DOI: 10.5817/FAI2015-2-1 No. 2/2015
7
Application of GARCH-Copula Model in Portfolio
Optimization
Aleš Kresta
Vysoká škola Báňská – TU Ostrava
Faculty of Economics, Department of Finance
Sokolská tř. 33, Ostrava
E-mail: ales.kresta@vsb.cz
Abstract: Although the cornerstone of modern portfolio theory was set by
Markowitz in 1952, the portfolio optimization problem is a never-ending research
topic for both academics and practitioners. In this problem the future prediction
of time series evolution plays an important role. However, it is rarely addressed in
research. In the paper we analyze the applicability of the GARCH-copula model.
To be more concrete we assume the investor maximizing Sharpe ratio while the
future evolution of the time series is simulated by means of the AR(1)GARCH(1,1)
model using the copula modelling approach. The bootstrapping
technique is applied as a benchmark. From the empirical results we found out
that the GARCH-copula model provides better forecasts of future financial time
series evolution than the bootstrapping method. Assuming the investor is
maximizing the Sharpe ratio, both the final wealth increases and maximum
drawdown decreases when we apply the GARCH-copula model compared to the
application of bootstrapping technique.
Keywords: portfolio optimization, Sharpe ratio, GARCH, copula function
JEL codes: G11, G17
Introduction
Although the cornerstone of modern portfolio theory was set by Markowitz in
1952, the portfolio optimization problem is a never-ending research topic for both
academics and practitioners. In this problem, the prediction of future time series
evolution plays an important role. However, when we model the financial time
series, there are some characteristic features which have to be taken into
consideration.
First, empirically observed returns of a financial time series are characterized by
fatter tails compared to the Gaussian (normal) distribution, see e.g. Mandelbrot
(1963). Next, empirical volatility of returns is not constant over time, but is
rather clustered. Thus, for the same asset, periods with high volatility (high
gains/losses) can be seen as well as periods in which volatility is low (the
gains/losses are close to zero). This issue can be tackled by volatility modeling. A
typical tool which can be utilized is the GARCH model, see Bollerslev (1986). The
Financial Assets and Investing
8
last issue one has to deal with is dependency within a particular time series.
Generally, the returns are not correlated strongly when they are around zero,
however in the tails the correlation increases. An appropriate tool for dependency
modelling is the copula function, see Sklar (1973). Thus, the sound model of
financial returns should be composed of a GARCH model and a copula function
accompanied with some heavy-tailed marginal distributions.
While in most papers, see e.g. Farinelli et al. (2008) among others, the
parametric stochastic process for financial returns is not assumed and
performance ratio is optimized on historically observed returns (bootstrapping
technique), in our paper we apply the GARCH-copula model to simulate future
returns. The results obtained by means of the bootstrapping method are utilized
only as a benchmark.
The aim of this paper is to analyze the applicability of the GARCH-copula model in
portfolio optimization. To be more concrete, we assume an investor maximizing
Sharpe ratio while the future time series is simulated by means of the GARCHcopula
model and by means of the bootstrapping technique. Applying the
procedures on a rolling window basis, we compare the values of final wealth at
the end of the analyzed period and maximum (percentage) drawdown during the
analyzed period for both approaches.
The paper is structured as follows: In the next section, the portfolio optimization
model in the Markowitz mean-variance framework is described. Then, in the
second section we present the application of the GARCH-copula model. In third
and fourth sections the utilized dataset and obtained results are described. In the
last section the discussion and conclusion are provided.
1 Portfolio Optimization Problem
The cornerstone of modern portfolio theory was established by pioneer work of
Harry Markowitz in 1952 by his well know paper, see Markowitz (1952). Under
the proposed assumptions, he assumed the returns to be normally distributed
and the investor to be risk-averse, and that the rational investor wants to
maximize the portfolio expected return and minimize its variance. However, the
relationship between these two characteristics is generally positive – by
decreasing the variance also the expected return decreases, see e.g. Lundblad
(2007), and thus without the knowledge of the investor's level of risk aversion we
can find only the set of (Pareto) efficient portfolios. The portfolio is identified as
efficient if and only if there is no other portfolio with a lower risk delivering higher
or equal expected return and no other portfolio with the higher expected return
No. 2/2015
9
and lower or equal risk. For further details of modern portfolio theory see e.g.
Elton et al. (2014).
