001) because investors prefer simple allocation
rules and the implementation of the equally-weighted portfolio is easy. To measure the
4
potential actual benefits that can be realized by investors through the proposed port-
folios, we analyze the out-of-sample performance of the strategies using mean return
and the information ratio, etc.
The West perturbation method is similar to the portfolio optimization approach of
Roll (1992) and Jorion (2003), in the sense that a benchmark is utilized in the opti-
mization step to construct a portfolio while controlling total portfolio risk. However,
the perturbation method is more favorable than that of Roll (1992) and Jorion (2003)
from two financial standpoints. First, the perturbation method constructs a portfolio
with sparse and stable asset selection that can diminish transaction and management
costs (Shen et al. 2014), which cannot be achieved by the methods of Roll (1992) and
Jorion (2003). A sparse portfolio tends to set negligent portfolio weights to actual zero,
so that it optimizes asset allocation by focusing on components that are anticipated
to make substantial contributions to the portfolio. Second, the perturbation method
utilizes the benchmark more directly, in the sense that the obtained asset allocation
depends on the benchmark. On the other hand, Roll (1992) and Jorion (2003) did
not fully exploit the benchmark portfolio; their constructed portfolio is completely
independent of the benchmark portfolio, but only depends on the overall investment
opportunity set.
The North perturbation method is related to portfolio optimization for enhanced
indexation (EI) (Ruszczynski and Vanderbei 2003; Kuosmanen 2004a,b; Luedtke 2008;
Canakgoz and Beasley 2008; F´abi´an et al. 2011; Guastaroba and Speranza 2012; Ro-
man et al. 2013; Kopa and Post 2014; Hodder et al. 2015; Bruni et al. 2015; Longarela
2016), in that it attempts to amplify the returns of an underlying portfolio or index
fund while controlling risk. One recent promising strategy for EI is the selection of
portfolios that stochastically dominate the benchmark. Optimization models for EI
based on stochastic dominance criteria have been examined in several studies (Levy
1992, 2006; Gotoh and Konno 2000). Specifically, Bruni et al. (2017) proposed a new
type of approximate stochastic dominance rule, in which the model can be formulated
as a linear program so that it can be efficiently solved in practice. Comparisons be-
tween the perturbation method and EI methods can be found in empirical analysis
5
part (see Sections 4 and 5).
The remainder of this paper is organized as follows. In the rest of this section, we
summarize the contributions of our work, introduce the notations used in the paper,
and review the perturbation methods (Lee and Park 2016). In Sections 2 and 3, we
develop the proposed portfolio optimization models, MAD and Dantzig, respectively.
In Sections 4 and 5, we describe the implementation details of these models and eval-
uate several out-of-sample performances of the proposed strategies using 11 real data
and simulated data. In Section 6, we conclude the paper. The additional plots and
information on datasets used are available as Supplementary materials.
1.1. Contribution
From the financial optimization perspective, we show that the Dantzig perturbation
model is a new stochastic optimization model that uses the linear form of the variance
of returns as a measure of risk so that it admits linear programming. This model can
be a possible choice to an investor who considers the variance as a risk measure via
fast computation without considering quadratic terms due to a covariance matrix.
Our second contribution is to analytically characterize the properties of the proposed
portfolios. Specifically, under some settings, North-MAD tend to assign non-negative
weights on the components if their in-sample rates of returns are large, and give non-
positive weights to the components if those of rates are small. Under different settings,
West-MAD and North-MAD are no-short-sale portfolios (Jagannathan and Ma 2003),
i.e. they limit the shorting of the portfolio without any constraints regarding shorting
of individual assets. Note that holding the short-sale constraint actually improves the
empirical performance of portfolios (Jagannathan and Ma 2003). MAD perturbation
models are simple models without the short-sale constraint, but they enjoy such advan-
tageous property under some conditions. We also prove that, under some conditions
the Dantzig perturbation model also enjoys no-short-sale properties without holding
such constraints.
Our third contribution is to show that Dantzig perturbation model can be rewritten
as the global minimum variance portfolio with certain regularized covariance matrices,
6
which convey a novel interpretation. The new covariance matrix reflects generalized
similarities between assets, that makes use of information of all of the assets in the
market, which is an innovative concept in portfolio optimization. Utilizing only pair-
wise similarity does not incorporate a comprehensive structure between the two assets
and other assets in the market. The similarity between two assets may be better de-
fined using the information of all of the other assets related to this pair of assets in
the market as in the new regularized covariance matrices. In addition, under some
conditions, the Dantzig portfolio possesses desirable stochastic properties, in that the
selected assets in the current time are not arbitrary in the sense that it consists of the
components that have relatively significant weights at the previous time point. This
suggests that Dantzig could yield stable asset allocation throughout time.
Fourth, the regularization parameters used in the perturbed portfolio are easily
interpretable. For sparse and stable model selection in the perturbed portfolio, the
proposed models minimize the weighted `1penalty that usually appears in the objec-
tive function with a regularization parameter in most of Markowitz’s framework. The
parameters presented in the West portfolios (i.e. c3and c5) define relative risk of the
portfolios with respect to that of the benchmark, while the parameters in the North
portfolios (i.e. c4and c6) correspond to the relative excess return with respect to that
of the benchmark. Therefore, decision makers can specify their target relative risk and
excess return with respect to that of the benchmark by specifying these parameters.
Note that in our empirical analysis, we determined these regularization parameters
via 10-fold cross-validation using the mean return or Sharpe ratio as criteria func-
tions. The main reason for considering these two different cross validation criteria is
to show that the regularization parameters in the proposed perturbation methods can
be determined successfully with reasonable criteria via cross-validation.
Our fifth contribution is to compare the behavior of the proposed portfolios to that
of the benchmark, as well as EI portfolios, on various simulated and empirical data.
Specifically, we consider dynamic setting, such that the constituents are changed by
time in the considered period, motivated by a realistic setting, in which the considered
component sets may change over time. The results confirm that the proposed portfolios
7
attain better out-of-sample performances than other competitors in many cases.
1.2. Limitation of method
First, the proposed LP perturbation methods involve two regularization parameters, α
and ci, to be updated for each time. In the current work, we update these parameters
via 10-fold cross validation. It would be helpful to update them automatically without
cross-validated grid-search in the optimization procedure. Second, the performance of
the proposed LP depends on the initial perturbed portion ˜
δiat initial time t= 1. To
diminish the effect due to this arbitrary initial portion, we use the strategy that omits
a first few years when computing performance measures. One can set an initial portion
based on financial information immediately prior to the initial time, which potentially
reduces the negative effect of this arbitrary initialization.
1.3. Notations
We select assets by certain investment considerations. For example, if we consider the
perturbation method utilizing the Financial Times Stock Exchange 100 Index (FTSE
100) as a benchmark index, we select 100 companies listed on the London Stock
Exchange as components of FTSE 100. Throughout the paper, we denote the number
of assets in an investment list by p. For i= 1,· · · , p, let ribe the rate of return of
asset iand µithe expected return of asset i. Additionally, we denote the weight vector
of the portfolio by δ= (δ1,· · · , δp)0with Piδi= 1, which is the investor’s budget
constraint. The vector of the expected returns by µ= (µ1, . . . , µp)0, the vector of the
rate of returns by r= (r1, . . . , rp)0, and the covariance matrix of rby Σ . We use ˆµi,
ˆµ= (ˆµ1,...,ˆµp)0, and ˆ
Σ to represent the estimates of µi,µ, and Σ, respectively. Hence,
we can express the portfolio return, estimate of the mean of the portfolio return, and
estimate of the variance of the portfolio return as rδ:= δ0r, ˆµδ:= δ0ˆµ, and ˆσ2
δ:= δ0ˆ
Σδ,
respectively. For any vector v∈Rp, let |v|∞= maxj∈{1,··· ,p}|vj|.
8
1.4. Basic idea of the perturbation methods
Lee and Park (2016) proposed two perturbation methods, namely, West and North
perturbations. For each method, α∈(0,1) is invested into a benchmark v, and 1 −α
into a set of selected components of the benchmark v. Specifically, the constructed
portfolio is αv +δ, where δ∈Rprepresents the perturbed weight of the passets.
We start with the case in which the investor is interested in the risk of the portfolio.
The first perturbation method (Lee and Park 2016), West perturbation, is formulated
as follows: for fixed α6= 1 and 0 < c1≤1,
minδ∈RpPp
i=1
|δi|
|˜
δi|,
subject to δ01= 1 −α,
(αv +δ)0ˆ
Σ(αv +δ)≤c1ˆσ2
v,
(1)
where ˆσ2
vis the estimated risk (variance) of the benchmark v,1is a vector of all ones,
and ˜
δiis the weight of the asset iat a previous time. For ˆσ2
v, they used the sample
variance of the benchmark’s return, calculated from the past 12-month data. The
optimization (1) updates the portfolio from time to time, while controlling variations
from the previous time step. Sparse and stable asset selection is mainly achieved by
using the weighted `1penalty in the objective function.
Note that investors usually want to hold a portfolio with a few assets to control
transaction costs, especially to avoid fixed transaction costs that should be paid for
each component. An investor may also prefer to avoid portfolios with very large de-
viations in assets across times. The weighted Lasso penalty in (1) plays a role in
attaining these two preferences simultaneously without holding two different penalty
types, which is one of the main benefit of the perturbation method.
The main consideration is the trade-off between the performance (e.g., the mean
return or Sharpe ratio) of the perturbed portfolio and the management and transaction
costs. The penalty parameter c1controls this trade-off: the higher the c1value, the
worse the performance (higher risk) of the perturbed portfolio, but the lower the
transaction costs. Conversely, with a lower c1value, the perturbed portfolio has lower
9
risk, but the transaction costs may increase.
The second perturbation method, North perturbation, is formulated as follows: for
fixed α6= 1 and c2≥0,
minδ∈RpPp
i=1
|δi|
|˜
δi|,
subject to δ01= 1 −α,
(αv +δ)0ˆ
Σ(αv +δ)≤ˆσ2
v,
ˆµvα+ ˆµ0δ≥ˆµv+c2|ˆµv|,
(2)
where ˆµvis the estimated expected return of the benchmark v. For ˆσ2
v, the sample
variance of the benchmark’s return was used, calculated from the past 12-month data.
