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# A novel algorithm for uncertain portfolio selection

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a

b,c,*

### Chorng-Shyong Ong

a

a

Department of Information Management, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei 106, Taiwan

b

Institute of Management of Technology and Institute of Traﬃc and Transportation College of Management, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu 300, Taiwan

Taoyuan 338, Taiwan

Abstract

In this paper, the conventional mean–variance method is revised to determine the optimal portfolio selection under the uncertain situation. The possibilistic area of the return rate is ﬁrst derived using the possibisitic regression model. Then, the Mellin trans-formation is employed to obtain the mean and the risk by considering the uncertainty. Next, the revised mean–variance model is proposed to deal with the problem of uncer-tain portfolio selection. In addition, a numerical example is used to demonstrate the proposed method. On the basis of the numerical results, we can conclude that the pro-posed method can provide the more ﬂexible and accurate results than the conventional method under the uncertain portfolio selection situation.

*

Corresponding author.

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Keywords: Mean–variance method; Portfolio selection; Possibilistic regression; Mellin transfor-mation

1. Introduction

The mean–variance approach was proposed by Markowitz to deal with the portfolio selection problem [1]. A decision-maker can determine the optimal investing ratio to each security based on the sequent return rate. The formula-tion of the mean–variance method can be described as follows [1–3]:

min X n i¼1 Xn j¼1 rijxixj; s.t. X n i¼1 lixiP E; Xn i¼1 xi¼ 1; xiP0 8i ¼ 1; . . . ; n; ð1Þ

where li denotes the expected return rate of the ith security, rij denotes the

covariance coeﬃcient between the ith security and the jth security, and E denotes the acceptable least rate of the expected return.

It is clear that the accuracy of the mean–variance approach depends on the accurate value of the expected return rate and the covariance matrix. Several methods have been proposed to forecast the appropriate expected return rate and variance matrix such as the arithmetic mean method[1–3]and the regres-sion-based method [4]. However, these methods only derive the precise ex-pected return rate and covariance matrix and do not consider the problem of uncertainty. That is, since the decision-maker try to determine the optimal portfolio strategy to gain the maximum proﬁts in the future, how can we ignore the future uncertainty. We should highlight that the possible area of the return rate and the covariance matrix should be derived for the decision-maker to determine the future optimal portfolio selection strategy. In addition, these methods are based on the large sample theory and cannot obtain a satisfactory solution in the small sample situation[5].

In this paper, the possible area of the return rate and the covariance matrix are derived using the asymmetrical possibilistic regression. Then, the Mellin transformation is employed to calculate the uncertain return rate and the var-iance with the speciﬁc distribution. Finally, the optimal portfolio selection model can be reformulated based on the concepts above. In addition, a numer-ical example is used to illustrate the proposed method and compared with the

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conventional mean–variance method. On the basis of the simulated results, we can conclude that the proposed method can provide the better portfolio selec-tion strategy than the convenselec-tional mean–variance method by considering the situation of uncertainty.

The remainder of this paper is organized as follows. The possibilistic regres-sion model is discussed in Section 2. The Mellin transformation and the pro-posed method are presented in Section 3. A numerical example, which is used to illustrate the proposed method and compare with the mean–variance method, is in Section 4. The discussions of the numerical results are presented in Section 5 and the conclusions are presented in the last section.

2. Possibilistic regression

The possibilistic regression model was ﬁrst proposed by Tanaka and Guo[6]

to reﬂect the fuzzy relationship between the dependent and the independent variables. The upper and the lower regression boundaries are used in the pos-sibilistic regression to reﬂect the pospos-sibilistic distribution of the output values. By solving the linear programming (LP) problem, the coeﬃcients of the possi-bilistic regression can easy be obtained.

Next, we describe the possibilistic regression model[6]to obtain the uncer-tain return rate and the variance as follows. In order to exactly obuncer-tain the results, we extend the symmetrical fuzzy numbers to the asymmetrical fuzzy numbers. The general form of a possibilistic regression can be expressed as

y¼ A0þ A1x1þ    þ Anxn¼ A0x; ð2Þ

where Aiis a asymmetrical possibilistic regression coeﬃcient which is denoted

as (ai ciL, ai, ai+ ciR). In order to obtain the minimum degree of uncertainty,

the ﬁtness function of the possibilistic regression can be deﬁned as min

a;c J¼

X

j¼1;...;m

c0Ljxjj þ c0Rjxjj. ð3Þ

In addition, the dependent variable should be restricted to satisfy the follow-ing two equations:

yjP a0x

j c0Ljxjj; ð4Þ

yj6a0xjþ c0

Rjxjj. ð5Þ

On the basis of the concepts above, we can obtain the formulation of a pos-sibilistic regression model as follows:

