Discrete Dynamics in Nature and Society
Volume 2019 (2019), Article ID 6903679, 19 pages
https://doi.org/10.1155/2019/6903679

Research Article

New Safe Approximation of Ambiguous Probabilistic Constraints for Financial Optimization Problem

NingNing Du,¹ Yan-Kui Liu,² and Ying Liu¹

¹Risk Management & Financial Engineering Laboratory, College of Mathematics and Information Science, Hebei University, Baoding 071002, China
²Key Laboratory of Machine Learning & Computational Intelligence, College of Mathematics and Information Science, Hebei University, Baoding 071002, China

Correspondence should be addressed to Ying Liu; yingliu@hbu.edu.cn

Received 18 November 2018; Revised 25 February 2019; Accepted 4 April 2019; Published 2 May 2019

Academic Editor: Seenith Sivasundaram

Copyright © 2019 NingNing Du et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In financial optimization problem, the optimal portfolios usually depend heavily on the distributions of uncertain return rates. When the distributional information about uncertain return rates is partially available, it is important for investors to find a robust solution for immunization against the distribution uncertainty. The main contribution of this paper is to develop an ambiguous value-at-risk (VaR) optimization framework for portfolio selection problems, where the distributions of uncertain return rates are partially available. For tractability consideration, we deal with new safe approximations of ambiguous probabilistic constraints under two types of random perturbation sets and obtain two equivalent tractable formulations of the ambiguous probabilistic constraints. Finally, to demonstrate the potential for solving portfolio optimization problems, we provide a practical example about the Chinese stock market. The advantage of the proposed robust optimization method is also illustrated by comparing it with the existing optimization approach via numerical experiments.

1. Introduction

The core of modern portfolio theory is how to formulate a tradeoff between the return and risk, specifically to determine an effective investment strategy for allocating capital over a number of available assets so as to maximize the return while minimizing the risk. The first systematic approach to the problem of asset allocation, called mean-variance (MV) one, was developed by Markowitz [1]. To overcome the limitation of variance as a risk measure, some useful realistic risk measures were proposed to separate undesirable downside movements from desirable upside movements. In this respect, the interested reader may also refer to the absolute semi-deviation [2, 3], moment [4, 5], value-at-risk (VaR) [6–8], and conditional value-at-risk (CVaR) [9, 10].

The researches mentioned above need the precise estimates of the return and risk based on the explicit distribution extracted from considerable current market data. Due to the noise and outliers in the data, or the imperfect data of newly listed stocks, it is often impossible to determine the exact distributions of future returns [11]. As a result, distributionally robust optimization is a useful approach for decision under uncertainty, where uncertain model data are governed by a family of probability distributions. Using this approach, decision makers can find a robust solution for immunization against the distribution uncertainty.

The first research of distributionally robust optimization was proposed by Scarf [12] to solve a linear inventory problem with a min–max solution. Since then, the approach has been well developed in the literature [13–17]. Among them, Sun et al. [16] developed a distributionally robust joint chance constrained optimization model for a dynamic network design problem, and Wiesemann et al. [17] proposed a unifying framework for modeling and solving a distributionally robust multiproduct newsvendor problem.

In financial field, distributionally robust optimization method has proved useful in constructing the portfolio selection model. Shen and Zhang [18] proposed a robust optimization resolution to stochastic programming based on a scenario tree, with the system parameters being modeled to reside in ellipsoidal ambiguity sets. Natarajan et al. [19] proposed a computationally tractable approximation method for minimizing the VaR of a portfolio based on robust optimization techniques, where the uncertainty set ranged from polytopes to more advanced conic-representable uncertainty sets. Huang et al. [20] considered the relative robust conditional value-at-risk portfolio selection problem, where the underlying probability distribution of portfolio return belonged to a certain set. Pınar and ç [21] derived the distributionally robust portfolio rules, addressed potential uncertainty in the mean return vector, and derived a closed-form portfolio rule when the uncertainty in the return vector was modeled via an ellipsoidal uncertainty set. Liu et al. [22] adopted an uncertainty set in distributionally robust portfolio selection problems, where the considered uncertainties are in terms of both the distribution and the first two order moments.

The choice of uncertainty set is important for the tractability of distributionally robust portfolio selection model. Box uncertainty and ellipsoidal uncertainty are the most often used uncertainty structures in robust optimization formulations [23]. The obtained solution based on single uncertainty set like box is usually too conservative. In addition, when chance constraints exist in distributionally robust optimization model, the feasible region is usually nonconvex; thus it is difficult to solve such hard optimization model. In this case, a common technique is to find the safe approximation of ambiguous chance constraint under a given uncertainty set [13, 24].