It is reasonable to think that the investor would like to select the parameter c2that
maximizes the expected return while controlling the risk. The penalty parameter c2
in (2) controls the trade-off between the performance of the perturbed portfolio and
the management and transaction costs.
2. Mean-absolute deviation risk model
In this section, we introduce our first perturbation model, the MAD risk model. We
consider the following West-MAD perturbation optimization: for fixed α6= 1 and
0< c3≤1, an investor determines a portfolio to solve
minδ∈RpPp
i=1
|δi|
|˜
δi|,
subject to δ01= 1 −α,
E[|(αrv+Pp
i=1 riδi)−E[αrv+Pp
i=1 riδi]|]≤c3ˆσv,
(3)
where ˆσvis the estimated risk of the benchmark’s return. For ˆσv, we use the sample
MAD of the benchmark return, calculated from the past 12-month data.
At the initial time, we set ˜
δi= 1/p for all i; that is, we minimize Pp
i=1 |δi|, subject to
the constraints in (3). The constraint E[|(αrv+Pp
i=1 riδi)−E[αrv+Pp
i=1 riδi]|]≤c3ˆσv
controls the total portfolio risk, measured by MAD. For example, when c3= 1, we
10
consider the set of portfolios with smaller risk than the benchmark, and thus we push
the benchmark to the west in the risk-return plane.
When implementing at time τ, we replace the second constraint of (3) with the
following constraint based on an empirical measure:
1
T
τ−1
X
t=τ−T
α˜r(t)
v+
p
X
i=1
˜r(t)
iδi
≤c3ˆσv,(4)
where ˜r(t)
v:= r(t)
v−¯rv, ˜r(t)
i:= r(t)
i−¯ri, ¯rv:= 1
TPτ−1
t=τ−Tr(t)
v, ¯ri:= 1
TPτ−1
t=τ−Tr(t)
i, and
r(t)
vand r(t)
iare the returns of benchmark vand asset iat time t, respectively. We
use the time length T= 250, i.e., we use the past 12-month data to calculate the
sample mean and risk. Hence, we solve (3) by solving the following linear program
(West-MAD):
minδ∈RpPp
i=1
|δi|
|˜
δi|,
subject to δ01= 1 −α,
1
TPτ−1
t=τ−Tyt≤c3ˆσv,
yt≥α˜r(t)
v+Pp
i=1 ˜r(t)
iδi,
yt≥ −(α˜r(t)
v+Pp
i=1 ˜r(t)
iδi).
(5)
In the above linear program, the vector of the optimal perturbed portfolio weights
ˆ
δhas at most 2T+ 2 nonzero components due to the properties of LP (Konno and
Yamazaki 1991). However, ˆ
δmay not be sufficiently sparse, because one may use a
large T(e.g., T= 250). Therefore, a sparse solution is achieved by using the `1penalty
in the objective function. Specifically, the objective function of (5) plays an important
role in sparse and stable asset selection by penalizing the weights adaptively based on
a previous time value.
Let us consider an investor who chooses a portfolio weights to minimize the following
11
North perturbation problem: for fixed α6= 1 and c4≥0,
minδ∈RpPp
i=1
|δi|
|˜
δi|,
subject to δ01= 1 −α,
E[|(αrv+Pp
i=1 riδi)−E[αrv+Pp
i=1 riδi]|]≤ˆσv,
ˆµvα+ ˆµ0δ≥ˆµv+c4|ˆµv|,
(6)
where ˆµvis the estimated expected return of benchmark v. The additional constraint
ˆµvα+ ˆµ0δ≥ˆµv+c4|ˆµv|aims to push benchmark vto the north by c4|ˆµv|for the
perturbed portfolio to achieve a better expected return than the benchmark v. For ˆµv,
we use the sample mean of the benchmark’s return, calculated from the past 12-month
data, i.e., we replace the third constraint of (6) with:
¯rvα+ ¯r0δ≥¯rv+c4|¯rv|,(7)
where ¯r:= (¯r1,· · · ,¯rp)0∈Rp. Similarly, we replace the second constraint of (6) with
(4). Hence, we solve the following linear program (North-MAD):
minδ∈RpPp
i=1
|δi|
|˜
δi|,
subject to δ01= 1 −α,
1
TPτ−1
t=τ−Tyt≤ˆσv,
yt≥(α˜r(t)
v+Pp
i=1 ˜r(t)
iδi),
yt≥ −(α˜r(t)
v+Pp
i=1 ˜r(t)
iδi),
¯rvα+ ¯r0δ≥¯rv+c4|¯rv|.
(8)
The parameters c3and c4are typically given by investors. To choose these in a
data-driven manner, we use the cross-validation procedure through in-sample period
by utilizing two different criterion functions.
12
2.1. Properties of MAD model
In this subsection, we investigate some properties of the West and North perturbations
of MAD model. By the Lagrange multiplier method, the West perturbation problem
(5) is to minimize:
X
i
|δi|/|˜
δi|+λ1 1
T
τ−1
X
t=τ−T
yt−c3ˆσv!−
τ−1
X
t=τ−T
λ2t(yt− α˜r(t)
v+
p
X
i=1
˜r(t)
iδi!)
−
τ−1
X
t=τ−T
λ3t(yt+ α˜r(t)
v+
p
X
i=1
˜r(t)
iδi!)+λ4(1 −α−δ01).
Let g∈Rpbe the sub-gradient vector of Pi|δi|/|˜
δi|, i.e., the i-th element of gis:
gi= 1/|˜
δi|if δi>0
gi=−1/|˜
δi|if δi<0
gi∈[−1/|˜
δi|,1/|˜
δi|] if δi= 0.
Then, the Karush-Kuhn-Tucker (KKT) optimality conditions are:
gi−Pτ−1
t=τ−T(λ2t+λ3t)˜r(t)
i−λ4= 0 for all i= 1,· · · , p
λ1
T−λ2t−λ3t= 0 for all t=τ−T , · · · , τ −1
λ11
TPτ−1
t=τ−Tyt−c3ˆσv= 0
λ2tnyt−α˜r(t)
v+Pp
i=1 ˜r(t)
iδio= 0, λ3tnyt+α˜r(t)
v+Pp
i=1 ˜r(t)
iδio= 0
1
TPτ−1
t=τ−Tyt≤c3ˆσv
yt≥α˜r(t)
v+Pp
i=1 ˜r(t)
iδi, yt≥ −(α˜r(t)
v+Pp
i=1 ˜r(t)
iδi)
δ01= 1 −α, λ1, λ2t, λ3t≥0.
(9)
Proposition 2.1. Let ˆ
δbe West-MAD, i.e., the solution to (5). Then ˆ
δj≥0for all
j.
Proof. The proof is deferred to the Supplementary materials.
Proposition 2.1 reveals that West-MAD is a no-short-sale portfolio (Jagannathan
13
and Ma 2003). West-MAD limits the shorting of the portfolio without the short-sale
constraint. Jagannathan and Ma (2003) reported that a short-sale constraint actually
improves the empirical performance of portfolios. The West-MAD enjoys this good
property without holding the constraint. The no-short-selling property of West-MAD
can be viewed as a more realistic condition on the portfolio selection problem, since
it is often difficult for investors to short sell an asset because short selling is usually
expensive for individual investors and is generally prohibited for most institutional
investors.
Similarly, by the Lagrange multiplier method, the North perturbation problem (8)
can be reformulated. See Section 3 of the Supplementary materials for details.
Proposition 2.2. Let ˆ
δbe North-MAD, i.e., the solution to (8). If λ5is non-zero
(i.e. return constraint is binding), then it holds that if Pτ−1
t=τ−T˜r(t)
i/T ≥ −λ4/λ5, then
ˆ
δi≥0, else if Pτ−1
t=τ−T˜r(t)
i/T ≤ −λ4/λ1, then ˆ
δi≤0. But if λ5= 0, then ˆ
δj≥0for
all j.
Proof. The proof is deferred to the Supplementary materials.
Proposition 2.2 reveals the interesting properties of the North-MAD. When the
return constraint is binding (i.e., λ5>0), the sign of the weight ˆ
δifor the stock i
depends on whether or not its past rate of return Pτ−1
t=τ−T˜r(t)
iis larger than −λ4/λ5.
This suggests that North-MAD depends highly on the momentum of the past rate of
return. On the other hand, when λ5= 0, North-MAD enjoys no-short-sale property.
2.2. Choosing regularization parameters
For MAD optimization problems (5) and (8), we choose regularization parameters
α,c3, and c4using 10-fold cross-validation. We divide the daily returns of the past
12 months into 10 subsets of approximately equal size, where each subset serves as
a validation set, respectively. For each validation set, we construct a portfolio using
the remaining nine subsets as a training set, and the mean return or Sharpe ratio of
the constructed portfolio is computed over the validation set. The main reason for
considering these two different cross validation criteria is to show that the regulariza-
14
tion parameters in the proposed perturbation methods can be determined successfully
with reasonable criteria via cross-validation. These values are averaged over 10 trials,
and a particular regularization parameter set (α, c3) or (α, c4) with the highest score
is chosen for West-MAD and North-MAD optimization, respectively. The parameter
αis selected from the set {0.86,0.88,...,0.98}, while parameters c3and c4are se-
lected from the sets {0.6,0.7,...,1.00}and {0,0.2,...,0.8}, respectively. Note that
the optimization problems could be infeasible for the selected c3and c4. If West-MAD
optimization (5) is infeasible for the selected c3, we then increase c3until (5) becomes
feasible. Similarly, if North-MAD optimization (8) is infeasible for the selected c4, we
then decrease c4until (8) becomes feasible. In most cases, we observe that (5) and (8)
are feasible when c3≥0.8 and c4<0.4, respectively.
