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Analysis of a portfolio selection model at three-moments
Análise de um modelo de seleção de portfólio em três momentos
Análisis de un modelo de selección de portafolio en tres momentos
DOI: 10.55905/oelv22n2-085
Receipt of originals: 01/04/2024
Acceptance for publication: 01/26/2024
Patricia Reis Martins
PhD in Computer Science
Institution: Centro Federal de Educação Tecnológica Celso Suckow da Fonseca
(CEFET- RJ)
Address: R. Gen. Canabarro, 485, Maracanã, Rio de Janeiro - RJ, CEP: 20271-204
E-mail: patricia.martins@pos.ime.uerj.br
Patrícia Nunes da Silva
PhD in Applied Mathematics
Institution: Universidade do Estado do Rio de Janeiro
Address: Rua São Francisco Xavier, 524, Pavilhão João Lyra Filho, Maracanã, Rio de
Janeiro - RJ, CEP: 20550-900
E-mail: nunes@ime.uerj.br
Carlos Frederico Vasconcellos
PhD in Mathematics
Institution: Universidade do Estado do Rio de Janeiro
Address: Rua São Francisco Xavier, 524, Pavilhão João Lyra Filho, Maracanã, Rio de
Janeiro - RJ, CEP: 20550-900
E-mail: cfredvasc@ime.uerj.br
Regina Serrão Lanzillotti
PhD in Transportation Engineering
Institution: Universidade do Estado do Rio de Janeiro
Address: Rua São Francisco Xavier, 524, Pavilhão João Lyra Filho, Maracanã, Rio de
Janeiro - RJ, CEP: 20550-900
E-mail: reginalanzillotti@ime.uerj.br
ABSTRACT
This work performs a numerical simulation for an investment portfolio selection model
that considers the three first moments of asset returns distribution – mean return, variance,
and skewness. The application of the model, based on data collected on the platform of a
Brazilian stockbroker, allowed obtaining portfolios of maximum skewness for fixed
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values of expected return and weighted variance of the portfolio. The results are analyzed
and presented graphically, giving rise to an optimal surface for triples of moments
associated with portfolios of maximum skewness. Furthermore, this experiment allowed
us to confirm the relevance of considering higher-order moments in the selection of
investment portfolios and verifying the efficiency of the Three-Moments model, having
as reference the Markowitz solution in his Mean-Variance model.
Keywords: portfolio selection, maximum skewness, numerical simulations.
RESUMO
Este trabalho realiza uma simulação numérica para um modelo de seleção de portfólio de
investimentos que considera os três primeiros momentos da distribuição de retorno de
ativos - retorno médio, variação e assimetria. A aplicação do modelo, com base em dados
coletados na plataforma de um corretor acionário brasileiro, permitiu a obtenção de
carteiras de assimetria máxima para valores fixos de retorno esperado e variância
ponderada da carteira. Os resultados são analisados e apresentados graficamente, dando
origem a uma superfície ideal para triplos de momentos associados com carteiras de
máxima assimetria. Além disso, este experimento nos permitiu confirmar a relevância de
considerar momentos de ordem superior na seleção de carteiras de investimento e
verificar a eficiência do modelo de Três Momentos, tendo como referência a solução de
Markowitz em seu modelo de Variância Média.
Keywords: seleção de portfólio, máxima simetria, simulações numéricas.
RESUMEN
En este trabajo se realiza una simulación numérica para un modelo de selección de
portafolio de inversión que considera los tres primeros momentos de la distribución de
los retornos de los activos: retorno medio, varianza y asimetría. La aplicación del modelo,
basado en datos recolectados en la plataforma de un corredor de bolsa brasileño, permitió
obtener carteras de máxima asimetría para valores fijos de retorno esperado y varianza
ponderada de la cartera. Los resultados se analizan y presentan gráficamente, dando lugar
a una superficie óptima para triples de momentos asociados a portafolios de máxima
asimetría. Además, este experimento permitió confirmar la relevancia de considerar
momentos de orden superior en la selección de carteras de inversión y verificar la
eficiencia del modelo de Tres Momentos, teniendo como referencia la solución de
Markowitz en su modelo de Varianza Media.
Palabras clave: selección de portafolio, máxima asimetría, simulaciones numéricas.
