Available online at www.sciencedirect.com
ScienceDirect Procedia Economics and Finance 22 (2015) 610 – 617
2nd International Conference ‘Economic Scientific Research - Theoretical, Empirical and Practical Approaches’, ESPERA 2014, 13-14 November 2014, Bucharest, Romania
Modeling Financial Data Using Risk Measures with Interval Analysis Approach Silvia Dedua,d,*, Florentin Şerbanb,c,d a Institute of National Economy, Calea 13 Septembrie nr. 13, Bucharest, 050711, Romania University of Bucharest, Faculty of Mathematics and Computer Science, Academiei nr. 14, Bucharest, 010014, Romania c Romanian Academy, Calea Victoriri 125, Bucharest, 010071, Romania d Bucharest University of Economic Studies, Department of Applied Mathematics, Piata Romana nr. 6, Bucharest, 010374, Romania b
Abstract In this paper we construct some new measures which can be used for risk assessment and optimization. Due to the random character of economic phenomena, modeling financial data by real numbers does not perform accurately in decision making problems under uncertainty. First we introduce some concepts related to interval analysis, by replacing real numbers with interval numbers. Using these concepts, some risk measures are defined in this new framework. The theoretical results obtained are used to solve a case study. Computational results are provided. © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license © 2015 The Authors. Published by 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 management; Uncertainty; Optimization; Interval Analysis.
1. Introduction The foundations of interval analysis theory were stated by Moore in 1979. A lot of contributions in optimization using interval analysis approach have been developed by (Hansen and Walter, 2004 and Ishibuchi and Tanaka, 1990). Later, the improving effectiveness of interval analysis for dealing with a wide range of real life problems enhanced the extension of its main concepts in a probabilistic framework. In this way, the classical probabilistic concept of random variable was extended to the corresponding concept of interval random variable, which enables
* Corresponding author. Tel.: +4-072-258-0269. 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)00271-3
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us to model not only the randomness character, by the mean of probability theory, but also imprecision and nonspecificity, by using the concepts defined in the interval analysis framework. The approach based on interval analysis provides realistic mathematical models and effective computational tools for financial data modeling and for dealing with optimization problems under uncertainty. The randomness character of the parameters of the model can be captured by using stochastic programming models. Stochastic programming techniques are widely used in a lot of real decision making problems which arise in various fields, such as: economy, social sciences, engineering and many other domains. We can mention here the contributions of (Barik et al., 2012, Filip, 2012, Istudor and Filip, 2014, Preda, 2014 and Ştefănoiu et al., 2014). Modeling the trend of financial indices and portfolio selection topics have caught the interest of the researchers, see, for example, (Costea and Bleotu, 2012, Costea et al., 2009, Georgescu, 2014, Lupu and Tudor, 2008, Moinescu and Costea, 2014, Nastac et al., 2009, Şerban et al., 2011, Toma and Leoni-Aubin, 2013 and Tudor, 2012. Recently, Toma and Dedu, 2014, Tudor, 2012 and Şerban et al., 2011) used risk measures and data analysis techniques for solving decision making problems under uncertainty. In this paper we present some quantitative risk management methods developed in the interval analysis framework, which can be used for modeling data uncertainty. Some fundamental concepts defined using interval analysis approach are presented in Section 2. It is known that the performance of the theoretical models is improved by correctly choosing the risk measures involved. For this purpose we will use the approach from (Artzner et al., 1999, Ogryczak, 2002 and Dedu et al., 2014.) The theoretical basis is completed in Section 3 by defining some new risk measures using the interval analysis approach. The results presented are used to solve decisional making problems under uncertainty in Section 4. The conclusions of the paper are presented in Section 5.
2. Interval analysis 2.1. Interval numbers L
Let x , x
U
be real numbers, with x
L
U
x .
Definition 2.1. An interval number is a set defined by: [x]
x
R |x
We will denote by numbers. Remark 2.1. If x we will say that
L
x ,x
x
U
L
x ; x ,x
the interval number
x
L
L
U
R . L
x ,x
U
L
, with x , x
U
U
R . We will denote by IR the set of all interval
x , then we say that the interval number x L , x U U
is a degenerate interval number. Otherwise,
is a proper interval number.