A set of constraints is called a safe approximation of the chance constraint if the feasible set of the safe approximation is a subset of the feasible set of the chance constraint [25]. For the time being, there are several approximation methods for solving chance constraint problems, such as sample approximation [26–28], scenario approximation [29, 30], Bernstein approximation [31, 32], and robust counterpart approximation [33, 34]. To get a further understanding about the approximation method of chance constraint, the interested readers may refer to the following references. Ben-Tal and Nemirovski [35] developed a computationally tractable approximation of chance constrained linear matrix inequalities based on measure concentration results. Ding et al. [36] presented a tractable convex programming approximation for distributionally robust individual chance constrained problem under interval sets of mean and covariance information. Yang and Xu [37] derived sufficient conditions to guarantee tractability and investigated tractable approximations of joint probabilistic envelope constraints. Postek et al. [38] presented three new safe tractable approximations of chance constraints under partial information of increasing computational complexity and quality.

Among all the approximation methods of chance constraint, the method on approximating the ambiguous chance constraint with its robust counterpart has received an increasing attention recently [39]. Various robust counterpart optimization formulations are derived based on different types of uncertainty set. Ben-Tal et al. [13] discussed some robust counterpart approximations of chance constraints under box, ellipsoid, and the intersection of box and ellipsoid uncertainty sets. Yuan et al. [39] proposed a two-layer algorithm to investigate the tractable robust optimization approximation framework for solving the joint chance constrained problem under the intersection of interval & polyhedral and interval & ellipsoidal type uncertainty sets. Li et al. [40] investigated different types of uncertainty set for robust optimization, including box, ellipsoidal, polyhedral, interval & ellipsoidal, and interval & polyhedral. To the best of our knowledge, no discussion of the robust counterpart approximation is under ball & budget and ellipsoid & generalized budget uncertainty sets.

In this paper, we will study the distributionally robust portfolio selection problem from a new perspective, where the uncertain model parameters are described by two new combinations of perturbation sets of ball & budget and ellipsoid & generalized budget. The proposed safe approximation method incorporated the robust counterpart, can refine the approximation set, and further improve approximation capacity via changing the intersection mode of perturbation sets. For tractability consideration, the safe approximations of ambiguous chance constraint are also derived under the new combined perturbation sets.

Our contributions in this paper include the following several aspects:

We develop an ambiguous VaR optimization framework for distributionally robust linear optimization problem, where we optimize the worst case VaR over an ambiguity set. The obtained robust solution is capable of providing immunization against the distribution uncertainty.

To obtain tractable formulation, we approximate the ambiguous probabilistic constraints under two new combinations of perturbation sets: one is the intersection of ball with budget, and the other is the intersection of ellipsoid with generalized budget. Depending on the choice of two new ambiguity sets, the resulting framework of safe approximation includes a system of efficiently computable convex constraints, which can be solved efficiently by general-purpose commercial-grade solvers.

We demonstrate our approach for addressing a practical case about the Chinese stock market. To show the advantage of the proposed approach, we compare the proposed robust optimization approach with nominal distribution optimization approach via numerical experiments.

The paper is organized as follows: In Section 2, we develop an ambiguous VaR optimization framework for distributionally robust linear optimization problem. In Section 3, we approximate the ambiguous probabilistic constraints under two new combinations of perturbation sets. In Section 4, we demonstrate our approach for addressing a practical case study. Section 5 briefly concludes the paper. All proofs are given in the appendix.

2. Stochastic Optimization with Ambiguous Probabilistic Constraints

In this section, we consider a class of uncertain linear optimization problems of the form

where is the vector of decision variables, is a collection of deterministic constraints, and in the objective are uncertain, and the problem data are also uncertain.

In stochastic optimization, the uncertain model data are assumed to be random. In the simplest case, these random data follow known in advance probability distributions [41, 42], while in more advanced settings, these distributions are only partially known. In this case, all we know is that belongs to a prescribed family of distributions on the space of the data [43, 44].

