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 1/N 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 r=(r1,…,rn)T∈ℝn denote the return vector of the n risk assets, where ri is the gross rate of return for the ith risky asset. The investor’s position is described by vector w=(w1,…,wn)T∈ℝn, where the ith component wi represents the proportion invested to the ith risky asset. Let wb∈ℝn denote a fixed portfolio that investor seeks to outperform. wb also is called benchmark portfolio in the literature. Define the tracking error of portfolio w relative to benchmark portfolio wb as Δw=(w-wb)Tr. In order to obtain a good tracking error portfolio w, Roll [3] considered the solution of the following variance tracking error (VTE) problem
(1)VTE:max{μTw:(w-wb)TΣ(w-wb)≤σvte,∑i=1nwi=1},
where μ and Σ are the expectation and covariance matrix of return vector r, respectively, and σmv is the preset tracking error level. When μ and Σ 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
(2)LPMm(ρ)=𝔼[((ρ-X)+)m],
where X is the random variable, for example, the asset return of risky asset; ρ may be a target that investors want to outperform, m is a parameter, which can take any nonnegative value to model the risk attitude of an investor, (a)+=max(a,0), 𝔼[·], and ℙ{·} below expresses the expectation and probability of random variable. For the case of m=0,1,2, we have
(3)LPMm(ρ)=𝔼[((ρ-X)+)m]={ℙ{X≤ρ},m=0;𝔼[(ρ-X)+],m=1;𝔼[((ρ-X)+)2],m=2.
That is to say that LPM0 is nothing but the probability of the asset return falling below the benchmark index LPM1 is the expected shortfall of the investment falling below the benchmark index and LPM2 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 μ and Σ 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 1/N 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 ℐ be a subset of {1,2,…,n}. If assets in ℐ are restricted to a limited weight, we call ℐ the set of restricted assets. For given constant q, weights constraint requires
(4)eℐTw=(≤)q,
where eℐ=(e1,…,en)T∈ℝn is an index vector with ei=1 when i∈ℐ, and ei=0 when i∉ℐ. The notation “=(≤)” means the equality only (or inequality only) weights constraint. Under the estimation of μ and Σ is exactly obtained and selling short is allowed, Bajeux-Besnainou et al. [36] considered the following active portfolio optimization problem with single weight constraint:
(5)max{𝔼[wTr]:(w-wb)TΣ(w-wb)≤σ,eℐTw=(≤)q,[wTr]eTw=1}
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 r. Instead, we assume that we only partially know some knowledge about the statistical properties of returns r and the underlying distribution function is only known to belong to a certain set 𝒟 of distribution functions ℱ of the returns vector r. Under the uncertainty set 𝒟 of the underlying distribution, we have the worst-case LPMm.
Definition 1.
For any m≥0 and real number ρ, the worst-case lower-partial moment (WCLPM for short) of random variable X with respect to ρ under ℱ∈𝒟 is defined by
(6)WCLPMm(ρ)=supℱ∈𝒟𝔼[((ρ-X)+)m].
In tracking error portfolio selection problems, investors want in fact to find a portfolio w, such that the return of portfolio w can be close to or outperform the return of benchmark wbTμ. For this, we call wbTμ-wTμ the loss of portfolio w relative to wbTμ and control the risk of portfolio by restricting the loss. Based on this idea, we maximize on one hand the return wTμ and on the other hand control the loss by using worst-case LPMm. Thus, the mean-WCLPMm robust tracking error problem with multiple weights constraints can be written as
(7)maxwwTμs.t.supℱ∈𝒟𝔼[((wbTr-wTr)+)m]≤σmeℐiTw=(≤)qi,i=1,…,p,eTw=1,
where σm(m=0,1,2) are the preset constants. The second class of inequalities constraints; p weights constraints express that several classes of assets are restricted to the limited weights, where ℐi⊆{1,2,…,n} and qi(i=1,…,p) are constants. We call these constraints multiple weights constraints. When p=1, it is the single weight constraint. In this paper, we only consider the cases of m=0,1,2 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 𝒟 is the set of allowable distributions of returns r with the known mean and covariance; that is,
(8)𝒟={r∣𝔼[r]=μ,Cov(r)=Σ≻0}.