3. Dantzig-type model
In this section, we introduce the second perturbation method, a Dantzig-type model
(or simply Dantzig), which uses variance as a measure of risk, but does not involve any
quadratic terms and can be solved via LP. We define the West-Dantzig perturbation
(West-Dantzig) as follows: for fixed α6= 1 and c5>0,
minδ∈RpPp
i=1
|δi|
|˜
δi|,
subject to δ01= 1 −α,
|ˆ
Σ0(αv +δ)|∞≤c5ˆσ2
v,
(10)
where ˆσ2
vis the estimated risk of benchmark v. For ˆσ2
v, we use the sample variance of
the expected benchmark return, calculated from the past 12-month data. Note that
the second constraint |ˆ
Σ0(αv +δ)|∞≤c5ˆσ2
vis a linear version of quadratic constraint
(αv +δ)0ˆ
Σ(αv +δ)≤˜c5for some ˜c5>0. This idea is essentially the same as for the
Dantzig selector (Candes and Tao 2007), which used different optimization problems
under the linear regression model, and hence the use of “Dantzig”.
We add an additional constraint that aims to achieve a higher portfolio return for
the West-Dantzig optimization (10), and construct a corresponding North-Dantzig
15
perturbation (North-Dantzig) as follows: for fixed α6= 1 and c6≥0,
minδ∈RpPp
i=1
|δi|
|˜
δi|,
subject to δ01= 1 −α,
|ˆ
Σ0(αv +δ)|∞≤c5ˆσ2
v,
ˆµvα+ ˆµ0δ≥ˆµv+c6|ˆµv|,
(11)
where ˆµvand ˆµare the sample mean of the benchmark vand the sample mean vector
of the assets based on the past 12-month data, respectively. In (11), we set c5= 1 for
simplicity, which works well in our empirical setting. In the implementation of (11), we
replace the third constraint with (7). Note that (10) and (11) are not bona fide, because
they involve an unknown benchmark weight through the term ˆ
Σ0v. However, this can
be substituted by utilizing the estimate of the cross-covariance of the benchmark and
other assets. Specifically, since Cov(v0r, r) = v0Cov(r, r) = Σ0v, we replace ˆ
Σ0vwith
the sample cross-covariances of the benchmark return and asset returns.
3.1. Properties of Dantzig perturbation models
In this subsection, we show that the West and North perturbations of the Dantzig
model are the global minimum variance portfolios (DeMiguel et al. 2009) of certain
regularized sample covariance matrices. By the Lagrange multiplier method, the West
perturbation problem (10) is to minimize:
X
i
|δi|/|˜
δi|+X
j
λ1j
2(αv +δ)0ˆ
Σ0eje0
jˆ
Σ(αv +δ)−c2
5ˆσ2
v+λ2(1 −α−δ01),
where ejis the p-dimensional unit vector with the jth component 1. Let g∈Rpbe
the sub-gradient vector of Pi|δi|/|˜
δi|, i.e., the i-th element of gis:
gi= 1/|˜
δi|if δi>0
gi=−1/|˜
δi|if δi<0
gi∈[−1/|˜
δi|,1/|˜
δi|] if δi= 0.
16
Then, the Karush-Kuhn-Tucker (KKT) optimality conditions are:
Pp
j=1 λ1jnˆ
Σ0eje0
jˆ
Σδ+ˆ
Σ0eje0
jˆ
Σαvo+g−λ21= 0,
λ1j(αv +δ)0ˆ
Σ0eje0
jˆ
Σ(αv +δ)−c2
5ˆσ2
v= 0,|ˆ
Σ0(αv +δ)|∞≤c5ˆσ2
v,
δ01= 1 −α, λ1j≥0.
(12)
Proposition 3.1. The solution ˆ
δto (10) is also the solution to the following
minimum-variance problem:
min β0˜
Σ1β,
s.t. β01= 1 −α,
where the regularized sample covariance matrix ˜
Σ1is given by
˜
Σ1=ˆ
ΣDˆ
Σ + η110+1η0
1,(13)
where η1=g
1−α+ˆ
Σ0Dˆ
Σαv
1−α,and D=diag(λ11,· · · , λ1p),λ11 ,· · · , λ1pare the Lagrangian
multipliers, and gis the sub-gradient of Pi|δi|/|˜
δi|evaluated at ˆ
δ.
The regularized covariance matrix ˜
Σ1in Proposition 3.1 is the weighted sum of the
transformed covariance estimate ˆ
ΣDˆ
Σ and the structured rank-two matrix. Note that
covariance ˆ
Σ characterizes the similarity of each asset pair. We can see that the (i, j)
component of the matrix ˆ
ΣDˆ
Σ is related to a correlation between ˆ
Σi,·and ˆ
Σj,·. In other
words, if two assets iand jare both highly correlated with the same set of assets,
the new covariance [ˆ
ΣDˆ
Σ]ij is high. The ˆ
ΣDˆ
Σ can be understood as a “correlation
of correlation between assets”, a new similarity measure of assets that makes use of
information of all of the assets in the market. Therefore, ˆ
ΣDˆ
Σ may better explain
asset pairs than those of the regular covariance matrix ˆ
Σ through the incorporation
of information from other assets.
The matrix ˜
Σ1may be interpreted in a similar spirit to Jagannathan and Ma (2003)
17
and DeMiguel et al. (2009). Note that in the unconstrained portfolio variance mini-
mization problem, those stocks jwith low covariances with other stocks tend to receive
high portfolio weights ˆ
δj(Jagannathan and Ma 2003). Note also that the effect of g
1−α
in η1is that whenever ˆ
δi>0 for stock i, its variance was raised by 2
(1−α)|˜
δi|, and its
covariance with another stock jwas increased by 1
(1−α)|˜
δi|. Since the weight tends to
be large (i.e., ˆ
δi>0), for those stocks iwith low covariances with other stocks in the
unconstrained variance minimization problem, and low estimated covariances tend to
suffer from an under-estimating error (Jagannathan and Ma 2003), this may reduce
sampling error and achieve a shrinkage-like effect (Jagannathan and Ma 2003).
Proposition 3.2. Suppose α < 1. Let ˆ
δbe the solution to (10). Let ˆ
S={j|ˆ
δj6= 0}.
If |ˆ
Σ0(αv +ˆ
δ)|∞< c5ˆσ2
v, then ˆ
δj≥0for all jand ˆ
S={j| |˜
δj| ≥ γ}for some γ > 0.
Proof. The proof is deferred to the Supplementary materials.
Proposition 3.2 implies that the deviations part ˆ
δin West-Dantzig does not have
any negative components when ˆ
δlies strictly in the interior of the feasible region of the
risk constraint, which suggests that West-Dantzig would allow for a natural handling
of negative δvalues. Note that the selected assets in ˆ
δare not arbitrary in the sense
that it consists of the assets that have relatively significant weights at the previous
time point.
By using the same argument, the Karush-Kuhn-Tucker (KKT) optimality conditions
of the North perturbation problem (11) can be also reformulated. See Section 4 of the
Supplementary materials for details. The following Proposition 3.3 shows that North-
Dantzig is the GMVP of the regularized covariance matrix:
Proposition 3.3. The solution ˆ
δto (11) is also the solution to the following
minimum-variance problem:
min β0˜
Σ2β,
s.t. β01= 1 −α,
18
where the regularized sample covariance matrix ˜
Σ2is given by
˜
Σ2=ˆ
ΣDˆ
Σ + η210+1η0
2,(14)
where η2=g
1−α+ˆ
Σ0Dˆ
Σαv
1−α−λ3
1−αˆµ, and D=diag(λ11,· · · , λ1p),λ11,· · · , λ1p, λ3are
non-negative Lagrangian multipliers, and gis the sub-gradient of Pi|δi|/|˜
δi|evaluated
at ˆ
δ.
The matrix ˜
Σ2can be interpreted as that of ˜
Σ1in the West perturbation method.
Since the first and the second terms in η2are basically the same as η1, we focus on
the role of the last term −λ3
1−αˆµin η2when interpreting ˜
Σ2. Related to this last term,
notice that for the stock i, its variance is raised by −2λ3
1−αˆµi, and its covariance with
another stock jis raised by −λ3
1−αˆµi. Since ˆµirepresents the expected return of the
stock i, this means that for the stock iwith high expected return, its variance and
covariance with other stocks are decreased, and thus the stock itends to be selected
in the deviations part ˆ
δ. This is consistent with the goal of the North perturbation, as
it aims to outperform the benchmark in terms of return.
The following Proposition 3.4 shows that when the risk constraint in (11) is un-
binding, i.e., λ1i= 0, the weight ˆ
δifor the asset iis determined based on ˆµior ˜
δi,
depending on whether the return constraint is binding or unbinding:
Proposition 3.4. Suppose α < 1. Let ˆ
δbe the solution to (11) satisfying |ˆ
Σ0(αv +
ˆ
δ)|∞< c5ˆσ2
v. If λ36= 0 (i.e., the return constraint is binding), then there exists the
stork jwith ˆµj>−λ2/λ3, and we have
ˆ
δi≥0if ˆµi>−λ2/λ3
ˆ
δi≤0if ˆµi<−λ2/λ3.
Or else if λ3= 0 (i.e., the return constraint can be unbinding), then ˆ
δj≥0for all j
and ˆ
S={j| |˜
δj| ≥ γ}for some γ > 0.
19
Proof. The proof is deferred to the Supplementary materials.
Proposition 3.4 shows that when the risk constraint in (11) is unbinding, but the
return constraint is binding, the stock ihas a non-negative weight ˆ
δiin the North
perturbation when its estimated expected return is greater than −λ2/λ3, while it
has a non-positive weight when its estimated expected return is less than −λ2/λ3.
Here, −λ2/λ3determines the signs of the weights in ˆ
δ. The existence of the stork j
with ˆµj>−λ2/λ3implies that −λ2/λ3is not large enough under the first case in
Proposition 3.4. This demonstrates that when the risk constraint is unbinding but
the return constraint is binding for the optimal ˆ
δ, North-Dantzig tends to give high
weights in ˆ
δto those stocks with a high expected return. However, when λ3= 0, i.e.,
the return constraint can also potentially be unbinding, then the selected assets in ˆ
δ
consists of those with relatively significant weights at the previous time point. This
implies that Dantzig could yield stable asset allocation throughout time.
