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Procedia Economics and Finance 22 (2015) 531 – 537
2nd International Conference ‘Economic Scientific Research - Theoretical, Empirical and
Practical Approaches’, ESPERA 2014, 13-14 November 2014, Bucharest, Romania
An Integrated Risk Measure And Information Theory Approach
For Modeling Financial Data And Solving Decision Making
Problems
Silvia Dedua,b,*, Aida Tomaa,c
a
Bucharest University of Economic Studies, Department of Applied Mathematics, Piata Romana nr. 6, Bucharest, 010374, Romania
b
Institute of National Economy, Calea 13 Septembrie nr. 13, Bucharest, 050711, Romania
c
”Gh. Mihoc-C. Iacob”Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, Bucharest, Romania
Abstract
In this paper we build some integrated techniques for modeling financial data and solving decision making problems, based on
risk theory and information theory. Several risk measures and entropy measures are investigated and compared with respect to
their analytical properties and effectiveness in solving real problems. Some criteria for portfolio selection are derived combining
the classical risk measure approach with the information theory approach. We analyze the performance of the methods proposed
in case of some financial applications. Computational results are provided.
©
B.V.
This is an open access article under the CC BY-NC-ND license
© 2015
2015The
TheAuthors.
Authors.Published
PublishedbybyElsevier
Elsevier
B.V.
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Selection and/or peer-review under responsibility of the Scientific Committee of ESPERA 2014.
Selection and/or peer-review under responsibility of the Scientific Committee of ESPERA 2014
Keywords: Risk measure; Information theory; Financial data; Decision making problems.
* Corresponding author. Tel.: +40-72-258-0269.
E-mail address: [email protected]
2212-5671 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Selection and/or peer-review under responsibility of the Scientific Committee of ESPERA 2014
doi:10.1016/S2212-5671(15)00252-X
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Silvia Dedu and Aida Toma / Procedia Economics and Finance 22 (2015) 531 – 537
1. Introduction
The concept of entropy provides a measure of the degree of uncertainty corresponding to a random variable. In
the last decades entropy measures have been introduced in the literature for solving real problems which arise in
finance and econometrics, see, for example,( Maasoumi, 1993 and Ullah, 1996).
The approach based on risk measures and entropy measures provides realistic techniques and efficient
computational tools for risk assessment and for solving optimization problems under uncertainty. Various
mathematical techniques are used for dealing with decision problems from economy, social sciences and many other
fields. In this respect can be mentioned the contributions of (Ştefănoiu et al., 2014; Istudor and Filip, 2014;
Moinescu and Costea, 2014; Barik et al., 2012; Filip, 2012; Costea and Bleotu, 2012; Nastac et al., 2009 and Costea
et al., 2009). Modeling the trend of financial indices and portfolio selection topics have caught the interest of the
researchers, see, for example (Georgescu, 2014; Toma and Dedu, 2014; Tudor, 2012; Tudor and Popescu-Duţă,
2012;Şerban et al., 2011 and Lupu and Tudor, 2008. Recently, (Toma, 2014; Preda et al., 2014; Toma and LeoniAubin, 2013 and Toma, 2012) investigate a variety of methods for financial data modeling using entropy measures.
Mean-risk models represent widely used techniques for solving portfolio selection problems. Recently, many
research papers are devoted to the topic of decision making using different risk measures, see for example,
(Ogryczak, 2002; Fulga and Dedu, 2010; Şerban et al., 2011; Tudor and Dedu, 2012).
In this paper we develop some integrated techniques for modeling financial data and solving decision making
problems, based on risk theory and information theory. Some fundamental concepts of information theory are
presented in Section 2. The most important risk measures used in financial data fodeling are introduced in Section 3.
Some criteria for portfolio selection are derived combining the classical risk measure approach with the information
theory approach in Section 4. The results obtained are used to solve decisional problems under uncertainty in
Section 5. The performance of the methods proposed is analysed in case of some financial applications The
conclusions of the paper are presented in Section 6.
2. Information measures applied to financial data modeling
The concept of entropy evaluates the degree of disorder in a system or the expected information in a probability
distribution. First we introduce the concept of entropy.
Let Ω, K , P be a probability space and X : Ω o R be a discrete random variable, given by
§ xi
X : ¨¨
© pi
·
¸¸
¹i
, with
xi  R , xi xi 1 , i 1,n 1
1,n
and
n
¦p
i
1 . Let D  [0,1] .
i 1
Definition 2.1. The Shannon entropy of the interval random variable
X is defined by the number
n
H( X ) ¦ pi ln pi .
i 1
1
, for all i 1,n , the Shannon entropy attains its maximum value: H( X ) ln N .
n
If there exists k 1,n such that pk 1 and pi 0 for all i 1,n , i z k , then the Shannon entropy attains the
minimal value: H( X ) 0 .
