Mathematical Problems in Engineering

Volume 2014 (2014), Article ID 912389, 13 pages

http://dx.doi.org/10.1155/2014/912389

Research Article

A Numerical Study for Robust Active Portfolio Management with Worst-Case Downside Risk Measure

Aifan Ling¹ and Le Tang²

¹School of Finance, Jiangxi University of Finance & Economics, Nanchang 330013, China

²Jiangxi University of Technology, Nanchang 330098, China

Received 9 December 2013; Accepted 22 January 2014; Published 1 April 2014

Academic Editor: Chuangxia Huang

Copyright © 2014 Aifan Ling and Le Tang. 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.

Recently, active portfolio management problems are paid close attention by many researchers due to the explosion of fund industries. We consider a numerical study of a robust active portfolio selection model with downside risk and multiple weights constraints in this paper. We compare the numerical performance of solutions with the classical mean-variance tracking error model and the naive portfolio strategy by real market data from China market and other markets. We find from the numerical results that the tested active models are more attractive and robust than the compared models.

1. Introduction

The choice of an optimal portfolio of assets has become a major research topic in financial economics. The mean-variance model proposed by Markowitz (1952, 1956) [1, 2] provided a fundamental basis of portfolio selection for the theoretical and practical applications today. Analytical expression of the mean-variance efficient frontier could be derived by solving convex quadratic programs and the optimal portfolio can be found when the expected returns and covariance matrix of risk assets are exactly estimated. Based on Markowitz’s mean-variance model, Roll (1992) [3] proposed an active portfolio management model which is called tracking error portfolio model in the literature. Roll [3] used the variance of tracking error to measure how closely a portfolio follows the index to which it is benchmarked. Motivated by Roll’s seminal paper, many researches pay close attention to the active portfolio selection problems; see [4–9] and recent papers [10–18].

Let denote the return vector of the risk assets, where is the gross rate of return for the th risky asset. The investor’s position is described by vector , where the th component represents the proportion invested to the th risky asset. Let denote a fixed portfolio that investor seeks to outperform. also is called benchmark portfolio in the literature. Define the tracking error of portfolio relative to benchmark portfolio as . In order to obtain a good tracking error portfolio , Roll [3] considered the solution of the following variance tracking error (VTE) problem

where and are the expectation and covariance matrix of return vector , respectively, and is the preset tracking error level. When and are known exactly, similar to Markowitz’s mean-variance portfolio, problem (1) can be solved as a convex quadratic programming.

Since the gain and the loss are symmetric on the mean value in the variance, Markowitz (1959) [19] proposed to use the semivariance of portfolio to control risk. Bawa (1975) [20], Bawa and Lindenberg (1977) [21], and Fishburn (1977) [22] later introduced a class of downside risk measure known as the lower partial moment (LPM) to better suit different risk profits of the investors. Because LPM mainly control the loss of portfolio, it becomes a popular risk controlling tool in theory and practice; see [23–28].

Generally, LPM can be expressed as

where is the random variable, for example, the asset return of risky asset; may be a target that investors want to outperform, is a parameter, which can take any nonnegative value to model the risk attitude of an investor, , , and below expresses the expectation and probability of random variable. For the case of , we have

That is to say that is nothing but the probability of the asset return falling below the benchmark index is the expected shortfall of the investment falling below the benchmark index and is an analog of the semivariance; here, however, the deviation is in reference to a preset target or benchmark return instead of the mean.

In the last five to ten years, robust portfolio optimization problems based on the robust optimization technique developed by Ben-Tal and Nemirovski (1998) [29] are the focus of many financial and economic researchers. Based on robust portfolio optimization theory, one can deal with models with uncertainty parameters or uncertainty distribution. For instance, Costa and Paiva (2002) [30] considered a robust framework of model (1) with and in a polytopic uncertainty set described by its vertices. El Ghaoui et al. (2003) [31] proposed a worst-case Value-at-Risk (VaR) robust optimization model in which they assumed that the distribution of returns is partially known. Only bounds on the mean and covariance matrix are available in El Ghaoui et al.’s model and the proposed model can be solved by a second order cone programming approach. Goldfarb and Iyengar (2003) [32] proposed a robust factor model and later their models were developed and extended by Erdoğan et al. (2008) [33] to a robust index tracking and active portfolio management problem; see also Ling and Xu (2012) [34] for a similar research. Zhu et al. (2009) [28] proposed a robust framework based on LPM constraints with uncertainty discrete distribution. Glabadanidis (2010) [11] considered a robust and efficient strategy to track and outperform a benchmark in which he proposed a sequential stepwise regression and relative method based on factor models of security returns. Chen et al. (2011) [23] developed some tight bounds on the expected values of LPM under the framework of robust optimization models arising from portfolio selection problems. More robust portfolio models based on different parameters uncertainty or distribution uncertainty can be found in the recent survey given by Fabozzi et al. (2010) [35].