3.2. Choosing regularization parameters
For Dantzig optimization problems (10) and (11), we choose regularization parameters
α,c5, and c6using 10-fold cross-validation similar to the MAD model. The parameter
αis selected from the set {0.86,0.88,...,0.98}, while the parameters c5and c6are
selected from the sets {0.1,...,0.5}and {0,0.2,...,0.8}, respectively. If West-Dantzig
optimization (10) is infeasible for the selected c5, we then increase c5until (10) becomes
feasible. Similarly, if North-Dantzig optimization (11) is infeasible for the selected c6,
we then decrease c6until (11) becomes feasible. In most cases, we observe that (10)
and (11) are feasible when c5≥0.2 and c6<0.5, respectively, and that Dantzig
optimization problems tend to achieve feasibility easier compared with MAD problems.
3.3. Estimation of the covariance matrix
When implementing Dantzig model, the optimization problems (10) and (11) require
ˆ
Σ, an estimate of the covariance matrix. We use the daily returns of the past 12-month
data at every first trading day of each month. To estimate the covariance matrix of the
20
Table 1. Perturbation methods, performance measures, and port-
folio types.
Perturbation method Performance measure Portfolio type
west perturbation Mean return WR
Sharpe ratio WS
north perturbation Mean return NR
Sharpe ratio NS
returns, we apply the shrinkage method of Ledoit and Wolf (2004a,b), which takes an
optimal weighted average of the sample covariance matrix towards the identity matrix.
The shrinkage method for the covariance matrix estimation is suitable for large data
sets, and both well-conditioned and more accurate than the sample covariance matrix
(Ledoit and Wolf 2004a,b). Moreover, as in Becker et al. (2015), if the variance of
estimators is large, for example, for short observation periods or large samples, it is
recommendable to additionally implement constraints or to use the estimator of Ledoit
and Wolf (2004a,b).
4. Implementation details
In this section, we present an extensive empirical analysis of the proposed models by
giving the implementation details and analyzing the out-of-sample performance. For
MAD and Dantzig models, we consider four portfolio types, as listed in Table 1, based
on the West or North perturbation methods and the two performance measures (e.g.,
the mean return or Sharpe ratio) used in the cross-validation. We compare the MAD
and Dantzig based on the four perturbed portfolios by calculating various performance
measures.
We compare the seven portfolio selection models listed below:
(1) West-MAD: the proposed MAD model; consider two types of portfolios “WR-
MAD” and “WS-MAD” corresponding to when the mean return and Sharpe
ratio are considered for cross-validation criteria, respectively.
(2) North-MAD: the proposed MAD model; consider two types of portfolios “NR-
MAD” and “NS-MAD” corresponding to when the mean return and Sharpe
ratio are considered for cross-validation criteria, respectively.
(3) West-Dantzig: the proposed Dantzig model; consider two types of portfolios
21
“WR-Dantzig” and “WS-Dantzig” corresponding to when the mean return and
Sharpe ratio are considered for cross-validation criteria, respectively.
(4) North-Dantzig: the proposed Dantzig model; consider two types of portfolios
“NR-Dantzig” and “NS-Dantzig” corresponding to when the mean return and
Sharpe ratio are considered for cross-validation criteria, respectively.
(5) EI-linear: enhanced indexation (EI) model proposed by Bruni et al. (2015); it
utilizes a linear bi-objective optimization approach to EI that maximizes average
excess return and minimizes underperformance over a learning period.
(6) EI-mixed: enhanced indexation (EI) model proposed by Canakgoz and Beasley
(2008); it presents related mixed-integer linear programming formulations and
includes transaction costs and a constraint limiting the number of assets that
can be selected.
(7) EI-CZSD: enhanced indexation (EI) model proposed by Bruni et al. (2017); it
is based on requirement that the cumulative performance of the selected portfolio
on all subsets of past observations outperforms that of the index up to an
tolerance.
Note that the aforementioned three EI methods “EI-linear”,“EI-mixed”, and “EI-
CZSD” can be also computed via LP. When implementing the EI methods, we used
the regularization parameters suggested by the original papers. Note that EI-mixed
method (Bruni et al. 2017) often suffers from infeasibility due to its constraint related
to transaction cost.
In the analysis, we construct an optimal portfolio by solving an optimization prob-
lem for each month. In line with DeMiguel et al. (2009), Bruni et al. (2015), and
Besslera et al. (2017), we employ a rolling time window approach (RTW). i.e., we
shift the in-sample window and consequently the out-of-sample window all over the
time, representing investors who construct a portfolio using recent historical data.
The RTW approach allows for rebalancing to capture dynamic market conditions,
making it more suitable for real applications. In particular, the optimal tracking port-
folio with regularization parameters selection via cross validation is determined using
a 250-observation (one year) in-sample window and kept used for the next 21 out-
22
of-sample days (one month), following the approach in Jegadeesh and Titman (2001)
and Bruni et al. (2017). Then, the in-sample window is moved forward by 21 days,
and the new optimal portfolio is constructed using 250 days and used for the next 21
out-of-sample days, and so forth, i.e., the portfolio is updated every month. Note that
the performances of the portfolio are only computed from the out-of-sample part.
4.1. Datasets
We test the proposed portfolios on two types of data. First, we consider 11 real-world
data sets from major stock markets across the world listed below.
(1) DAX30 (Deutscher Aktienindex, Germany): containing 30 assets from the period
January 2004 - December 2015.
(2) FTSE100 (Financial Times Stock Exchange 100, UK): containing 81 assets from
the period January 2004 - December 2015.
(3) FTSE250 (Financial Times Stock Exchange 250, UK): containing 170 assets
from the period January 2004 - December 2015.
(4) DOW30 (Dow Jones Industrial Average, USA): containing 29 assets from the
period January 2004 - December 2015.
(5) S&P100 (Standard & Poor’s 100 Stock Index, USA): containing 89 assets from
the period January 2004 - December 2015.
(6) S&P500 (Standard & Poor’s 500 Stock Index, USA): containing 445 assets from
the period January 2004 - December 2015.
(7) Hang Seng (Hong Kong): containing 43 assets from the period November 2005
- November 2016.
(8) NASDAQ100 (National Association of Securities Dealers Automated Quotation,
USA): consisting of 83 assets from the period March 2004 - November 2016.
(9) Euro Stoxx50 (Eurozone): containing 50 assets from the period May 2001 - April
2016.
(10) NASDAQ3000 (National Association of Securities Dealers Automated Quota-
tion, USA): consisting of 2760 assets from the period January 2004 - December
23
2015.
(11) RUSSELL2000: consisting of 1920 assets from the period January 2004 - De-
cember 2015.
For the Hang Seng, Euro Stoxx50, and NASDAQ100, we use the publicly available
daily data from Carleo et al. (2017). For the first six and the last two data sets, we
collected the daily data of the benchmark and its components from the time-period
January 2004 - December 2015 obtained from the Bloomberg terminal (Bloomberg
2006). For the choice of constituents of the indices, stocks with at least 10 years of
observations were included, following Bruni et al. (2017) and Carleo et al. (2017).
Second, we consider partially simulated data generated from the aforementioned 11
data. Motivated by a realistic setting, in which the considered component sets may
change over time, we generate a dynamic setting such that the constituents are changed
by time in the considered period. Specifically, for each of the 11 data, we consider
ten additional simulated assets i=p+ 1,· · · , p + 10 to the existing pcomponents
set, with the rate of return of the asset p+iat time t(say rt
p+i) being rt
p+i=
Q10(i−p−1)%({rt
1,· · · , rt
p})+zt
i. Here, the first part represents the 10(i−p−1)% quantile
of the set of prates of returns at time t(i.e. {rt
1,· · · , rt
p}), and zt
i∼N(0, σ2
t) are
independent random noise, where σtis the standard deviation of the {rt
1,· · · , rt
p}.
For example, rt
p+1 is the minimum rate of return at time tplus zt
1. For a dynamic
setting, we set p+ 5 assets indexed by {1,2,· · · , p + 5}as a component set at the
time 0, while the five simulated assets, p+ 6,· · · , p + 10, are added in the component
sets at times T/6,2T /6,· · · ,5T /6, respectively, and are contained in the sets until
the last time point T. On the other hands, the five simulated assets, p+ 1,· · · , p + 5,
are eliminated from the component set at times T/7,2T /7,· · · ,5T /7, respectively.
This dynamic setting reflects the case in which assets with low returns are likely to
be removed from a component set, while assets with high returns are added in the
component set during the time period. For each of the 11 generated data, at each
time t, we considered the equally-weighted portfolio constructed from the components
corresponding to the time t.
We calculate monthly optimized portfolios at the first trading day of each month,
24
based on the different optimization approaches. Since the initial performances of the
perturbed portfolios depend on the arbitrary initial perturbed portion ˜
δi= 1/p at
initial time t= 1, we omit the first five years’ performance measures. For example,
when considering January 2004 to December 2015, which span 12 years, we record the
last seven years’ performances. In other words, to calculate performance measures, we
set τ= 12 months×5 years = 60 months, and T= 12 months×12 years = 144 months;
see Subsection 4.2 for the considered performance measures that depend on τand T.
4.2. Performance measures
Let wt=αtvt+δtbe a portfolio at time t. Let At={1≤j≤p|δt,j 6= 0}be the set
of nonzero components in the perturbed portion δt. Let rt∈Rpbe a vector of asset
returns at time t. To evaluate the performance of portfolio wtover the period of time
from τ+ 1 to T, we use the following out-of-sample performance measures:
(i) Mean return: µw=1
T−τPT
t=τ+1 w0
trt,
(ii) Risk: σw=q1
T−τ−1PT
t=τ+1(w0
trt−µw)2,
(iii) Sharpe ratio (Sharpe 1966, 1994): the ratio between the mean return and risk:
µw
σw. The larger the Sharpe ratio, the better the risk-adjusted returns.
(iv) Information ratio (Goodwin 1998): the ratio of the expected annualized active
return (µw−v) to annualized standard deviation of the active return (σw−v) :
µw−v
σw−v. Information ratio is used to assess the ability of considered portfolios
to consistently generate excess returns relative to a benchmark portfolio. The
median manager typically provides an Information ratio of near or below zero
(Grinold and Kahn 2000).