In this section, we deal with uncertain model (1) by introducing risk levels , and enforcing the constraints in the probability sense. To make model (1) meaningful, we introduce an artificial variable , and we build the following VaR optimization problem of the form

When the distribution of random data is only partially known, we replace problem (2) with the following ambiguous VaR optimization problem

where the distribution of belongs to the distribution family , and the distribution of belongs to the family Model (3) is called an ambiguous probabilistic constraint model because we do not have any information about the probability distribution except for the fact that it belongs to a family

Model (3) is a semi-infinite programming problem with infinite probability constraints. It is usually difficult to evaluate with high accuracy the probability in the left hand side of (3), even in the case when is simple [45]. Due to the severe computational difficulties associated with ambiguous probabilistic constraints, a natural course of action is to replace the constraints with their computationally tractable safe approximations.

In the next section, we will deal with the scheme for new safe approximations of the ambiguous probabilistic constraints.

3. New Safe Approximation Methods

It has been well known that the safe approximations of chance constraints are highly dependent on the choice of the uncertainty set. In this section, we will introduce two kinds of new uncertainty sets.

For the sake of clarity, we assume that the uncertainty set is parameterized in an affine fashion by perturbation vector varying in a given perturbation set

Ben-Tal et al. [13] reported that the values of uncertain data are obtained from the nominal values of the data by random perturbation. The affine formula of the uncertainty set has been used in many works [25, 34, 38].

In the following, we approximate the ambiguous probabilistic constraints of (3) under two new combinations of perturbation sets: one is the intersection of ball with budget, and the other is the intersection of ellipsoid with generalized budget.

Case I (safe approximations under ball and budget perturbation set). We first address the perturbation set in (4), and (5) is the intersection of and balls, i.e.,

where and are given parameters. In this case, the perturbation set has the following conic representation:

where

(i) and , whence its dual conic ;(ii) and , whence its dual conic .

In the light of Remark 1.3.6 in [13], the semi-infinite constraint (4) can be represented by the system of conic constraints in variables :

Setting , with -dimensional , , and one-dimensional , , and then substituting into system (8), we have the following system of conic quadratic constraints in variables :

Similarly, in virtue of the conic representation of (5), the following system of conic constraints in variables can be obtained:

Eliminating from above two systems the variables for every feasible solution to these two systems, we have , and the solution obtained when replacing with is still feasible. The reduced system in variables reads

which is an equivalent representation of (4) and (6).

On the other hand, the following system

is an equivalent representation of (5) and (6).

Next we present a scheme for the safe approximation of ambiguous probabilistic constraints. For this purpose, suppose perturbation vectors included in (4) and (5) possess the following properties:

On the basis of the above analysis, we obtain the following observation.

Theorem 1. The RC of the uncertain linear constraint (4) with the perturbation set (6) is equivalent to the system of conic quadratic constraints (11)-(12).
In the case of (15), the component of every feasible solution to this system satisfies the randomly perturbed inequality (4) with probability of at least

Theorem 2. The RC of the uncertain linear constraint (5) with the perturbation set (6) is equivalent to the system of conic quadratic constraints (13)-(14).
In the case of (15), the component of every feasible solution to this system satisfies the randomly perturbed inequality (5) with probability of at least

Case II (safe approximations under ellipsoid and generalized budget perturbation set). Next, consider the perturbation set in (4) and (5) is the intersection of ellipsoid and generalized budget, i.e.,

where and are given parameters. In this case, the perturbation set has the following conic representation:

where

(i), with and , whence its dual conic ;(ii), with and , whence its dual conic .

Set with L-dimensional , , and one-dimensional , , where and In virtue of the conic representation of (4), the semi-infinite constraint (4) can be represented by the following system of conic constraints in variables :

Similarly, according to the conic representation of (5), the following system of conic constraints in variables can be obtained:

Eliminating from above two systems the variables for every feasible solution to these two systems, we have , and the solution obtained when replacing with is still feasible. The reduced system in variables reads

which is an equivalent representation of (4) and (16).

Furthermore, the following system

is an equivalent representation of (5) and (16).

Next we focus on a scheme for the safe approximation of ambiguous probabilistic constraints. For this purpose, we assume the perturbation vector affecting (4) and (5) possesses the following properties:

P.1 are mutually independent random variables;P.2the distributions of the components satisfy the following conditions

with known constants

In virtue of the above analysis, we obtain the following observations.

Theorem 3. Let the random perturbations affecting (4) obey ; the RC of the uncertain linear constraint (4) with the perturbation set (16) is equivalent to the system of conic quadratic constraints (20)-(21).
In addition, the component of every feasible solution to this system satisfies the randomly perturbed inequality (4) with probability of at least ,

Theorem 4. Let the random perturbations affecting (5) obey ; the RC of the uncertain linear constraint (5) with the perturbation set (16) is equivalent to the system of conic quadratic constraints (22)-(23).
In addition, the component of every feasible solution to this system satisfies the randomly perturbed inequality (5) with probability of at least ,

4. Financial Application

Portfolio selection is the problem of allocating capital over a number of available assets in order to maximize the return on the investment while minimizing the risk. In this section, we consider the situation that the distribution of random return is partially known and model the portfolio optimization problem with an ambiguous VaR method.

