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Mean–variance models for portfolio selection subject to experts’ estimations
Expert Systems with Applications 39 (2012) 5887–5893
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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Mean–variance models for portfolio selection subject to experts’ estimations Xiaoxia Huang ⇑ School of Economics and Management, University of Science and Technology Beijing, Beijing 100083, China
a r t i c l e
i n f o
Keywords: Portfolio selection Mean–variance model Mean–semivariance model Uncertain programming Uncertain variable
a b s t r a c t Since the security market is complex, sometimes the future security returns are available mainly based on experts judgements. This paper discusses a portfolio selection problem in which security returns are given subject to experts’ estimations. The use of uncertain measure is justified, and two new mean–variance and mean–semivariance models are proposed. In addition, a hybrid intelligent algorithm for solving the optimization models is given. To illustrate the application of the new models, the method to obtain the uncertainty distributions of the security returns based on experts’ evaluations is given, and two selection examples are provided. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Portfolio selection is concerned with optimal allocation of one’s capital to a number of securities such that one can obtain the maximum investment return with risk control. Since the introduction of mean–variance models by Markowitz (1952), variance has been a very popular risk measure. In his models, Markowitz proposed that expected return could be regarded as the investment return and variance the risk. It was proposed that for a given level of variance, an optimal portfolio could be obtained when the expected return was maximized; or for a given expected return, the optimal portfolio could be obtained when the variance value was minimized. Since Markowitz, numerous models and algorithms have been proposed for portfolio selection based on the risk measurement of variance, e.g. papers by Deng, Li, and Wang (2005), Oh, Kim, Min, and Lee (2006), Chen, Hou, Wu, and Chang-Chien (2009a) and Soleimani, Golmakani, and Salimi (2009), etc. Since variance considers high return extremes as equally undesirable as low return extremes, variance will become an unreasonable risk measure when security returns are asymmetrical. Thus, semivariance was proposed to replace variance as an improvement of risk measure, and many scholars such as Choobineh and Branting (1986), Markowitz (1993), Kaplan and Alldredge (1997) and Grootveld and Hallerbach (1999) researched the properties and computation problem of mean–semivariance models. In these studies, security returns were all assumed to be random variables. However, the security market is complex, and randomness is not the only type of uncertainty. It has been found that sometimes security returns cannot be well reflected by historical data. The prediction of returns in the situation relies heavily on ⇑ Corresponding author. Tel.: +86 10 82376260; fax: +86 10 62333582. E-mail address: [email protected] 0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.11.119
experts’ estimation and contains much subjective imprecision rather than randomness. Therefore, many scholars argued that we should find another way to model people’s imprecise estimation of the returns other than random variables. With the introduction of fuzzy set theory by Zadeh (1965), scholars have tried employing fuzzy numbers and fuzzy set theory to manage portfolio since 1990s. Much work has been done on extending the mean–variance selection idea to fuzzy environment in different ways. For example Tanaka, Guo, and Türksen (2000) quantified mean and variance of a portfolio through fuzzy probability. Carlsson, Fullér, and majlender (2002) used their definition of mean and variance of fuzzy numbers (Carlsson & Fullér, 2001) to find the optimum portfolio. Chen, Liao, and Wang (2009b) discussed a cutting plane algorithm for a possibilistic mean–variance model with trapezoidal fuzzy returns, and Chen and Huang (2009) proposed a portfolio selection model for equity mutual funds with triangular fuzzy returns. Based on credibility measure Huang (2007) proposed two mean–variance models for fuzzy portfolio selection Qin, Li, and Ji (2009) developed mean– variance-cross entropy models, and Zhang, Zhang, and Cai (2010) discussed an adjusting problem for an existing portfolio. To deal with portfolios with fuzzy asymmetrical security returns, Huang (2008) proposed credibilistic mean–semivariance models, and Li, Qin, and Ka (2010) developed mean–variance–skewness models. These models developed portfolio selection theory and broadened the way to deal with securities whose returns are subject to experts’ estimations. However, as we research the problem deeper, we find that paradoxes will appear if we use fuzzy variables to describe the subjective estimations of security returns. For example, if a security return is regarded as a fuzzy variable, then we should have a membership function to characterize it. Suppose it is a triangular fuzzy variable n = (0.4, 0.3, 1.0) (see Fig. 1). Based on the membership function, it is known from possibility theory or credibility theory that the return is exactly 0.3 with belief degree
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X. Huang / Expert Systems with Applications 39 (2012) 5887–5893
(i) (Normality) MfCg ¼ 1: (ii) (Self-duality) MfKg þ MfKc g ¼ 1: (iii) (Countable subadditivity) For every countable sequence of events {Ki}, we have
( 1 [
M
)
Ki
6
i¼1
Fig. 1. Membership function of a security return n = (0.4, 0.3, 1.0).
1 in possibility measure or 0.5 in credibility measure. However, this conclusion is unacceptable because the belief degree of exactly 0.3 is almost zero. In addition, the return being exactly 0.3 and not exactly 0.3 have the same belief degree in either possibility measure or credibility measure, which implies that the two events will happen equally likely. This conclusion is quite astonishing and hard to accept. Recently Liu (2007) proposed an uncertain measure and developed an uncertainty theory which can be used to handle subjective imprecise quantity. Much research work has been contributed to the development of uncertainty theory and related theoretical work. For example Gao (2009) studied some properties of continuous uncertain measure. You (2009) proved some convergence theorems of uncertain sequences. Peng and Iwamura (2010) proved a sufficient and necessary condition of uncertainty distribution. Liu (2008) defined uncertain process, and Chen and Liu (2010) proved the existence and uniqueness theorem for uncertain differential equations, etc. When we use uncertainty theory to model subjective estimations of security returns, the above mentioned paradoxes will disappear immediately. Based on uncertain measure Zhu (2010) solved an optimal control problem of portfolio selection, and Huang (2011) proposed a mean-risk portfolio selection method. In this paper, we will follow Markowitz’s mean–variance idea and use uncertainty theory to help select portfolios whose returns are given mainly by experts’ estimations. As there do exist cases that security returns are asymmetrical (Prakash, Chang, & Pactwa, 2003; Simkowitz & Beedles, 1978), we will further define semivariance and develop a mean–semivariance selection model. The rest of the paper is organized as follows. For better understanding of the paper, some necessary knowledge about uncertain variable will be introduced in Section 2. Then, mean–variance and mean–semivariance models will be proposed in Section 3. After that, a hybrid intelligent algorithm will be presented for solving the proposed model problems in Section 4. To illustrate the application of the new models, the method for determining the uncertainty distributions of security returns based on experts’ estimations will be given in Section 5, and two selection examples will be provided in the same section. Finally, conclusion remarks will be given in Section 6. 2. Fundamentals of uncertainty theory Liu (2007) in 2007 proposed an uncertain measure and further developed an uncertainty theory which is an axiomatic system of normality, self-duality, countable subadditivity and product measure. Definition 1 Liu, 2007. Let C be a nonempty set, and L a r-algebra over C. Each element K 2 L is called an event. A set function MfKg is called an uncertain measure if it satisfies the following three axioms:
1 X
MfKi g:
i¼1