As we know, some policies are commonly found in the contracts between investors and portfolio managers and require that the weights of certain types of assets should be smaller, higher, or equal to a given percentage in some funds. We call this type of restriction in this paper weights constraint(s) of a portfolio. For instance, weights constraint appears frequently in some index funds (ETF), stock style funds (the weights invested in stocks is not less than a preset percentage of the market value of fund), bond style funds (the weights invested in stocks is not larger than a preset value), QFII (Qualified foreign institutional investors) funds, and QDII (Qualified domestic institutional investor) funds require that the weights invested in foreign markets not be larger than a preset value. However, it is to be regretted that weights constraint is rarely used in the current tracking error (robust) portfolio literature. Recently, Bajeux-Besnainou et al. (2011) [36] first introduce this kind of constraint into a mean-variance index tracking portfolio model. In their paper, Bajeux-Besnainou et al. described a comparison with and without weights constraint and showed that the influence of weights constraint is very remarkable and cannot be ignored for the returns of portfolio.

Motivated by the topics of active portfolio management that are paid close attention by many researchers and the fact that there exists the rare model with explicit solutions in the robust active portfolio management literature, in this paper, we further consider a numerical study of robust active portfolio selection model that is proposed by Ling et al. in [37]. In the current framework, we test the models in [37] using the real market data which includes ten stock indexes from Shanghai Stock Exchange (SHH) and Shenzhen Stock Exchange (SHZ), which are called domestic assets, and four stock indexes from other stock exchanges which are called foreign assets. Our numerical studies give the comparisons with the classical (VTE) and the naive portfolio strategy considered by DeMiguel et al. (2009) [38]. Numerical results indicate that the proposed active portfolio selection model can obtain better numerical performance for real market data. Most of proofs of theorems in the current paper can be found in [37], but, in order to keep the completeness of reading, we still give these main proofs in Appendix.

This paper is arranged as follows. In Section 2, we give the robust active portfolio problem with multiple weights constraints and establish the robust active portfolio models. We explore the explicit solutions of the proposed models in Section 3. In Section 4, we do some numerical tests and comparisons based on real market data.

2. Robust Active Portfolio Problems

Let be a subset of . If assets in are restricted to a limited weight, we call the set of restricted assets. For given constant , weights constraint requires

where is an index vector with when , and when . The notation “” means the equality only (or inequality only) weights constraint. Under the estimation of    and    is exactly obtained and selling short is allowed, Bajeux-Besnainou et al. [36] considered the following active portfolio optimization problem with single weight constraint:

and get a closed solution of this problem. As pointed in Section 1, the classical mean-variance framework may not be appropriate since it controls not only the loss of portfolio, but also the gain of portfolio.

In our framework, we use LPM to control the risk of portfolio by which we in fact only control the loss of portfolio. Additionally, we make no assumption on the distribution of returns . Instead, we assume that we only partially know some knowledge about the statistical properties of returns and the underlying distribution function is only known to belong to a certain set of distribution functions of the returns vector . Under the uncertainty set of the underlying distribution, we have the worst-case .

Definition 1. For any and real number , the worst-case lower-partial moment (WCLPM for short) of random variable with respect to under is defined by

In tracking error portfolio selection problems, investors want in fact to find a portfolio , such that the return of portfolio can be close to or outperform the return of benchmark . For this, we call the loss of portfolio relative to and control the risk of portfolio by restricting the loss. Based on this idea, we maximize on one hand the return and on the other hand control the loss by using worst-case . Thus, the mean- robust tracking error problem with multiple weights constraints can be written as

where are the preset constants. The second class of inequalities constraints; weights constraints express that several classes of assets are restricted to the limited weights, where and are constants. We call these constraints multiple weights constraints. When , it is the single weight constraint. In this paper, we only consider the cases of and multiple equality weights constraints. The corresponding results with multiple inequality weights constraints can be obtained by the similar methods.