(v) Sparsity at time t: the proportion of selected assets in the perturbed portfolio
for each time t:|At|/p, where |At|is the size of the selected assets at time t.
Sparse asset selection in the perturbed portfolio is critical for investors to control
management costs and plays a key role in formulating portfolios.
(vi) Stability at time t: the relative size of set differences of the selected assets be-
tween adjacent time points: |At4At−1|/p, where |At4At−1|is the size of the
set difference of the selected assets at times t−1 and t. Stable asset selection
25
is directly related to transaction costs, and thus it is one of the main factors to
be considered in the perturbed portfolio.
(vii) Turnover at time t: the sum of the absolute value of the rebalancing trades
across the p assets and over the T−τ−1 trading dates, normalized by the total
number of trading dates: 1
T−τ−1PT−1
t=τPp
j=1 |δt+1
i−δt
i|. This measure serves as
a proxy for transaction costs (Gerhard and Hess 2003).
These performance measures have been commonly used in analyses (Fernandes et al.
2011; Xing et al. 2014; Fastrich et al. 2015). For comparisons of the perturbation
methods and the EI methods, we use mean return, Sharpe ratio, Information ratio,
and Turnover.
5. Performances of the portfolios
In this section, we evaluate several out-of-sample performances utilizing the two data
types described in Subsection 4.1. Any results using the partially simulated data are
included in the Supplementary materials due to space limitations.
26
5.1. Risk-return plane
-0.2 -0.1 0 0.1 0.2
Risk
-0.1
-0.05
0
0.05
0.1
Return
Index;West
-0.1 -0.05 0 0.05 0.1
Risk
-0.1
-0.05
0
0.05
0.1
Return
Index;North
-0.2 -0.1 0 0.1 0.2
Risk
-0.05
0
0.05
Return
Equal;West
-0.2 -0.1 0 0.1 0.2
Risk
-0.1
-0.05
0
0.05
0.1
Return
Equal;North
Return
Sharpe
Figure 1. Relative positions of MAD perturbed portfolios with respect to the benchmark when the 11 real
data are considered. The two different symbols are used in the plots based on the cross-validation criterion; the
cross and circle symbol represent the portfolios that use mean return and Sharpe ratio as criterion functions,
respectively. Since we consider the 11 data sets for the market index and equally-weighted portfolio cases, each
subplot in the first and second rows contains 11 ×2 = 22 points.
-0.1 -0.05 0 0.05 0.1
Risk
-0.05
0
0.05
Return
Index;West
-0.2 -0.1 0 0.1 0.2
Risk
-0.1
-0.05
0
0.05
0.1
Return
Index;North
-0.05 0 0.05
Risk
-0.1
-0.05
0
0.05
0.1
Return
Equal;West
-0.1 0 0.1
Risk
-0.1
-0.05
0
0.05
0.1
Return
Equal;North
Return
Sharpe
Figure 2. Relative positions of Dantzig perturbed portfolios with respect to the benchmark when the 11
real data are considered. The symbols are the same as those of Figure 1.
Figures 1-2 display the relative positions of the MAD and Dantzig portfolio, respec-
tively, in the risk-return plane with respect to the benchmark when the real data are
27
considered. The left and right plots correspond to the West and Morth perturbations,
respectively, and from top to bottom, the index and the equally-weighted portfolio are
considered as a benchmark, respectively. Each subplot corresponds to a specific port-
folio type (West or North) and specific benchmark type (index and equally-weighted).
Each point in each subplot corresponds to a specific data set with a specific criterion
function (the mean return or Sharpe ratio). The horizontal and vertical axes corre-
spond to the relative risk (i.e., variance) and the average excess return of the proposed
portfolios with respect to the benchmark, respectively.
When the index is used as a benchmark, we observe that the West-MAD and West-
Dantzig portfolios consistently reduce the risk and push the benchmark to the west
in the risk-return plane, respectively. On average, the West-MAD portfolio and West-
Dantzig reduce the variance over the benchmark by 0.072 and 0.041 (p-value=2.11 ×
10−6and 5.11 ×10−4), respectively. The North-MAD and North-Dantzig portfolios
have higher variances than the benchmark by 0.034 and 0.043 (p-value=1.03 ×10−5
and 6.23×10−5), respectively. On the other hand, the North-MAD and North-Dantzig
portfolios have higher returns than the benchmark by 0.045 and 0.052 (p-value=0.001
and 3.22 ×10−4), respectively. However, the West portfolios do not improve the mean
return over the benchmark, in that the difference is not statistically significant (p-value
>0.5), which is anticipated because main goals of West and North perturbed portfolio
are to achieve lower risks and higher returns than the benchmark, respectively.
When the equally-weighted portfolio is used as a benchmark, the main patterns of re-
sults are similar to those of when the index is used as a benchmark. We observe that the
West portfolios consistently reduce the risk and push the benchmark to the west, while
the North portfolios improve the mean returns in the risk-return plane. Specifically,
the West-MAD and West-Dantzig portfolio reduce the variance over the benchmark
by 0.091 and 0.039 (p-value=4.10 ×10−7and 3.02 ×10−4) on average, respectively.
However, the West portfolios do not improve the mean return over the benchmark
as we expected (p-value >0.5). The North-MAD and North-Dantzig portfolios have
higher variances than the benchmark by 0.031 and 0.033 (p-value=1.33 ×10−4and
1.20 ×10−4), respectively, while they produce higher returns than the benchmark by
28
0.035 and 0.040 (p-value=1.34 ×10−4and 2.93 ×10−4), respectively.
Table 2 shows the average returns of strategies when the 11 real data are considered.
For the average returns of strategies generated from dynamic settings, see Table 1
of the Supplementary materials. If the mean returns of the proposed portfolio are
larger than those of the benchmark and their difference is statistically significant (p-
value <0.05), the corresponding mean returns are marked in bold. Overall, when
the 11 real data are considered, it is seen that the West-MAD, West-Dantzig, North-
MAD, and North-Dantzig improve the mean return over the benchmark in about
68.2%, 54.5%, 77.3%, and 68.2% of cases, respectively, and these improvements are
statistically significant in 34.1%, 29.5%, 68.2%, and 61.4% of cases, respectively. On
the other hand, EI methods significantly improve in 45.5%, 63.6%, and 54.5% of cases,
respectively, showing that the North perturbation methods consistently produce higher
returns than the benchmark compared to those of EI methods.
Table 2. Mean return comparison
Mean return with market index as a benchmark (real data)
Data Bench EI MAD Datzig
Index EI-linear EI-mixed EI-CZ SD WR WS NR NS WR WS NR NS
DOW30 0.0953 0.1357 0.1225 0.1294 0.0936 0.0979 0.1158 0.1165 0.0960 0.0991 0.1021 0.1052
DAX30 0.1164 0.1189 0.0964 0.0962 0.1351 0.1585 0.1525 0.1364 0.1250 0.1265 0.1720 0.1672
FTSE100 0.0331 0.0812 0.0868 0.0788 0.0586 0.0160 0.0832 0.0843 0.0597 0.0606 0.0809 0.0831
SP100 0.1045 0.1364 0.1287 0.1311 0.1063 0.1050 0.1099 0.1330 0.1040 0.1042 0.0963 0.1315
FTSE250 0.1168 0.1701 0.0913 0.1398 0.1350 0.1312 0.1337 0.1425 0.1156 0.1162 0.1043 0.1042
SP500 0.1105 0.1371 0.1699 0.1375 0.1203 0.1129 0.0929 0.1135 0.1447 0.1223 0.0880 0.0768
Hang Seng 0.0328 0.0362 0.0090 0.0312 0.0427 0.0585 0.0716 0.0719 0.0379 0.0310 0.0709 0.0718
NASDAQ100 0.1644 0.1766 0.2168 0.1820 0.1580 0.1840 0.1950 0.2015 0.1590 0.1626 0.2185 0.2091
Euro Stoxx50 -0.0084 0.0120 0.0370 -0.0012 -0.0330 -0.0063 -0.0247 -0.0090 0.0102 0.0076 -0.0337 -0.0080
NASDAQ3000 0.1454 0.1590 0.1231 0.1477 0.1702 0.2026 0.1726 0.1594 0.1713 0.1688 0.1862 0.1783
RUSSELL2000 0.1179 0.1289 0.1800 0.1625 0.1116 0.0805 0.1661 0.1394 0.1167 0.1159 0.1286 0.1267
Mean return with the equally-weighted portfolio as a benchmark (real data)
Data Bench EI MAD Datzig
Equally EI-linear EI-mixed EI-CZ SD WR WS NR NS WR WS NR NS
DOW30 0.1449 0.1447 0.2230 0.1447 0.1470 0.1462 0.1462 0.1835 0.1430 0.1419 0.1426 0.1442
DAX30 0.1242 0.1245 0.1813 0.1245 0.144 0.1475 0.1810 0.1474 0.1291 0.1294 0.1734 0.1751
FTSE100 0.1544 0.1526 0.1715 0.1526 0.1255 0.1228 0.1950 0.1799 0.1690 0.1687 0.2005 0.2000
SP100 0.1398 0.1388 0.1309 0.1388 0.1400 0.1340 0.1528 0.1551 0.1363 0.1362 0.1556 0.1671
FTSE250 0.1787 0.1770 0.1288 0.1735 0.1792 0.1790 0.1772 0.1858 0.1744 0.1768 0.1762 0.1767
SP500 0.1581 0.1558 0.1641 0.1467 0.1631 0.1605 0.1523 0.1557 0.1815 0.1583 0.1256 0.1240
Hang Seng 0.0454 0.0409 -0.0076 0.0409 0.0556 0.0344 0.0778 0.0776 0.0474 0.0420 0.0810 0.0745
NASDAQ100 0.2051 0.2031 0.2263 0.2031 0.2062 0.2008 0.2417 0.2331 0.1974 0.2040 0.2527 0.2322
Euro Stoxx50 0.0513 0.0497 0.0688 0.0510 0.0371 0.0579 0.0208 0.0423 0.0601 0.0622 0.0176 0.0325
NASDAQ3000 0.1504 0.1300 0.1197 0.1356 0.1873 0.1873 0.1621 0.168 0.1730 0.1801 0.1996 0.1928
RUSSELL2000 0.1557 0.1419 0.1867 0.1562 0.1386 0.1293 0.1490 0.1542 0.1541 0.1467 0.1771 0.1733
29
5.2. Sparse and stable portfolio selection
In this section, we investigate sparse and stable asset selection in the deviations part
ˆ
δ(t). Sparse and stable asset selection in the portfolio is critical for investors to control
management and transaction costs, and plays a key role in formulating portfolios. In
practice, individual investors usually hold only a small number of assets. To avoid
the costs of monitoring and portfolio re-weighting, investors usually follow common
practice to limit the number of components that can be selected from the portfolio.