4.1. The Equivalent Conic Quadratic Financial Optimization Model

Suppose there exist stocks that can be chosen by an investor in the financial market. The goal is to determine the capital allocation among the stocks in order to maximize the VaR of the resulting portfolio. Let random vector denote uncertain return rates of the stocks, and denote the amount of the investments in the stocks. Each component of uncertain return rate vector is represented as

where is the nominal value of , and , , are the basic shifts data of . , , are independent random perturbations, and is varying in a given perturbation set Thus an ambiguous VaR portfolio optimization model can be built as

where is the family of potential probability distributions of random perturbation vector .

In model (26), the uncertain return rate of each stock is an affine function of bounded and mutually independent perturbation variables When historical data for stocks are available, some statistical tools such as principal component analysis and linear regression can be used to calibrate the values of from the data [46]. The perturbation set of provides the support information of , while the moment information of characterizes the random nature of the return . In this section, we consider the following two cases in our numerical experiments:

(i)Case I. Perturbation vector fluctuates in a ball and budget perturbation set depicted in (6), and the distribution family of , satisfies the case of (15).(ii)Case II. Perturbation vector fluctuates in an ellipsoid and generalized budget perturbation set depicted in (16), and the distribution family of , satisfies the assumptions

When fluctuates in a ball and budget perturbation set as depicted in Case I, it follows from Theorem 1 that ambiguous VaR model (26) is equivalent to the following conic quadratic optimization model:

On the other hand, when fluctuates in an ellipsoid and generalized budget perturbation set as depicted in Case II, by Theorem 3, ambiguous VaR model (26) is equivalent to the following conic quadratic optimization model:

The above conic quadratic optimization models (27) and (28) are established optimization formats supported by commercial solvers that scale well with the number of variables in the problem. In our numerical experiments, we employ the LINGO 11.0.0.20 solver to find the robust optimal portfolio. All the experiments are conducted in a personal computer with Inter(R) Core(TM) i5-4200U CPU 1.60GHz and RAM 4.00GB.

4.2. Data Description

We select ten stocks with different return rates from Chinese stock market on October 24, 2018, as the candidate. Each stock has a daily return rate , These data are obtained through stock software, named Straight Flush, which can provide stock market display, analysis, and trading. Since the stock return rates are affected by various factors, the daily return rate is varying in a segment around its daily expected return rate. The daily expected return rate is viewed as the nominal value of the random return rate . For instance, the return rate of stock 601116 takes its value in , where and represent the minimum value and the maximum value of the return rate, respectively, and its nominal value is Figure 1 shows the fluctuations of the daily return rate of stock 601116, from which it is not difficult to see that daily return rate fluctuates around the nominal value.

Figure 1: The return rate of stock 601116 on October 24, 2018.

The nominal values of ten return rates and the variation ranges of the return rates are summarized in Table 1. In the following, we will conduct some numerical experiments to find the optimal portfolio policy based on these data.

Table 1: Descriptive statistics for the return rates of 10 stocks.

Stock code
Nominal value (%)
Range of (%)

600408
0.33
[-1.68,4.72]

601116
1.20
[-0.03,2.32]

000750
1.93
[-1.52,10.08]

000750
2.49
[-1.20,7.53]

300444
3.19
[0.39,7.07]

603607
3.82
[-1.68,7.24]

300392
4.53
[1.49,8.11]

600844
5.82
[-1.94,10.11]

600695
6.61
[3.33,10.06]

000153
7.99
[4.47,10.04]

4.3. The Computational Results under Case I

In order to find the optimal portfolio selection, we solve the distributionally robust portfolio optimization problem under Case I. First of all, we need to analyze the connections between perturbation parameters and risk level to ensure that the selected values of parameters are reasonable. From the proofs of Theorems 1 and 3, it is known that the relationship between the risk level and perturbation set parameters is

Note that the values of parameters and reflect the relationship between ball and budget perturbation set. When , the intersection of ball and budget perturbation set is illustrated in Figure 2(a); when , the intersection of ball and budget perturbation set is illustrated in Figure 2(b). Based on this observation, the parametric values should be chosen from the interval .