Sharpe (1966) continued in the framework established by Markowitz and
proposed the well-known Sharpe ratio (Sharpe index, the Sharpe measure or the
reward-to-variability ratio) which he first defined as the ratio between the excess
expected return (i.e. the expected return minus risk-free rate, also known as risk
premium) and its volatility,
, (1)
where is the observed (or predicted) distribution of returns or equiprobable
realizations of this distribution and is risk-free rate. The original ratio was
revised by Sharpe (1994) substituting the risk-free rate by an applicable
benchmark , which can change in time,
. (2)
Further in this paper we assume the original version of Sharpe ratio (1), which in
fact is a special case of the revised version (2) in which . The Sharpe
ratio defines the profile of an investor who prefers titles with higher expected
excess returns for unity of volatility (standard deviation). When comparing two
assets versus a common benchmark (in our case risk-free rate ), the one
with a higher Sharpe ratio provides a better return for the same risk (or,
equivalently, the same return for a lower risk).
The Sharpe ratio is closely related to the Markowitz mean-variance framework as
it focuses only on the first two moments of the probability distribution. However,
as it is known, the empirical distribution of financial asset returns is characterized
by heavy-tails and skewness. Thus, many researchers have proposed their own
ratios, which take into account the kurtosis and skewness of the probability
distribution. Among others, see for instance the Gini ratio (Shalit and Yitzhaki,
1984), mean absolute deviation ratio (Konno and Yamazaki, 1991), mini-max
ratio (Young, 1998), Rachev ratio (Biglova et al., 2004) and others. For the
summary see e.g. Farinelli et al. (2008).
In this paper we assume the investor maximizes Sharpe ratio, to solve the
following portfolio optimization problem, i. e. he solves the following portfolio
optimization problem,
( )
( ) ( )
RF
RF RF
R R R
E R R E R R
SR R
σ σ−
− −
= =
% %
% %
%
R%
RFR
BR%
( )
( )
B
B
R R
E R R
SR R
σ −
−
=
% %
% %
%
B RFR R=%
RFR
Financial Assets and Investing
10
(3)
in which represents the vector of weights (portfolio composition) and is the
matrix of random realizations of returns (rows represent realizations with equal
probability and columns represent particular assets the investor can include in the
portfolio). The matrix contains the random realizations of future returns, and
thus is not directly observable but must be simulated. We describe the simulation
procedure in the following section. Furthermore, the constraints of the
optimization problem bound the weight of each asset between 0% (short selling is
not allowed) and 25% (the portfolio is composed of at least four assets).
2 Financial Asset Returns Modelling
The evolution of financial asset returns over time is specific in the following ways,
for further details see e.g. Cont (2001). Empirical volatility of returns is not
constant over time, but is rather clustered. Thus, for the same asset, periods of
high volatility (high gains/losses) can be seen as well as periods in which volatility
is low (the gains/losses are low). This issue can be tackled using volatility
modeling. In this paper, we apply the GARCH model for this purpose (Bollerslev,
1986). Even after the correction of returns for volatility clustering, the residual
time series still exhibits heavy tails. The conditional distribution, however, is less
heavy-tailed than the unconditional distribution. In our paper we utilize joint
Student distributions for residuals. Due to the estimation and simulation
requirements, this joint distribution is decomposed into Student marginal
distributions and the Student copula function in line with Sklar’s theorem (Sklar,
1973). We address the obtained model as a GARCH-copula model, which was
already applied in risk management by Huang et al. (2009), Wang et al. (2010)
and others.
Assume that we want to model the future returns of n assets. For each asset we
assume AR(1)-GARCH(1,1) process, i.e. i-th asset returns can be modelled as
follows,
, (4)
, (5)
( )
1
max
1
0, 1,...,
0.25, 1,...,
N
i
i
i
i
SR
x
x i N
x i N
=



 =


≥ =
 ≤ =
∑
x
R×x%
x R%
R%
tititiiiti rR ,,1,1,0,,
~εσµµ ⋅+⋅+= −
( )0,1~~
, νε tti
No. 2/2015
11
where and are parameters of the conditional mean equation, is the
standard deviation (volatility) modelled by the GARCH model and is a random
number from Student probability distribution (henceforth filtered residual).