We use the |A(t)|/p and |A(t)4A(t−1)|/p for the measure of sparsity and stability,
respectively. See Section 4.2 for details of these measures.
Tables 2-4 of the Supplementary materials show the average sparsity (|A(t)|/p) over
time tfor each portfolio and benchmark when the real data and partially simulated
data are considered, respectively. Note that one standard deviation of the sparsity
measures is less than 0.08 for all of the cases. We observe that the perturbed portfolios
consistently select less than 15% of the assets on average in the market for most cases.
This suggests that the perturbed portfolios select a few components in the perturbed
portion, as we expected.
Tables 5-7 of the Supplementary materials show the average stability measure
(|A(t)4A(t−1)|/p) over time twhen the two data types are considered, respectively.
Note that one standard deviation of the stability measures is less than 0.08 for all cases.
We see that the stability measure is less than 20% in most cases, suggesting that se-
lected assets in ˆ
δ(t)are stable across time t. In other words, the portfolio weights of the
proposed strategy are stable, making the proposed portfolio more credible, especially
for investors who are reluctant to implement an unstable portfolio.
30
Table 3. Turnover comparison
Turnover with market index as a benchmark (real data)
Data EI MAD Datzig
EI-linear EI-mixed EI-CZ SD WR WS NR NS WR WS NR NS
DOW30 0.231 0.694 0.304 0.101 0.138 0.132 0.126 0.027 0.039 0.131 0.137
DAX30 0.120 0.612 0.230 0.180 0.165 0.156 0.154 0.023 0.016 0.149 0.168
FTSE100 0.256 0.230 0.266 0.182 0.297 0.033 0.042 0.02 0.017 0.025 0.047
SP100 0.386 0.673 0.380 0.156 0.199 0.103 0.092 0.041 0.037 0.125 0.103
FTSE250 0.626 0.293 0.412 0.043 0.022 0.038 0.039 0.030 0.033 0.104 0.106
SP500 0.662 0.482 0.422 0.039 0.054 0.050 0.056 0.017 0.042 0.070 0.090
Hang Seng 0.259 0.719 0.419 0.149 0.127 0.070 0.063 0.035 0.035 0.061 0.059
NASDAQ100 0.342 0.516 0.332 0.168 0.245 0.079 0.082 0.045 0.053 0.078 0.078
Euro Stoxx50 0.238 0.628 0.138 0.121 0.117 0.139 0.154 0.026 0.033 0.121 0.139
NASDAQ3000 0.611 0.675 0.402 0.162 0.103 0.065 0.101 0.031 0.039 0.068 0.094
RUSSELL2000 0.872 0.794 0.422 0.081 0.156 0.074 0.089 0.017 0.024 0.052 0.068
Turnover with the equally-weighted portfolio as a benchmark (real data)
Data EI MAD Datzig
EI-linear EI-mixed EI-CZ SD WR WS NR NS WR WS NR NS
DOW30 0.000 0.843 0.125 0.170 0.162 0.126 0.205 0.027 0.040 0.197 0.188
DAX30 0.000 0.573 0.231 0.111 0.289 0.209 0.187 0.030 0.026 0.170 0.171
FTSE100 0.000 0.148 0.323 0.240 0.125 0.141 0.150 0.020 0.016 0.119 0.131
SP100 0.000 0.649 0.392 0.170 0.240 0.141 0.143 0.034 0.045 0.144 0.132
FTSE250 0.184 0.270 0.222 0.046 0.035 0.043 0.051 0.017 0.023 0.132 0.115
SP500 0.819 0.461 0.509 0.037 0.046 0.050 0.053 0.031 0.045 0.090 0.098
Hang Seng 0.000 0.800 0.153 0.122 0.141 0.095 0.073 0.026 0.017 0.076 0.083
NASDAQ100 0.000 0.413 0.512 0.082 0.090 0.101 0.092 0.031 0.030 0.106 0.076
Euro Stoxx50 0.000 0.673 0.312 0.152 0.133 0.174 0.149 0.022 0.026 0.154 0.156
NASDAQ3000 0.897 0.355 0.517 0.120 0.186 0.096 0.101 0.024 0.019 0.086 0.091
RUSSELL2000 0.917 0.844 0.610 0.160 0.106 0.094 0.109 0.031 0.049 0.095 0.082
As can be seen in Table 3, the turnovers of the proposed perturbed portfolios are
generally less than 0.15, while the three EI methods have unstable turnover, which
suggests that the perturbed portfolios do not allow much variations in components
compared to EI methods. We also observe that turnovers for the dynamic settings (i.e.,
partially simulated model) are also fair (see Table 8 of the Supplementary materials),
but are generally larger than those of the other two data cases (i.e. static components
setting), which can be explained by the observation that the perturbed portfolios
usually produce more turnover when the eliminated or added components are selected
in the deviations part ˆ
δ(t)during the considered period.
31
5.3. Sharpe ratio comparison
Table 4. Sharpe Ratio comparison
Sharpe Ratio with market index as a benchmark (real data)
Data Bench EI MAD Datzig
Index EI-linear EI-mixed EI-CZ SD WR WS NR NS WR WS NR NS
DOW30 0.2219 0.3170 0.2347 0.2893 0.2301 0.2318 0.2604 0.2683 0.2254 0.2336 0.2291 0.2385
DAX30 0.1717 0.1732 0.1319 0.1359 0.2282 0.2917 0.2176 0.201 0.193 0.1958 0.2407 0.2311
FTSE100 0.0661 0.1615 0.1379 0.1449 0.1242 0.0326 0.1632 0.1664 0.1208 0.1232 0.1564 0.1647
SP100 0.2413 0.3215 0.2310 0.2698 0.2668 0.2814 0.225 0.2996 0.2419 0.2416 0.1938 0.3029
FTSE250 0.2250 0.3239 0.1508 0.2567 0.2668 0.2645 0.2478 0.2633 0.2333 0.2332 0.1889 0.1931
SP500 0.2485 0.3085 0.3247 0.2927 0.2867 0.2683 0.2078 0.2458 0.3091 0.2741 0.1726 0.1547
Hang Seng 0.0483 0.0517 0.0118 0.0472 0.0806 0.1079 0.0978 0.0985 0.0589 0.0483 0.0971 0.0981
NASDAQ100 0.3381 0.3612 0.3743 0.3721 0.3485 0.3907 0.3432 0.3642 0.3153 0.3259 0.3717 0.3757
Euro Stoxx50 -0.0119 0.0171 0.0430 0.0102 -0.0590 -0.0108 -0.0358 -0.0138 0.0151 0.0112 -0.0497 -0.012
NASDAQ3000 0.2850 0.2978 0.1672 0.2717 0.3724 0.4362 0.3002 0.2813 0.3447 0.3411 0.3389 0.3326
RUSSELL2000 0.1887 0.2048 0.2188 0.2483 0.1877 0.1442 0.2393 0.2124 0.1894 0.1885 0.1655 0.1689
Sharpe Ratio with the equally-weighted portfolio as a benchmark (real data)
Data Bench EI MAD Datzig
Equally EI-linear EI-mixed EI-CZ S D WR WS NR NS WR WS NR NS
DOW30 0.3377 0.3375 0.4362 0.3271 0.3661 0.3573 0.3095 0.4054 0.3358 0.3332 0.3024 0.3234
DAX30 0.1846 0.1851 0.2655 0.1840 0.2410 0.2512 0.2536 0.2197 0.1998 0.2008 0.2433 0.2456
FTSE100 0.2988 0.2963 0.3050 0.2931 0.2512 0.2629 0.3521 0.3346 0.3287 0.3291 0.3659 0.3511
SP100 0.3013 0.2991 0.1900 0.2905 0.3341 0.3503 0.2914 0.3002 0.2972 0.2957 0.3167 0.3675
FTSE250 0.3607 0.3582 0.2298 0.3503 0.3766 0.3771 0.3397 0.3687 0.3637 0.3714 0.3338 0.3484
SP500 0.3237 0.3236 0.2741 0.3022 0.3454 0.3563 0.3033 0.3186 0.3575 0.3296 0.2278 0.2342
Hang Seng 0.0675 0.0609 -0.0103 0.0611 0.0988 0.059 0.1083 0.1085 0.0744 0.0658 0.1112 0.1032
NASDAQ100 0.3902 0.3864 0.3703 0.3864 0.3968 0.3935 0.3698 0.3992 0.3696 0.3865 0.3885 0.4026
Euro Stoxx50 0.0721 0.0695 0.0907 0.0800 0.0658 0.1018 0.0301 0.0634 0.0881 0.0912 0.0253 0.0483
NASDAQ3000 0.2392 0.1994 0.1568 0.2216 0.3515 0.3501 0.2374 0.2504 0.2917 0.3036 0.3051 0.300
RUSSELL2000 0.2352 0.2147 0.2281 0.2399 0.2346 0.2227 0.2108 0.2164 0.2324 0.2214 0.2309 0.2323
In this subsection, we investigate the Sharpe ratio of the proposed portfolios. Note that
maximizing the return to risk ratios of portfolios is a well-established principle. Table
4 (and Table 9 of the Supplementary materials) shows the Sharpe ratio comparisons
for the strategies, when the two data types are considered, respectively. To measure
the statistical significance of the difference between the Sharpe ratios of the returns of
the benchmark vand the perturbed portfolios w, we use the bootstrapping methods
proposed in Ledoit and Wolf (2008). Specifically, to test the hypothesis that the Sharpe
ratio of the return of the perturbed portfolio wis equal to that of the benchmark v,
i.e., H0:µw/σw=µv/σv, we compute a two-sided p-value using the studentized
circular block bootstrap proposed in Ledoit and Wolf (2008) with B= 1000 bootstrap
resamples and a block size equal to b= 5, following the setting in DeMiguel et al.