(a)

(b)

Figure 2: The relationship between ball and budget perturbation set.

According to the discussion above, we give the corresponding values of perturbations set parameters and in Table 2. Then, we solve model (27) to find the optimal portfolio strategy under the values of these parameters. Note that the value of perturbation parameter can be different even at the same risk. Taking into account the different values of , we select some of the computational results on portfolio strategy and report them in Table 3. For example, when is set to be 0.93 and equals 7.293, we take as 5.2, 6.4, and 7.2. In order to better demonstrate the variation of optimal portfolio strategies in Table 3 at different parameter values of , , and , it is plotted in Figures 3(a)–3(j). The table shows the variation of allocation proportion of each stock with respect to , and the corresponding values of perturbation parameter are marked in Figure 3.

Table 2: The corresponding values between risk parameter and perturbation parameters .

budget parameters
ball parameters

0.91
6.940
(4.907,6.940)

0.92
7.107
(5.026,7.107)

0.93
7.293
(5.157,7.293)

0.94
7.501
(5.304,7.501)

0.95
7.740
(5.473,7.740)

0.96
8.024
(5.674,8.024)

0.97
8.374
(5.922,8.374)

0.98
8.845
(6.255,8.845)

0.99
9.597
(6.786,9.597)

Table 3: The optimal portfolio strategies under ball and budget perturbation set (%).

0.91
6.940
5.3
2.984554
13.012910
6.135631
3.185021
12.655810
13.610000
17.450490
18.766460
9.504393
2.694748

6.2
2.984527
13.012800
6.135777
3.184999
12.655910
13.609980
17.450390
18.766480
9.504414
2.694729

6.8
2.975392
12.991980
6.127129
3.176417
12.680580
13.618360
17.461880
18.778560
9.501712
2.687999

0.92
7.107
5.1
2.984560
13.012930
6.135599
3.185026
12.655790
13.610000
17.450520
18.766450
9.504388
2.694753

6.2
2.984526
13.012800
6.135777
3.184999
12.655910
13.609980
17.450390
18.766480
9.504414
2.694729

7.0
2.975386
12.991980
6.127086
3.176410
12.680590
13.618370
17.461920
18.778570
9.501704
2.687995

0.93
7.293
5.2
2.984557
13.012920
6.135615
3.185023
12.655800
13.610000
17.450500
18.766450
9.504390
2.694750

6.4
2.984520
13.012780
6.135810
3.184994
12.655930
13.609980
17.450370
18.766490
9.504419
2.694724

7.2
2.975385
12.991980
6.127073
3.176408
12.680590
13.618380
17.461930
18.778580
9.501702
2.687993

0.94
7.501
5.4
2.984551
13.012890
6.135648
3.185018
12.655820
13.610000
17.450480
18.766460
9.504395
2.694746

7.0
2.984502
13.012700
6.135907
3.184979
12.656000
13.609970
17.450300
18.766500
9.504434
2.694711

7.4
2.975387
12.991980
6.127092
3.176411
12.680580
13.618370
17.461910
18.778570
9.501705
2.687995

0.95
7.740
6.1
2.995548
13.063250
6.263475
3.197506
12.593050
13.586360
17.342490
18.742500
9.512746
2.703092

6.8
2.984508
13.012730
6.135875
3.184984
12.655980
13.609970
17.450320
18.766500
9.504429
2.694716

7.6
2.975396
12.991980
6.127152
3.176420
12.680580
13.618350
17.461860
18.778560
9.501717
2.688002

0.96
8.024
6.3
2.984523
13.012790
6.135794
3.184996
12.655920
13.609980
17.450380
18.766480
9.504417
2.694727

7.5
2.984487
13.012640
6.135989
3.184966
12.656060
13.609960
17.450240
18.766520
9.504446
2.694700

7.9
2.975394
12.991980
6.127137
3.176418
12.680580
13.618360
17.461870
18.778560
9.501714
2.688000

0.97
8.374
6.7
3.107925
13.246470
6.249412
3.300299
12.367660
13.505560
17.298960
18.604680
9.533868
2.785177

7.4
2.984490
13.012650
6.135973
3.184969
12.656050
13.609960
17.450260
18.766510
9.504444
2.694703

8.2
2.984465
13.012560
6.136104
3.184949
12.656140
13.609950
17.450170
18.766540
9.504463
2.694685

0.98
8.845
7.4
2.984490
13.012650
6.135974
3.184969
12.656050
13.609960
17.450260
18.766510
9.504444
2.694702