For convenience, we sometimes denote r∈𝒟 by r~(μ,Σ). 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 wb;that is, eTwb=1 and introduce a new variable y=w-wb. Notice the wTμ=yTμ+wbTμ; then optimization problem (7) with multiple equality weights constraints can be written into the following form:
(9)(RSm):maxyyTμs.t.supr∈𝒟𝔼[((-yTr)+)m]≤σmeℐiTy=qi-eℐiTwb,i=1,…,p,eTy=0.
We call y a self-financing portfolio and w a fully investing portfolio. We mention that qi-eℐiTwb, i=1,…,p can be positive or negative, most likely between -1 and 1 since qi and eℐiTwb are in most cases between 0 and 1. In the rest of the paper, we suppose that p+1 vectors e and eℐi(i=1,…,p) are linearly independent and n>p+1 always holds. If there exists certain eℐk, such that eℐk can be expressed as the linear combination of eℐi(i=1,…,p,i≠k), then eℐkTy=qk-eℐkTwb becomes a redundant constraint. Additionally, the linear equality constraints will lead to a unique feasible solution y if n=p+1.
The following lemmas indicate that a tight upper bound of the worst-case LPMm can be obtained explicitly, which will be helpful for our analysis later. We extend the proof in [23] for WCLPM0 to a general case.
Lemma 2 (see [<xref ref-type="bibr" rid="B9">23</xref>, <xref ref-type="bibr" rid="B3">39</xref>]).
Let ξ be a random variable with mean μ and variance σ but have unknown distribution and ρ is an arbitrary real number. Then, we have the following.
For the case of m=0,
(10)supξ~(μ,σ)ℙ{ξ≤ρ}={11+(ρ-μ)2/σ2,ifρ≤μ;1,ifρ≥μ.
For the case of m=1,
(11)supξ~(μ,σ)𝔼[(ρ-ξ)+]=ρ-μ+σ2+(ρ-μ)22.
For the case of m=2,
(12)supξ~(μ,σ)𝔼[((ρ-ξ)+)2]=[(ρ-μ)+]2+σ2={σ2,ρ≤μ;(ρ-ξ)2+σ2,ρ≥μ.
Lemma 3 (see [<xref ref-type="bibr" rid="B31">40</xref>]).
For any a∈ℝn, denote
(13)S1={aTr∣r∈𝒟},S2={η∣𝔼(η)=aTμ,Var(η)=aTΣa}.
Then S1=S2.
Lemma 4 (see [<xref ref-type="bibr" rid="B26">37</xref>]).
For any r∈𝒟, η∈S2, and any real vector a∈ℝn, the following equality
(14)supr∈𝒟𝔼[((ρ-aTr)+)m]=supη~(aTμ,aTΣa)𝔼[((ρ-η)+)m]
holds.
Lemma 4 establishes the relationship of single variable η and multivariable r, 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. 0 is a zero matrix or zero vector by context. For simplicity, we denote eℐk by ek, k=1,…,p and qi-eℐiTwb by di and thus di∈[-1,1]. Let
(15)M=[e1,…,ep]∈ℝn×p,d=(d1,…,dp)T∈ℝp.
Then, from the assumption that e1,…,ep and e are linearly independent, matrix M has column full rank. Let A=MTΣ-1M∈ℝp×p. Then matrix A is positive definite via Σ-1≻0, where the notation Z⪰Y (or Z≻Y) means matrix Z-Y is positive semidefinite (or positive definite). Further, we denote
(16)B=I-MA-1MTΣ-1∈ℝn×n,C=Σ-1MA-1∈ℝn×p,
where I is a unit matrix with reasonable dimensions. Then, for these matrices, we have the following results.
Lemma 5.
(a) BM=0.
(b) MTΣ-1B=0.
(c) BTC=0.
(d) BTΣ-1B=Σ-1B.
(e) CTΣC=A-1.
(f) Matrix B is positive semidefinite.
The conclusions are straightforward for (a)–(e). For (f), we have from (d), Σ-1B=BTΣ-1B⪰0 since Σ-1≻0. Hence B is a positive semidefinite matrix.