Because we are interested in developing robust tracking error portfolio policies and exploring the explicit solutions of problems (7), in the rest of this paper, we assume that is the set of allowable distributions of returns with the known mean and covariance; that is,

For convenience, we sometimes denote by . Our model can be extended to the case of parameters uncertainty, for which parameters are not estimated exactly.

Similar to Erdoğan et al. [33] and Bajeux-Besnainou et al. [36], we normalize the benchmark portfolio ;that is, and introduce a new variable . Notice the ; then optimization problem (7) with multiple equality weights constraints can be written into the following form:

We call a self-financing portfolio and a fully investing portfolio. We mention that , can be positive or negative, most likely between and 1 since and are in most cases between 0 and 1. In the rest of the paper, we suppose that vectors and are linearly independent and always holds. If there exists certain , such that can be expressed as the linear combination of , then becomes a redundant constraint. Additionally, the linear equality constraints will lead to a unique feasible solution if .

The following lemmas indicate that a tight upper bound of the worst-case can be obtained explicitly, which will be helpful for our analysis later. We extend the proof in [23] for to a general case.

Lemma 2 (see [23, 39]). Let be a random variable with mean and variance but have unknown distribution and is an arbitrary real number. Then, we have the following.

(a)For the case of ,

(b)For the case of ,

(c)For the case of ,

Lemma 3 (see [40]). For any , denote

Then .

Lemma 4 (see [37]). For any , , and any real vector , the following equality

holds.

Lemma 4 establishes the relationship of single variable and multivariable , which is useful for our analysis later. We end this section by introducing some notations that will be used frequently. Unless otherwise specified, in the rest of paper, bold letter expresses a vector, uppercase letter is a matrix, and lowercase is a real number. is a zero matrix or zero vector by context. For simplicity, we denote by , and by and thus . Let

Then, from the assumption that and are linearly independent, matrix has column full rank. Let . Then matrix is positive definite via , where the notation (or ) means matrix is positive semidefinite (or positive definite). Further, we denote

where is a unit matrix with reasonable dimensions. Then, for these matrices, we have the following results.

Lemma 5. (a) .

(b) .

(c) .

(d) .

(e) .

(f) Matrix is positive semidefinite.

The conclusions are straightforward for (a)–(e). For (f), we have from (d), since . Hence is a positive semidefinite matrix.

Based on matrices , , and , we denote additionally

Without loss the generality, we assume ; then we have that

and if and only if via .

3. The Solutions

In this section, we will explore the explicit solution of problem (RSm). Notice that ; then we can rewrite separably problem (RSm) into the following three problems:

which is corresponding to the case of in the worst-case .

3.1. Maximization of Information Ratio: A Variation of WCLPM₀

Let us first consider problem (19). The inequality is usually called chance constraint or Roy’s safety-first rule in the literature. Generally speaking, is taken far less than 1/2 which leads to . Hence, from Lemma 2(a) and Lemma 4, we have that, when ,

The last inequality means that information ratio (or sharpe ratio) of the portfolio is not less than a preset constant. By this characteristic, we consider a variation version of problem (19), in which we minimize the ; that is,

which results from (22) the maximization of information ratio (IR for short):

Problem (24) is related to the mean-variance with multiple weights constraints (MVMWC, for short) for a given parameter :

If , that is, there is not weights constraint, then problem (25) is nothing but Roll’s VTE problem. If , that is, , then problem (25) becomes the case of single weight constraint problem which has been considered by Bajeux-Besnainou et al. [36].

The explicit solution of problem (24) without weights constraints has been obtained in the literature (e.g., see [16, 23] for detail). But the methods in the literature cannot be used directly for problem (24) because of the existence of multiple weights constraints. Thus, we need to find the different method from the literature to obtain the explicit solution. To this end, we deal with this problem in two stages to get the explicit solution of (24). We introduce a parameter at first stage and parameterize the solution of (24), and then maximize IR of the solution with respect to the parameter in the second stage.

Theorem 6. There exists a real number, say , such that the explicit solution of problem (24) can be expressed as

Proof. See Theorem  1 in [37]; see also Appendix for detail.