The triplet ðC; L; MÞ is called an uncertainty space. In order to define product uncertain measure Liu (2009) proposed the fourth axiom as follows: (iv) (Product measure) Let ðCk ; Lk ; Mk Þ be uncertainty spaces for k = 1, 2, . . . , n. The product uncertain measure is M ¼ M1 ^ M2 ^ ^ Mn :
Definition 2 Liu, 2007. An uncertain variable is a measurable function n from an uncertainty space ðC; L; MÞ to the set of real numbers, i.e., for any Borel set B of real numbers, the set
fn 2 Bg ¼ fc 2 CjnðcÞ 2 Bg is an event. Theorem 1 Liu, 2010. Any uncertain measure M is increasing, i.e., for any events K1 K2, we have
MfK1 g 6 MfK2 g: In application, a random variable is usually characterized by a probability density function or probability distribution function. Similarly, an uncertain variable can be characterized by an uncertainty distribution function. Definition 3 Liu, 2007. The uncertainty distribution U : R ! ½0; 1 of an uncertain variable n is defined by
UðtÞ ¼ Mfn 6 tg: For example, by a normal uncertain variable, we mean the variable that has the following normal uncertainty distribution
UðtÞ ¼ 1 þ exp
pðl tÞ pffiffiffi 3r
1 t 2 R;
;
where l and r are real numbers and r > 0. For convenience, it is denoted in the paper by n N ðl; rÞ. We call an uncertain variable a lognormal uncertain variable if it has the following lognormal uncertainty distribution
UðtÞ ¼ 1 þ exp
1
pðl ln tÞ pffiffiffi 3r
;
t 2 R;
where l and r are real numbers and r > 0. For convenience, it is denoted in the paper by n LOGN ðl; rÞ. When the uncertain variables n1, n2, . . . , nn are represented by uncertainty distributions, the operational law can be expressed as follows: Theorem 2 Liu, 2010. Let n be an uncertain variable with continuous uncertainty distribution U, and let f be a strictly increasing function. Then the uncertainty distribution W of f(n) can be obtained via
WðtÞ ¼ Uðf 1 ðtÞÞ
ð1Þ
which can also be expressed by
W1 ðaÞ ¼ f ðU1 ðaÞÞ;
0 < a < 1:
ð2Þ
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X. Huang / Expert Systems with Applications 39 (2012) 5887–5893
To tell the size of an uncertain variable, Liu defined the expected value of uncertain variables. Definition 4 Liu, 2007. Let n be an uncertain variable. Then the expected value of n is defined by
E½n ¼
Z
1
Mfn P tgdt
Z
0
0
Mfn 6 tgdt
ð3Þ
1
provided that at least one of the two integrals is finite. It can be calculated that the expected value of the normal uncertain variable n N ðl; rÞ is E[n] = l, and the expected value of the variable n LOGN ðl; rÞ is pffiffiffi lognormal puncertain ffiffiffi pffiffiffi E½n ¼ 3r expðlÞ cscð 3rÞ if r < p= 3:. Theorem 3 Liu, 2010. Let n be an uncertain variable with continuous uncertainty distribution U. If its expected value exists, then
E½n ¼
Z
1
U1 ðaÞda:
2
V½n ¼ E½ðn eÞ :
ð5Þ
Let n be an uncertain variable with continuous uncertainty distribution U. Then þ1
0
6e ¼
Mfðn eÞ2 g P tdt ¼
Z
þ1
Mfðn P e þ
pffiffi tÞ [ ðn
0
Z pffiffi t Þgdt 6
þ1
ðMfn P e þ
pffiffi pffiffi t g þ Mfn 6 e tgÞdt
0
Z
þ1
ð1 Uðe þ
pffiffi pffiffi t Þ þ Uðe tÞÞdt
¼
þ1
2ðt eÞð1 UðtÞ þ Uð2e tÞÞdt:
e
In this case, it is always assumed that the variance is
V½n ¼ 2
> > > > > > :
E½x1 n1 þ x2 n2 þ þ xn nn P b x1 þ x 2 þ þ x n ¼ 1
xi P 0;
ð8Þ
i ¼ 1; 2; . . . ; n
Z
3.2. Mean–semivariance models When we employ the uncertain mean–variance models (7) and (8), it implies that the uncertain security returns are symmetrical. Since when eliminating variance value, high return deviations from the expected return are eliminated equally as low return deviations from the expected returns, yet high return deviation implies the potential high return of the investment and is what the investors welcome. Thus, when security returns are asymmetrical, variance becomes an unreasonable risk measure. To measure only the low deviations from the expected return, we define the semivariance for uncertain variables as follows: Definition 6. Let n be an uncertain variable with finite expected value e. Then the semivariance of n is defined by SV[n] = E [[(n e)]2], where
0
Z
8 min V½x1 n1 þ x2 n2 þ þ xn nn > > > > > > < subject to :
where b denotes the minimum return that the investor can accept.
Definition 5 Liu, 2007. Let n be an uncertain variable with finite expected value e. Then the variance of n is defined by
Z
ð7Þ
where E denotes the expected value operator, V the variance operator of the uncertain variables, and a the preset maximum tolerable risk level. When the investor presets an expected return level that he/she feels satisfactory, and wants to minimize the risk, the selection model becomes:
ð4Þ
0
V½n ¼
8 max E½x1 n1 þ x2 n2 þ þ xn nn > > > > > > < subject to : V½x1 n1 þ x2 n2 þ þ xn nn 6 a > > > x > 1 þ x2 þ þ xn ¼ 1 > > : xi P 0; i ¼ 1; 2; . . . ; n
ðn eÞ ¼
0;
þ1
ðt eÞð1 UðtÞ þ Uð2e tÞÞdt:
ð6Þ
e
It can be calculated that the variance value of a normal uncertain variable n N ðl; rÞ is V[n] = r2.
n e; if n 6 e
ð9Þ
if n > e:
When the uncertain variable n has continuous uncertainty distribution U, then
SV½n ¼ 3. Uncertain portfolio selection models
¼
Z
þ1
Z0 e
Mfððn eÞ Þ2 P tgdt ¼
Z
þ1
Mfn 6 e
pffiffi t gdt
0
2ðe tÞUðtÞdt:
ð10Þ
1
3.1. Mean–variance models As discussed in introduction, when security returns are given mainly by experts’ estimations, it is better to use uncertain variables to describe the security returns. Let xi denote the investment proportions in securities i, and ni the uncertain returns of the ith securities given by experts. The returns are defined as ni ¼ ðp0i þ di pi Þ= pi ; i ¼ 1; 2; . . . ; n; respectively, where p0i are the estimated closing prices of the securities i in a future time, pi the closing prices of the securities i at present, and di the estimated dividends of the securities i during the period. According to Markowitz’s mean–variance idea, when making investment decision, the investor should strike a balance between maximizing the return and minimizing the risk. Then in case when the investor pre-gives a tolerable level of risk, and wants to maximize the expected return, we have the mean–variance selection model as follows:
Thus, in the situation that the security returns are asymmetrical, when the investor wants to maximize the expected return at a given specific level of risk, the uncertain mean–semivariance model can be built as follows:
8 max E½x1 n1 þ x2 n2 þ þ xn nn > > > > > > < subject to : > > > > > > :
SV½x1 n1 þ x2 n2 þ þ xn nn 6 u
ð11Þ
x1 þ x 2 þ þ x n ¼ 1 xi P 0;
i ¼ 1; 2; . . . ; n
where SV is the semivariance operator of the uncertain variable, and u the maximum semivariance level the investor can tolerate. When the investor wants to minimize the risk for a given level of expected return w, the mean–semivariance model is as follows:
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X. Huang / Expert Systems with Applications 39 (2012) 5887–5893
8 min SV½x1 n1 þ x2 n2 þ þ xn nn > > > > > > < subject to : E½x1 n1 þ x2 n2 þ þ xn nn P w > > > x > 1 þ x2 þ þ xn ¼ 1 > > : xi P 0; i ¼ 1; 2; . . . ; n:
when security returns are symmetrical, according to Theorem 2, we can obtain the expected value directly via
ð12Þ
" n X
E
# xi ni ¼ W1 ð0:5Þ ¼
i¼1
n X
xi U1 i ð0:5Þ:
ð14Þ
i¼1
4.3. Calculating variance via inverse uncertainty distribution 4. Hybrid intelligent algorithm Since uncertain mean–variance models (7) and (8) and mean– semivariance models (11) and (12) are complex, it is difficult to use analytical method to solve them. Genetic algorithm (GA) is a good method for solving complex optimization problems and has solved many complex portfolio selection problems (Chen et al., 2009a; Huang, 2007). Here, we introduce a method for calculating expected value, variance and semivariance values and integrate the results into the GA (Huang, 2007) to produce a hybrid intelligent algorithm for solving the new model problems in general cases. The introduction of the algorithm is as follows.
Pn
From Eq. (6), we know that the variance of the portfolio return can be calculated via
i¼1 xi ni
V
" n X
#
xi ni ¼ 2
Z
þ1
ðt eÞð1 WðtÞ þ Wð2e tÞÞdt:
e
i¼1
Since when we employ mean–variance models to select the optimal portfolio, the uncertainty distributions of the securities should be symmetrical, we have
2
Z
þ1
ðt eÞð1 WðtÞ þ Wð2e tÞÞdt
e
¼4
4.1. Obtaining inverse uncertainty distribution of portfolio return
Z
þ1
ðt eÞð1 WðtÞÞdt:
ð15Þ
e
Let Ui be the continuous uncertainty distributions of uncertain security returns ni, i = 1, 2, . . . , n, respectively, and W the uncertainty distribution of the uncertain portfolio return n X
Thus, we design the process for computing the variance value
V
" n X
# xi ni
i¼1
xi n i :
as follows:
i¼1
It is clear that W is also continuous. Since the investment proportions xi are non-negative numbers, then, according to Theorem 2, the uncertainty distribution W of the uncertain portfolio return can be obtained via
W1 ðaÞ ¼
n X
xi U1 i ðaÞ
i¼1
where a 2 [0, 1].