8.3
3.075341
13.108540
6.426811
3.273873
12.499240
13.483630
17.175920
18.633460
9.561397
2.761813

8.7
2.984450
13.012500
6.136186
3.184936
12.656200
13.609940
17.450110
18.766550
9.504475
2.694674

0.99
9.597
7.4
2.984489
13.012650
6.135975
3.184969
12.656050
13.609960
17.450250
18.766510
9.504444
2.694702

8.0
2.984471
13.012580
6.136072
3.184954
12.656120
13.609950
17.450190
18.766530
9.504458
2.694689

9.4
2.967558
12.953210
6.191147
3.170496
12.715800
13.608250
17.415440
18.783990
9.511675
2.682452

(a) Allocation proportion

(b) Allocation proportion

(c) Allocation proportion

(d) Allocation proportion

(e) Allocation proportion

(f) Allocation proportion

(g) Allocation proportion

(h) Allocation proportion

(i) Allocation proportion

(j) Allocation proportion

Figure 3: The variation of optimal portfolio strategies in Case I under different risks.

Further, we plot Figure 4 to illustrate the effect of parameter on the value of the objective function. When equals 0.92 and takes values in the corresponding interval (5.026,7.107), the changeable range of optimal value is smaller. However, when equals 0.97 and takes values in (5.922,8.374), the changeable range of optimal value on the objective function is larger. Thus the optimal objective value is sensitive to the parameter . The reason why different portfolio strategies and objective values at the same risk level are obtained is that the size of the perturbation sets is different. When the perturbation set is larger, the safe approximation is more conservative. From the computational results, the decentralized investment decisions can be obtained when uncertain return fluctuates under Case I. Taking into account the difference of portfolio under different allocations, the relationship between the optimal objective value and perturbation set parameters is given; see Figure 5.

Figure 4: The comparison of the optimal values under different ball parameters-.

Figure 5: The optimal values under ball and budget perturbation set.

4.4. The Computational Results under Case II

To find the optimal portfolio selection, we also solve the distributionally robust portfolio optimization problem in Case II.

The situation of Case II considers the fluctuation in the intersection of ellipsoid and generalized budget, and the values on the radius of ellipsoid and generalized budget are set as

Here it is required to solve model (28). Similarly, the relationship between risk level and perturbation set parameter is shown in Table 2, and the computational results on portfolio strategies with different values of parameters are reported in Table 4. Further, to illustrate the variation of optimal portfolio strategies in Table 4, it is depicted in Figures 6(a)–6(j). From these figures, we can know how to change the allocation proportion of each stock at risk parameter , and the values of perturbation parameter are marked in Figure 6.

Table 4: The optimal portfolio strategies under ellipsoid and generalized budget perturbation set (%).

0.91
6.940
5.0
2.984245
13.011700
6.137194
3.184769
12.656970
13.609830
17.449420
18.766740
9.504627
2.694527

6.0
2.984154
13.011350
6.137643
3.184694
12.657310
13.609780
17.449110
18.766820
9.504694
2.694462

6.8
2.984081
13.011070
6.137998
3.184634
12.657580
13.609740
17.448870
18.766890
9.504748
2.694410

0.92
7.107
5.1
2.984236
13.011670
6.137242
3.184761
12.657000
13.609820
17.449380
18.766740
9.504634
2.694521

6.1
2.984145
13.011310
6.137690
3.184686
12.657340
13.609770
17.449080
18.766830
9.504701
2.694455

7.0
2.984063
13.011000
6.138089
3.184619
12.657650
13.609730
17.448810
18.766900
9.504761
2.694397

0.93
7.293
5.2
2.984227
13.011630
6.137290
3.184754
12.657040
13.609820
17.449350
18.766750
9.504641
2.694514

6.3
2.984127
13.011240
6.137782
3.184672
12.657410
13.609760
17.449020
18.766840
9.504715
2.694442

7.1
2.984054
13.010960
6.138136
3.184612
12.657680
13.609720
17.448770
18.766910
9.504768
2.694390

0.94
7.501
5.4
2.984209
13.011560
6.137383
3.184739
12.657100
13.609800
17.449290
18.766770
9.504655
2.694501

6.3
2.984127
13.011240
6.137785
3.184672
12.657410
13.609760
17.449010
18.766840
9.504715
2.694442

7.3
2.984036
13.010890
6.138227
3.184598
12.657750
13.609710
17.448710
18.766930
9.504782
2.694378

0.95
7.740
5.5
2.984200
13.011520
6.137432
3.184732
12.657140
13.609800
17.449250
18.766780
9.504662
2.694495