To obtain the portfolio with the maximum information ratio, that is, the optimal solution of problem (24), what is then the value of parameter ? As the second stage, we replace into the objective function (24). Then we get the information ratio of portfolio , denoted by , which is a function of parameter . Thus, we can get the maximum information ratio by maximizing with respect to . That is, we have

Let be the optimal solution of (27); then portfolio is the optimal solution of (24), for which it has the maximum information ratio. Hence, we have the results below and its proof is straightforward.

Theorem 7. The maximizer of problem (27) is given by

when and the optimal solution of (24) is obtained by

Theorem 8 (see [37]). If in (26) is obtained for certain positive number , then, there exists a , such that both problems (24) and (25) have the same optimal solution.

Let be the optimal solution of (24); in view of the proof of Theorem 6 in Appendix, if is chosen and satisfies

then the optimal portfolio for problem (25) is also the optimal solution of (24). The relationship between problems (24) and (25) can be presented as follows. Solving (24) will get a portfolio with the maximum IR. However, the optimal portfolio of problem (25) is not necessarily the one with the maximum IR. If we solve (25) with given by (30), then the optimal portfolio of (25) will have the maximum IR, too.

3.2. The Mean-WCLPM₁

The constraint in problem (20) from Lemmas 2 and 4 can be written as

In fact, it can verify that

Hence, inequality constraint is redundant. Thus, for any preset , we can rewrite problem (20) as

Here, we call the upper bound of tracking error under constraint and denote the upper bound of tracking error by . The following lemma gives the feasible condition of problem (33).

Lemma 9. If is chosen to satisfy

then problem (33) is feasible, where

Proof. Let

and , . Then this lemma is to verify that the set is nonempty for any . Now, for any , consider the following subproblem:

and denote its optimal value by which is a quadratic function of parameter . Thus, (for the computation of and , see Appendix for detail.),

gives a quadratic inequality with respect to . Solving (38), we have that the equality in (38) holds when , where is given by (35). Hence, set to be nonempty and has at least an element when . This implies problem (33) is feasible. The proof is completed.

Lemma 9 indicates that problem is well defined and meaningful. The following results give the explicit solution of problem .

Theorem 10. Let the condition of Lemma 9 be satisfied. Then the optimal solution of problem (33) can be expressed explicitly as

where

Proof. See Theorem  4 in [37]; see also Appendix for detail.

3.3. The Mean-WCLPM₂

In this subsection, we consider mean- portfolio problem. In view of Lemma 2(c) and Lemma 4, the constraint can be formulated as

Then we can write problem (7) with as

If we add constraint , then problem (42) is nothing but the classical M-V model with weights constraints. On the other hand, if we add constraint , then constraint (41) is

which is more conservative than the variance of portfolio for the same preset value since is a positive semidefinite matrix. We call , instead of , the upper bound of tracking error under constraint and denote the upper bound by . Similar to Lemma 9, we can also give the feasible condition of problem (42).

Lemma 11. Let

Then problem (42) is feasible.

Proof. Let

It follows from solving straightforwardly the problem that its optimal solution is

where

Then, we get that

Hence, the problem (42) is feasible when satisfies . This completes the proof.

By Lemma 11, problem (42) is well-defined and meaningful. The following results give the explicit expression of solution of problem (42).

Theorem 12. Let the condition of Lemma 11 be satisfied. Then problem (42) has the explicit solution as follows:

where

Proof. See Theorem  5 in [37]; see also Appendix for detail.

We mention that the optimal solution, , of problem (42) is consistent with problem (25) when or . This is not surprising. Because the term vanishes in problem (42) in these cases.

4. Numerical Results

In this section, we consider the performance of models , , and MV using real market data. We choose ten stock indexes from Shanghai Stock Exchange (SHH) and Shenzhen Stock Exchange (SHZ), which are called domestic assets, and four stock indexes from other stock exchanges which are called foreign assets. All fourteen risky assets are listed in Table 1. Our data covers the period from January 1, 2000 to March 18, 2011 and includes 2663 samples for each asset. We group the whole period into two subperiods; that is, the first subperiod is from January 1, 2000 to April 21, 2006 and the second subperiod is from April 24, 2006 to March 18, 2011. The data in the first subperiod includes 1663 samples of each asset and is used as the in-sample test set. Other 1000 samples are used as the out-of-sample test set. The statistic properties of samples of all assets are reported in Table 2. We take the benchmark as the naive portfolio [38] with . We fix in the next numerical experiment.