Based on matrices B, C, and Σ-1, we denote additionally
(17)a=eTΣ-1e,b=eTΣ-1μ,c=μTΣ-1μ,a0=eTCd,b0=μTCd,c0=dTA-1d,a1=eTΣ-1Be,b1=eTΣ-1Bμ,c1=μTΣ-1Bμ,a2=a1c1-b12,b2=a1b0-a0b1,c2=a1c0+a02.
Without loss the generality, we assume Σ-1B≠0; then we have that
(18)ac>b2,a1>0,c1>0,a2>0,c2≥0,
and c2=0 if and only if d=0 via A-1≻0.
3. The Solutions
In this section, we will explore the explicit solution of problem (RSm). Notice that y=w-wb; then we can rewrite separably problem (RSm) into the following three problems:
(19)maxy{yTμ:supr∈𝒟ℙ{yTr≤0}≤σ0,MTy=d,eTy=0},(20)maxy{yTμ:supr∈𝒟𝔼[(-yTr)+]≤σ1,MTy=d,eTy=0},(21)maxy{yTμ:supr∈𝒟𝔼[((-yTr)+)2]≤σ2,MTy=d,eTy=0}
which is corresponding to the case of ρ=0 in the worst-case LPMm.
3.1. Maximization of Information Ratio: A Variation of WCLPM<sub>0</sub>
Let us first consider problem (19). The inequality ℙ{yTr≤0}≤σ0 is usually called chance constraint or Roy’s safety-first rule in the literature. Generally speaking, σ0 is taken far less than 1/2 which leads to (1/σ0)-1>0. Hence, from Lemma 2(a) and Lemma 4, we have that, when yTμ≥0,
(22)supr∈𝒟ℙ{yTr≤0}≤σ0⟺11+(-yTμ)2/yTΣy≤σ0⟺(-yTμ)2yTΣy≥(1σ0-1)⟺yTμyTΣy≥(1σ0-1).
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 WCLPM0; that is,
(23)miny{supr∈𝒟ℙ{yTr≤0}∣s.t.MTy=d,eTy=0},
which results from (22) the maximization of information ratio (IR for short):
(24)MAXIR=maxy{yTμyTΣy:MTy=d,eTy=0}.
Problem (24) is related to the mean-variance with multiple weights constraints (MVMWC, for short) for a given parameter σ0′:
(25)(MVMWC):maxy{yTμ:yTΣy≤σ0′,MTy=d,eTy=0}.
If p=0, that is, there is not weights constraint, then problem (25) is nothing but Roll’s VTE problem. If p=1, that is, M=e1, 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 τ0, such that the explicit solution of problem (24) can be expressed as
(26)yτ0*=1τ0(Σ-1Bμ-b1a1Σ-1Be)+a0(Cda0-Σ-1Bea1).
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 τ0? As the second stage, we replace yτ0* into the objective function (24). Then we get the information ratio of portfolio yτ0*, denoted by MAXτ0, which is a function of parameter τ0. Thus, we can get the maximum information ratio by maximizing MAXτ0 with respect to τ0. That is, we have
(27)MAXIR=maxτ0>0MAXτ0=maxτ0>0(yτ0*)Tμ(yτ0*)TΣyτ0*.
Let τ0* be the optimal solution of (27); then portfolio yτ0** 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
(28)τ0*=a1b0-a0b1a1c0+a02
when a1b0-a0b1>0 and the optimal solution of (24) is obtained by
(29)y
IR
*=(a1c0+a02)a1b0-a0b1(Σ-1Bμ-b1a1Σ-1Be)+a0(Cda0-Σ-1Bea1).
Theorem 8 (see [<xref ref-type="bibr" rid="B26">37</xref>]).
If yτ0* in (26) is obtained for certain positive number τ0, then, there exists a σ0′>0, such that both problems (24) and (25) have the same optimal solution.
Let y0* be the optimal solution of (24); in view of the proof of Theorem 6 in Appendix, if σ0′ is chosen and satisfies
(30)a2(a1σ0′-c2)a2=(y0*)TΣy0*(μTy0*),
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 σ0′ given by (30), then the optimal portfolio of (25) will have the maximum IR, too.