Remark. In the case pi
Silvia Dedu and Aida Toma / Procedia Economics and Finance 22 (2015) 531 – 537
533
In modern portfolio theory, entropy is regarded as a measure of portfolio diversification. In financial applications,
portfolios are first constructed using different selection techniques and in the second stage their degree of
diversification is evaluated using different entropy measures.
3. Risk measures applied to financial data modeling
irst we recall the definitions of the most important risk measures used in financial data modeling.
The Value-at-Risk measure is used to assess the maximal loss which can occur in a time period with a given
probability level. Let X : Ω → R be a random variable defined on the probability space (Ω, K, P), with cumulative
distribution function FX(x) = P(X d x), x  R. Let D  (0, 1). The Value-at-Risk measure corresponding to the
random variable X at the probability level D is given by:
VaR D ( X )
inf ^x  R | P( X d x) t D `.
Value-at-Risk is used for computing the capital adequacy limits for insurance companies, banks and other financial
institutions. It plays an important role in risk management and also in regulatory control activity of financial
institutions.
The Conditional Tail Expectation corresponding to the random variable X and the probability level D is defined by:
CTED ( X )
E>X | X t VaR D ( X )@ .
The Conditional Value-at-Risk measure of the random variable X at the probability level D(0,1) is defined by:
f
³ x dF
CVaR D ( X )
D
X
( x ) , where
f
FXD ( x )
0,
x VaR D ( X )
­
°
.
® FX ( x ) - D
°̄ 1 D , x t VaR D ( X )
In recent literature there were developed several representation formulas for CVaR. One of the most used, proposed
by Acerbi, 2002, is given by:
CVaR D ( X )
1
D
D
³ VaR E ( X ) dE
0
Conditional Value-at-Risk measure assesses the mean value of the total losses greater than Value-at-Risk which can
occur in a given time period. If we take into account a fixed probability level D, we remark that VaRD(X) represents
the inferior bound of CVaRD(X) measure.
4. An integrated risk measure and information theory approach for portfolio optimization
Mean-risk models represent a widely used technique for solving portfolio optimization problems. Recently, a lot
of research papers deal with the topic of portfolio selection using risk measures. Mean-risk models are based on
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Silvia Dedu and Aida Toma / Procedia Economics and Finance 22 (2015) 531 – 537
estimating and comparing financial returns using two statistics: the mean value and the value of a risk measure.
Thus, mean-risk models have a realistic interpretation of the results and are convenient from the computational point
of view. The risk measure used in the optimization model plays an important role in the decision making process.
For solving the portfolio selection problem, decision is made on the proportion of capital which must invested in
each of a number of available assets in order to obtain maximal return at the end of the investment period.
We consider a set of n assets and denote by Rj the random variable which models the return corresponding to the
asset j at the end of the investment period, j  1, n . We denote by c the total amount of the capital which will be
invested and by cj the capital to be invested in the asset j, j  1, n . Then the proportion of the capital invested in the
cj
n
asset j is x j
, j  1, n . Let x x1 , x 2 ,..., x n  R be the decision vector or the portfolio resulting from the
c
choice. The portfolio return is the random variable R x
n
¦x R
j
j
. A feasible set A of decision vectors consists of the
j 1
weights
x
x1 , x2 ,..., xn that must satisfy a set of constraints. The simplest way to define the feasible set is by
requiring that the sum of the weights be equal to 1 and short selling is not allowed. For this basic version of the
problem, the set of feasible decision vectors is given by:
n
T
­
A ® x1 ,..., xn  R n ¦ x j
j 1
¯
½
1, x j t 0, i 1, n ¾ .
¿
Let l(x, p) be the random variable which models the loss associated with the decision vector x  A  Rn and the
random vector p. The vector x is interpreted as a portfolio, A as the set of available portfolios subject to various
constraints and p stands for the uncertainties, for example future prices of the assets in portfolio, which can affect
the loss. The loss equals to the difference between the initial wealth W0 and the final random wealth W = xTp and it
can be expressed by:
l(x, p) = W0 - xTp.
We note that loss might be negative and in this case it constitutes a gain. We will recall the classical optimization
models in the mean-risk framework based on Value-at-Risk measure. Let VaRD l ( x , p be the Value-at-Risk
corresponding to the portfolio.
The first model aims to minimize the Value-at-Risk of the portfolio, provided that its expected return has a lower
bound P 0 :
min VaRD l ( x , p) xA
n
¦ P j x j t P0 .
(1)
j 1
The second model searches for a portfolio x which maximizes the expected return, for a given upper
bound v0 of the Value-at-Risk corresponding to the loss random variable:
n
max ¦ P j x j
xA
j 1
(2)
VaRD l ( x , p) d Q 0 .