Table 1: Chosen 14 indexes as the risky assets.

Number
Asset name
Symbols

Domestic assets

1
SSE Composite Index
001.SS

2
SSE A-share Index
002.SS

3
SSE B-share Index
003.SS

4
SSE Industrial Index
004.SS

5
SSE Commercial Index
005.SS

6
SSE Properties Index
006.SS

7
SSE Utilities Index
007.SS

8
SSE Component Index
001.SZ

9
SSE Component A
002.SZ

10
SSE Composite Index
106.SZ

Foreign Assets

11
BSE SENSEX
BSESN

12
FTSE Bursa Malaysia KLCI
KLSE

13
Hang Seng Index
HSI

14
NIKKEI 225
N225

Table 2: Asset market daily returns from January 1, 2000 to March 18, 2011.

Asset
Mean
Standard deviation
Minimum
Maximum

Panel A: in-sample data: from January 01, 2000 to April 21, 2006

001.SS
0.4263
0.2019
−0.0654
0.1096

002.SS
0.4194
0.1970
−0.0651
0.1098

003.SS
0.8951
0.4687
−0.1029
0.1840

004.SS
0.4381
0.1997
−0.0631
0.1137

005.SS
0.3720
0.2390
−0.0753
0.1039

006.SS
0.2563
0.3519
−0.0974
0.1054

007.SS
0.5023
0.1960
−0.0634
0.1089

001.SZ
0.4986
0.2282
−0.0693
0.1163

002.SZ
0.4399
0.2284
−0.0680
0.1208

106.SZ
0.2936
0.2172
−0.0682
0.1056

BSESN
0.5748
0.2476
−0.1181
0.1046

KLSE
0.2094
0.0862
−0.0634
0.0649

HSI
0.1150
0.1749
−0.0929
0.0601

N225
−0.0510
0.2112
−0.0901
0.0722

Panel B: out-of-sample data: from April 24, 2006 to March 18, 2011

001.SS
0.0172
0.4691
−0.1130
0.0903

002.SS
0.0138
0.4692
−0.1128
0.0903

003.SS
0.6245
0.6482
−0.1572
0.0937

004.SS
0.1737
0.4982
−0.1186
0.0895

005.SS
0.5640
0.5404
−0.1261
0.0918

006.SS
0.1804
0.9067
−0.1817
0.0951

007.SS
0.1058
0.5745
−0.1365
0.0938

001.SZ
0.4622
0.5780
−0.1210
0.0916

002.SZ
0.4951
0.5776
−0.1206
0.0916

106.SZ
0.6488
0.5608
−0.1339
0.0852

BSESN
0.2246
0.4266
−0.1268
0.1599

KLSE
0.2428
0.2111
−0.1925
0.1986

HSI
0.0759
0.4654
−0.1358
0.1341

N225
−0.6399
0.3917
−0.1272
0.1323

Here, we compare the performance of three models: , MVMWC, and IRMWC (information ratio problem with multiple weights constraints, i.e., problem (24)). We consider the following three cases of weights constraints:

The wealth invested to foreign assets is restricted by the weights constraint. Table 3 gives the return, variance, and of optimal portfolios obtained by these models. Figures 1, 2, and 3 give the comparisons of portfolio value obtained in these models. The portfolio value in these figures is computed by

where is the obtained optimal portfolios and is the value of the th market index at the th exchange day. Some interesting results from our numerical comparisons can be found as follows.

Table 3: The comparisons of return, variance, and information ratio of optimal solutions.