1 INTRODUCTION
According to the Modern Portfolio Theory, selecting efficient portfolios consists
of determining the weights referring to the contribution of the capital invested in assets
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that make up the investment portfolio in order to optimize the expected result of
diversification, which corresponds to obtaining the desired return of the portfolio under
the lower risk conditions. Markowitz (1952) solves this problem by the Mean-Variance
model. However, there is still much discussion about the fact that the distribution of asset
returns presents a significant skewness and does not always follow a Normal Probability
Distribution. In this context, the influence of moments of order higher than order two of
assets in the portfolio selection has been widely investigated. Studies such as Athayde e
Flôres (2004), Barone-Adesi (1985), and Harley and Siddique (2000) argue about the
relevance of considering higher-order moments when choosing a more efficient portfolio.
Central moments characterize the form of the distribution. Scott and Horvath (1980) show
that, in the context of optimization, to maximize the utility function, odd moments must
be maximized, while even moments must be minimized, which translates into investor
satisfaction. In this sense, Athayde e Flôres (2004) propose a model that incorporates the
third central moment – here treated as skewness – of asset returns to the portfolio
selection, extending the Mean-Variance model of Markowitz (1952).
In Athayde e Flôres (2004), a solution to the problem of minimizing the variance
by fixing the return and skewness of the portfolio is indicated, obtaining an optimal
configuration of the portfolio with minimum variance, thus suggesting a way to obtain
even more efficient results. After a long study of the model proposed by Athayde e Flôres
(2004), because of a result of duality, we inverted the parameters in the problem, taking
skewness as an objective function in Author (2015), proposing the maximization of
skewness, when fixed the first ordinary moment and the second central moment. This
new perspective made it possible to obtain results regarding the existence of a solution to
the optimization problem and to determine the optimal configuration for a maximum
skewness portfolio. In both problems, we present a solution as a system of nonlinear
implicit equations as a function of the optimal weights.
In order to investigate the relationships between the three moments considered in
the problem, as well as the relevance of incorporating higher-order moments in the
selection of investment portfolios, through the numerical application of the portfolio
selection model proposed in Author (2015), we selected a sample of nine assets present
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in the Brazilian stock market, where some had significant skewness in the distribution of
their monthly returns. The data on the investment funds were obtained through the
platform of a Brazilian stockbroker, which allowed obtaining the matrices of the means,
covariances, and coskewness for monthly returns of these assets using an electronic
spreadsheet. The optimal skewness of the portfolio was calculated from the model that
considers the first three moments from the perspective of maximizing the skewness, as
seen in Author (2015), solving an optimization problem with two constraints. From the
perspective of Economic Sciences, the most expressive skewness leads to greater
variability. That is, there is an increase in the portfolio risk that will cause the most
expressive skewness. From the extreme thresholds of the tail on the right referring to the
expressive positive skewness of the portfolio’s assets, it is possible to observe the
possibility of a rare event, translated by low probability, which will positively modify the
expectation of gain in the portfolio. Therefore, the increase in positive skewness suggests
a more efficient portfolio for bolder investors. It is possible to obtain the optimal
configuration of an investment portfolio by maximizing the skewness under the
conditions of fixing the expected return and the weighted variance of the portfolio,
defining the admissible set by the intersection of these two constraints. In order to
guarantee a non-empty and non-unitary admissible set, it is necessary to establish the
appropriate return and variance values initially.
This work analyses an investment portfolio selection model that considers the first
three moments of asset returns: mean return, variance, and skewness. The following
section presents the portfolio selection model proposed by Markowitz (1952) that
considers only the first two moments and then presents the model proposed by Athayde
and Flôres (2004) that incorporates the third central moment to the optimization problem
and the dual problem proposed by Author (2015). In Section 3, the model is applied to
obtain a portfolio of maximum skewness for pre-established values of expected return
and weighted variance of the portfolio, using the data collected regarding investment
funds from the platform of a Brazilian stockbroker. Section 4 analyses the results obtained
for five different combinations of assets. Section 5 expands the results by obtaining triples
of moments for portfolios of maximum skewness at each new configuration
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of the set admissible, defined by the pair . The visualization of these results refers
to a three-dimensional graph of the triples referring to the central moments, obtaining a
structure for the optimal surface of maximum skewness. The last section adds additional
considerations.
2 PORTFOLIO SELECTION – AN OPTIMIZATION PROBLEM
To Markowitz (1952), choosing a portfolio is directly related to the risk and return
of the assets present in this portfolio. The Mean-Variance model considers these two
factors. The portfolio’s expected return corresponds to the combination of the means of
the observed returns of each asset. The risk, measured by the degree of volatility
associated with the expected returns, is represented by the variability of the values of the
assets according to the portfolio’s covariance matrix, which reveals the joint dispersion
associated with investment risk uncertainty.