6.5
2.984109
13.011170
6.137877
3.184657
12.657480
13.609750
17.448950
18.766860
9.504729
2.694429

7.5
2.984019
13.010820
6.138319
3.184583
12.657820
13.609700
17.448650
18.766940
9.504796
2.694365

0.96
8.024
5.7
2.984182
13.011450
6.137526
3.184717
12.657210
13.609790
17.449190
18.766790
9.504676
2.694482

6.8
2.984082
13.011060
6.138015
3.184635
12.657580
13.609740
17.448850
18.766880
9.504750
2.694410

7.6
2.984010
13.010790
6.138367
3.184576
12.657850
13.609700
17.448610
18.766950
9.504803
2.694358

0.97
8.374
6.1
2.984145
13.011310
6.137710
3.184687
12.657340
13.609770
17.449060
18.766830
9.504704
2.694456

6.9
2.984073
13.011030
6.138064
3.184628
12.657620
13.609730
17.448820
18.766890
9.504757
2.694404

7.9
2.983983
13.010680
6.138504
3.184554
12.657950
13.609680
17.448520
18.766970
9.504823
2.694339

0.98
8.845
6.4
2.984112
13.011180
6.137884
3.184660
12.657470
13.609750
17.448940
18.766860
9.504730
2.694432

7.3
2.984037
13.010890
6.138248
3.184599
12.657750
13.609710
17.448690
18.766920
9.504784
2.694378

8.3
2.979340
12.993050
6.159839
3.180717
12.675100
13.607610
17.434070
18.771330
9.507953
2.691004

0.99
9.597
6.8
2.984075
13.011040
6.138075
3.184630
12.657610
13.609730
17.448810
18.766890
9.504758
2.694406

7.9
2.983975
13.010650
6.138563
3.184548
12.657980
13.609670
17.448480
18.766980
9.504832
2.694334

8.9
2.979341
12.993050
6.159864
3.180718
12.675100
13.607610
17.434050
18.771320
9.507958
2.691005

(a) Allocation proportion

(b) Allocation proportion

(c) Allocation proportion

(d) Allocation proportion

(e) Allocation proportion

(f) Allocation proportion

(g) Allocation proportion

(h) Allocation proportion

(i) Allocation proportion

(j) Allocation proportion

Figure 6: The variation of optimal portfolio strategies in Case II under different risks.

From the computational results, the decentralized investment decisions can be obtained when uncertain return fluctuates under Case II. Taking into account the difference of portfolio under different allocations, the relationship between optimal objective value and perturbation set parameters is given in Figure 7.

Figure 7: The optimal values under ellipsoid and generalized budget perturbation set.

4.5. Comparison Study

In this subsection, the distributionally robust portfolio optimization model is compared with nominal distribution optimization model, in which uncertain return vector has a nominal probability distribution.

According to uncertain portfolio optimization model (26), we have the stochastic portfolio optimization model with a nominal distribution of random return. To compare the distributionally robust optimization model with nominal distribution optimization model, we analyze the normal distribution information under Case II.

In Case II, the perturbation vector fluctuates in an ellipsoid and generalized budget perturbation set depicted in (16), the distribution family in model (26) satisfies assumptions P.1-2, and the mutually independent random variables are only partially known with expectations and variances We next consider the scenario of , i.e., In this situation, the family of normal distributions in Case II degenerates into a single normal distribution. As a result, model (26) reduces to the following nominal distribution model:

To compare the ambiguous VaR model with nominal distribution model, we analyze the computational results at the same risk level. Based on the data in Table 2, we solve model (31) and obtain the computational results under different risks as shown in Table 5. Further, Figure 8 is plotted to show the evolution of the optimal portfolio values of nominal distribution model at risk parameter .

Table 5: The optimal portfolio strategies produced by nominal distribution model (31) (%).

0.91
-
-
-
-
-
-
-
-
23.805630
76.194370

0.92
-
-
-
-
-
-
-
-
25.430380
74.569620

0.93
-
-
-
-
-
-
-
0.341380
26.757220
72.901400

0.94
-
-
-
-
-
-
-
3.227114
27.332420
69.440470

0.95
-
-
-
-
-
-
-
5.835254
27.852130
66.312620

0.96
-
-
-
-
-
-
-
8.179620
28.319590
63.500790

0.97
-
-
-
-
-
-
-
10.621300
28.806190
60.572500

0.98
-
-
-
-
-
-
-
13.112000
29.302940
57.585060

0.99
-
-
-
-
-
-
-
16.123170
29.902920
53.973910

Figure 8: The optimal values generated by nominal distribution model (31) under risk parameter.