(2008). We use the code available at http://www.iew.uzh.ch/chairs/wolf.html.
When perturbed portfolios or EI methods have a significantly higher Sharpe ratio
(i.e., corresponding p-value is less than 0.05), the corresponding values are marked in
32
bold in Table 4.
We notice that when the index is used as a benchmark for real data, the MAD
significantly improves benchmark in 58.6% of cases, while Dantzig and the three EI
methods improve in 46.6%, and 27.3%, 40.9%, and 31.8% of cases, respectively. We
observe that for 48.9% of the equally-weighted in real data cases, the MAD and Dantzig
have significantly higher Sharpe ratios than those of the corresponding benchmarks,
while the EI methods significantly improve Sharpe ratios in 0%, 27.3%, and 0% cases,
respectively, showing that the proposed perturbation methods are favorable in terms
of risk-adjusted return measure in this case. When the partially simulated are used as
benchmarks, it is relatively difficult to obtain a higher Sharpe ratio than the benchmark
(MAD and Dantzig dominate benchmark in 50% of cases, while EI methods dominate
in 9%, 36.6%, and 36.6%, respectively). However, the perturbation methods dominates
other EI methods, suggesting that performances of perturbation methods are less
sensitive to those of EI methods.
5.4. Information ratio
As noted in Jorion (2001), meaningful risk measures should account for the possibility
that the portfolio underperforms the benchmark. Information ratio is one such metric,
which measures the active return of the portfolio divided by the risk relative to the
benchmark.
33
Table 5. Information ratio comparison
Information ratio with market index as a benchmark (real data)
Data EI MAD Datzig
EI-linear EI-mixed EI-CZ SD WR WS NR NS WR WS NR NS
DOW30 0.549 0.097 0.437 -0.011 0.022 0.146 0.162 0.070 0.220 0.140 0.220
DAX30 0.082 -0.065 -0.154 0.081 0.159 0.237 0.146 0.210 0.310 0.630 0.620
FTSE100 1.135 0.157 0.441 0.172 -0.125 0.379 0.393 0.420 0.690 1.230 1.360
SP100 0.729 0.064 0.235 0.011 0.003 0.041 0.303 -0.040 -0.030 -0.040 0.710
FTSE250 0.669 -0.081 0.222 0.191 0.158 0.179 0.231 -0.030 -0.030 -0.030 -0.090
SP500 0.559 0.224 0.339 0.121 0.045 -0.086 0.015 0.550 0.180 -0.080 -0.050
Hang Seng 0.050 -0.091 -0.014 0.040 0.107 0.217 0.220 0.230 -0.020 0.730 0.740
NASDAQ100 0.177 0.140 0.147 -0.085 0.195 0.131 0.176 -0.030 -0.080 0.270 0.310
Euro Stoxx50 0.254 0.114 0.005 -0.067 0.006 -0.059 -0.003 0.140 0.260 -0.050 0.020
NASDAQ3000 0.150 -0.049 0.015 0.106 0.234 0.132 0.081 0.220 0.270 0.270 0.490
RUSSELL2000 0.101 0.149 0.283 -0.077 -0.221 0.235 0.161 -0.020 -0.030 0.250 0.190
Information ratio with the equally-weighted portfolio as a benchmark (real data)
Data EI MAD Datzig
EI-linear EI-mixed EI-CZ SD WR WS NR NS WR WS NR NS
DOW30 -0.0553 0.3000 -0.0600 0.0150 0.0100 0.0090 0.2510 -0.0400 -0.0500 -0.0400 -0.0400
DAX30 0.0317 0.1710 0.0300 0.0860 0.0890 0.2820 0.1520 0.2600 0.2100 0.3400 0.3800
FTSE100 -0.2601 0.0670 -0.2600 -0.1420 -0.1380 0.2400 0.1620 0.3800 0.3700 0.6300 0.6400
SP100 -0.1917 -0.0200 -0.1900 0.0020 -0.0290 0.0840 0.1010 -0.0200 -0.0300 0.2900 0.4800
FTSE250 -0.0768 -0.1470 -1.0100 0.0080 0.0040 -0.0140 0.0600 -0.0300 -0.0200 -0.0200 -0.0300
SP500 -0.0438 0.0220 -0.2400 0.0930 0.0320 -0.0280 -0.0120 0.3400 0.0200 -0.0200 -0.0200
Hang Seng -0.3724 -0.1960 -0.3700 0.0680 -0.0570 0.1820 0.1850 0.1400 -0.0200 0.3700 0.4600
NASDAQ100 -0.2584 0.0500 -0.2600 0.0400 -0.0580 0.1400 0.1910 -0.0500 -0.0300 0.4700 0.3200
Euro Stoxx50 -0.1131 0.0530 -0.0210 -0.0420 0.0220 -0.1080 -0.0430 0.3100 0.3900 -0.0700 -0.0400
NASDAQ3000 -0.1565 -0.0930 -0.1600 0.1200 0.1170 0.0560 0.0880 0.4800 0.5300 0.6500 0.6300
RUSSELL2000 -0.1200 0.0880 0.0040 -0.1150 -0.1220 -0.0510 -0.0110 -0.0800 -0.1100 0.3800 0.3600
Tables 5 (and Table 10 of the Supplementary materials) report the information
ratio for the two data types, respectively. When the index is used as a benchmark in
real data, “EI-linear”, “NR-Dantzig”, and “NS-Dantzig” generally outperform other
methods. We see that the information ratios of these three methods are above 0.25 in
54.5%, 45.5%, and 54.5% cases, respectively, which is known to be a good performance
level. On the other hand, “NR-Dantzig” and “NS-Dantzig” clearly outperform other
methods when the equally-weighted portfolio is used as a benchmark in real data.
They are above 0.25 in 63.6% and 63.6% cases, respectively. This demonstrates that
the North-Dantzig perturbations could be favorable portfolio compared to the other
perturbations in terms of this risk-adjusted measure compared to the benchmark.
34
6. Conclusion
In this paper, we proposed two computationally efficient portfolio optimization models
for the construction of portfolios by utilizing a benchmark portfolio. The proposed
portfolios can outperform the benchmark portfolio in various performance measures,
including the mean return and Sharpe ratio. The two models, the MAD and Dantzig
models, are both computationally and empirically efficient. The MAD model does not
depend on the variance risk, while the Dantzig model essentially utilizes the variance
risk and requires covariance matrix estimation. By using the 11 real data sets and
their simulated dynamic settings in which the constituents are changed by time, we
observe that the proposed portfolios consistently select sparse and stable assets across
time, and produce less turnovers compared to EI methods. Moreover, we found that
the North perturbed methods are comparable with these EI methods and outperform
them in many cases. Overall, the West perturbation empirically reduces the risk, while
the North perturbation usually outperforms the benchmark in terms of mean return.
Geolocation information
Paper’s study areas (Latitude, Longitude) are Seoul (37.568260, 126.977830), South
Korea and New Haven (41.3255, -72.93826), USA.
Acknowledgments
This work was supported by National Institutes of Health (NIH) under Grants
GM59507, CA154295, and CA196530; National Research Foundation of Korea under
Grant NRF-2017R1A2B20005661.
References
Alexander, G. J. (2009). From markowitz to modern risk management. The European Journal
of Finance 15 (5-6), 451–461.
35
Alexander, G. J. and A. M. Baptista (2004). A comparison of var and cvar constraints on
portfolio selection with the meanvariance model. Management Science 50 (9), 1261–1273.
Becker, F., M. G¨urtler, and M. Hibbeln (2015). Markowitz versus michaud: portfolio opti-
mization strategies reconsidered. The European Journal of Finance 21 (4), 269–291.
Benartzi, S. and R. H. Thaler (2001). Naive diversification strategies in defined contribution
saving plans. American Economic Review 91 (1), 79–98.
Benati, S. (2003). The optimal portfolio problem with coherent risk measure constraints.
European Journal of Operational Research 150 (3), 572–584.
Benati, S. and R. Rizzi (2007). A mixed integer linear programming formulation of the optimal
mean/value-at-risk portfolio problem. European Journal of Operational Research 176, 423–
434.
Bernard, C., L. R¨uschendorf, S. Vanduffel, and J. Yao (2017). How robust is the value-at-risk
of credit risk portfolios? The European Journal of Finance 23(6), 507–534.
Besslera, W., H. Opferb, and D. Wolffa (2017). Multi-asset portfolio optimization and outof-
sample performance: an evaluation of black–litterman, mean-variance, and na¨ıve diversifi-
cation approaches. European Journal of Finance 23 (1), 1–30.
Best, M. J. and R. R. Grauer (1991). On the sensitivity of mean-variance-efficient portfolios
to changes in asset means: some analytical and computational results. Review of Financial
Studies 4, 315–342.
Bloomberg, L. P. (2006). Return on assets. Bloomberg.
Britten-Jones, M. (1999). The sampling error in estimates of mean-variance ecient portfolio
weights. Annals of Operations Research 54 (2), 655–671.
Broadie, M. (1993). Computing efficient frontiers using estimated parameters. Annals of
Operations Research 45 (1), 21–58.
Bruni, R., F. Cesarone, A. Scozzari, and F. Tardella (2017). On exact and approximate
stochastic dominance strategies for portfolio selection. European Journal of Operational
Research 259, 322–329.
Bruni, R., F. Cesarone, A. F. Scozzari, and F. Tardella (2015). A linear risk-return model for
enhanced indexation in portfolio optimization. OR Spectrum 37, 735–759.