2.1 MODEL CONFIGURATION
According to Athayde and Flôres (2004), the investment portfolio is composed of
risky assets and a risk-free asset, allowing for short selling, that is, the weights can be
negative. The -dimensional vector , a point in the , represents the weights:
.
Given the matrix of the observed returns of the risky assets during
months,
.
the mean return of each asset gives rise to the mean returns’ matrix :
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.
Thus, by defining the risk-free asset return rate , it is possible to calculate the
matrix of excess returns , according to the notation in Athayde and Flôres (2004):
.
Furthermore, from the date of the matrix, the second and third central co-
moments are calculated, giving rise to the matrices and of covariances and
skewness, respectively:
and
The weighted statistics referring to the moments are obtained through matrix
products, giving rise to the weighted return, variance, and skewness of the set of assets:
• : weighted return on risky assets;
• : return weighted by the complementary weight of the risk-
free asset;
• : weighted variance of the portfolio;
• : weighted skewness of the portfolio.
, , and are, respectively, the matrices containing the mean returns,
covariances, and coskewness calculated from the observed returns of each of the risky
assets that make up the portfolio, is the -dimensional vector composed of ’s and
refers to the Kronecker product.
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The expected return on the portfolio, , is a composite of the returns on risky
and risk-free assets:
.
Express by , , the excess return of the portfolio, and the matrix
of excess returns of risky assets, by . Thus, we have the constraint on the
expected excess return:
.
This configuration guarantees that the sum of the weights of risky and riskless
assets will equal 1, according to the expected return .
2.2 MARKOWITZ MEAN-VARIANCE MODEL
Let be a fixed value for a portfolio’s excess return and let and be
the matrices containing the means and covariances of the observed returns in ,
respectively. Then, according to the composition defined for the portfolio, the Markowitz
Mean-Variance model is structured as follows:
(1)
which corresponds to a constrained optimization problem, which minimizes the
variance of the portfolio under the constraint that the excess return equals .
The configuration for the optimal portfolio of minimum variance of the
Markowitz model is easily calculated using the Lagrange multipliers method that
provides the solution:
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(2)
From (2),
(3)
Where
, is the minimum variance associated with the optimal portfolio .
Mandelbrot (1963) reported that the distribution of asset returns in the financial
market rarely follows the Normal Distribution. In this sense, several studies such as
Barone-Adesi (1985), Harvey and Siddique (2000), and Kraus and Litzemberg (1976),
seek to consolidate the idea that higher-order moments can significantly contribute to the
portfolio selection problem.
It is possible, from the solution obtained by Markowitz in the Mean-Variance
problem, to calculate a weighted skewness value for the minimum variance
portfolio through a matrix product:
in order to obtain an expression for the skewness associated with the Markowitz
solution as a function of and the return fixed in the problem:
Where
.
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2.3 THE PORTFOLIO SELECTION MODEL THAT CONSIDERS THE THREE
FIRST CENTRAL MOMENTS
Athayde and Flôres (2004) propose a notation that allows working with higher-
order moments through matrix calculus. They also indicate a solution to the problem of
minimizing the variance that considers the first three moments, extending the Markowitz
model with the inclusion of skewness in the optimization problem, considering the returns
of risky assets.
Athayde and Flôres (2004) adopted and , fixed values of excess
return and portfolio skewness, respectively, and , , and , the matrices that contain
the means, covariances, and skewness of the returns observed in , respectively. The
optimal portfolio selection problem will be
(4)
a constrained optimization problem, which minimizes the portfolio variance, and
whose constraints are the excess return equal to and the skewness equal to .
Considering the dual nature of the optimization problem in (4), the proposal in
Author (2015) is a new perspective of the problem, which corresponds to changing the
objective function, to maximize the skewness having return and variance constants as
constraints to the problem.
In this case, we adopted and , fixed values of excess return and
portfolio variance, respectively. The optimal portfolio selection problem will be
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a constrained optimization problem, which maximizes the portfolio skewness,
whose constraints are the excess return equal to and the variance, .
Author (2015) established the existence of a solution to the optimization problem
and obtained the configuration of the optimal portfolio through the Lagrange multipliers
method when the gradients of the constraints are linearly independent, that is when
. Obtained also an expression for the solution :
(5)
and determined the maximum skewness
associated with this optimal portfolio
:
(6)
Where
.
Remark 1. When
, we saw that the admissible set is unitary and that the
solution is Markowitz’s. On the other hand, when
, the admissible set is empty,
and there is no possible solution; that is, the adoption of parameters in this scenario makes
unfeasible the optimization problem.