Method 2: Step 1. Set v = 0. P Step 2. set e ¼ ni¼1 xi U1 i ð0:5Þ: P Step 3. Set b ¼ ni¼1 xi U1 i ð0:9999Þ: Step 4. Randomly generate an a value from (0.5, 0.9999). P n 1 Step 5. Let v ¼ v þ i¼1 xi Ui ðaÞ e ð1 aÞ: Step 6. Repeat the 4th to 5th steps for N times, where N is a sufficiently large number. Step 7. Return v = v 4(b e)/N.
4.2. Calculating expected value via inverse uncertainty distribution 4.4. Calculating semivariance via inverse uncertainty distribution With the inverse uncertainty distribution W1 of the uncertain portfolio return, according to Theorem 3, the expected value of the P uncertain portfolio return ni¼1 xi ni can be calculated by
E
" n X
#
xi ni ¼
i¼1
9999 X
n X
j¼1
i¼1
1 i ð0:0001jÞ=9999:
xi U
ð13Þ
Thus, we design the process for computing the expected value
E
" n X
# xi ni
Pn
SV
Method 1: Step 1. Set e = 0. Step 2. Set j = 1. P Step 3. Let yj ¼ ni¼1 xi U1 i ð0:0001jÞ: Step 3. Let e = e + yj. Step 4. If j < 9999, let j = j + 1. Then turn back to Step 3. Step 5. Return e = e/9999. Please note that when we employ mean–variance models to select the optimal portfolio, the uncertainty distributions of the security returns should be symmetrical. Then, it is clear that in this case the expected return should be the value of W1(0.5). Thus,
" n X
#
xi ni ¼ 2
Z
e
ðe tÞWðtÞdt
ð16Þ
1
i¼1
where e is the expected return of the uncertain portfolio. Thus, we design the process for computing the semivariance value
i¼1
as follows:
According to Eq. (10), the semivariance of the portfolio return can be calculated via
i¼1 xi ni
SV
" n X
# xi ni
i¼1
as follows: Method 3: Step 1. Set sv = 0. P Pn 1 Step 2. Set e ¼ 9999 j¼1 i¼1 xi Ui ð0:0001jÞ=9999: Step 3. Use bisection search method to find the b value such that Pn 1 i¼1 xi UP i ðbÞ ¼ e: Step 4. Set a ¼ ni¼1 xi U1 i ð0:0001Þ: Step 5. Randomly generate an a value from (0.0001, b). P Step 6. Let sv ¼ sv þ e ni¼1 xi U1 i ðaÞ a: Step 7. Repeat the 5th to 6th steps for N times, where N is a sufficiently large number.
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X. Huang / Expert Systems with Applications 39 (2012) 5887–5893
Step 8. Return sv = sv 2(e a)/N. Please note that in all the three methods, when using inverse uncertainty distribution function, we let U value, i.e., a value, increase 0.0001 each time. In fact, this number can also be 0.00001 or even smaller according to the precision requirement of the problem. 4.5. Hybrid intelligent algorithm When the expected, variance and semivariance values of portfolio returns have been calculated, we can integrate the calculation results into the GA presented in paper (Huang, 2007) to find the optimal portfolio. Below is the representation structure of the solutions by the chromosomes and the summary of the hybrid intelligent algorithm. For the detailed description of the GA, the interested readers can refer to paper (Huang, 2007). Representation structure: The solution x = (x1, x2, . . . , xn) is represented by the chromosome C = (c1, c2, . . . , cn), where the genes c1, c2, . . . , cn are restricted in the interval [0, 1]. The mapping between a chromosome and a solution has the following form,
xi ¼
ci ; c1 þ c2 þ þ cn
i ¼ 1; 2; . . . ; n
ð17Þ
which ensures that x1 + x2 + + xn = 1 always holds. Hybrid intelligent algorithm: Step 1. Determine the parameters of the GA, i.e., the population size pop_size, the parameter in the rank-based evaluation function m, the probability of crossover operation Pc, and the probability of mutation operation Pm. Step 2. Initialize pop_size feasible chromosomes. Use the proposed Methods 1–3 to calculate the constraint values and check the constraint. Step 3. Use the proposed Methods 1–3 to calculate the objective values for all the chromosomes. Step 4. Give the rank order of the chromosomes according to the objective values, and compute the values of the rankbased evaluation functions of the chromosomes. Step 5. Compute the fitness of each chromosome according to the rank-based-evaluation function. Step 6. Select the chromosomes by spinning the roulette wheel. Step 7. Update the chromosomes by crossover and mutation operations. Use the proposed Methods 1–3 to calculate the constraint values when checking the constraint. Step 8. Repeat the third to the seventh steps for a given number of cycles. Step 9. Take the best chromosome as the solution of portfolio selection.
evaluate the occurrence degree of a series of uncertain return events. Independently, m sets of estimation data for the security return are given by experts. That is, from the experts the data set of
ðt 1 ; a1=j Þ; ðt2 ; a2=j Þ; . . . ; ðt k ; ak=j Þ;
j ¼ 1; 2; . . . ; m
were obtained, where
t1 < t2 < < tk ;
0 6 a1=j 6 a2=j 6 ak=j 6 1:
Since the experts were regarded equally knowledgable, we allocate the same weight to each expert’s estimation and aggregate the m experts’ estimations about the occurrence degree of a return event n 6 ti as follows:
Uðt i Þ ¼ ai ¼
m 1 X ai=j ; m j¼1
i ¼ 1; 2; . . . ; k:
ð18Þ
Then we get a new set of data for the security return as follows:
ðt 1 ; a1 Þ; ðt 2 ; a2 Þ; . . . ; ðt k ; ak Þ;
t1 < t2 < < tk ;
0 6 a1 6 a2
6 ak 6 1; Supposed it is believed that the security return is a normal uncertain variable N ðl; rÞ: Then we need to find the parameters l and r. To find the parameters, we adopt the principle of least squares proposed by Liu (2010) which says that the unknown parameter h of a known continuous uncertainty distribution form U(tjh) can be obtained via solution of either of the following two minimization problems
min h
k X ðUðti jhÞ ai Þ2 i¼1
and
min h
k X ðU1 ðai jhÞ t i Þ2 : i¼1
In our case one, since the uncertainty distribution of the security return is continuous, we can solute either
min l;r
k X ðUðti jl; rÞ ai Þ2
ð19Þ
i¼1
where
Uðt i jl; rÞ ¼ 1 þ exp
1
pðl ti Þ pffiffiffi 3r
;
or
min l;r
k X ðU1 ðai jl; rÞ t i Þ2
ð20Þ
i¼1
where
pffiffiffi 3r
ai
5. Two examples
U1 ðai jl; rÞ ¼ l þ
In order to illustrate the application of the mean–variance and mean–semivariance models and the effectiveness of the proposed algorithm, examples are provided in two cases when security returns are symmetrically and asymmetrically distributed. In the examples, security returns can not be well reflected by the historical data and are given by experts’ evaluations. The investor wants to select portfolios from 6 individual securities in each case. In the following, without losing generality, we will first introduce via one security the method for determining the uncertainty distribution of return based on experts’ evaluations, and then select portfolios in the above mentioned two cases. To evaluate the uncertainty distribution of the security return, first, questionnaires are given to m experts who are asked to
to find the parameters l and r. By solving the optimization problem (20), we can obtain the ^ and r ^ via the following equations estimated l
l^ ¼ t
p
ln
1 ai
pffiffiffi k ^ X 3r ai ln kp i¼1 1 ai
ð21Þ
and k k P P t pffiffiffi ln 1aia t i ln 1aia i i 3p k i¼1 i¼1 r^ ¼ 2 ; 2 3 P k k P ai ai ln 1a n ln 1a i¼1
i
i¼1
i
ð22Þ
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X. Huang / Expert Systems with Applications 39 (2012) 5887–5893
Pk 1
where t ¼ n i¼1 t i : If the experts think that the distribution of the security return is a lognormal uncertain variable LOGN ðl; rÞ; we obtain the ^ and r ^ by solving the following minimization problem estimated l
min l ;r
k X
ðUðti jl; rÞ ai Þ2
i¼1
1
1
2
3
4
5
6
Capital proportion
0.00
0.00
0.00
0.00
65.38
34.62
Table 3 Lognormal uncertain returns of 6 securities (%).
