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An Integrated Risk Measure and Information Theory Approach for Modeling Financial Data and Solving Decision Making Problems
Available online at www.sciencedirect.com
ScienceDirect 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
532
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 , x t VaR D ( X ) ° ¯ 1D
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
534
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) xA
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 xA
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)) xA
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 xA
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.
References Acerbi, C., 2002. Spectral measures of risk: A coherent representation of subjective risk aversion, Journal of Banking and Finance 26, 1505-1518.
Silvia Dedu and Aida Toma / Procedia Economics and Finance 22 (2015) 531 – 537 Costea, A., Bleotu, V., 2012. A new fuzzy clustering algorithm for evaluating the performance of non-banking financial institutions in Romania, Economic Computation and Economic Cybernetics Studies and Research, 46, 4, 179-199. Costea, A., Eklund, T., Karlsson, J., Voineagu, V., 2009. Financial Performance Analysis of the Scandinavian Telecommunications Companies using Statistical and Neural Networks Techniques, Economic Computation and Economic Cybernetics Studies and Research, 43, 4,87-104. Filip, F.G., 2012. Decision-Making Perspective for Designing and Building Information Systems. International Journal of Computers, Communications and Control 7, 2, 264-272. Fulga, C., Dedu, S., 2010. A New Approach in Multi-Objective Portfolio Optimization Using Value-at-Risk Based Risk Measure, Proceedings of The IEEE International Conference on Information and Financial Engineering, p. 765. Georgescu, G., 2014. Public Debt, Sovereign Risk and Sustainable Development of Romania. Procedia Economics and Finance 8, 353-361, Elsevier. Gulko, L., 1995. The Entropy Theory of Bond Option Pricing, Working Paper, Yale School of Management, October, New Haven, CT. Guo, W.Y., 2001. Maximum entropy in option pricing: a convex-spline smoothing method, Journal of Futures Markets 21, 819-832. Istudor, I., Filip, F.G., 2014. The Innovator Role of Technologies in Waste Management towards the Sustainable Development. Procedia Economics and Finance 8, 420-428. Kaniadakis, G., 2001. Non-linear kinetics underlying generalized statistics, Physica A: Statistical Mechanics and its Applications 296(3), 405— 425. Lupu, R., Tudor, C., 2008. Direction of change at the Bucharest Stock Exchange. Romanian Economic Journal 11 (27), 165-185. Maasoumi, E., 1993. A compendium to information theory in economics and econometrics. Econometric Review 12, 137-181. Moinescu, B., Costea, A., 2014. Towards an early-warning system of distressed non-banking financial institutions, Economic Computation and Economic Cybernetics Studies and Research, 48, 2, 75-90. Nastac, I., Bacivarov, A., Costea, A., 2009. A Neuro-Classification Model for Socio-Technical Systems, Romanian Journal of Economic Forecasting 11, 3, 100-109. Ogryczak, W., 2002. Multiple criteria optimization and decisions under risk. Control and Cybernetics 31, 4, 975-1003. Preda, V., Dedu, S., Sheraz, M., 2014. New measure selection for Hunt and Devolder's semi-Markov regime switching interest rate model , Physica A 407, 350-359. Şerban, F., Ştefănescu, M.V., Dedu, S., 2011. Building an Optimal Portfolio Using a Mean-VaR Framework. Mathematical Methods and Techniques in Engineering and Environmental Science, 126-131. Şerban, F., Ştefănescu M.V., Dedu, S., 2011. The Relationship Profitability - Risk for an Optimal Portfolio Building with Two Risky Assets and a Risk-Free Asset. International Journal of Applied Mathematics and Informatics, 5, 4, 299 – 306. Ştefănoiu, D., Borne, P., Popescu, D., Filip, F.G., El Kamel, A., 2014. Optimisation en science de l’ingénieur; méthaheuristiques, méthods stochastiques et aide à la décision. Hermes Science - Lavoisier, Paris. Toma, A., 2012. Robust Estimations for Financial Returns: An approach based on pseudodistance minimization, Economic Computation and Economic Cybernetics Studies and Research, 46, 1, p. 117. Toma, A., 2014. Model selection criteria using divergences, Entropy 16(5), 2686-2698, 2014. Toma, A., Dedu, S., 2014. Quantitative techniques for financial risk assesment: a comparative approach using different risk measures and estimation methods. Procedia Economics and Finance 8, 712-719, Elsevier. Toma, A., Leoni-Aubin, S., 2013. Portfolio selection using minimum pseudodistance estimators, Economic Computation and Economic Cybernetics Studies and Research, 47, 1, 97-113. Tsallis, C., 1988. Possible generalization of Boltzmann-Gibbs statistics, Journal of Statistical Physics 52,479-487. Tudor, C., Popescu-Duţă, C., 2012. On the causal relationship between stock returns and exchange rates changes for 13 developed and emerging markets. Procedia-Social and Behavioral Sciences 57, 275-282. Tudor, C., 2012. Active portfolio management on the Romanian Stock Market, Procedia-Social and Behavioral Sciences 58, 543-551. Tudor, M., Dedu, S., 2012. A new risk management model using quantile-based risk measure, with applications to non-normal distributions, Recent Researches in Applied Information Science, Recent Advances in Computer Engineering Series, 2, 183-186. Ullah, A., 1996. Entropy, divergence and distance measures with econometric applications, Journal of Statistical Planning and Inference 49, 137162.
537
ScienceDirect 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
532
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 , x t VaR D ( X ) ° ¯ 1D
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
534
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) xA
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 xA
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.
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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)) xA
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 xA
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.
References Acerbi, C., 2002. Spectral measures of risk: A coherent representation of subjective risk aversion, Journal of Banking and Finance 26, 1505-1518.
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