3.2. The Mean-WCLPM<sub>1</sub>
The WCLPM1 constraint in problem (20) from Lemmas 2 and 4 can be written as
(31)supr∈𝒟𝔼[(-yTr)+]≤σ1⟺-yTμ+yTΣy+(-yTμ)22≤σ1⟺{yTΣy-4σ1yTμ≤4σ12,yTμ≥-2σ1.
In fact, it can verify that
(32)miny{yTμ:yTΣy-4σ1yTμ≤4σ12}>-2σ1.
Hence, inequality constraint yTμ≥-2σ1 is redundant. Thus, for any preset σ1>0, we can rewrite problem (20) as
(33)WCLPM1:maxy{yTμ:yTΣy-4σ1yTμ≤4σ12,MTy=d,eTy=0}.
Here, we call 2σ1 the upper bound of tracking error under WCLPM1 constraint and denote the upper bound of tracking error by TE1=2σ1. The following lemma gives the feasible condition of problem (33).
Lemma 9.
If σ1 is chosen to satisfy
(34)σ1>σ1*,
then problem (33) is feasible, where
(35)σ1*=(a1b0-a0b1)2+(a1c1-b12+a1)(a1c0+a02)-(a1b0-a0b1)2(a1c1-b12+a1).
Proof.
Let
(36)f(y)=yTΣy-4σ1yTμ,
and 𝒮11={y:f(y)≤4σ12}, 𝒮12={y∈ℝn:MTy=d,eTy=0}. Then this lemma is to verify that the set 𝒮1=𝒮11⋂𝒮12 is nonempty for any σ1>σ1*. Now, for any σ1>0, consider the following subproblem:
(37)miny{f(y)=yTΣy-4σ1yTμ:MTy=d,eTy=0},
and denote its optimal value by fσ1 which is a quadratic function of parameter σ1. Thus, (for the computation of fσ1 and σ1*, see Appendix for detail.),
(38)fσ1≤4σ12
gives a quadratic inequality with respect to σ1. Solving (38), we have that the equality in (38) holds when σ1=σ1*>0, where σ1* is given by (35). Hence, set 𝒮1 to be nonempty and has at least an element when σ1>σ1*. This implies problem (33) is feasible. The proof is completed.
Lemma 9 indicates that problem WCLPM1 is well defined and meaningful. The following results give the explicit solution of problem WCLPM1.
Theorem 10.
Let the condition of Lemma 9 be satisfied. Then the optimal solution of problem (33) can be expressed explicitly as
(39)y1*=(λ1+2σ1)(Σ-1Bμ-b1a1Σ-1Be)+a0(Cda0-Σ-1Bea1),
where
(40)λ1=4(a1c1-b12+a1)σ12+4(a1b0-a0b1)σ1-(a1c0+a02)a1c1-b12.
Proof.
See Theorem 4 in [37]; see also Appendix for detail.
3.3. The Mean-WCLPM<sub>2</sub>
In this subsection, we consider mean-WCLPM2 portfolio problem. In view of Lemma 2(c) and Lemma 4, the WCLPM2 constraint can be formulated as
(41)((-yTμ)+)2+yTΣy≤σ2.
Then we can write problem (7) with m=2 as
(42)WCLPM2:maxy{yTμ:(41),MTy=d,eTy=0}.
If we add constraint yTμ≥0, then problem (42) is nothing but the classical M-V model with weights constraints. On the other hand, if we add constraint yTμ<0, then constraint (41) is
(43)(yTμ)2+yTΣy=yT(Σ+μμT)y≤σ2,
which is more conservative than the variance of portfolio for the same preset value σ2 since (Σ+μμT)-Σ=μμT is a positive semidefinite matrix. We call σ2, instead of σ2, the upper bound of tracking error under WCLPM2 constraint and denote the upper bound by TE2=σ2. Similar to Lemma 9, we can also give the feasible condition of problem (42).
Lemma 11.
Let
(44)σ2>a1c0+a02a1+((a0b1-a1b0)+)2a12+a1(a1c1-b12).
Then problem (42) is feasible.
Proof.