:
Example 1. Suppose in case one, the security returns are believed to be normal uncertain variables. Ten experts give their individual judgements about security returns independently. By using Eqs. (18)–(22), the distributions of security returns are obtained and given in Table 1. If the investor requires that the minimum expected return should not be less than 5.0% and wants to minimize the variance value, then according to the discussion in Section 3, the uncertain mean–variance model is built as follows:
8 min V½x1 n1 þ x2 n2 þ þ x6 n6 > > > > > > < subject to :
E½x1 n1 þ x2 n2 þ þ x6 n6 P 0:05
ð24Þ
x1 þ x 2 þ þ x 6 ¼ 1 xi P 0;
Security i
ð23Þ
ffiffi ti Þ where Uðt i jl; rÞ ¼ 1 þ exp pðlpln 3r
> > > > > > :
Table 2 Allocation of money to 6 securities with normal uncertain return rates (%).
i ¼ 1; 2; . . . ; 6
where E represents the expected value operator, V the variance of the uncertain variable, ni the ith uncertain security returns, and xi the investment proportions in the ith security. Since it has been proven (Liu, 2010) from Theorem 2 that if ni N ðli ; ri Þ; i ¼ 1; 2; . . . ; n; are normal uncertain variables, Pthen P g ¼P ni¼1 ki ni is also a normal uncertain variable g N ni¼1 ki li ; ni¼1 ki ri Þ for ki > 0, i = 1, 2, . . . , n. Then the model (24) can be converted into the following linear programming model: 8 min 0:058x1 þ 0:061x2 þ 0:073x3 þ 0:084x4 þ 0:065x5 þ 0:098x6 > > > > > > < subject to : 0:031x1 þ 0:033x2 þ 0:044x3 þ 0:051x4 þ 0:041x5 þ 0:067x6 P 0:05 > > > x1 þ x2 þ þ x6 ¼ 1 > > > : xi P 0; i ¼ 1;2;... ;6: ð25Þ
By solving the model (25), the optimal portfolio is given in Table 2. The objective value is 0.0764, which means the variance value of the selected portfolio is 0.0058. Example 2. Suppose in case two, the security prices are regarded to be lognormal uncertain variables. Ten experts give their individual judgements about the security prices independently. By using Eq. (18), data set about the security prices are got. Then by solving minimization problem (23), the distributions of the security prices are obtained. Next, by calculating ni ¼ ðp0i þ di pi Þ=pi ; the distributions of the security returns are available and are given in Table 3. Since the security returns are asymmetrical now, we use the mean–semivariance model to select the portfolio. If the investor requires that the minimum expected return should not be less than 4.5% and wants to minimize the risk value, then the portfolio selection model is built as follows:
Table 1 Normal uncertain returns of 6 securities (%). Security i
Uncertain return
Security i
Uncertain return
1 2 3
N (3.1, 5.8) N (3.3, 6.1) N (4.4, 7.3)
4 5 6
N (5.1, 8.4) N (4.1, 6.5) N (6.7, 9.8)
Security i
Uncertain return
Security i
Uncertain return
1 2 3
LOGN ð4:65; 16:60Þ 1 LOGN ð3:95; 7:66Þ 1 LOGN ð3:68; 18:10Þ 1
4 5 6
LOGN ð3:02; 5:84Þ 1 LOGN ð2:10; 3:30Þ 1 LOGN ð4:02; 5:08Þ 1
Table 4 Allocation of money to 6 securities with lognormal uncertain prices (%). Security i
1
2
3
4
5
6
Capital proportion
8.77
91.23
0.00
0.00
0.00
0.00
8 min SV½x1 n1 þ x2 n2 þ þ x6 n6 > > > > > > < subject to : > > > > > > :
E½x1 n1 þ x2 n2 þ þ x6 n6 P 0:045 x1 þ x 2 þ þ x 6 ¼ 1
xi P 0;
ð26Þ
i ¼ 1; 2; . . . ; 6:
Since model (26) is complex, and is difficult to convert it into crisp form, we use Methods 1 and 3 introduced in Section 4 to calculate the expected and semivariance values, respectively, and then imbed the calculation results into the GA (Huang, 2007) to find the optimal solution. Each time, we increase the a value by 0.000001. The parameters in the GA are set as follows: the population size is 30, the probability of crossover Pc = 0.4, the probability of mutation Pm = 0.3, and the parameter in the rank-based evaluation function m = 0.05. A run of the algorithm with 1000 generations shows that to minimize semivariance at the constraint that the expected return should not be lower than 4.5%, the investor should allocate his/her money according to Table 4. The minimum semivariance value is 0.0036.
6. Conclusion In this paper, we have discussed a portfolio selection problem in which security returns are given mainly by experts’ estimations. Two new mean–variance and mean–semivariance models have been given, and a hybrid intelligent solution algorithm has been provided. To illustrate the application of the models, we presented a method for determining the uncertainty distributions of the security returns based on experts’ evaluations, and provided two examples. The results of the examples have shown that the algorithm is good for solving the new models. Acknowledgments This work was supported by National Natural Science Foundation of China Grant Nos. 70871011 and 71171018, Program for New Century Excellent Talents in University, and the Fundamental Research Funds for the Central Universities. References Carlsson, C., & Fullér, R. (2001). On possibilistic mean value and variance of fuzzy numbers. Fuzzy Sets and Systems, 122, 315–326. Carlsson, C., Fullér, R., & majlender, P. (2002). A possibilistic approach to selecting portfolios with highest utility score. Fuzzy Sets and Systems, 131, 13–21.
X. Huang / Expert Systems with Applications 39 (2012) 5887–5893 Chen, J. S., Hou, J. L., Wu, S. M., & Chang-Chien, Y. W. (2009a). Constructing investment strategy portfolios by combination genetic algorithms. Expert Systems with Applications, 36, 3824–3828. Chen, L. H., & Huang, L. (2009). Portfolio optimization of equity mutual funds with fuzzy return rates and risks. Expert Systems with Applications, 36, 3720–3727. Chen, G., Liao, X., & Wang, S. (2009b). A cutting plane algorithm for MV portfolio selection model. Applied Mathematics and Computation, 215, 1456–1462. Chen, X. W., & Liu, B. (2010). Existence and uniqueness theorem for uncertain differential equations. Fuzzy Optimization and Decision Making, 9, 69–81. Choobineh, F., & Branting, D. (1986). A simple approximation for semivariance. European Journal of Operational Research, 27, 364–370. Deng, X. T., Li, Z. F., & Wang, S. Y. (2005). A minimax portfolio selection strategy with equilibrium. European Journal of Operational Research, 166, 278–292. Gao, X. (2009). Some properties of continuous uncertain measure. International Journal of Uncertainty. Fuzziness and Knowledge-Based Systems, 17, 419–426. Grootveld, H., & Hallerbach, W. (1999). Variance vs downside risk: Is there really that much difference? European Journal of Oprational Research, 114, 304–319. Huang, X. (2007). Portfolio selection with fuzzy returns. Journal of Intelligent and Fuzzy Systems, 18, 383–390. Huang, X. (2008). Mean–semivariance models for fuzzy portfolio selection. Journal of Computational and Applied Mathematics, 217, 1–8. Huang, X. (2011). Mean-risk model for uncertain portfolio selection. Fuzzy Optimization and Decision Making, 10, 71–89. Kaplan, P. D., & Alldredge, R. H. (1997). Semivariance in risk-based index construction: Quantidex Global Indexes. The Journal of Investing, 6, 82–87. Li, X., Qin, Z. F., & Ka, S. (2010). Mean–variance–skewness model for portfolio selection with fuzzy returns. European Journal of Operational Research, 202, 239–247. Liu, [5] B. (2007). Uncertainty Theory (2nd ed.). Berlin: Springer-Verlag [Chapter 5]. Liu, B. (2008). Fuzzy process, hybrid process and uncertain process. Journal of Uncertain Systems, 2, 3–16. Liu, B. (2009). Some research problems in uncertainty theory. Journal of Uncertain Systems, 3, 3–10.