Remark 2.2. A real number x
R can be regarded as the interval number x , x .
Definition 2.2. An interval number U 0; negative, if x L 0; positive, if x L 0; nonnegative, if x U 0. nonpositive, if x
x
L
x ,x
U
is said to be:
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2.2. Interval arithmetic Many relations and operations defined on sets or pairs of real numbers can be extended to relations and operations between intervals. Next we will introduce the basic operations needed to estimate some financial indices in the interval analysis framework. Let [ x ] x L , x U and [ y ] y L , y U be interval numbers. Definition 2.3. The equality between interval numbers is defined by: y if x
x
L
y
L
and x
U
y
U
.
Definition 2.4. The median of the interval number x
m [x]
L
x
U
L
x
x ,x
Let
x
L
L
y
x
U
L
x
x ,x
U
is the interval number defined by:
.
an interval number and a
U
is the number defined by:
L
y :x
x ,x
U
x .
Definition 2.6. The absolute value of the interval number x
is the number defined by:
U
.
Definition 2.5. The length of the interval number x
x ,x
U
2
l [x]
L
x
R a real number.
Definition 2.7. The product between the real number a and the interval number [x] is defined by: L
U
,
if a
0
U
L
,
if a
0 .
if a
0
a x ,a x a
x
a x|x
x
a x ,a x 0 ,
Definition 2.8. The summation of two interval numbers is defined by: x
y
x
L
L
y ,x
U
y
U
.
The subtraction of two interval numbers is defined by: x
y
x
L
U
y ,x
U
y
L
.
The product between two interval numbers is defined by:
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x
y
L
min
L
L
x y ,x y
U
,x
U
L
y ,x
U
y
U
L
, max
L
L
x y ,x y
U
,x
U
L
y ,x
U
y
613
U
y , then the division between two interval numbers is defined by:
If 0
x
x
L
y
y
U
,
x
U
y
L
.
2.3. Inequalities between interval numbers Next we will introduce the definitions which extend the classical concepts of inequality relation between real numbers to the concept of inequality relation between interval numbers. Let [ x ] x L , x U and [ y ] y L , y U be L U L U R . interval numbers, with x , x , y , y Definition 2.9. We say that: 1)
x
y
if x
2)
x
y
if
L
x
L
x
U
if x
L
3)
x
4)
x ≼ y if m x
5)
x ≺ y if x
y
L
L
y
and x
y
L
y
U
L
y
m
U
or
and y
y
x
L
x
U
U
U
;
y
L
y
U
or
x
L
y
L
x
U
y
U
;
U
x .
, according to Ishibuchi and Tanaka,1990.
y
U
y .
Next we will introduce the following interval inequalities in the framework defined by the inequality “≼”. If x
U
y
L
the interval inequality relation x ≼ y is said to be optimistic satisfactory and we will denote this
relation by “≼o”. If x
U
y , the interval inequality relation x ≼ y L
is said to be pessimistic satisfactory and we
will be denote it by “≼P”. 3. Risk measures defined using interval analysis approach 3.1. Interval random variables Let Ω , K , P be a probability space and IR be the set of all interval numbers. Definition 3.1. An interval random variable is an application X : Ω [X ]
X
L
,X
U
,
IR ,
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L
where X , X
U
IR are random variables, with X
:Ω
L
U
X
almost surely.
IR is said to be a discrete interval random variable if it Definition 3.2. The interval random variable X : Ω takes values in a discrete subset of the interval real numbers. 1 ,2 ,....., n is a finite set, then the probability distribution of the discrete interval random variable If I IR can be expressed as:
X :Ω
L
U
L
x1 , x1
X :
U
x2 ,x2
p1
L
.......... ..........
p2
U
xn ,xn
.......... .......... ...
,
pn
n
U
L
where x i
xi 1 , i
1, n
1 and
1.
pi i 1
Definition 3.3. The cumulative distribution function of the interval random variable X : Ω
IR is defined by:
F[X]:IR [0,1], F[X]([x]) = P([X] ≼o [x]).