Let
(45)σ2*=miny{((-yTμ)+)2+yTΣy:MTy=d,eTy=0}.
It follows from solving straightforwardly the problem that its optimal solution is
(46)y^σ2*=[a1v-(a1b0-a0b1)]a1c1-b12(Σ-1Bμ-b1a1Σ-1Be)+a0(Cda0-Σ-1Bea1),
where
(47)v=(a1c1-b12)(a0b1-a1b0)+a1(a1+a1c1-b12)+a1b0-a0b1a1.
Then, we get that
(48)σ2*=a1c0+a02a1+((a0b1-a1b0)+)2a12+a1(a1c1-b12).
Hence, the problem (42) is feasible when σ2 satisfies σ2>σ2*. 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:
(49)y2*=a1v*-(a1b0-a0b1)a1c1-b12[Σ-1Bμ-b1a1Σ-1Be]+a0[Cda0-Σ-1Bea1],
where
(50)v*={a1a2(a1+a2)(a1σ2-c2)-((-b2)+)2a1(a1+a2)+a2(a1σ2-c2)a1(1-a1a1+a2)(b2)+b2+(a1+a2)b2+a2(-b2)+a1(a1+a2),b2a1+a2(a1σ2-c2)a1≥0,σ2*≤σ2≤b22a1a2+c2a1;b2a1+a2(a1σ2-c2)a1≥0,σ2>b22a1a2+c2a1.
Proof.
See Theorem 5 in [37]; see also Appendix for detail.
We mention that the optimal solution, y2*, of problem (42) is consistent with problem (25) when b2≥0 or σ2>b22/a1a2+c2/a1. This is not surprising. Because the term ((-yTμ)+)2 vanishes in problem (42) in these cases.
4. Numerical Results
In this section, we consider the performance of models WCLPM1, WCLPM2, 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 1/N portfolio [38] with N=14. We fix TE=0.05 in the next numerical experiment.
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
Asset market daily returns from January 1, 2000 to March 18, 2011.
Asset
Mean (10-3)
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: WCLPM1, MVMWC, and IRMWC (information ratio problem with multiple weights constraints, i.e., problem (24)). We consider the following three cases of weights constraints:
(51)p=1:w11+w12+w13+w14≤120.p=2:{w11+w12+w13+w14≤120,w1+w3+w5+w11+w14≤114;p=4:{w11+w12+w13+w14≤120,w1+w3+w5+w11+w14≤114,w7+w12+w14≤114w8+w2+w13+w14≤120.
The wealth invested to foreign assets is restricted by the weights constraint. Table 3 gives the return, variance, and IR 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
(52)PVt=∑i=114wi×Assetit,
where w=(w1,…,w14)T is the obtained optimal portfolios and Asseti is the value of the ith market index at the tth exchange day. Some interesting results from our numerical comparisons can be found as follows.
The comparisons of return, variance, and information ratio of optimal solutions.
Model
Mean return (10-3)
Standard deviation
Information ratio
Panel A: in-sample data
Benchmark
0.3850
1.1993e-04
—
VTE
0.4304
0.0527
0.0782
p=1
WCLPM_{1}
0.4522
0.0548
0.0818
MVMWC
0.4256
0.0510
0.0759
IRMWC
0.1300
0.0170
0.0781
p=2
WCLPM_{1}
0.4412
0.0548
0.0813
MVMWC
0.4204
0.0510
0.0755
IRMWC
0.1365
0.0173
0.0786
p=4
WCLPM_{1}
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
3.3696e-004
—
MV
0.3925
0.0631
0.0102
p=1
WCLPM_{1}
0.4282
0.0642
0.0104
MVMWC
0.4033
0.0598
0.0099
IRMWC
0.1237
0.0235
0.0103
p=2
WCLPM_{1}
0.3851
0.0637
0.0102
MVMWC
0.3542
0.0593
0.0101
IRMWC
0.1032
0.0237
0.0102
p=4
WCLPM_{1}
0.4268
0.0649
0.0101
MVMWC
0.4006
0.0611
0.0101
IRMWC
0.1260
0.0208
0.0101
The comparison of portfolio values of models WCLPM1, MVMWC, IRMWC, and benchmark for in-sample and out-of-sample data when there is single weight constraint.