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Liu, B. (2010). Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty. Berlin: Springer-Verlag [Chapter 1]. Markowitz, H. (1952). Portfolio selection. Journal of Finance, 7, 77–91. Markowitz, H. (1993). Computation of mean–semivariance efficient sets by the critical line algorithm. Annals of Operations Research, 45, 307–317. Oh, K. J., Kim, T. Y., Min, S. H., & Lee, H. Y. (2006). Portfolio algorithm based on portfolio beta using genetic algorithm. Expert Systems with Applications, 30, 527–534. Peng, Z., & Iwamura, K. (2010). A sufficient and necessary condition of uncertainty distribution. Journal of Interdisciplinary Mathematics, 13, 277–285. Prakash, A. J., Chang, C. H., & Pactwa, T. E. (2003). Selecting a portfolio with skewness: Recent evidence from US, European, and Latin American equity markets. Journal of Banking & Finance, 27, 1375–1390. Qin, Z. F., Li, X., & Ji, X. (2009). Portfolio selection based on fuzzy cross-entropy. Journal of Computational and Applied Mathematics, 228, 139–149. Simkowitz, M., & Beedles, W. (1978). Diversification in a three moment world. Journal of Financial and Quantitative Analysis, 13, 927–941. Soleimani, H., Golmakani, H. R., & Salimi, M. H. (2009). Markowitz-based portfolio selection with minimum transaction lots, cardinality constraints and regarding sector capitalization using genetic algorithm. Expert Systems with Applications, 36, 5058–5063. Tanaka, H., Guo, P., & Türksen, I. B. (2000). Portfolio selection based on fuzzy probabilities and possibility distributions. Fuzzy Sets and Systems, 111, 387–397. You, C. (2009). Some convergence theorems of uncertain sequences. Mathematical and Computer Modelling, 49, 482–487. Zadeh, L. (1965). Fuzzy sets. Information and Control, 8, 338–353. Zhang, X., Zhang, W. G., & Cai, R. (2010). Portfolio adjusting optimization under credibility measures. Journal of Computational and Applied Mathematics, 234, 1458–1465. Zhu, Y. (2010). Uncertain optimal control with application to a portfolio selection model. Cybernetics and Systems, 41, 535–547.
Contents lists available at SciVerse ScienceDirect
Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Mean–variance models for portfolio selection subject to experts’ estimations Xiaoxia Huang ⇑ School of Economics and Management, University of Science and Technology Beijing, Beijing 100083, China
a r t i c l e
i n f o
Keywords: Portfolio selection Mean–variance model Mean–semivariance model Uncertain programming Uncertain variable
a b s t r a c t Since the security market is complex, sometimes the future security returns are available mainly based on experts judgements. This paper discusses a portfolio selection problem in which security returns are given subject to experts’ estimations. The use of uncertain measure is justified, and two new mean–variance and mean–semivariance models are proposed. In addition, a hybrid intelligent algorithm for solving the optimization models is given. To illustrate the application of the new models, the method to obtain the uncertainty distributions of the security returns based on experts’ evaluations is given, and two selection examples are provided. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Portfolio selection is concerned with optimal allocation of one’s capital to a number of securities such that one can obtain the maximum investment return with risk control. Since the introduction of mean–variance models by Markowitz (1952), variance has been a very popular risk measure. In his models, Markowitz proposed that expected return could be regarded as the investment return and variance the risk. It was proposed that for a given level of variance, an optimal portfolio could be obtained when the expected return was maximized; or for a given expected return, the optimal portfolio could be obtained when the variance value was minimized. Since Markowitz, numerous models and algorithms have been proposed for portfolio selection based on the risk measurement of variance, e.g. papers by Deng, Li, and Wang (2005), Oh, Kim, Min, and Lee (2006), Chen, Hou, Wu, and Chang-Chien (2009a) and Soleimani, Golmakani, and Salimi (2009), etc. Since variance considers high return extremes as equally undesirable as low return extremes, variance will become an unreasonable risk measure when security returns are asymmetrical. Thus, semivariance was proposed to replace variance as an improvement of risk measure, and many scholars such as Choobineh and Branting (1986), Markowitz (1993), Kaplan and Alldredge (1997) and Grootveld and Hallerbach (1999) researched the properties and computation problem of mean–semivariance models. In these studies, security returns were all assumed to be random variables. However, the security market is complex, and randomness is not the only type of uncertainty. It has been found that sometimes security returns cannot be well reflected by historical data. The prediction of returns in the situation relies heavily on ⇑ Corresponding author. Tel.: +86 10 82376260; fax: +86 10 62333582. E-mail address: [email protected] 0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.11.119
experts’ estimation and contains much subjective imprecision rather than randomness. Therefore, many scholars argued that we should find another way to model people’s imprecise estimation of the returns other than random variables. With the introduction of fuzzy set theory by Zadeh (1965), scholars have tried employing fuzzy numbers and fuzzy set theory to manage portfolio since 1990s. Much work has been done on extending the mean–variance selection idea to fuzzy environment in different ways. For example Tanaka, Guo, and Türksen (2000) quantified mean and variance of a portfolio through fuzzy probability. Carlsson, Fullér, and majlender (2002) used their definition of mean and variance of fuzzy numbers (Carlsson & Fullér, 2001) to find the optimum portfolio. Chen, Liao, and Wang (2009b) discussed a cutting plane algorithm for a possibilistic mean–variance model with trapezoidal fuzzy returns, and Chen and Huang (2009) proposed a portfolio selection model for equity mutual funds with triangular fuzzy returns. Based on credibility measure Huang (2007) proposed two mean–variance models for fuzzy portfolio selection Qin, Li, and Ji (2009) developed mean– variance-cross entropy models, and Zhang, Zhang, and Cai (2010) discussed an adjusting problem for an existing portfolio. To deal with portfolios with fuzzy asymmetrical security returns, Huang (2008) proposed credibilistic mean–semivariance models, and Li, Qin, and Ka (2010) developed mean–variance–skewness models. These models developed portfolio selection theory and broadened the way to deal with securities whose returns are subject to experts’ estimations. However, as we research the problem deeper, we find that paradoxes will appear if we use fuzzy variables to describe the subjective estimations of security returns. For example, if a security return is regarded as a fuzzy variable, then we should have a membership function to characterize it. Suppose it is a triangular fuzzy variable n = (0.4, 0.3, 1.0) (see Fig. 1). Based on the membership function, it is known from possibility theory or credibility theory that the return is exactly 0.3 with belief degree
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X. Huang / Expert Systems with Applications 39 (2012) 5887–5893
(i) (Normality) MfCg ¼ 1: (ii) (Self-duality) MfKg þ MfKc g ¼ 1: (iii) (Countable subadditivity) For every countable sequence of events {Ki}, we have
( 1 [
M
)
Ki
6
i¼1
Fig. 1. Membership function of a security return n = (0.4, 0.3, 1.0).