The Student distribution is applied for its ability to model fat tails (higher
kurtosis) of a probability distribution, which are usually present in financial time
series of returns. The volatility is modelled by means of the GARCH model
(Bollerslev, 1986), an extension of the ARCH model (Engle, 1982). The applied
model takes the following form,
, (6)
where , and are the parameters that must be estimated. Positive
variance is assured if all the parameters are equal or greater than zero. The
model is stationary if .
In order to preserve the mutual dependence among the asset returns, the filtered
residuals are joined together applying copula function modelling. Copula functions
are projections of the dependency among particular distribution functions into
,
. (7)
Basic reference for the theory of copula functions is Nelsen (2006), while Rank
(2007) and Cherubini et al. (2004, 2011) target mainly on the application issues
in finance. Actually, any copula function can be regarded as a multidimensional
distribution function with marginals in the form of a standardized uniform
distribution. Following the Sklar's theorem (Sklar, 1959), any joint distribution
function, in our case the joint distribution function of filtered residuals
, can be decomposed into marginal distributions and a selected
copula function,
. (8)
The formulation above should be understood such that the copula function C
specifies the dependency, nothing less, nothing more. In the paper we apply the
Student copula function, which belongs to the family of elliptical copula functions,
, (9)
,0iµ ,1iµ ti,σ
ti,
~ε
( )0,1νt
2
1,1,
2
1,1,0,
2
, −− ⋅+⋅+= tiitiiiti εβσαασ
0,iα 1,iα 1,iβ
11,1, <+ ii βα
[ ]1,0
[ ] [ ] { },...3,2,o1,01,0: ∈→ nRnC nn
( )nuuF n
,...,1~,...,~
1 εε
( ) ( ) ( )( )nn uFuFCuuF nn εεεε ~1~1~,...,~ ,...,,..., 11
=
( )
( )
( ) ( )
z
Qzz'
Q
Q d
n
uuC
nut nut
n
n
t
∫∫
−−
∞−
+
−
∞−






+






Γ





 +
Γ
=
1
1
1
2
1, 1
2
2
,...,
νν
ν
ν
ν
νπ
ν
ν
L
Financial Assets and Investing
12
where is gamma function, stands for degrees of freedom both in marginals
and Student copula function and Q is a correlation matrix. Note that by
accompanying Student copula function with Student marginals we obtain the
joint-Student distribution.
2.1 Parameters Estimation and Subsequent Returns Simulation
In order to model the future evolution of financial time series by means of
GARCH-copula model the following procedure should be undertaken, see Figure 1.
First, the parameters of GARCH model are estimated for each particular return
time series from past observations. When GARCH models are estimated, the
residuals (observed in the past) can be obtained. These are put together into a
matrix and the parameters of the copula function are estimated. There exist three
main approaches to estimate parameters for copula function based dependency
modelling: exact maximum likelihood method (EMLM), inference function for
margins (IFM), and canonical maximum likelihood (CML). While for the EMLM all
the parameters are estimated within one step, which might be very time
consuming (mainly for high dimensional problems or complicated marginal
distributions), the other two methods are based on the estimation of the
parameters for the marginal distributions and parameters for the copula function
separately – marginal distributions are estimated in the first step and the copula
function in the second step. Following IFM the estimated marginals are utilized in
the second step. For CML instead of estimated marginals the empirical
distributions are utilized. In this paper we apply the IFM estimation method as it
provides a reasonable trade-off between the accuracy and computational
requirements.
For the simulation the sequence is inverse. First, random numbers are simulated,
while the dependency among them is maintained by means of the estimated
copula function. Then, these simulated random numbers are transformed to the
filtered residuals (by inverse distribution function), which are converted to the
returns by means of estimated GARCH models. These returns can be then easily
utilized for computation of expected portfolio return and/or its risk.
Figure 1 GARCH-Copula model estimation/simulation procedure
Source: Author
Γ ν
GARCH
models
estimation
Calculation of
filtered
residuals
Normalization
of residuals
Copula model
estimation
Random
numbers
simulation
Calculation
of simulated
residuals
Calcualtion
of simulated
returns
No. 2/2015
13
3 Dataset
The utilized dataset consists solely of the stocks incorporated in one of the
American stock market indices – Dow Jones Industrial Average (henceforth DJIA).
We assumed all the components of the index as of October 6, 2014, except the
stocks of The Goldman Sachs Group, Inc. (Yahoo Finance ticker GS) and Visa Inc.