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min a;c J¼ X j¼1;...;m c0Ljxjj þ c0Rjxjj; s.t. yjP a0x j c0Ljxjj; yj6a0xjþ c0 Rjxjj; j¼ 1; . . . ; m; cL;cRP0. ð6Þ

By solving the mathematical programming model above, we can obtain the uncertain return rate and the variance of the security with the speciﬁc distribu-tion in the future.

Next, we depict a graph, as shown inFig. 1, to describe the concept of the proposed method. Suppose we have six period return rates of stocks and we want to determine the optimal investing rate to each stock in period 7. Let the broken line denotes the trend of the return rate of a stock. Then, we can obtain the upper, the lower, and the center possibilistic regressions using Eq.

(6) to derive the possibilistic area of the return rate of period 7. Note that the triangular possibilistic distribution is used in this example. However, other possibilistic distributions can be employed using the same concepts.

We should highlight that the triangular area in period 7 because it denotes the distribution of the possible return rate and variance of the stock. That is, the decision-maker should incorporate the information above to determine the optimal investing rate to each stock. However, since the possibilistic area may be triangular, uniform, or other distributions, the problem is how to eﬃ-ciently and eﬀectively calculate the possible return rate and the variance. Next, the Mellin transformation is described to overcome this problem.

3. The Mellin transformation

Given a random variable, x2 R+, the Mellin transformation, M(s), of a probability density function (pdf) (f(x)) can be deﬁned as

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Mff ðxÞ; sg ¼ MðsÞ ¼ Z 1

0

fðxÞxs1dx. ð7Þ

Let h is a measurable function on R into R and Y = h(x) is a transformed random variable. Then, some properties of the Mellin transformation can be described as shown inTable 1. For example, if Y = ax then the scaling property can be expressed as Mff ðaxÞ; sg ¼ Z 1 0 fðaxÞxs1dx¼ as Z 1 0 fðaxÞðaxÞs1dx¼ asMðsÞ.

Next, let a continuous non-negative random variable, X, the nth moment of X can be deﬁned as

EðXnÞ ¼

Z 1 0

xnfðxÞ dx. ð8Þ

Then, by setting n = 1, the mean of X can be expressed as EðX Þ ¼

Z 1 0

xfðxÞ dx ð9Þ

and the variance of X can be calculated by r2x ¼ EðX2Þ  ½EðX Þ2

. ð10Þ

Since the relationship between the nth moment and the Mellin transforma-tion of X can be linked using the equatransforma-tion

EðXnÞ ¼

Z 1 0

xðnþ1Þ1fðxÞ dx ¼ Mff ðxÞ; n þ 1g; ð11Þ

the mean and the variance of X can be calculated by

EðX Þ ¼ Mff ðxÞ; 2g; ð12Þ

r2

x ¼ Mff ðxÞ; 3g  fMff ðxÞ; 2gg 2

. ð13Þ

From Eqs.(12) and (13), we can eﬃciently calculate the mean and the var-iance of any distribution using the Mellin transformation. In practice, the uni-form, the triangular and the trapezoidal distributions are usually used and their

Table 1

The properties of the Mellin transformation

Properties of Mellin transformation Y = h(x) M(s)

Scaling property ax asM(s)

Multiplication by xa xaf(x) M(s + a)

Rising to a real power f(xa) a1s

aÞ; ða > 0Þ Inverse x1f(x1) M(1s) Multiplication by ln x ln x f(x) d dsMðsÞ Derivative dk dskfðxÞ CðsÞ CðskÞ

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corresponding Mellin transformations can be summarized as shown inTable 2. More Mellin transformation for other probability density functions can refer to[7].

On the basis ofTable 2, we can eﬃciently derived the values of the mean and the variance respect to the speciﬁc distribution by calculating M(2) and M(3). Next, we can reformulate the conventional mean–variance method as shown in the following mathematical programming model to consider the impact of uncertainty: min X n i¼1 xixi ½Mið3Þ  Mið2Þ 2  þX n i¼1 Xn j¼1 xixjrij; s.t. X n i¼1 xiMið2Þ P E; Xn i¼1 xi¼ 1; xiP08i ¼ 1; . . . ; n; ð14Þ

where the ﬁrst part of the objective function denotes the next period risk of the security, the second part of the objective function denotes the unsystematic risk which is considered in the mean–variance model. Next, we use a numerical example to illustrate the proposed method and to compare with the conven-tional method.