1 in possibility measure or 0.5 in credibility measure. However, this conclusion is unacceptable because the belief degree of exactly 0.3 is almost zero. In addition, the return being exactly 0.3 and not exactly 0.3 have the same belief degree in either possibility measure or credibility measure, which implies that the two events will happen equally likely. This conclusion is quite astonishing and hard to accept. Recently Liu (2007) proposed an uncertain measure and developed an uncertainty theory which can be used to handle subjective imprecise quantity. Much research work has been contributed to the development of uncertainty theory and related theoretical work. For example Gao (2009) studied some properties of continuous uncertain measure. You (2009) proved some convergence theorems of uncertain sequences. Peng and Iwamura (2010) proved a sufficient and necessary condition of uncertainty distribution. Liu (2008) defined uncertain process, and Chen and Liu (2010) proved the existence and uniqueness theorem for uncertain differential equations, etc. When we use uncertainty theory to model subjective estimations of security returns, the above mentioned paradoxes will disappear immediately. Based on uncertain measure Zhu (2010) solved an optimal control problem of portfolio selection, and Huang (2011) proposed a mean-risk portfolio selection method. In this paper, we will follow Markowitz’s mean–variance idea and use uncertainty theory to help select portfolios whose returns are given mainly by experts’ estimations. As there do exist cases that security returns are asymmetrical (Prakash, Chang, & Pactwa, 2003; Simkowitz & Beedles, 1978), we will further define semivariance and develop a mean–semivariance selection model. The rest of the paper is organized as follows. For better understanding of the paper, some necessary knowledge about uncertain variable will be introduced in Section 2. Then, mean–variance and mean–semivariance models will be proposed in Section 3. After that, a hybrid intelligent algorithm will be presented for solving the proposed model problems in Section 4. To illustrate the application of the new models, the method for determining the uncertainty distributions of security returns based on experts’ estimations will be given in Section 5, and two selection examples will be provided in the same section. Finally, conclusion remarks will be given in Section 6. 2. Fundamentals of uncertainty theory Liu (2007) in 2007 proposed an uncertain measure and further developed an uncertainty theory which is an axiomatic system of normality, self-duality, countable subadditivity and product measure. Definition 1 Liu, 2007. Let C be a nonempty set, and L a r-algebra over C. Each element K 2 L is called an event. A set function MfKg is called an uncertain measure if it satisfies the following three axioms:
1 X
MfKi g:
i¼1
The triplet ðC; L; MÞ is called an uncertainty space. In order to define product uncertain measure Liu (2009) proposed the fourth axiom as follows: (iv) (Product measure) Let ðCk ; Lk ; Mk Þ be uncertainty spaces for k = 1, 2, . . . , n. The product uncertain measure is M ¼ M1 ^ M2 ^ ^ Mn :
Definition 2 Liu, 2007. An uncertain variable is a measurable function n from an uncertainty space ðC; L; MÞ to the set of real numbers, i.e., for any Borel set B of real numbers, the set
fn 2 Bg ¼ fc 2 CjnðcÞ 2 Bg is an event. Theorem 1 Liu, 2010. Any uncertain measure M is increasing, i.e., for any events K1 K2, we have
MfK1 g 6 MfK2 g: In application, a random variable is usually characterized by a probability density function or probability distribution function. Similarly, an uncertain variable can be characterized by an uncertainty distribution function. Definition 3 Liu, 2007. The uncertainty distribution U : R ! ½0; 1 of an uncertain variable n is defined by
UðtÞ ¼ Mfn 6 tg: For example, by a normal uncertain variable, we mean the variable that has the following normal uncertainty distribution
UðtÞ ¼ 1 þ exp
pðl tÞ pffiffiffi 3r
1 t 2 R;
;
where l and r are real numbers and r > 0. For convenience, it is denoted in the paper by n N ðl; rÞ. We call an uncertain variable a lognormal uncertain variable if it has the following lognormal uncertainty distribution
UðtÞ ¼ 1 þ exp
1
pðl ln tÞ pffiffiffi 3r
;
t 2 R;
where l and r are real numbers and r > 0. For convenience, it is denoted in the paper by n LOGN ðl; rÞ. When the uncertain variables n1, n2, . . . , nn are represented by uncertainty distributions, the operational law can be expressed as follows: Theorem 2 Liu, 2010. Let n be an uncertain variable with continuous uncertainty distribution U, and let f be a strictly increasing function. Then the uncertainty distribution W of f(n) can be obtained via
WðtÞ ¼ Uðf 1 ðtÞÞ
ð1Þ
which can also be expressed by
W1 ðaÞ ¼ f ðU1 ðaÞÞ;
0 < a < 1:
ð2Þ
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X. Huang / Expert Systems with Applications 39 (2012) 5887–5893
To tell the size of an uncertain variable, Liu defined the expected value of uncertain variables. Definition 4 Liu, 2007. Let n be an uncertain variable. Then the expected value of n is defined by
E½n ¼
Z
1
Mfn P tgdt
Z
0
0
Mfn 6 tgdt
ð3Þ
1
provided that at least one of the two integrals is finite. It can be calculated that the expected value of the normal uncertain variable n N ðl; rÞ is E[n] = l, and the expected value of the variable n LOGN ðl; rÞ is pffiffiffi lognormal puncertain ffiffiffi pffiffiffi E½n ¼ 3r expðlÞ cscð 3rÞ if r < p= 3:. Theorem 3 Liu, 2010. Let n be an uncertain variable with continuous uncertainty distribution U. If its expected value exists, then
E½n ¼
Z
1
U1 ðaÞda:
2
V½n ¼ E½ðn eÞ :
ð5Þ
Let n be an uncertain variable with continuous uncertainty distribution U. Then þ1
0
6e ¼
Mfðn eÞ2 g P tdt ¼
Z
þ1
Mfðn P e þ
pffiffi tÞ [ ðn
0
Z pffiffi t Þgdt 6
þ1
ðMfn P e þ
pffiffi pffiffi t g þ Mfn 6 e tgÞdt
0
Z
þ1
ð1 Uðe þ
pffiffi pffiffi t Þ þ Uðe tÞÞdt
¼
þ1
2ðt eÞð1 UðtÞ þ Uð2e tÞÞdt:
e
In this case, it is always assumed that the variance is
V½n ¼ 2
> > > > > > :
E½x1 n1 þ x2 n2 þ þ xn nn P b x1 þ x 2 þ þ x n ¼ 1
xi P 0;
ð8Þ
i ¼ 1; 2; . . . ; n
Z
3.2. Mean–semivariance models When we employ the uncertain mean–variance models (7) and (8), it implies that the uncertain security returns are symmetrical. Since when eliminating variance value, high return deviations from the expected return are eliminated equally as low return deviations from the expected returns, yet high return deviation implies the potential high return of the investment and is what the investors welcome. Thus, when security returns are asymmetrical, variance becomes an unreasonable risk measure. To measure only the low deviations from the expected return, we define the semivariance for uncertain variables as follows: Definition 6. Let n be an uncertain variable with finite expected value e. Then the semivariance of n is defined by SV[n] = E [[(n e)]2], where
0
Z
8 min V½x1 n1 þ x2 n2 þ þ xn nn > > > > > > < subject to :
where b denotes the minimum return that the investor can accept.
Definition 5 Liu, 2007. Let n be an uncertain variable with finite expected value e. Then the variance of n is defined by
Z
ð7Þ
where E denotes the expected value operator, V the variance operator of the uncertain variables, and a the preset maximum tolerable risk level. When the investor presets an expected return level that he/she feels satisfactory, and wants to minimize the risk, the selection model becomes:
ð4Þ
0
V½n ¼
8 max E½x1 n1 þ x2 n2 þ þ xn nn > > > > > > < subject to : V½x1 n1 þ x2 n2 þ þ xn nn 6 a > > > x > 1 þ x2 þ þ xn ¼ 1 > > : xi P 0; i ¼ 1; 2; . . . ; n
ðn eÞ ¼
0;
þ1
ðt eÞð1 UðtÞ þ Uð2e tÞÞdt:
ð6Þ
e
It can be calculated that the variance value of a normal uncertain variable n N ðl; rÞ is V[n] = r2.