VaRD l ( x , p be the Value-at-Risk and H(l ( x , p)) be the Shannon entropy
Let O  [0,1] . Let
corresponding to the portfolio. We propose the following portfolio optimization models which are constructed by
taking into consideration the return of the portfolio, the total risk and the entropy.
Silvia Dedu and Aida Toma / Procedia Economics and Finance 22 (2015) 531 – 537
535
The objective of the first model aims is to minimize an integrated risk measure, which represents a linear convex
combination of Value-at-Risk and Shannon entropy corresponding to the portfolio, provided that its expected return
has a lower bound P 0 :
min OVaRD l ( x , p (1 - O )H(l ( x , p))
xA
n
¦P x
j
j
t P0 .
(3)
j 1
The objective of the second model is to maximize the expected return, for a given upper bound v0 of the
integrated Value-at-Risk-Shannon entropy corresponding to the loss random variable:
n
max ¦ P j x j
xA
j 1
(4)
OVaRD l ( x , p (1 - O )H(l ( x , p)) d Q 0 .
5. Application: portfolio optimization problem
In this section we will use the concepts defined in the previous section to model financial data. We consider the
case of a portfolio p p1 , p2 ,..., pn composed by n assets, with
pi t 0, k
n
1, n , ¦ pi
1.
i 1
We will evaluate the outcome of the assets using the daily log-return, since it is widely used in financial analysis.
For each j, j  1, s , let Q j (t ) be the closing price of the asset j at the moment t , t t 0 . The log-return of the
asset j corresponding to the time horizon [t , t 1] is defined as follows:
R j (t,t 1) ln Q j (t 1) ln Q j (t ), j 1, s.
Similarly, we define the loss random variable L j (t , t 1) of the asset j corresponding to the time horizon
[t , t 1] as follows:
L j (t,t 1) R j (t,t 1) ln Q j (t ) ln Q j (t 1), j 1, s.
Let D  (0,1) .
The Value-at-Risk of the loss random variable L j (t , t 1) of the asset j corresponding to the probability
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Silvia Dedu and Aida Toma / Procedia Economics and Finance 22 (2015) 531 – 537
level
D and the time horizon [t , t 1] is defined by:
VaRα L j (t , t 1) min ^ z  R P( L j (t , t 1) d z) t α`.
We have used data available on the www.bvb.ro website corresponding to 300 days for a number of ten
assets from Bucharest Stock Exchange, denoted by S1 to S10. In table 1 we present a selection of the results
obtained solving the optimization problem (4) for O 0.5 and various values of the threshold Q0. For each value of
the upper bound Q0 we provide the optimal portfolio and the value of the maximal return.
Table 1. The results obtained solving the optimization problem
Risk
threshold
Optimal portfolio
Optimal
Return
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
0.029
0.000
0.000
0.000
0.730
0.000
0.100
0.000
0.000
0.170
0.000
- 0.017
0.021
0.000
0.000
0.000
0.720
0.000
0.040
0.000
0.000
0.240
0.000
- 0.004
0.022
0.000
0.000
0.000
0.650
0.000
0.000
0.000
0.000
0.350
0.000
0.065
0.023
0.040
0.000
0.000
0.530
0.000
0.000
0.000
0.000
0.430
0.000
0.104
0.024
0.070
0.000
0.000
0.450
0.000
0.000
0.000
0.000
0.480
0.000
0.117
0.025
0.120
0.000
0.000
0.410
0.000
0.000
0.000
0.000
0.470
0.000
0.138
0.026
0.180
0.000
0.000
0.390
0.000
0.000
0.000
0.000
0.430
0.000
0.152
0.027
0.290
0.000
0.000
0.320
0.000
0.000
0.000
0.000
0.390
0.000
0.164
0.028
0.380
0.000
0.000
0.270
0.000
0.000
0.000
0.000
0.350
0.000
0.178
0.029
0.510
0.000
0.000
0.200
0.000
0.000
0.000
0.000
0.290
0.000
0.191
0.030
0.670
0.000
0.000
0.120
0.000
0.000
0.000
0.000
0.210
0.000
0.217
(Q0)
Conclusions
In this paper we have built some integrated techniques for modeling financial data and solving decision making
problems, based on combining risk theory and information theory. Some criteria for portfolio selection are derived
combining the classical risk measure approach with the information theory approach. More precisely, we have
proposed a new risk measure given by the convex combination of the two measures: Value-at-Risk and Shannon
entropy. The new approach provides new mathematical techniques and computational tools for modeling the
imprecision of financial data and for solving decision making problems under uncertainty. Using this new approach
enables us to obtain different results of the optimization problem, depending on the proportion the two measures are
used.
Acknowledgements
This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS –
UEFISCDI, project number PN-II-RU-TE-2012-3-0007.
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