(Yahoo Finance ticker V). These two stocks were excluded from the dataset as we
were not able to obtain sufficiently long historical data. Thus, the dataset consists
of only the remaining 28 stocks.
Historical data of the stocks were obtained from Yahoo Finance website1
over the
period December 1, 1997 until December 31, 2014 (4,298 daily observations for
each stock). However, we estimated the parameters from 250 observations, thus
the backtesting was performed in the period from November 30, 1998 until
December 31, 2014, leaving the first year of data for initial parameter estimation.
The evolution of DJIA price index2
in the analyzed period is depicted in Figure 2.
The index took the value of 9,116.55 on November 30, 1998 and 17,983.07 on
December 31, 2014. Thus the average annual return (to be more specific the
average return of 250 trading days) in the analysed period was 4.26% whereas
the maximum drawdown over the analyzed period was 53.78%.
Figure 2 Evolution of DJIA index in the analysed period
Source: Author
1
http://finance.yahoo.com
2
Price index considers only price movements in the components, dividends are not
considered.
Financial Assets and Investing
14
4 Empirical Part and Results
In the previous sections we proposed an optimization problem which can be
applied to find the optimal portfolio. In this section we apply this portfolio
optimization problem on a moving window basis. Starting with the initial wealth
at the beginning of the analyzed period, we can recursively compute the
ex-post wealth path over the analyzed period,
, (10)
where are ex-post observed returns and are the weights of particular
assets at time t (portfolio composition). These weights were obtained by means
of maximizing the Sharpe ratio, see problem (3). Under this set-up the matrix of
random realizations of future returns ( ) was simulated for each day assuming
the following two approaches:
1. 100,000 simulated trials (rows) were calculated utilizing GARCH-copula
model, which was estimated from 250 days prior to the examined day,
2. taking last 250 observed returns prior to the examined day
(bootstrapping method).
Our goal is to analyse the soundness of GARCH-copula model to describe the
return time series and moreover to predict their future evolution. The
bootstrapping method, which represents a rather naïve model, is analysed only as
a benchmark to the proposed GARCH-copula approach.
In this section we present the results obtained by applying two above mentioned
approaches for simulation of future returns. In our application we assumed the
following values of risk-free rate: 0%, 1%, …, 6% and 10% p.a. for the Sharpe
ratio (1). The reason to assume more values of the risk-free rate is to analyze the
robustness of the proposed approach.
All the computations were performed in Matlab. While doing so we utilized some
algorithms already presented in Kresta (2015) while most of them had to be
programmed. Nevertheless, all the algorithms, by which the results were
obtained, are freely available upon an e-mail request to the author.
4.1 Bootstrapping Method
The ex-post wealth paths obtained by means of the bootstrapping method are
depicted in Figure 3. In order to keep the clarity of the graph we plotted only the
wealth paths for selected risk-free rates. As can be seen from the graph, the final
wealth ranges between 2.5 (risk-free rate of 0% p.a.) and 3.6 (risk-free rate of
0 1W =
tW
1 , ,
1
1
N
t t i t i t
i
W W r w+
=
 
= ⋅ + ⋅ 
 
∑
,i tr ,i tw
R%
No. 2/2015
15
10% p.a.) and there is a clear relationship – the higher the risk-free rate the
higher the final wealth. The relationship can be extended to the whole wealth
paths – the wealth paths with higher risk-free rates dominate the ones with lower
risk-free rates, except the period from December 2008 until April 2009 (in which
there was a big drop in portfolio wealth due to the subprime crisis). Note that the
risk-free rate is applied only in calculation of the Sharpe ratio, however, the
investor is not allowed to invest into a risk-free asset, i.e. he always invests his
whole wealth into risky assets, see portfolio optimization problem (3). Due to this
reason, the wealth paths dropped in the period 2008-2009 as there was general
decline of stocks' prices due to the subprime crisis.
Figure 3 Ex-post wealth paths obtained by means of bootstrapping method
Source: Author
On the other hand, from the figure we can see that the higher the risk-free rate,
the higher the volatility of the wealth paths and the drawdowns. However this is
true only for the absolute values of the drawdowns. If we compare the maximum
drawdowns (which actually took place in 2008-2009) stated relatively, we find out
that there is no relationship between the values of risk-free rate and the
maximum drawdown (in %), see Table 1.