Model
Mean return
Standard deviation
Information ratio

Panel A: in-sample data

Benchmark
0.3850

—

VTE
0.4304
0.0527
0.0782

WCLPM₁
0.4522
0.0548
0.0818

MVMWC
0.4256
0.0510
0.0759

IRMWC
0.1300
0.0170
0.0781

WCLPM₁
0.4412
0.0548
0.0813

MVMWC
0.4204
0.0510
0.0755

IRMWC
0.1365
0.0173
0.0786

WCLPM₁
0.4526
0.0543
0.0666

MVMWC
0.4278
0.0512
0.0626

IRMWC
0.1325
0.0139
0.0607

Panel B: out-of-sample data

Benchmark
0.2278

—

MV
0.3925
0.0631
0.0102

WCLPM₁
0.4282
0.0642
0.0104

MVMWC
0.4033
0.0598
0.0099

IRMWC
0.1237
0.0235
0.0103

WCLPM₁
0.3851
0.0637
0.0102

MVMWC
0.3542
0.0593
0.0101

IRMWC
0.1032
0.0237
0.0102

WCLPM₁
0.4268
0.0649
0.0101

MVMWC
0.4006
0.0611
0.0101

IRMWC
0.1260
0.0208
0.0101

Figure 1: The comparison of portfolio values of models , MVMWC, IRMWC, and benchmark for in-sample and out-of-sample data when there is single weight constraint.

Figure 2: The comparison of portfolio values of models , MVMWC, IRMWC, and benchmark for in-sample and out-of-sample data when there are two weights constraints.

Figure 3: The comparison of portfolio values of models , MVMWC, IRMWC, and benchmark for in-sample and out-of-sample data when there are four weights constraints.

(1) The tracking error of in-sample data from Figures 1–3 is less than that of out-of-sample data, and the tracking error of model IRMWC is the least among these models whatever the in-sample and out-of-sample data is. But model from Table 3 has the best expectation return among these considered models for all , and in-sample and out-of-sample data. This is understandable since model has an excess MV region where it also exceeds the return of MVMWC. Moreover, from the view of portfolio value, can not only exceed benchmark, but can also get greater terminal wealth than that of models MVMWC and IRMWC.

(2) It seems to be not understandable that the tracking errors of all models from Figures 4, 5, and 6 increases as the number of restricted assets increases. But it is in fact not surprising, because the feasible sets of all models reduce as increases; this clearly leads to a larger error relative to the case of without restricted assets. Additionally, from Figures 4–6, we also find that models and MVMWC are more sensible than model IRMWC with respect to the number of restricted assets.

Figure 4: The comparison of portfolio values of model for in-sample and out-of-sample data and different values of .

Figure 5: The comparison of portfolio values of model MVMWC, for in-sample and out-of-sample data and different values of .

Figure 6: The comparison of portfolio values of model IRMWC for in-sample and out-of-sample data and different values of .

5. Conclusions

We considered a numerical extension of three active portfolio selection problems with the worst-case -, - and, -order lower partial moment risk and multiple weights constraints which are proposed in [37]. Using ten stocks from China market and four stocks from other market, we compared the numerical performance with VTE and “1/N” strategy. Clearly, from the numerical results can obtain the better expectation excess return and information ratio than when the tracking error taken is small. Model MVMWC will be another good choice when a large tracking error is required and sell-shorting is forbidden.

Appendix

Technical Proofs

Proof of Theorem 8. Let be an optimal solution of problem (24). Then, must satisfy the first order optimal condition of problem (24). Thus, we have

where , are the Lagrange multipliers. Let and ; then equality (A.1) can be reduced as

where is a constant. Thus, combining (A.2) and the equality constraints and , we get that

This completes the proof.

The Computation of and . Consider the following subproblem:

and denote its optimal value by . Let be its optimal solution. Then, by the KKT optimal condition, we have

where and are the Lagrange multiplier. Solving the equality system of KKT conditions, we get

Substituting it into the objective function and using some results in Lemma 5, it follows that

and it is a quadratic function of parameter . Solving directly the quadratic equation in

we get a positive root of this equation as

This obtains equality (35). Thus, inequality holds when . This is the conclusion of Lemma 9.

Proof of Theorem 10. Let be the optimal solution of problem (33). Then from KKT condition, we have

where , , and are the Lagrange multipliers. Solving directly KKT system (A.10), it gets the conclusions of Theorem 10. The proof is finished.

Proof of Theorem 12. Let . Then we can rewrite problem (42) as

here, we view as an auxiliary variable. The Lagrange function of this problem is

where , , , and are the Lagrange multipliers. Let the pair be the optimal solution of (A.11). Then the pair satisfies the first optimal condition; that is,

By the first equation of (A.13), it gets that

Substituting into equation , we obtain that

where . Substituting into and using Lemma 5, we further get

Combining equation , it gets that

Notice that . Thus, can be reduced as

Substituting into the last equation of (A.13), we get

On the other hand, substituting into the third equation of (A.13), we can obtain another expression of as

Hence, combining (A.19) and (A.20) and noticing the notations in Lemma 5, we have that satisfies the quadratic equation

Next, we discuss the solutions of (A.21) by the different sign of .