Thus, after selecting the assets that will compose the portfolio in our experiment
and calculating the respective matrices , , and , it is enough to solve the system
of nonlinear equations (5) to determine the optimized weights of the maximum
skewness portfolio.
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2.4 APPLICATION OF THE THREE-MOMENTS MODEL
For the application of the Three-Moments model, a combination of three assets
was considered, with corresponding matrices , , and obtained, for the monthly
means, covariances, and coskewness, respectively. Then, the expected return and
weighted variance parameters are fixed to define the problem’s constraints.
While selecting portfolios at three moments to obtain an optimal portfolio by
maximizing skewness, the imposed initial constraints on the first-order and second-
central moments define the admissible set for seeking a maximum. In order to define this
admissible set for the three-moments problem, the optimal portfolio with minimum
variance was initially obtained using the Markowitz Mean-Variance model, in which only
the two first moments are considered, for which the solution is given by as in (2), and
the minimum variance for , associated to , is given by as in (3). Geometrically,
the equations of the two first moments (1), when fixed in values obtained through the
Mean-Variance model, define a hyperplane of return that touches a hyper ellipsoid of
variance in the space of the weights , determining the Markowitz solution at the point
of tangency. In this scenario, the intersection set
has only one element, with being the smallest variance for . Defining the
admissible set of the problemat three moments requires fixing variance value at
greater than , keeping the same value for the return, in order to produce an
admissible set, the intersection between the constraints, non-empty and non-unitary.
We defined the initial return as the mean of the means excess returns, that is,
the mean of the values resulting from the subtraction of the mean return of each asset by
the rate of the risk-free asset . Then the minimum variance is obtained for
by applying the Mean-Variance model. The parameters to be set as constraints in the
three-moment problem are , taken as the same return set in the Mean-Variance model,
and the variance , taken greater than the minimum obtained for according to
the Mean-Variance model.
According to the Markowitz Mean-Variance model, the minimum variance
portfolio for was obtained using the fsolve tool in Octave by solving the equation
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The weights, components of the vector, obtained through the Mean-Variance
model for , define the optimal portfolio, and the minimum variance is calculated
from the optimal portfolio , with no other possible configuration for the weights that
provide lower variance with the same fixed return.
The optimal portfolio with minimum variance and return has a weighted
skewness, calculated from :
The return was maintained, and the variance was initially raised by for the
admissible set for the problem at three moments, creating an admissible set — the
intersection of constraints – non-empty and non-unitary. Thus, among the different
portfolios with a return equal to and variance equal to , it is possible to select
one with greater skewness.
To solve this equation, we use the SQP method as implemented in the routines of
the Octave program. In addition, the program also provides the maximum skewness
associated with the optimal portfolio , which corresponds to the skewness calculated
from the solution , using the expression (6), whose discriminant root is positive.
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3 RESULTS
The selection of nine assets corresponding to Investment Funds constituted a
sample considered diversified concerning skewness, which meets the proposal of this
study, obtained from the website referring to the XP brokerage from January to December
2017, Table 1. The choice of this period aims to minimize the influence of external events
that could alter the portfolio’s volatility, avoiding periods of political and social
disturbance. Since the proposed method demands a risk-free asset, savings were chosen,
with a rate of return , for monthly income, which occurred in December 2017,
given the portfolio’s composition in the model proposed in Athayde and Flôres (2004).
Table 1. Classification and risk according to XP brokerage
ASSETS: Investment funds
Classification
Risk assessment
(0-100)
XP Corporate Plus FIC FIM CP
XP Debentures Incentivadas Crédito Privado
Kinea Chronos FIM
Multimarket
26
10
6
Selection RF Light FIC FI CP LP
Fixed Income
6
IP Value Hedge FIC
Indie FIC FIA
Shares
15
39
Kiron FIC FIA
Leblon Ações FIC FIA
Alaska Black FIC FIA - BDR Nível I
Variable income Long Only Free
41
41
68
Source: Platform of the Brazilian stockbroker XP Investimentos, Sep/2021,
(https://www.xpi.com.br/investimentos/fundos-de-investimento/lista/).
The values of the monthly returns of the assets allowed us to obtain the third
central moment to verify if they have skewed distribution. Among the nine, two tended
towards symmetry, Assets and , presenting and hundredths of the standard
deviation, respectively. Five assets indicated right-skewed with values , , , and
hundredths of the standard deviation, for Assets , , and , and , respectively,
with Assets and presented values very close to hundredths of the standard
deviation. Assets and showed negative skewness of the order of and hundredths
of the standard deviation to the left, respectively. The statistics can be found in Table 2.