Financial Assets and Investing
16
Table 1 Final wealth and maximum drawdown of particular wealth paths
Risk-free rate 0% 1% 2% 3% 4% 5% 6% 10%
Final wealth 2.50 2.59 2.63 2.75 2.91 3.00 3.03 3.06
Maximum
drawdown
44.1% 44.1% 44.0% 43.9% 43.6% 43.4% 43.7% 47.1%
Source: Author
Table 1 summarizes also the values of final wealth for all the risk-free rates
assumed. We can see that the above discussed relationship between the values of
risk-free rate and final wealth is valid for all analyzed risk-free rates.
To sum up, we can conclude that the final wealth ranges between 2.5 (average
annual return of 5.82%) and 3.06 (average annual return of 7.16%) and
maximum drawdown over the analysed period fluctuates around 44%, except for
10% risk-free rate. Although the strategies outperformed the passive investment
into Dow Jones Industrial Average index (average annual return of 4.26% and
maximum drawdown 53.78%) both in terms of profitability and maximum
drawdown, the profitability after accounting for transaction costs is questionable.
4.2 GARCH-Copula Model
The ex-post wealth paths obtained by means of GARCH-copula model are
depicted in Figure 4. From the graph we can observe two findings: the values of
final wealth are higher than applying previous approach and the ex-post wealth
paths evolved more closely to each other for different risk-free rates. Also in the
case of GARCH copula model, the higher the risk-free rate the higher the value of
final wealth. It is difficult to analyze the volatility of the wealth paths as they are
close to each other.
Table 2 summarizes the values of final wealth and maximum drawdown. We can
see that the values of both the final wealth and the maximum drawdown grow as
applied risk-free rate increases (this relationship is not strictly true for final
wealth, but the general trend is obvious). Concerning the values, we can sum up
that the final wealth ranges between 11.33 (average annual return of 16.18%)
and 12.48 (average annual return of 16.88%) and maximum drawdown over the
analyzed period is in range of 33%–35.3%. As it is obvious, the results are not
significantly sensitive to applied values of the risk-free rate and the proposed
methodology is robust.
No. 2/2015
17
Figure 4 Ex-post wealth paths obtained by means of GARCH-copula model
Source: Author
We can see that when applying the GARCH-copula model, the values of maximum
drawdown are smaller and the values of final wealth are higher compared to the
bootstrapping method. This approach is thus clearly superior to the previous one.
However, note that the results are provided without deduction of transaction
costs.
Table 2 Final wealth and maximum drawdown of particular wealth paths
Risk-free rate 0% 1% 2% 3% 4% 5% 6% 10%
Final wealth 11.38 11.33 11.55 11.47 11.61 12.15 12.30 12.48
Maximum
drawdown
33.2% 33.4% 33.7% 33.9% 34.2% 34.3% 34.5% 35.3%
Source: Author
Financial Assets and Investing
18
Conclusions
The cornerstone of modern portfolio theory was set by Markowitz in 1952 and the
portfolio optimization problem is in the constant focus of both academics and
practitioners. Although, the prediction of future time series evolution plays an
important role in the problem, it is rarely addressed in research. In the paper we
analysed the applicability of the GARCH-copula model. To be more concrete we
assumed the investor maximizing Sharpe ratio while the future time series are
simulated by means of GARCH-copula model and by means of bootstrapping
technique.
In our paper we did not subtract the transaction costs. The reason is twofold.
Firstly, they differ significantly – although they would represent high fraction of
the gains for small private investors, for large institutional investors they would
be of smaller values. Secondly, also the considered investments into DJIA index is
connected with transaction costs which are caused by the changes in stock prices
(and thus also changes in relative weights). We are also aware of another
drawback of our analysis when considered for practical purposes – survivorship
bias. Loosely speaking, by the survivorship bias we address the situation in which
the decision making is influenced by the information which are not known at the
moment we are making decision. In our analysis, the problematic point is the
selection of dataset – we took the components of DJIA index as of October 6,
2014, however this composition was not known during the whole analysed period
(years 1998-2014). We didn't address this feature as we were mainly focused on
the analysis of GARCH-copula model applicability for financial time series
predictions. We compared the GARCH-copula approach to bootstrapping
technique and applied the same dataset for both methods. By doing so, the
results can be compared, however it would be tricky to make conclusions about
profitability of strategies.