(1) . Notice that is the expectation excess return of self-finance portfolio . Thus, generally speaking, is increasing as the preset tracking error increases. Thus, (A.21) has only an efficient positive solution as follows. (Equation (A.21) has in this case another solution

when and is taken to satisfy , but it is not an efficient solution since is decreasing as increases)

(2) . If , then (A.21) has the solution

when is chosen to satisfy . If , then the side of right hand of equality (A.24) is positive and therefore equality (A.24) is not a solution of (A.21). In the case of , the solution equation (A.21) has still the form of (A.23).

Summarizing the two cases above, we get that the solution of (A.21) can be expressed as (50). This completes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are in debt to two anonymous referees and Lead Guest Editor, Professor Chuangxia Huang, for their constructive comments and suggestions to improve the quality of this paper. This work is supported by National Natural Science Foundations of China (nos. 71001045 and 71371090), Science Foundation of Ministry of Education of China (13YJCZH160), Natural Science Foundation of Jiangxi Province of China (20114BAB211008), and Jiangxi University of Finance and Economics Support Program Funds for Outstanding Youths.

References

H. M. Markowitz, “Portfolio selection,” Journal of Finance, vol. 7, pp. 77–91, 1952.

H. Markowitz, “The optimization of a quadratic function subject to linear constraints,” Naval Research Logistics Quarterly, vol. 3, pp. 111–133, 1956.

R. Roll, “A mean-variance analysis of the tracking error,” Journal of Portfolio Management, vol. 18, no. 1, pp. 13–22, 1992.

G. J. Alexander and A. M. Baptista, “Active portfolio management with benchmarking: adding a value-at-risk constraint,” Journal of Economic Dynamics & Control, vol. 32, no. 3, pp. 779–820, 2008.

P. Jorion, “Portfolio optimization with tracking-error constraints,” Financial Analysts Journal, vol. 59, no. 5, pp. 70–82, 2003.

M. Rudolf, H. J. Wolter, and H. Zimmermann, “A linear model for tracking error minimization,” Journal of Banking and Finance, vol. 23, no. 1, pp. 85–103, 1999.

F. Wen and Z. Liu, “A copula-based correlation measure and its application in chinese stock market,” International Journal of Information Technology and Decision Making, vol. 8, no. 4, pp. 787–801, 2009.

F. Wen and X. Yang, “Skewness of return distribution and coefficient of risk premium,” Journal of Systems Science & Complexity, vol. 22, no. 3, pp. 360–371, 2009.

Y. Zhao, “A dynamic model of active portfolio management with benchmark orientation,” Journal of Banking and Finance, vol. 31, no. 11, pp. 3336–3356, 2007.

H. H. Chena, H. T. Tsaib, and D. K. J. Linc, “Optimal mean-variance portfolio selection using Cauchy-Schwarz maximization,” Applied economics, vol. 43, pp. 2795–2801, 2011.

P. Glabadanidis, “Robust and efficient strategies to track and outperform a benchmark,” working paper, 2010, http://papers.ssrn.com/sol3/papers.cfm?abstract-id=1571250

I. Guedj and J. Huang, “Are ETFs replacing index mutual funds?” Manuscript, 2009, http://papers.ssrn.com/sol3/papers.cfm?abstract-id=1108728

V. Glode, “Why mutual funds ‘Underperform’,” Manuscript, 2010, http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1121436

C. Huang, C. Peng, X. Chen, and F. Wen, “Dynamics analysis of a class of delayed economic model,” Abstract and Applied Analysis, vol. 2013, Article ID 962738, 12 pages, 2013.

C. X. Huang, X. Gong, X. H. Chen, and F. H. Wen, “Measuring and forecasting volatility in Chinese stock market using HAR CJM model,” Abstract and Applied Analysis, vol. 2013, Article ID 143194, 13 pages, 2013.

S. Liu and R. Xu, “The effects of risk aversion on optimization,” MSCI Barra Research, 2010, http://ssrn.com/abstract=1601412

M. Wang, C. Xu, F. Xu, and H. Xue, “A mixed 0-1 LP for index tracking problem with CVaR risk constraints,” Annals of Operations Research, vol. 196, pp. 591–609, 2012.