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Table 2. Monthly returns for assets
Assets
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Mean
Variance
Coef. skewness
Asset 1
XP Corpo-
rate Plus
FIC FIM
CP
1.38
1.27
1.30
0.76
0.90
0.95
1.02
0.86
0.76
0.71
0.57
0.70
0.93
0.06
0.01
27.0%
0.516
2.008
Asset 2
XP Deben-
tures Incen-
tivadas
Crédito Pri-
vado FIC
1.64
2.02
1.64
0.31
-1.17
0.51
3.29
0.93
1.03
-0.39
-0.34
0.78
0.85
1.34
0.41
135.4%
0.263
2.753
Asset 3
Kinea
Chronos
FIM
1.55
1.78
0.91
1.23
-0.31
0.85
2.83
1.07
1.70
0.35
0.21
0.53
1.06
0.65
0.22
76.1%
0.421
2.915
Asset 4
IP Value
Hedge FIC
3.12
3.49
0.24
1.49
1.10
0.81
1.37
1.82
1.07
1.72
0.58
1.34
1.51
0.84
0.71
60.4%
0.931
3.061
Asset 5
Selection
RF Light
FIC FI CP
LP
1.17
0.91
1.08
0.69
0.91
0.85
0.82
0.86
0.59
0.61
0.44
0.48
0.79
0.05
0.00
27.7%
0.046
2.065
Asset 6
Indie FIC
FIA
7.89
4.95
2.29
1.79
-3.43
1.45
5.70
6.60
6.44
1.12
-2.36
5.92
3.20
12.15
-21.85
109.1%
-0.516
2.115
Asset 7
Kiron FIC
FIA
7.59
4.45
-1.00
1.32
-3.98
-1.48
4.19
6.65
7.05
-0.42
-2.25
4.40
2.21
14.64
-3.50
173.1%
-0.062
1.572
Asset 8
Leblon
Ações FIC
FIA
15.12
5.63
1.32
-0.60
-3.38
-2.18
6.17
9.03
9.01
0.57
-3.80
5.69
3.55
31.29
73.98
157.6%
0.423
2.257
Asset 9
16.28
18.45
-2.71
-3.89
-15.81
-1.18
24.19
17.60
4.58
-2.58
-2.65
12.52
5.40
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Alaska
Black FIC
FIA BDR
Nível I
134.64
10.76
214.9%
0.007
1.915
Source: Authors’ experiments, 2023.
Skewness and kurtosis characterize the distribution of returns on an asset.
Moreover, the kurtosis indices of these assets allow verifying their classification since the
leptokurtic form can be considered desired in the optimization of the risk scenario of the
asset portfolio.
The purpose of this article corresponds to combining investment assets with
diversified characteristics since the model’s objective is to maximize the third central
moment (skewness), considering increases in the minimum variance obtained through the
Mean-Variance model.
The diversity of asset characteristics encouraged the combination of three assets
to compose each portfolio to obtain five different configurations. First, combinations
were chosen in which Asset is part of each group, with the characteristic Leptokurtic
distribution.
In the portfolio whose combination includes Assets , , and , Assets and
presented right-skewed distribution, and Asset left-skewed. Only Asset presented
Leptokurtic distribution, and the others Platicurtic. For the simulation, we kept Asset
and included Assets and , both right-skewed Platicurtic. The third portfolio was
composed of Asset , adding Assets and , with Asset tending to symmetry and Asset
, right-skewed, both in a Platicurtic form.
All combinations of assets with increases of , , , and in
Markowitz’s weighted variance result in increases of , , , and in
their coefficient of variation.
In Assets , , and portfolio, with an increase of in Markowitz’s weighted
variance, there was an increase of in the coefficient in the Three-Moment model,
with weights according to the vector:
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.
Similarly, increases of , , and in Markowitz’s variance reflect
increases of , , and in the skewness coefficient of the Three-Moment
model, according to respective weights:
and
Increases in variance from to resulted in an increase of in the
skewness coefficient, from to , , from to , , in decreasing
trend. There is then a deceleration in the increase in skewness provided by the
implementation of the model.
For the combination of Assets , and , the increases of and
in the Markowitz variance reflected in increases of , , and
in the respective coefficient of skewness by implementing the model at three
moments, showing the same decreasing trend, having the following weights:
,
,
The same downward trend was observed in Assets , , and . In this combination,
for variance increases of , and , there were increases of ,
, , and in the skewness coefficient, whose weights were:
,
,
,
.