4. A numerical example

For simplicity, the possibilistic area of the return rate is represented as the triangular form in this numerical example. Suppose the historical sequent re-turn rates of the ﬁve securities from periods t-6 to t-1 can be represented as shown inTable 3. The corresponding time chart of the ﬁve securities can also be depicted as shown inFig. 2. Our concern here is to obtain the optimal port-folio selection strategy in the next period t.

First, we use the conventional mean–variance model to obtain the optimal portfolio selection strategy. To do this, the arithmetic mean and the covariance matrix can be calculated as shown inTables 4 and 5.

Table 2

The Mellin transformation of three probability density functions

Distribution Parameters M(s) Uniform UNI(a, b) bs as sðbaÞ Triangular TRI(l, m, u) 2 ðulÞsðsþ1Þ½ uðusmsÞ ðumÞ  lðmslsÞ ðmlÞ Trapezoidal TRA(a, b, c, d) 2 ðcþdbaÞsðsþ1Þ ðdsþ1csþ1Þ ðdcÞ  bsþ1asþ1Þ ðbaÞ h i

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Let the acceptable least rate of the expected return is equal to its average re-turn rate, we can obtain the optimal portfolio selection using Eq.(1)as shown inTable 8. 0.0800 0.0900 0.1000 0.1100 0.1200 0.1300 0.1400 0.1500 0.1600 0.1700 0.1800 t-6 t-5 t-4 t-3 t-2 t-1 Period Return rate Security 1 Security 2 Security 3 Security 4 Security 5

Fig. 2. The time chart of the ﬁve securities. Table 3

The historical return rates of the ﬁve securities Return rate t-6 t-5 t-4 t-3 t-2 t-1 Security 1 0.1686 0.1117 0.1149 0.1293 0.1397 0.1406 Security 2 0.1330 0.1466 0.1741 0.1131 0.1022 0.1552 Security 3 0.1698 0.1528 0.1302 0.1471 0.1139 0.1177 Security 4 0.1750 0.1026 0.1543 0.1475 0.1158 0.1148 Security 5 0.1291 0.1192 0.1491 0.1318 0.1377 0.1450 Table 5

The covariance matrix

Security 1 Security 2 Security 3 Security 4 Security 5 Security 1 0.00036 Security 2 0.00017 0.00060 Security 3 0.00010 0.00000 0.00039 Security 4 0.00024 0.00004 0.00027 0.00066 Security 5 0.00000 0.00009 0.00014 0.00004 0.00010 Table 4

Arithmetic mean of the expected return

Security 1 2 3 4 5 Average

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Next, we use the proposed method to obtain the optimal portfolio selection as follows. In order to obtain the possibilistic area of the ﬁve securities, the pos-sibilistic regression is employed. Then, using the Mellin transformation we can obtain the forecasting mean and risk of the securities by considering the situ-ation of uncertainty as shown inTable 6.

Furthermore, we incorporate the information of the forecasting mean to derive the second part of the objective function in Eq.(14), i.e. the covariance matrix, as shown inTable 7.

Finally, with the same acceptable least rate of the expected return rate we can obtain the optimal portfolio selection under the uncertain situation using Eq. (14). The comparison of the conventional and the proposed method can be described as shown inTable 8.

FromTable 8, it can be seen that the main diﬀerence is the portfolio selec-tion in Securities 1 and 4. In the next secselec-tion, we will discuss the irraselec-tional rea-son using the conventional method in our numerical example.

Table 6

The possibilistic area, the mean and the variance

Security 1 Security 2 Security 3 Security 4 Security 5 Possibilistic area (0.1117,0.1117, 0.1578) (0.0868,0.1407, 0.1741) (0.0890,0.0890, 0.1241) (0.0646,0.0646, 0.1294) (0.1500,0.1500, 0.1737) Mean 0.1306 0.1339 0.1007 0.0862 0.1579 Variance 0.000118 0.000323 0.000068 0.000233 0.000031 Table 7

The new covariance matrix

Security 1 Security 2 Security 3 Security 4 Security 5 Security 1 0.00031 Security 2 0.00015 0.00052 Security 3 0.00010 0.00001 0.00051 Security 4 0.00023 0.00006 0.00046 0.00086 Security 5 0.00001 0.00007 0.00022 0.00010 0.00015 Table 8