From the empirical results we found out that GARCH-copula model provides
better forecasts of future financial time series evolution than bootstrapping
method. Assuming the investor, who is maximizing the Sharpe ratio, both the
final wealth increased and maximum drawdown decreased.
Acknowledgments
This paper was supported by Operational Programme Education for
Competitiveness – project no. CZ.1.07/2.3.00/20.0296 and by GA ČR (Czech
Science Foundation – Grantová Agentura České Republiky) under the project no.
13-18300P. All the support is greatly acknowledged and appreciated.
No. 2/2015
19
References
Biglova, A. et al. (2004). Different approaches to risk estimation in portfolio
theory. The Journal of Portfolio Management, 31(1), pp. 103-112. DOI:
10.3905/jpm.2004.443328
Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity.
Journal of Econometrics, 31(3), pp. 307-327.
Cont, R. (2001). Empirical properties of asset returns: stylized facts and
statistical issues. Quantitative Finance, 1(2), pp. 223-236.
Elton, E. J. et al. (2014). Modern Portfolio Theory and Investment Analysis. 9th
Ed. Hoboken: Wiley.
Engle, R. F. (1982). Auto-regressive conditional heteroskedasticity with estimates
of the variance of United Kingdom inflation. Econometrica, 50(4), pp. 987-1007.
DOI: 10.2307/1912773
Farinelli, S. et al. (2008). Beyond Sharpe ratio: Optimal asset allocation using
different performance ratios. Journal of Banking & Finance, 32(10), pp. 2057-
2063. DOI: 10.1016/j.jbankfin.2007.12.026
Huang, J. J. et al. (2009). Estimating value at risk of portfolio by conditional
copula-GARCH method. Insurance: Mathematics and Economics, 45(3), pp. 315-
324.
Cherubini, U. et al. (2004). Copula Methods in Finance. Chichester: Wiley.
Cherubini, U. et al. (2011). Dynamic Copula Methods in Finance. Chichester:
Wiley.
Konno, H., and Yamazaki, H. (1991). Mean-absolute deviation portfolio
optimization model and its applications to Tokyo stock market. Management
Science, 37(5), pp. 519-531.
Kresta, A. (2015). Financial Engineering in Matlab: Selected Approaches and
Algorithms, SAEI, vol. 33. Ostrava: VSB-TU Ostrava.
Lundblad, C. (2007). The risk return tradeoff in the long run: 1836–2003. Journal
of Financial Economics, 85(1), pp. 123-150. DOI: 10.1016/j.jfineco.2006.06.003
Mandelbrot, B. (1963). The Variation of Certain Speculative Prices. The Journal of
Business, 36(4), pp. 394-419. DOI: 10.1086/294632
Markowitz, H. M. (1952). Portfolio selection. Journal of Finance, 7(1), pp. 77-91.
Financial Assets and Investing
20
Nelsen, R. B. (1997). Dependence and order in families of Archimedean copulas.
Journal of Multivariate Analysis, 60(1), pp. 111-122. DOI:
10.1006/jmva.1996.1646
Rank, J. (2006). Copulas: From Theory to Application in Finance. London: Risk
Books.
Shalit, H., and Yitzhaki, S. (1984). Mean‐Gini, Portfolio Theory, and the Pricing of
Risky Assets. The journal of Finance, 39(5), pp. 1449-1468.
Sharpe, W. F. (1966). Mutual Fund Performance. The Journal of Business, 39(1),
pp. 119-138. DOI: 10.1086/294846
Sharpe, W. F. (1994). The Sharpe Ratio. The Journal of Portfolio Management,
21(1), pp. 49-58. DOI: 10.3905/jpm.1994.409501
Sklar, A. (1959). Fonctions de repartition à n dimensions et leurs marges.
Publications de l'Institut de statistique de l'Université de Paris, 8, pp. 229-231.
Wang, Z.-R. et al. (2010). Estimating risk of foreign exchange portfolio: Using
VaR and CVaR based on GARCH–EVT-Copula model. Physica A: Statistical
Mechanics and its Applications, 389(21), pp. 4918-4928. DOI:
10.1016/j.physa.2010.07.012
Young, M. R. (1998). A minimax portfolio selection rule with linear programming
solution. Management Science, 44(5), pp. 673-683.