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In addition to these three combinations containing Asset with Leptokurtic
distribution, two more simulations considering only Assets of the Platicurtic form were
made varying the skewness characteristic: the combination of Assets , , and , inwhich
Assets and presented left-skewed and Asset presented right-skewed; and the
combination of Assets , , and , Assets and with a tendency to symmetry and Asset
with more significant left-skewed.
Although the portfolio performance focuses on leptokurtic distributions, we chose
to simulate the proposed model for groupings of assets in the Platicurtic form to compare
the model’s efficiency in different scenarios and verify if similar behaviors are perceived
in the other portfolios.
When Markowitz variance increases by , and , it results in
, , , and , respectively, increments in the coefficients of variation. It
causes the increase of , , , and 159.66%, respectively, in the
skewness coefficients for Assets , , and , and , , , and
, for Assets , , and . For these last two combinations, the same decreasing
trend was observed in the increase of the coefficient of skewness.
However, in addition to a more expressive increase in the skewness coefficient,
there was a change in shape. Initially left-skewed, with the implementation of the model,
they became right-skewed, Table 3.
Table 3. Comparison of results Mean-Variance and Three-Moments models.
Mean-Variance model
Three-Moments model
Com-
bina-
tion
a
b
c
d
e
f
g
h
i
j
k
l
m
Asset
2
Asset
4
Asset
6
1.3544
-0.3985
0.9962
0.1806
1.327
0.558
0.85
0.36
-
0.1161290
1.1855070
0.0723870
1.394
1.214
0.87
0.74
2.5%
102.18%
Asset
2
Asset
4
Asset
6
1.3544
-
0.0019298
1.2664000
0.0270160
1.460
1.565
0.89
0.89
4.9%
143.04%
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Asset
2
Asset
4
Asset
6
1.3544
0.0866110
1.3278000
-
0.0076613
1.526
1.868
0.91
0.99
7.2%
171.47%
Asset
2
Asset
4
Asset
6
1.3544
0.161958
1.378666
-0.036660
1.593
2.148
0.93
1.07
9.5%
192.78%
Asset
1
Asset
3
Asset
4
0.6675
1.366748
-
0.040390
0.098836
0.145
0.039
0.57
0.72
1.144836
-0.059181
0.203808
0.152
0.060
0.58
1.01
2.5%
41.62%
Asset
1
Asset
3
Asset
4
0.6675
1.045199
-0.061830
0.247747
0.159
0.072
0.60
1.13
4.9%
57.70%
Asset
1
Asset
3
Asset
4
0.6675
0.969861
-0.063653
0.280872
0.166
0.081
0.61
1.20
7.2%
67.67%
Asset
1
Asset
3
Asset
4
0.6675
0.906500
-0.065313
0.308801
0.173
0.090
0.62
1.25
9.5%
74.26%
Asset
4
Asset
5
Asset
6
1.3311
0.419014
2.857043
0.035218
0.799
0.377
0.67
0.53
0.659093
2.206986
0.013578
0.839
0.671
0.69
0.87
2.5%
65.47%
Asset
4
Asset
5
Asset
6
1.3311
0.759980
1.945500
0.003249
0.879
0.810
0.70
0.98
4.9%
86.35%
Asset
4
Asset
5
1.3311
0.837810
1.749700
-0.005339
0.919
0.922
0.72
1.05
7.2%
98.38%
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Asset
6
Asset
4
Asset
5
Asset
6
1.3311
1.122971
1.541341
-0.027814
1.300
1.657
0.76
1.12
9.5%
105.71%
Asset
6
Asset
7
Asset
8
2.485
1.758259
-
1.424110
0.058652
3.963
-
4.367
0.80
-
0.55
1.884300
-1.703900
0.104100
4.161
-
0.654
0.82
-
0.08
2.5%
86.08%
Asset
6
Asset
7
Asset
8
2.485
1.952864
-1.799843
0.097268
4.360
0.974
0.84
0.11
4.9%
119.34%
Asset
6
Asset
7
Asset
8
2.485
2.008345
-1.868159
0.086511
4.558
2.282
0.86
0.23
7.2%
142.36%
Asset
6
Asset
7
Asset
8
2.485
2.055833
-1.923710
0.075664
4.756
3.425
0.88
0.33
9.5%
159.66%
Asset
5
Asset
6
Asset
9
2.6269
6.769982
0.401643
-
0.077551
3.315
-
0.497
0.69
-
0.08
7.600807
0.201955
-0.015837
3.481
0.435
0.71
0.07
2.5%
181.25%
Asset
5
Asset
6
Asset
9
2.6269
7.591702
0.149107
0.013775
3.647
0.765
0.73
0.11
4.9%
233.38%
Asset
5
Asset
6
Asset
9
2.6269
7.561608
0.115364
0.034091
3.812
1.068
0.74
0.14
7.2%
274.30%
Asset
5
2.6269
7.540403
0.087793
0.050494
3.978
1.369
0.76
0.17
9.5%
309.54%
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Asset
6
Asset
9
Note: (a) Fixed expected excess return (mean of monthly means of excess returns);
(b) Minimum variance portfolio (weights);
(c) Minimum weighted variance;
(d) Weighted skewness of the minimum variance portfolio;
(e) Coefficient of variance of the minimum variance portfolio;
(f) Coefficient of skewness of the minimum variance portfolio;
(g) Maximum skewness portfolio (weights);
(h) Fixed weighted variance (increased by 5%,10%,15% and 20%);
(i) Maximum weighted skewness of the portfolio;
(j) Coefficient of variance of the maximum skewness portfolio;