The comparisons of the portfolio selections

Portfolio strategy 1 2 3 4 5 Return rate Portfolio risk Conventional 0.000 0.069 0.195 0.303 0.433 0.136 0.000056 Proposed method 0.136 0.070 0.141 0.118 0.535 0.136 0.000073

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5. Discussions

The mean–variance method is widely used in the ﬁnance area to deal with the portfolio selection problem. However, the conventional method does not consider the situation of future uncertainty and usually fails under the small sample situation. We can describe the shortcomings of the conventional method from its purpose and theory, respectively, as follows.

The purpose of the mean–variance approach is to determine the t period optimal investing rate to each security based on the historical sequent return rate. The key is to forecast the t period return rate as accurately as possible. However, it is clear that the arithmetic mean can only reﬂect the average states of the past return rate instead of the future. Although many regression-based methods have been proposed to overcome the problem, these methods must obey the assumption of the large sample theory and cannot be used in the small sample situation theoretically.

In addition, these methods cannot reﬂect the degree of uncertainty. Since we want to determine the optimal portfolio selection in the future, the information of future uncertainty should not be ignore in the model. In this paper, the pos-sibilistic regression model is employed to derive the possible mean and the var-iance in the future. Then, the Mellin transformation is used to obtain the mean and the risk in the future by considering the uncertain situation. Finally, we can use the information above to reformulate the mean–variance method to obtain the optimal uncertain portfolio selection.

In order to highlight the shortcoming of the conventional method and to compare it with the proposed method, a numerical example is used in Section 4. Now, we can depict the time chart of Securities 1 and 4 to describe the irra-tional results using the convenirra-tional method as shown inFig. 3.

From the time chart, it can be seen from Security 1 that there is an increase in the period of t-4 to t-1. It is rational to suppose Security 1 also has the

0.0800 0.0900 0.1000 0.1100 0.1200 0.1300 0.1400 0.1500 0.1600 0.1700 0.1800 t-6 t-5 t-4 t-3 t-2 t-1 Period R eturn rate Security1 Security 4

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positive return rate in the period t. On the other hand, Security 4 shows the decrease since the period t-4, the optimal portfolio selection should eliminate the investing ratio in Security 4. On the basis of the numerical results, we can conclude that it is irrational to determine the uncertain portfolio selection using the conventional method. On the other hand, the proposed method can accurately reﬂect the deduction above. In addition, the proposed method can provide the more ﬂexible portfolio alternatives. That is, a decision-maker can determine the optimal possibilistic distribution based on his domain knowledge or the empirical results to obtain the exactly portfolio selection strategy.

6. Conclusions

Portfolio selection problem has been a popular issue in the ﬁnance area since 1950s. However, the conventional mean–variance method can not provide the satisfactory solution under the uncertain portfolio selection and the small sam-ple situations. In this paper, the possibilistic regression model is employed to derive the possibilistic area of the future return rate. The Mellin transforma-tion, then, is used to obtain the mean and the risk by considering the uncer-tainty. Using the information above, we propose the revised mean–variance model which incorporates the degree of uncertainty to deal with the problem of portfolio selection. A numerical example is used to demonstrate the pro-posed method. On the basis of the numerical results, we can conclude that the proposed method can provide the more ﬂexible and accurate results than the conventional method under the uncertain portfolio selection situation.

References

[1] H. Markowitz, Portfolio selection, J. Finance 7 (1) (1952) 77–91.

[2] H. Markowitz, Portfolio Selection: Eﬃcient Diversiﬁcation of Investments, Wiley, New York, 1959.

[3] H. Markowitz, Mean–Variance Analysis in Portfolio Choice and Capital Market, Basil Blackwell, New York, 1987.

[4] E.J. Elton, M.J. Gruber, Modern Portfolio Theory and Investment Analysis, Wiley, New York, 1995.

[5] E.J. Elton, M.J. Gruber, T.J. Urich, Are betas best? J. Finance 33 (5) (1978) 1357–1384. [6] H. Tanaka, P. Guo, Possibilistic Data Analysis for Operations Research, Physica-Verlag, New

York, 2001.

[7] K.P. Yoon, A probabilistic approach to rank complex fuzzy numbers, Fuzzy Sets Syst. 80 (2) (1996) 167–176.

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