Under the same risk level, we select the portfolio of ten stocks with positive return, by using the methods of distributionally robust optimization and stochastic optimization with nominal distribution, and obtain the following observations.

First of all, stochastic optimization method with nominal normal distribution is based on the assumption that the full distribution information of future return can be estimated exactly. When the full information on the probability distribution is unavailable, this optimization method will be invalid. In fact, due to the impacts of the economic environment and political factors, it is usually difficult to obtain the exact distribution information of future return. In such case, portfolio strategy under the nominal distribution cannot be recommended to the investors.

When only partial information on the underlying probability distribution is available, the investor should employ the distributionally robust portfolio optimization method. The distributionally robust optimal portfolio policy is the best choice for investors when the uncertain returns are fluctuating in a given ambiguity set. That is, the robust optimal portfolio provides immunization against the distribution uncertainty.

In order to compare the optimal values of distributionally robust optimization model and nominal distribution model, we firstly transform the values of risk parameter in model (31) to the corresponding values of and and then plot, in Figure 9, a comparison between the optimal values of distributionally robust model under ellipsoid and generalized budget perturbation set and nominal distribution model. In terms of optimal solution, we find that the optimal portfolio policy obtained from distributionally robust optimization model is more decentralized than that from nominal distribution model. For example, if is 0.98, based on nominal distribution model, the investor should select three stocks with number 8, 9, and 10 as a portfolio. However, solving distributionally robust model, the investor should allocate his/her money to ten stocks. The results show that the optimal portfolio strategies of the two methods are completely different, and the distributionally robust optimization method can better resist the distributional uncertainty.

Figure 9: The comparison of optimal values of nominal model and distributionally robust model in Case II.

5. Conclusions

In this paper, we studied portfolio optimization problem under distribution uncertainty from a new perspective. The obtained major results include the following three aspects.

First, to model the distribution of model parameters varying in a prescribed ambiguity set, we developed an ambiguous VaR optimization framework for distributionally robust linear optimization problem. The objective of the proposed model is to optimize the worst case VaR over an ambiguity set. As a result, the obtained robust solution provides immunization against the distribution uncertainty.

Second, we addressed the tractability of the ambiguous probabilistic constraints under two new combinations of perturbation sets. The first ambiguity set is the intersection of ball with budget, while the second one is the intersection of ellipsoid with generalized budget. Depending on the choice of the two new ambiguity sets, the resulting framework of the safe approximation can be reduced to a system of efficiently computable convex constraints.

Finally, to show the advantage of the proposed approach, we applied our approach to a practical case about the Chinese stock market. By comparison study, we concluded that the optimal solution to the proposed robust optimization approach provides immunization against distribution uncertainty, and even small perturbation of nominal probability distribution cannot be ignored in practical portfolio optimization problems.

Appendix

Proofs of Main Results

In this appendix, we only present the proofs of Theorems 1 and 3, and Theorems 2 and 4 can be proved similarly.

We first prove Theorem 1.

Proof. We already know that (11)-(12) represent the RC of (4), with the perturbation set being (6). Now let us prove that if , , satisfy hypothesis (15) and are feasible for (11)-(12), then is feasible for (4) with probability of at least ,
We pass from (4) to the following inequality

Thus, the following equivalent representation holds:

Indeed, when , are mutually independent and , we have the following.

Let be feasible for (11)-(12). Then we have

Thus,

As a result, we have

Setting , we have

According to the triangle inequality

we have

On the basis of the conclusion of proposition 2.3.1 in [13], we have the following result:

with , and thus

The proof of theorem is complete.

Next we prove Theorem 3.

Proof. We already know that (20)-(21) represent the RC of (4), with the perturbation set being (16). Now let us prove that if the independent random variables , , satisfy hypotheses P.1-2 and are feasible for (20)-(21), then is feasible for (4) with probability of at least ,
We can pass from (4) to the following inequality

Thus, the following equivalent representation holds:

Without loss of generality, when and are mutually independent, we have

Let be feasible for (20)- (21). Then we have

Thus, we have

As a result, we have

Setting , we have

According to the triangle inequality

we have

On the basis of the conclusion of proposition 2.4.2 in [13], we obtain the following result:

with thus

The proof of the theorem is complete.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Nos. 61773150 and 61374184) and the High-Level Innovative Talent Foundation of Hebei University (No. 801260201209).

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