Cai, X., K. Teo, X. Yang, and X. Zhou (2000). Portfolio optimization under a minimax rule.
Management Science 46, 957–972.
Canakgoz, N. A. and J. E. Beasley (2008). Mixed-integer programming approaches for index
tracking and enhanced indexation. European Journal of Operational Research 196, 384–399.
36
Candes, E. and T. Tao (2007). The dantzig selector: Statistical estimation when pis much
larger than n.Ann Stat 35, 2313–2351.
Carleo, A., F. Cesarone, A. Gheno, and J. M. Ricci (2017). Approximating exact expected
utility via portfolio efficient frontiers. Decisions Econ Finan 40, 115–143.
Costa, C. A. and J. O. Soares (2004). A multicriteria model for portfolio management. The
European Journal of Finance 10 (3), 198–211.
DeMiguel, V., L. Garlappi, F. Nogales, and R. Uppal (2009). A generalized approach to
portfolio optimization: Improving performance by constraining portfolio norms. Manag
Sci 55, 798–812.
DeMiguel, V., L. Garlappi, F. J. Nogales, and R. Uppal (2008). A generalized approach to
portofolio optimization: Improving performance by constraining portfolio norms. Manage-
ment Science 55, 798–812.
DeMiguel, V., L. Garlappi, and R. Uppal (2009). Optimal versus naive diversification: How
inefficient is the 1/n portfolio strategy? Rev Financ Stud 22, 1915–1953.
Deng, X.-T., Z.-F. Li, and S.-Y. Wang (2005). A minimax portfolio selection strategy with
equilibrium. European Journal of Operational Research 166 (1), 278–292.
Durand, R. B., J. Gould, and R. Maller (2011). On the performance of the minimum var
portfolio. The European Journal of Finance 17 (7), 553–576.
F´abi´an, C. I., G. Mitra, D. Roman, and V. Zverovich (2011). An enhanced model for portfolio
choice with ssd criteria: A constructive approach. Quantitative Finance 11, 1525–1534.
Fastrich, B., S. Paterlini, and P. Winker (2015). Constructing optimal sparse portfolios using
regularization methods. Comput Manag Sci 12, 417–434.
Fernandes, M., G. Rocha, and T. Souza (2011). Regularized minimum-variance portfolio with
asset group information.
Fishburn, P. C. (1977). Mean-risk analysis with risk associated with below-target returns.
American Economics Review 67, 116–126.
Gaivoronski, A. A., S. Krylov, and N. van der Wijst (2005). Optimal portfolio selection and
dynamic benchmark tracking. European Journal of Operational Research 163 (1), 115–131.
Gerhard, F. and D. Hess (2003). Multivariate market risk estimators: reliability and transaction
costs in the context of portfolio selection. European Journal of Finance 9 (1), 1–18.
Gondzio, J. and T. Terlaky (1996). A computational view of interior point methods In J. E.
Beasley. Advances in linear and integer programming. New York: Oxford University Press.
Goodwin, T. H. (1998). The information ratio. Financial Analysts Journal 54, 34–43.
37
Gotoh, J. and H. Konno (2000). Third degree stochastic dominance and mean-risk analysis.
Management Science 46, 289–301.
Grinold, R. C. and R. N. Kahn (2000). Active Portfolio Management. McGraw-Hill, New
York.
Guastaroba, G. and M. G. Speranza (2012). Kernel search: An application to the index tracking
problem. European Journal of Operational Research 217, 54–68.
Hodder, J. E., J. C. Jackwerth, and O. Kolokolova (2015). Improved portfolio choice using
second-order stochastic dominance. Review of Finance 19, 1623–1647.
Ill´es, T. and T. Terlaky (2002). Pivot versus interior point methods: Pros and cons. European
Journal of Operational Research 140 (2), 170–190.
Jagannathan, R. and T. Ma (2003). Risk reduction in large portfolios: Why imposing the
wrong constraints helps. J Finance 58, 1651–1683.
Jegadeesh, N. and S. Titman (2001). Profitability of momentum strategies: An evaluation of
alternative explanations. The Journal of Finance 56, 699–720.
Jobson, J. and B. Korkie (1980). Estimation for markowitz efficient portfolios. Journal of the
American Statistical Association 75 (371), 544–554.
Jorion, P. (2001). Value at Risk: The New Benchmark for Managing Financial Risk, Volume
2nd edition. New York: McGraw-Hill.
Jorion, P. (2003). Portfolio optimization with tracking-error constraints. Financ Anal J 59,
70–82.
Kamil, A. A. and K. Ibrahim (2007). Mean-absolute deviation portfolio optimization problem.
Journal of Information and Optimization Sciences 28 (6), 935–944.
Konno, H. and A. Wijayanayake (1999). Mean-absolute deviation portfolio optimization model
under transaction costs. Journal of Operations Research Society of Japan 42, 422–435.
Konno, H. and A. Wijayanayake (2001). Portfolio optimization problem under concave trans-
action costs and minimal transaction unit constraints. Mathematical Programming 89,
233–250.
Konno, H. and H. Yamazaki (1991). Mean-absolute deviation portfolio optimization model
and its applications to tokyo stock market. Management Science 37, 519–531.
Kopa, M. and T. Post (2014). A general test for ssd portfolio efficiency. OR Spectrum 37,
703–734.
Kuosmanen, T. (2004a). Efficient diversification according to stochastic dominance criteria.
Management Science 50, 1390–1406.
38
Kuosmanen, T. (2004b). Efficient diversification according to stochastic dominance criteria.
Management Science 50, 1390–1406.
Ledoit, O. and M. Wolf (2004a). Honey, i shrunk the sample covariance matrix. J Portfolio
Manage 30, 110–119.
Ledoit, O. and M. Wolf (2004b). Well-conditioned estimator for large-dimensional covariance
matrices. J Multivariate Anal 88, 365–411.
Ledoit, O. and M. Wolf (2008). Robust performance hypothesis testing with the sharpe ratio.
Journal of Empirical Finance 15, 850–859.
Lee, S. and S. Park (2016). Portfolio selection using a perturbation method.
Levary, R. R. (1983). A practical lp model for equity portfolios. Omega 11 (1), 41–48.
Levy, H. (1992). Stochastic dominance and expected utility: Survey and analysis. Management
Science 38, 555–593.
Levy, H. (2006). Stochastic dominance: Investment decision making under uncertainty (2nd
ed.). New York: Springer.
Longarela, I. R. (2016). A characterization of the ssd-efficient frontier of portfolio weights by
means of a set of mixed-integer linear constraints. Management Science 62 (12), 3549–3554.
Luedtke, J. (2008). New formulations for optimization under stochastic dominance constraints.
SIAM Journal on Optimization 19, 1433–1450.
Lwin, K. T., R. Qu, and B. L. MacCarthy (2017). Mean-var portfolio optimization: A non-
parametric approach. European Journal of Operational Research 260 (2), 751–766.
Mansini, R., W. Ogryczak, and M.G.Speranza (2003a). Lp solvable models for portfolio op-
timization: a classification and computational comparison. IMA Journal of Management
Mathematics 14, 187–220.
Mansini, R., W. Ogryczak, and M.G.Speranza (2003b). On lp solvable models for portfolio
selection. Informatica 14 (1), 37–62.
Mansini, R., W. Ogryczak, and M. G. Speranza (2007). Conditional value at risk and related
linear programming models for portfolio optimization. Annals of Operations Research 152,
227–256.
Markowitz, H. (1959). Portfolio Optimization under Lower Partial Risk Measures. John Wiley,
Sons, New York.
Michalowski, W. and W. Ogryczak (2001a). Extending the mad portfolio optimization model
to incorporate downside risk aversion. Naval Research Logistics 48, 185–200.
Michalowski, W. and W. Ogryczak (2001b). Extending the mad portfolio optimization model
39
to incorporate downside risk aversion. Naval Research Logistics 48, 185–200.
Ogryczak, W. (2000). Multiple criteria linear programming model for portfolio selection.
Annals of Operations Research 97, 143–162.
O.Strub and P.Baumann (2018). Optimal construction and rebalancing of index-tracking
portfolios. European Journal of Operational Research 264 (1), 370–387.
Rockafellar, R. and S. Uryasev (2000). Optimization of conditional value-at-risk. Journal of
Risk 2, 21–41.
Roll, R. (1992). A mean/variance analysis of tracking error. J Portfolio Manage 18, 13–22.
Roman, D., G. Mitra, and V. Zviarovich (2013). Enhanced indexation based on second-order
stochastic dominance. European Journal of Operational Research 228, 273–281.
Ruszczynski, A. and R. J. Vanderbei (2003). Frontiers of stochastically nondominated portfo-
lios. Econometrica 71, 1287–1297.
Sawik, B. (2009). Lexicographic and weighting approach to multi-criteria portfolio optimiza-
tion by mixed integer programming, chapter in: Lawrence k.d., kleinman g. (eds.). Appli-
cations of Management Science, Vol. 13, Financial Modeling Applications and Data Envel-
opment Applications, Emerald Group Publishing Limited, Bingley, UK , 3–18.
Sawik, B. (2012). Conditional value-at-risk vs. value-at-risk to multi-objective portfolio op-
timization, chapter in: Lawrence k.d., kleinman g. (eds.). Applications of Management
Science, Volume 15, Emerald Group Publishing Limited, Bingley, UK , 277–305.
Sawik, B. (2016). Triple-objective models for portfolio optimisation with symmetric and per-
centile risk measures,. International Journal of Logistics Systems and Management 25 (1),
96–107.
Sharpe, W. F. (1966). Mutual fund performance. Journal of Business 39, 119–138.
Sharpe, W. F. (1994). The sharpe ratio. The Journal of Portfolio Management 21, 49–58.
Shen, W., J. Wang, and S. Ma (2014). Doubly regularized portfolio with risk minimization.
Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence 28, 1286–1292.
Xing, X., J. Hu, and Y. Yang (2014). Robust minimum variance portfolio with l-infinity
constraints. J Bank Financ 46, 107–117.
Young, M. (1998). A minimax portfolio selection rule with linear programming solution.
Management Science 44, 673–683.
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