3.2. Interval risk measures using quantiles of an interval random variable Let Ω , K , P be a probability space and X : Ω L
X :
IR be a discrete interval random variable, given by
U
n
xi ,xi pi
, with x i
U
R , xi
L
xi 1 , i
1, n
1 and
pi
i 1 ,n
Definition 3.4. The inferior -quantile of the interval random variable X q
[X]
min j
L
x j ,x
U j
L
F[ X ] x j , x
U j
[X]
min j
L
x j ,x
U j
L
F[ X ] x j , x
U j
Remark 3.1. The following inequality holds: q [X] Let Ω , K , P be a probability space and let X : Ω distribution function F [ X ] . Let
[0,1] .
is defined by the interval number
.
Definition 3.5. The superior -quantile of the interval random variable X q
1 . Let
i 1
is defined by the interval number
.
q
[X]
.
IR be a discrete interval random variable, with cumulative
[0,1] .
Definition 3.6. The measure -Interval Value-at-Risk ( -IVaR) of the interval random variable [X] corresponding to the probability level is defined by the interval number
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IVaR
[X]
L
min
x j ,x
j
Remark 3.2. Note that IVaR
U
L
F[ X ] x j , x
j
[X]
q
[X]
U j
615
.
.
4. Algorithm and application regarding the assessment of the mean interval return of a portfolio In this section we will use the concepts defined in the previous section for modeling data uncertainty and for x 1 , x 2 ,..., x N composed by N assets, assessing some financial indices. We consider the case of a portfolio x with N
xk
0, k
1, N ,
1.
xk k 1
k
k
We denote by rt the return of the asset k at the moment t and by P t the price of the asset k at the moment t . We propose the following algorithm for computing the mean interval return of the portfolio: 1.
Compute the return of the asset k at the moment t by using the formula:
rt
2.
Pt
k
k
k
P0
.
k
P0
Compute the mean return of the asset k corresponding to the last T days using the formula: T
rt k
t 1
rT
3.
k
.
T
Compute the trend of the return of the asset k corresponding to the last T
rt r
4.
k
1
The mean interval return of the asset k is given by: r
5.
t T
k
k
k
min
rT , r
k
, max
k
rT , r
k
.
The mean interval return of the portfolio is given by: N
r
xk r k 1
k
.
days using the formula:
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We have used data available on the www.bvb.ro website corresponding to the last T
123 days for two assets, FP
and SNP. We obtained the mean return of the two assets corresponding to the last T SNP
and r123 FP
r10
and the trend of the return of the two assets corresponding to the last
0 . 0536
0 . 0712
SNP
and r10
FP
The mean interval return of the asset SNP is given by: r The mean interval return of the portfolio x r
0 .5 r
0 .5 r
SNP
0 . 0565
10 days:
0 . 0873 .
The mean interval return of the asset FP is given by: r
FP
FP
123 days: r123
, so we obtain r
0 . 07925
0 . 0712 SNP
0 . 0873
0 .5 , 0 .5 ,
,
0 . 0565 ,
0 . 0536
. .
composed by these two asset is given by:
0 . 05505
.
5. Conclusions The impact of uncertainty or estimation errors when dealing with financial data and risk measures create difficulties in the case of decision making problems. Using the assumption that data vary within a bounded interval, interval analysis framework makes possible the extension of the classical theories to the approach characterized by bounded uncertainty. The new approach developed in the interval analysis framework provides mathematical methods and computational tools for modeling the imprecision of financial data and for solving portfolio optimization problems under uncertainty. The algorithm proposed in this paper, based on interval analysis modeling, has proved to provide a better estimation of the portfolio return corresponding to a time period, since it evaluates the mean interval return taking into accout the mean return of the portfolio corresponding to this period and also the trend of the return corresponding to a short period, very close to the final moment. This procedure captures the trend of the financial indices corresponding to a lond time period and also their recent trend. The advantage of using risk management methods based on interval random concepts and results consists in a more realistically modeling of the economic phenomena under risk.
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