(k) Coefficient of skewness of the maximum skewness portfolio;
(l) Percentage increase of the coefficient of variance;
(m) Percentage increase of the coefficient of skewness.
Source: Authors’ experiments, 2023.
In all combination combinations of assets, it can be noted that the optimization of
the portfolio according to the model that considers the first three moments allowed an
increase in the skewness considered significant from small increases in variance, making
the tail distributions heavier. On the other hand, there was also a decrease in the increase
in skewness as more significant increases were applied to the Markowitz variance.
Furthermore, it was verified that the behavior of the weights obtained by the proportional
increase of the Markowitz variance from the application of the proposed model does not
present regularity for the combinations of assets. The combination of Assets , , and
stood out: there was a change in the sign of skewness and the shape of the distribution
from an increase in the variance of .
In the next section, we change the return and variance fixed values to obtain
several triples
for the combination of Assets , , and .
4 EXPANDING THE RESULTS
In Section 3, it was verified, from the proportional increase at intervals of in
the Markowitz variance, that the result of applying the Three-Moments model was
favorable to the expected increase in skewness in the five proposed combinations of three
assets. However, in Assets , , and and Assets , , and , there was a change in the
skewness shape from left to right. Therefore, it shows an even greater relevance of the
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experiment. Skewness shape change occurred when increasing the Markowitz variance
by for Assets , , and , while for the combination of Assets , , and , this change
only occurred after the increase of in the Markowitz variance. However, for larger
percentages of increase in the Markowitz variance, the combination of Assets , , and
achieved a more expressive skewness to the right than the other combination. Because it
presents a more expressive change in the direction of the tail, we choose Assets , , and
to proceed with the experiment.
Previously, for Assets , , and , with return fixed at , according to
the Markowitz model, the minimum variance was , and the skewness
was calculated for this portfolio of minimum variance, obtaining a triple
of moments for this portfolio. Then the Markowitz
variance was increased by , obtaining the maximum skewness
by
applying the Three-Moments model for the pair and . Thus, a
new triple of maximum skewness is obtained.
Similarly, for increases of , , and of the Markowitz variance, keeping the
same return, new triples of maximum skewness
were obtained by applying
the Three-Moments model:
In order to better understand what happens to the shape of the skewness, by
applying the Three-Moments model when the fixed parameters are changed by the
systematic increase of the weighted variance for different returns, values for return and
variance were arbitrated in order to guarantee a non-empty admissible set, yielding 378
triples of maximum skewness. The return varied in the range as a function
of the mean excess returns of Assets , , and , where an amplitude equal to was
adopted, approximately of the mean of the mean excess returns. The range goes from
the smallest to the largest excess mean of returns of the assets in the portfolio.
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Figure 1. Table of maximum skewness triples
Source: Authors’ experiments, 2023.
The variance interval was defined by obtaining the minimum
variance according to the Markowitz model for each return in the interval to
start the experiment from the minimum variance of Markowitz with each of the returns
in the interval, and then apply the Three-Moments model for each return combined with
the Markowitz variances obtained for the following returns. Thus, for each pair ,
where and , we continued to obtain new values
of optimal skewness
, associated with each new configuration of the admissible set,
obtaining triples for the values of the three moments
, of which are
Markowitz triples. Applying the Three-Moments model allowed us to obtain the
maximum skewness for each pair of return and variance in the respective intervals, and
the results of this treatment appear in the table in Figure 1.