The comparison of portfolio values of models WCLPM1, MVMWC, IRMWC, and benchmark for in-sample and out-of-sample data when there are two weights constraints.
The comparison of portfolio values of models WCLPM1, 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 WCLPM1 from Table 3 has the best expectation return among these considered models for all p=1,2,4, and in-sample and out-of-sample data. This is understandable since model WCLPM1 has an excess MV region where it also exceeds the return of MVMWC. Moreover, from the view of portfolio value, WCLPM1 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 p increases. But it is in fact not surprising, because the feasible sets of all models reduce as p 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 WCLPM1 and MVMWC are more sensible than model IRMWC with respect to the number of restricted assets.
The comparison of portfolio values of model WCLPM1 for in-sample and out-of-sample data and different values of p.
The comparison of portfolio values of model MVMWC, for in-sample and out-of-sample data and different values of p.
The comparison of portfolio values of model IRMWC for in-sample and out-of-sample data and different values of p.
5. Conclusions
We considered a numerical extension of three active portfolio selection problems with the worst-case m=0-, 1- and, 2-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, WCLPM1 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.
AppendixTechnical ProofsProof of Theorem <xref ref-type="statement" rid="thm3.3">8</xref>.
Let y0* be an optimal solution of problem (24). Then, y0* must satisfy the first order optimal condition of problem (24). Thus, we have
(A.1)μ(y0*)TΣy0*-(μTy0*)(y0*)TΣy0*Σy0*(y0*)TΣy0*-λ01e-Mλ02=0,
where λ01∈ℝ, λ02∈ℝp are the Lagrange multipliers. Let λ^01=λ01(y0*)TΣy0* and λ^02=λ02(y0*)TΣy0*; then equality (A.1) can be reduced as
(A.2)μ-τ0Σy0*-λ^01e-Mλ^02=0,
where τ0=(μTy0*)/(y0*)TΣy0* is a constant. Thus, combining (A.2) and the equality constraints MTy0*=d and eTy0*=0, we get that
(A.3)yτ0*=b1τ0(Σ-1Bμb1-Σ-1Bea1)+a0(Cda0-Σ-1Bea1).
This completes the proof.
The Computation of fσ1 and σ1*. Consider the following subproblem:
(A.4)miny{f(y)=yTΣy-4σ1yTμ∣s.t.MTy=d,eTy=0},
and denote its optimal value by fσ1. Let yσ1* be its optimal solution. Then, by the KKT optimal condition, we have
(A.5)2Σyσ1*-4σ1μ+λ11e+Mλ12=0,MTyσ1*=d,eTyσ1*=0,
where λ11∈ℝ and λ12∈ℝp are the Lagrange multiplier. Solving the equality system of KKT conditions, we get
(A.6)yσ1*=2σ1(Σ-1Bμ-b1a1Σ-1Be)+a0(Cda0-Σ-1Bea1).
Substituting it into the objective function and using some results in Lemma 5, it follows that
(A.7)fσ1=-4(a1c1-b12)a1σ12-4(a1b0-a0b1)a1σ1+a1c0+a02a1
and it is a quadratic function of parameter σ1. Solving directly the quadratic equation in σ1(A.8)fσ1=4σ12,
we get a positive root of this equation as
(A.9)σ1*=((a1b0-a0b1)2+(a1c1-b12+a1)(a1c0+a02)-(a1b0-a0b1)(a1b0-a0b1)2+(a1c1-b12+a1)(a1c0+a02))×(2(a1c1-b12+a1))-1.
This obtains equality (35). Thus, inequality fσ1<4σ12 holds when σ1>σ1*. This is the conclusion of Lemma 9.
Proof of Theorem <xref ref-type="statement" rid="thm3.5">10</xref>.
Let y1* be the optimal solution of problem (33). Then from KKT condition, we have
(A.10)(1+4σ1λ1)μ-2λ11Σy1*-λ12e-Mλ13=0,(y1*)TΣy1*-4σ1y1*μ=4σ12,My1*=d,eTy1*=0,
where λ11≥0, λ12∈ℝ, and λ13∈ℝp are the Lagrange multipliers. Solving directly KKT system (A.10), it gets the conclusions of Theorem 10. The proof is finished.