F. Wen and Z. Dai, “Modified Yabe-Takano nonlinear conjugate gradient method,” Pacific Journal of Optimization, vol. 8, no. 2, pp. 347–360, 2012.

H. M. Markowitz, Porfolio Selection: Ecient Diversification of Investment, John Willey, New York, NY, USA, 1959.

V. S. Bawa, “Optimal rules for ordering uncertain prospects,” Journal of Financial Economics, vol. 2, no. 1, pp. 95–121, 1975.

V. S. Bawa and E. B. Lindenberg, “Capital market equilibrium in a mean-lower partial moment framework,” Journal of Financial Economics, vol. 5, no. 2, pp. 189–200, 1977.

P. C. Fishburn, “Mean-risk analysis with risk associated with below-target returns,” American Economics Review, vol. 67, no. 2, pp. 116–126, 1977.

L. Chen, S. He, and S. Zhang, “Tight bounds for some risk measures, with applications to robust portfolio selection,” Operations Research, vol. 59, no. 4, pp. 847–865, 2011.

H. Grootveld and W. Hallerbach, “Variance vs downside risk: is there really that much difference?” European Journal of Operational Research, vol. 114, no. 2, pp. 304–319, 1999.

W. V. Harlow, “Asset allocation in a downside-risk framework,” Financial Analysis Journal, vol. 475, pp. 28–40, 1991.

F. Wen, L. Zhong, c. Xie, and S. David, “Study on the fractal and chaotic features of the Shanghai composite index,” Fractals, vol. 20, no. 2, pp. 133–140, 2012.

L. Yu, S. Zhang, and X. Y. Zhou, “A downside risk analysis based on financial index tracking models,” in Stochastic Finance, part 1, pp. 213–236, 2006.

S. Zhu, D. Li, and S. Wang, “Robust portfolio selection under downside risk measures,” Quantitative Finance, vol. 9, no. 7, pp. 869–885, 2009.

A. Ben-Tal and A. Nemirovski, “Robust convex optimization,” Mathematics of Operations Research, vol. 23, no. 4, pp. 769–805, 1998.

O. L. V. Costa and A. C. Paiva, “Robust portfolio selection using linear-matrix inequalities,” Journal of Economic Dynamics & Control, vol. 26, no. 6, pp. 889–909, 2002.

L. El Ghaoui, M. Oks, and F. Oustry, “Worst-case value-at-risk and robust portfolio optimization: a conic programming approach,” Operations Research, vol. 51, no. 4, pp. 543–556, 2003.

D. Goldfarb and G. Iyengar, “Robust portfolio selection problems,” Mathematics of Operations Research, vol. 28, no. 1, pp. 1–38, 2003.

E. Erdoğan, D. Goldfarb, and G. Iyengar, “Robust active portfolio management,” The Journal of Computational Finance, vol. 11, no. 4, pp. 71–98, 2008.

A. F. Ling and C. X. Xu, “Robust portfolio selection involving options under a “marginal + joint” ellipsoidal uncertainty set,” Journal of Computational and Applied Mathematics, vol. 236, no. 14, pp. 3373–3393, 2012.

F. J. Fabozzi, D. Huang, and G. Zhou, “Robust portfolios: contributions from operations research and finance,” Annals of Operations Research, vol. 176, pp. 191–220, 2010.

I. Bajeux-Besnainou, R. Belhaj, D. Maillard, and R. Portait, “Portfolio optimization under tracking error and weights constraints,” Journal of Financial Research, vol. 34, no. 2, pp. 295–330, 2011.

A. Ling, X. Yang, and L. Tang, “Robust LPM active portfolio selection problems with multiple weights constraints,” Journal of Management Science in China, vol. 16, no. 8, pp. 32–46, 2013.

V. DeMiguel, L. Garlappi, and R. Uppal, “Optimal versus naive diversification: how inefficient is the portfolio strategy?” Review of Financial Studies, vol. 22, no. 5, pp. 1915–1953, 2009.

D. Bertsimas and I. Popescu, “On the relation between option and stock prices: a convex optimization approach,” Operations Research, vol. 50, no. 2, pp. 358–374, 2002.

I. Popescu, “Robust mean-covariance solutions for stochastic optimization,” Operations Research, vol. 55, no. 1, pp. 98–112, 2007.