n e; if n 6 e
ð9Þ
if n > e:
When the uncertain variable n has continuous uncertainty distribution U, then
SV½n ¼ 3. Uncertain portfolio selection models
¼
Z
þ1
Z0 e
Mfððn eÞ Þ2 P tgdt ¼
Z
þ1
Mfn 6 e
pffiffi t gdt
0
2ðe tÞUðtÞdt:
ð10Þ
1
3.1. Mean–variance models As discussed in introduction, when security returns are given mainly by experts’ estimations, it is better to use uncertain variables to describe the security returns. Let xi denote the investment proportions in securities i, and ni the uncertain returns of the ith securities given by experts. The returns are defined as ni ¼ ðp0i þ di pi Þ= pi ; i ¼ 1; 2; . . . ; n; respectively, where p0i are the estimated closing prices of the securities i in a future time, pi the closing prices of the securities i at present, and di the estimated dividends of the securities i during the period. According to Markowitz’s mean–variance idea, when making investment decision, the investor should strike a balance between maximizing the return and minimizing the risk. Then in case when the investor pre-gives a tolerable level of risk, and wants to maximize the expected return, we have the mean–variance selection model as follows:
Thus, in the situation that the security returns are asymmetrical, when the investor wants to maximize the expected return at a given specific level of risk, the uncertain mean–semivariance model can be built as follows:
8 max E½x1 n1 þ x2 n2 þ þ xn nn > > > > > > < subject to : > > > > > > :
SV½x1 n1 þ x2 n2 þ þ xn nn 6 u
ð11Þ
x1 þ x 2 þ þ x n ¼ 1 xi P 0;
i ¼ 1; 2; . . . ; n
where SV is the semivariance operator of the uncertain variable, and u the maximum semivariance level the investor can tolerate. When the investor wants to minimize the risk for a given level of expected return w, the mean–semivariance model is as follows:
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X. Huang / Expert Systems with Applications 39 (2012) 5887–5893
8 min SV½x1 n1 þ x2 n2 þ þ xn nn > > > > > > < subject to : E½x1 n1 þ x2 n2 þ þ xn nn P w > > > x > 1 þ x2 þ þ xn ¼ 1 > > : xi P 0; i ¼ 1; 2; . . . ; n:
when security returns are symmetrical, according to Theorem 2, we can obtain the expected value directly via
ð12Þ
" n X
E
# xi ni ¼ W1 ð0:5Þ ¼
i¼1
n X
xi U1 i ð0:5Þ:
ð14Þ
i¼1
4.3. Calculating variance via inverse uncertainty distribution 4. Hybrid intelligent algorithm Since uncertain mean–variance models (7) and (8) and mean– semivariance models (11) and (12) are complex, it is difficult to use analytical method to solve them. Genetic algorithm (GA) is a good method for solving complex optimization problems and has solved many complex portfolio selection problems (Chen et al., 2009a; Huang, 2007). Here, we introduce a method for calculating expected value, variance and semivariance values and integrate the results into the GA (Huang, 2007) to produce a hybrid intelligent algorithm for solving the new model problems in general cases. The introduction of the algorithm is as follows.
Pn
From Eq. (6), we know that the variance of the portfolio return can be calculated via
i¼1 xi ni
V
" n X
#
xi ni ¼ 2
Z
þ1
ðt eÞð1 WðtÞ þ Wð2e tÞÞdt:
e
i¼1
Since when we employ mean–variance models to select the optimal portfolio, the uncertainty distributions of the securities should be symmetrical, we have
2
Z
þ1
ðt eÞð1 WðtÞ þ Wð2e tÞÞdt
e
¼4
4.1. Obtaining inverse uncertainty distribution of portfolio return
Z
þ1
ðt eÞð1 WðtÞÞdt:
ð15Þ
e
Let Ui be the continuous uncertainty distributions of uncertain security returns ni, i = 1, 2, . . . , n, respectively, and W the uncertainty distribution of the uncertain portfolio return n X
Thus, we design the process for computing the variance value
V
" n X
# xi ni
i¼1
xi n i :
as follows:
i¼1
It is clear that W is also continuous. Since the investment proportions xi are non-negative numbers, then, according to Theorem 2, the uncertainty distribution W of the uncertain portfolio return can be obtained via
W1 ðaÞ ¼
n X
xi U1 i ðaÞ
i¼1
where a 2 [0, 1].
Method 2: Step 1. Set v = 0. P Step 2. set e ¼ ni¼1 xi U1 i ð0:5Þ: P Step 3. Set b ¼ ni¼1 xi U1 i ð0:9999Þ: Step 4. Randomly generate an a value from (0.5, 0.9999). P n 1 Step 5. Let v ¼ v þ i¼1 xi Ui ðaÞ e ð1 aÞ: Step 6. Repeat the 4th to 5th steps for N times, where N is a sufficiently large number. Step 7. Return v = v 4(b e)/N.
4.2. Calculating expected value via inverse uncertainty distribution 4.4. Calculating semivariance via inverse uncertainty distribution With the inverse uncertainty distribution W1 of the uncertain portfolio return, according to Theorem 3, the expected value of the P uncertain portfolio return ni¼1 xi ni can be calculated by
E
" n X
#
xi ni ¼
i¼1
9999 X
n X
j¼1
i¼1
1 i ð0:0001jÞ=9999:
xi U
ð13Þ
Thus, we design the process for computing the expected value
E
" n X
# xi ni
Pn
SV
Method 1: Step 1. Set e = 0. Step 2. Set j = 1. P Step 3. Let yj ¼ ni¼1 xi U1 i ð0:0001jÞ: Step 3. Let e = e + yj. Step 4. If j < 9999, let j = j + 1. Then turn back to Step 3. Step 5. Return e = e/9999. Please note that when we employ mean–variance models to select the optimal portfolio, the uncertainty distributions of the security returns should be symmetrical. Then, it is clear that in this case the expected return should be the value of W1(0.5). Thus,
" n X
#
xi ni ¼ 2
Z
e
ðe tÞWðtÞdt
ð16Þ
1
i¼1
where e is the expected return of the uncertain portfolio. Thus, we design the process for computing the semivariance value
i¼1
as follows:
According to Eq. (10), the semivariance of the portfolio return can be calculated via
i¼1 xi ni
SV
" n X
# xi ni
i¼1
as follows: Method 3: Step 1. Set sv = 0. P Pn 1 Step 2. Set e ¼ 9999 j¼1 i¼1 xi Ui ð0:0001jÞ=9999: Step 3. Use bisection search method to find the b value such that Pn 1 i¼1 xi UP i ðbÞ ¼ e: Step 4. Set a ¼ ni¼1 xi U1 i ð0:0001Þ: Step 5. Randomly generate an a value from (0.0001, b). P Step 6. Let sv ¼ sv þ e ni¼1 xi U1 i ðaÞ a: Step 7. Repeat the 5th to 6th steps for N times, where N is a sufficiently large number.
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X. Huang / Expert Systems with Applications 39 (2012) 5887–5893
Step 8. Return sv = sv 2(e a)/N. Please note that in all the three methods, when using inverse uncertainty distribution function, we let U value, i.e., a value, increase 0.0001 each time. In fact, this number can also be 0.00001 or even smaller according to the precision requirement of the problem. 4.5. Hybrid intelligent algorithm When the expected, variance and semivariance values of portfolio returns have been calculated, we can integrate the calculation results into the GA presented in paper (Huang, 2007) to find the optimal portfolio. Below is the representation structure of the solutions by the chromosomes and the summary of the hybrid intelligent algorithm. For the detailed description of the GA, the interested readers can refer to paper (Huang, 2007). Representation structure: The solution x = (x1, x2, . . . , xn) is represented by the chromosome C = (c1, c2, . . . , cn), where the genes c1, c2, . . . , cn are restricted in the interval [0, 1]. The mapping between a chromosome and a solution has the following form,
xi ¼
ci ; c1 þ c2 þ þ cn
i ¼ 1; 2; . . . ; n
ð17Þ
which ensures that x1 + x2 + + xn = 1 always holds. Hybrid intelligent algorithm: Step 1. Determine the parameters of the GA, i.e., the population size pop_size, the parameter in the rank-based evaluation function m, the probability of crossover operation Pc, and the probability of mutation operation Pm. Step 2. Initialize pop_size feasible chromosomes. Use the proposed Methods 1–3 to calculate the constraint values and check the constraint. Step 3. Use the proposed Methods 1–3 to calculate the objective values for all the chromosomes. Step 4. Give the rank order of the chromosomes according to the objective values, and compute the values of the rankbased evaluation functions of the chromosomes. Step 5. Compute the fitness of each chromosome according to the rank-based-evaluation function. Step 6. Select the chromosomes by spinning the roulette wheel. Step 7. Update the chromosomes by crossover and mutation operations. Use the proposed Methods 1–3 to calculate the constraint values when checking the constraint. Step 8. Repeat the third to the seventh steps for a given number of cycles. Step 9. Take the best chromosome as the solution of portfolio selection.