Proof of Theorem <xref ref-type="statement" rid="thm3.7">12</xref>.
Let v=yTμ. Then we can rewrite problem (42) as
(A.11)maxy,vμTys.t.((-v)+)2+yTΣy≤σ2,MTy=d,eTy=0,μTy=v;
here, we view v as an auxiliary variable. The Lagrange function of this problem is
(A.12)L(y,v;λ21,λ22,λ23)=μTy-λ21eTy-λ22T(MTy-d^)-λ23[((-v)+)2+yTΣy-σ2]-λ24(μTy-v),
where λ21∈ℝ, λ22∈ℝp, λ23≥0, and λ24∈ℝ are the Lagrange multipliers. Let the pair (y2*,v*) be the optimal solution of (A.11). Then the pair (y2*,v*) satisfies the first optimal condition; that is,
(A.13)∂L∂y=-2λ23Σy2*-λ21e-Mλ22+(1-λ24)μ=0,∂L∂v=-2λ23(-v*)++λ24=0,((-v*)+)2+(y2*)TΣy2*=σ2,MTy2*=d,eTy2*=0,μTy2*=v*.
By the first equation of (A.13), it gets that
(A.14)y2*=-12λ23[λ21Σ-1e+Σ-1Mλ22-(1-λ24)Σ-1μ].
Substituting y2* into equation MTy2*=d, we obtain that
(A.15)λ22=-[λ21A-1MTΣ-1e+2λ23A-1d-(1-λ24)A-1MTΣ-1μ],
where A=MTΣ-1M. Substituting λ22 into y2* and using Lemma 5, we further get
(A.16)y2*=12λ23[(1-λ24)Σ-1Bμ-λ21Σ-1Be+2λ23Cd].
Combining equation eTy2*=0, it gets that
(A.17)λ21=(1-λ24)b1a1+2a0a1λ23.
Notice that 1-λ24=1-2λ23(-v*)+. Thus, y2* can be reduced as
(A.18)y2*=(12λ23-(-v*)+)[Σ-1Bμ-b1a1Σ-1Bμ]+a0[Cda0-Σ-1Bea1].
Substituting y2* into the last equation μTy2*=v* of (A.13), we get
(A.19)12λ23=a1v*-(a1b0-a0b1)a1c1-b12+(-v*)+.
On the other hand, substituting y2* into the third equation of (A.13), we can obtain another expression of 1/2λ23 as
(A.20)12λ23=(a1c1-b12)[a1σ2-(a1c0+a02)-a1((-v*)+)2]a1c1-b12+(-v*)+.
Hence, combining (A.19) and (A.20) and noticing the notations in Lemma 5, we have that v* satisfies the quadratic equation
(A.21)a12(v*)2+a1a2((-v*)+)2-2a1b2v*+b22-a2(a1σ2-c2)=0.
Next, we discuss the solutions of (A.21) by the different sign of b2.
(1) b2≥0. Notice that v*=μTy2* is the expectation excess return of self-finance portfolio y2*. Thus, generally speaking, v* is increasing as the preset tracking error σ2 increases. Thus, (A.21) has only an efficient positive solution as follows. (Equation (A.21) has in this case another solution
(A.22)v*=b2a1+a2-a1a2(a1+a2)(a1σ2-c2)a1(a1+a2)<0,
when v*<0 and σ2 is taken to satisfy σ2>(b22/a1a2)+(c2/a1), but it is not an efficient solution since v* is decreasing as σ2 increases)
(A.23)v*=b2a1+a2(a1σ2-c2)a1≥0.
(2) b2<0. If v*≤0, then (A.21) has the solution
(A.24)v*=b2a1+a2+a1a2(a1+a2)(a1σ2-c2)-b22a1(a1+a2)≤0,
when σ2 is chosen to satisfy σ2*≤σ2≤(b22/a1a2)+(c2/a1). If σ2>(b22/a1a2)+(c2/a1), 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 σ2>(b22/a1a2)+(c2/a1), 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.
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