In this combination of Assets, the maximum skewness associated with the
Markowitz portfolios were negative in all returns in the range . In applying
the Three-Moments model, new weights are obtained when increasing the variance for
the samereturn that produces maximum skewness bigger and bigger. For example, the
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skewness shape changes from an increase of in the Markowitz variance when
and from an increase of in the Markowitz variance when .
The graph in the three-dimensional space of moments, with the triples,
indicates the optimal surface of maximum skewness by varying the pair that
defines the admissible set, Figure 2.
Figure 2. Surface of maximum skewness
Source: The Authors, 2023.
The curve of the triples of the moments obtained according to the Markowitz
model is shown on the same graph, which will touch the surface of maximum skewness
at the points of minimum variance for .
It was also possible to obtain a graph in three-dimensional space, as a function of
the return, standard deviation and cube root of the skewness, which evidenced the
expected results of the application of the Three-Moments model, Figure 3.
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Figure 3. Surface of maximum skewness
Source: The Authors, 2023.
In the table of Figure 1, it is noticed that a slight increase in the variance for the
same return produces a significant increase in the maximum skewness, with the most
expressive increase verified in the first iteration, close to the minimum Markowitz
variance. Similarly, it is crucial to analyse the behaviour of the maximum skewness in the
table in Figure 1 from the decrease in return for the same variance value. From a small
reduction in the expected return, keeping the same variance, a significant increase in the
maximum skewness can be seen, and a change in the shape of the skewness, from left to
right, from a reduction of in the return of the Markowitz portfolio.
Following the behaviour of the triples from the decrease in the return for the same
variance, in the table of Figure 1, there is still a region in which the skewness behaves
differently from the expected, presenting an unexpected reduction, growing again in the
following iterations. The skewness shows a very expressive initial growth from a decrease
of in the return in Markowitz. It continues to grow slowly with each decrease in the
return until a certain return when it presents a smaller maximum skewness for a smaller
return and the same variance. This behavior occurred for Assets , , and , for variance
values starting at , in the blue cells in the table in Figure 1. This occurrence should
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be better studied, and it may indicate a loss of duality since, for a lower return and the
same variance, a lower skewness would indicate a portfolio considered less efficient.
The portfolio selection model at three moments, which considers moments greater
than order , will allow adjusting the skewness of a portfolio from a controlled increase
in risk according to the variance, in relation to the minimum Markowitz variance for a
given return, of way to make it more efficient.
5 CONCLUSIONS
The behaviour of the optimal surface of maximum skewness evidenced the
efficiency of the Three-Moments model, having its relevance accentuated when there is,
in the portfolio, a predominance of assets with the left-skewed distribution. On the other
hand, we noticed that the presence of assets with expressive right-skewed in the
combinations could reduce the relevance of the Three-Moments model since, in these
cases, the application of the Markowitz model can already produce portfolios with the
desired positive skewness. Even in these cases, the Three-Moment model presents
positive results but is less relevant.
On the surface in Figures 2 and 3 obtained for the portfolio of Assets , and ,
it was found that in a region very close to the Markowitz curve, the gain with the skewness
proved to be highly relevant, which suggests the possibility of expressive gain from an
economic point of view for bold investors. In addition, this combination of assets presents
high values of variance and skewness in their assets individually, suggesting a greater
relevance in using the Three-Moment selection model for these cases.
While obtaining the triples from the SQP in the Octave program, we see that the
numerical solution of these problems requires care since the application also provides a
local maximum, depending on the initial value used. Furthermore, we know that the
Markowitz variance delimits the duality region in the optimization problems involved in
the feasible set of the problem in which the skewness is maximized. However, we need
to learn more about other constraints for the duality region, which should be an object of
future research.
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Although the quantitative treatment was based on the third central moment, it is
worth evaluating the fourth one that classifies the asset distribution’s kurtosis since it is
clear that a skewed distribution to the right leptokurtic could optimize the selection of the
portfolio as a mitigating factor for bold investors.
The results obtained here made it possible to confirm the relevance of considering
higher-order moments in the selection of investment portfolios and verify the efficiency
of the Three-Moment model from the perspective of maximizing the skewness proposed
by Author (2015).
ACKNOWLEDGEMENTS
The authors gratefully acknowledge the support of FAPERJ, grants E26/210.341/2018
and E-26/010.001143/2019.
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