evaluate the occurrence degree of a series of uncertain return events. Independently, m sets of estimation data for the security return are given by experts. That is, from the experts the data set of
ðt 1 ; a1=j Þ; ðt2 ; a2=j Þ; . . . ; ðt k ; ak=j Þ;
j ¼ 1; 2; . . . ; m
were obtained, where
t1 < t2 < < tk ;
0 6 a1=j 6 a2=j 6 ak=j 6 1:
Since the experts were regarded equally knowledgable, we allocate the same weight to each expert’s estimation and aggregate the m experts’ estimations about the occurrence degree of a return event n 6 ti as follows:
Uðt i Þ ¼ ai ¼
m 1 X ai=j ; m j¼1
i ¼ 1; 2; . . . ; k:
ð18Þ
Then we get a new set of data for the security return as follows:
ðt 1 ; a1 Þ; ðt 2 ; a2 Þ; . . . ; ðt k ; ak Þ;
t1 < t2 < < tk ;
0 6 a1 6 a2
6 ak 6 1; Supposed it is believed that the security return is a normal uncertain variable N ðl; rÞ: Then we need to find the parameters l and r. To find the parameters, we adopt the principle of least squares proposed by Liu (2010) which says that the unknown parameter h of a known continuous uncertainty distribution form U(tjh) can be obtained via solution of either of the following two minimization problems
min h
k X ðUðti jhÞ ai Þ2 i¼1
and
min h
k X ðU1 ðai jhÞ t i Þ2 : i¼1
In our case one, since the uncertainty distribution of the security return is continuous, we can solute either
min l;r
k X ðUðti jl; rÞ ai Þ2
ð19Þ
i¼1
where
Uðt i jl; rÞ ¼ 1 þ exp
1
pðl ti Þ pffiffiffi 3r
;
or
min l;r
k X ðU1 ðai jl; rÞ t i Þ2
ð20Þ
i¼1
where
pffiffiffi 3r
ai
5. Two examples
U1 ðai jl; rÞ ¼ l þ
In order to illustrate the application of the mean–variance and mean–semivariance models and the effectiveness of the proposed algorithm, examples are provided in two cases when security returns are symmetrically and asymmetrically distributed. In the examples, security returns can not be well reflected by the historical data and are given by experts’ evaluations. The investor wants to select portfolios from 6 individual securities in each case. In the following, without losing generality, we will first introduce via one security the method for determining the uncertainty distribution of return based on experts’ evaluations, and then select portfolios in the above mentioned two cases. To evaluate the uncertainty distribution of the security return, first, questionnaires are given to m experts who are asked to
to find the parameters l and r. By solving the optimization problem (20), we can obtain the ^ and r ^ via the following equations estimated l
l^ ¼ t
p
ln
1 ai
pffiffiffi k ^ X 3r ai ln kp i¼1 1 ai
ð21Þ
and k k P P t pffiffiffi ln 1aia t i ln 1aia i i 3p k i¼1 i¼1 r^ ¼ 2 ; 2 3 P k k P ai ai ln 1a n ln 1a i¼1
i
i¼1
i
ð22Þ
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X. Huang / Expert Systems with Applications 39 (2012) 5887–5893
Pk 1
where t ¼ n i¼1 t i : If the experts think that the distribution of the security return is a lognormal uncertain variable LOGN ðl; rÞ; we obtain the ^ and r ^ by solving the following minimization problem estimated l
min l ;r
k X
ðUðti jl; rÞ ai Þ2
i¼1
1
1
2
3
4
5
6
Capital proportion
0.00
0.00
0.00
0.00
65.38
34.62
Table 3 Lognormal uncertain returns of 6 securities (%).
:
Example 1. Suppose in case one, the security returns are believed to be normal uncertain variables. Ten experts give their individual judgements about security returns independently. By using Eqs. (18)–(22), the distributions of security returns are obtained and given in Table 1. If the investor requires that the minimum expected return should not be less than 5.0% and wants to minimize the variance value, then according to the discussion in Section 3, the uncertain mean–variance model is built as follows:
8 min V½x1 n1 þ x2 n2 þ þ x6 n6 > > > > > > < subject to :
E½x1 n1 þ x2 n2 þ þ x6 n6 P 0:05
ð24Þ
x1 þ x 2 þ þ x 6 ¼ 1 xi P 0;
Security i
ð23Þ
ffiffi ti Þ where Uðt i jl; rÞ ¼ 1 þ exp pðlpln 3r
> > > > > > :
Table 2 Allocation of money to 6 securities with normal uncertain return rates (%).
i ¼ 1; 2; . . . ; 6
where E represents the expected value operator, V the variance of the uncertain variable, ni the ith uncertain security returns, and xi the investment proportions in the ith security. Since it has been proven (Liu, 2010) from Theorem 2 that if ni N ðli ; ri Þ; i ¼ 1; 2; . . . ; n; are normal uncertain variables, Pthen P g ¼P ni¼1 ki ni is also a normal uncertain variable g N ni¼1 ki li ; ni¼1 ki ri Þ for ki > 0, i = 1, 2, . . . , n. Then the model (24) can be converted into the following linear programming model: 8 min 0:058x1 þ 0:061x2 þ 0:073x3 þ 0:084x4 þ 0:065x5 þ 0:098x6 > > > > > > < subject to : 0:031x1 þ 0:033x2 þ 0:044x3 þ 0:051x4 þ 0:041x5 þ 0:067x6 P 0:05 > > > x1 þ x2 þ þ x6 ¼ 1 > > > : xi P 0; i ¼ 1;2;... ;6: ð25Þ
By solving the model (25), the optimal portfolio is given in Table 2. The objective value is 0.0764, which means the variance value of the selected portfolio is 0.0058. Example 2. Suppose in case two, the security prices are regarded to be lognormal uncertain variables. Ten experts give their individual judgements about the security prices independently. By using Eq. (18), data set about the security prices are got. Then by solving minimization problem (23), the distributions of the security prices are obtained. Next, by calculating ni ¼ ðp0i þ di pi Þ=pi ; the distributions of the security returns are available and are given in Table 3. Since the security returns are asymmetrical now, we use the mean–semivariance model to select the portfolio. If the investor requires that the minimum expected return should not be less than 4.5% and wants to minimize the risk value, then the portfolio selection model is built as follows:
Table 1 Normal uncertain returns of 6 securities (%). Security i
Uncertain return
Security i
Uncertain return
1 2 3
N (3.1, 5.8) N (3.3, 6.1) N (4.4, 7.3)
4 5 6
N (5.1, 8.4) N (4.1, 6.5) N (6.7, 9.8)
Security i
Uncertain return
Security i
Uncertain return
1 2 3
LOGN ð4:65; 16:60Þ 1 LOGN ð3:95; 7:66Þ 1 LOGN ð3:68; 18:10Þ 1
4 5 6
LOGN ð3:02; 5:84Þ 1 LOGN ð2:10; 3:30Þ 1 LOGN ð4:02; 5:08Þ 1
Table 4 Allocation of money to 6 securities with lognormal uncertain prices (%). Security i
1
2
3
4
5
6
Capital proportion
8.77
91.23
0.00
0.00
0.00
0.00
8 min SV½x1 n1 þ x2 n2 þ þ x6 n6 > > > > > > < subject to : > > > > > > :
E½x1 n1 þ x2 n2 þ þ x6 n6 P 0:045 x1 þ x 2 þ þ x 6 ¼ 1
xi P 0;
ð26Þ
i ¼ 1; 2; . . . ; 6:
Since model (26) is complex, and is difficult to convert it into crisp form, we use Methods 1 and 3 introduced in Section 4 to calculate the expected and semivariance values, respectively, and then imbed the calculation results into the GA (Huang, 2007) to find the optimal solution. Each time, we increase the a value by 0.000001. The parameters in the GA are set as follows: the population size is 30, the probability of crossover Pc = 0.4, the probability of mutation Pm = 0.3, and the parameter in the rank-based evaluation function m = 0.05. A run of the algorithm with 1000 generations shows that to minimize semivariance at the constraint that the expected return should not be lower than 4.5%, the investor should allocate his/her money according to Table 4. The minimum semivariance value is 0.0036.
6. Conclusion In this paper, we have discussed a portfolio selection problem in which security returns are given mainly by experts’ estimations. Two new mean–variance and mean–semivariance models have been given, and a hybrid intelligent solution algorithm has been provided. To illustrate the application of the models, we presented a method for determining the uncertainty distributions of the security returns based on experts’ evaluations, and provided two examples. The results of the examples have shown that the algorithm is good for solving the new models. Acknowledgments This work was supported by National Natural Science Foundation of China Grant Nos. 70871011 and 71171018, Program for New Century Excellent Talents in University, and the Fundamental Research Funds for the Central Universities. References Carlsson, C., & Fullér, R. (2001). On possibilistic mean value and variance of fuzzy numbers. Fuzzy Sets and Systems, 122, 315–326. Carlsson, C., Fullér, R., & majlender, P. (2002). A possibilistic approach to selecting portfolios with highest utility score. Fuzzy Sets and Systems, 131, 13–21.
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