#### Abstract

Recently, by imposing the regularization term to objective function or additional norm constraint to portfolio weights, a number of alternative portfolio strategies have been proposed to improve the empirical performance of the minimum-variance portfolio. In this paper, we firstly examine the relation between the weight norm-constrained method and the objective function regularization method in minimum-variance problems by analyzing the Karush–Kuhn–Tucker conditions of their Lagrangian functions. We give the range of parameters for the two models and the corresponding relationship of parameters. Given the range and manner of parameter selection, it will help researchers and practitioners better understand and apply the relevant portfolio models. We apply these models to construct optimal portfolios and test the proposed propositions by employing real market data.

#### 1. Introduction

The mean-variance model for portfolio selection pioneered by Markowitz [1] is used to find a portfolio such that the return and risk of the portfolio have a favorable trade-off. Only expected returns and covariance matrix are two inputs in the Markowitz mean-variance model. However, in the portfolio selection literature, it has long been recognized that the mean-variance model used with the sample mean and the sample covariance matrix is suboptimal and usually delivers extremely poor out-of-sample performance. As pointed out by Black and Litterman [2], in the classical mean-variance model, the portfolio decision is very sensitive to the mean in particular and the covariance matrix.

Being aware of the importance on estimation errors, various efforts have been made to modify the Markowitz mean-variance model to obtain a stable portfolio which depends less sensitively on the inputs. Ledoit and Wolf [3, 4] proposed replacing the sample covariance matrix with a weighted average of the sample covariance matrix and a low-variance target estimator matrix, . Jagannathan and Ma [5] imposed a no-short-sale constraint to the minimum-variance portfolio (MVP) model and gave some insightful explanations and demonstrations why the “wrong” constraint helps find a solution with better out-of-sample performance.

DeMiguel et al. [6] proposed a general framework for determining a portfolio that gives better out-of-sample performance in the presence of estimation error. The framework provided by DeMiguel et al. [6] depends on solving the traditional minimum-variance model with the additional norm constraint on the portfolio-weight vector. In the study of DeMiguel et al. [6], the -norm constraint, the -norm constraint, or the -norm constraint on the portfolio-weight vector are used. Brodie et al. [7] reformulated the classical Markowitz mean-variance model as a constrained least-squares regression problem. Then, they added a -regularization term to the objective function. Both theoretical analysis and empirical studies show this penalty regularizes the optimization problem and encourages sparse portfolios (i.e., portfolios with only few active positions). Inspired by the motivation of DeMiguel et al. [6] and Brodie et al. [7], the regularization method and norm-constrained method have wide applications in constructing portfolio selection models to find sparse and stable optimal portfolios with better out-of-sample performance (see [8–16]), in which different norms are used. For some important articles that are closely related to portfolio selection, see [17–29].

From the perspective of optimization, quadratic programming-based portfolio models with -norm constraints on the portfolio-weight vector or regularization on objective function are closely related. Some natural questions are as follows: (1) What are the difference and relation between the two models? (2) What is the range of parameters for the two models? (3) What is the corresponding relationship of parameters? The answers to these questions obviously have important bearings on the debate about portfolio selection problems. We will give the range of parameters for the two models and the corresponding relationship of parameters. Our research is beneficial to researchers and practitioners to better apply the minimum-variance portfolio optimization model.

In addition, this paper has a bit of relevance to that of Dai and Wen [30]. Dai and Wen [30] found the portfolio weight norm-constrained method mainly tries to obtain stable portfolios and the objective function regularization method mainly aims to obtain sparse portfolios and then proposed some general sparse and stable minimum-variance portfolio models by imposing both portfolio weight -norm constraints and objective function -regularization term. However, in this paper, by analyzing the KKT conditions (necessary and sufficient ones) of Lagrangian functions, we investigate the relation between the portfolio weight -norm-constrained method and the objective function -regularization method in minimum-variance portfolio selection problems and give the range of parameters for the two models and the corresponding relationship of parameters.

The rest of this paper is organized as follows: In Section 2, we introduce some existing minimum-variance portfolio models. In Section 3, we give the explanation about the relation between norm-constrained models and regularization models. In Section 4, we report some numerical experiments to test these models.

#### 2. The Related Minimum-Variance Portfolio Models

Financial literature has largely shown that estimation errors in the expected return estimates are much larger than those in the covariance matrix estimates. For example, Jagannathan and Ma [5] reported the estimation error in the sample mean is so large and nothing much is lost in ignoring the mean. Hence, in this paper, we also focus on the minimum-variance portfolio model, which relies solely on the covariance structure and neglects the estimation of expected returns.

Suppose we have *n* assets to be managed. Let be the return vector, be the expected return vector, be its associated covariance matrix, and be its portfolio allocation vector, satisfying , where is a vector. Then, the expected return and variance, for the portfolio , are equal to and , respectively.

For the empirical implementation, we can replace expectations by sample averages, that is,where be the vector of asset returns at time , in which is the return of asset at time . We define *R* as the matrix, the row of which equals , where . Since , . Given this notation, similarly to the study of Brodie et al. [7], we can have the following equivalent definition of variance:where .

The minimum-variance portfolio (MVP) is the solution of the following quadratic programming problem:

From (2), the minimum-variance portfolio model has the following equivalent multivariate regression form (RMVP):

Jagannathan and Ma [5] proposed a no-short-sale-constrained minimum-variance portfolio model (CMVP). In the no-short-sale-constrained minimum-variance minimization problem, portfolio weights are constrained to satisfy

From (2), the no-short-sale-constrained minimum-variance portfolio model has the following equivalent multivariate regression form (RCMVP):

Adding an -norm constraint to the portfolio weights, DeMiguel et al. [6] proposed the following -norm-constrained minimum-variance portfolio model (-NCMVP):where and is a parameter.

Similarly to the study of Brodie et al. [7], we add an -regularization term to the objective function in (3) to obtain the following -regularization minimum-variance portfolio model (-RMVP) as follows:where and *τ* is a regularization parameter that allows us to adjust the relative importance of the penalization in our optimization.

#### 3. Relation between Portfolio Weight Norm-Constrained Model and Regularization Model

In this section, we will investigate the difference and relation between the -regularization minimum-variance portfolio model and the -norm-constrained minimum-variance portfolio. By analyzing the Karush–Kuhn–Tucker (KKT) conditions (necessary and sufficient ones) of their Lagrangian functions, we can obtain the following proposition.

For convenience, we give the following notations:(i)“” stands for the *i*-th row vector of corresponding to , if .(ii)“” stands for the *i*-th row vector of corresponding to , if .(iii)“” stands for the *i*-th row vector of corresponding to , if .(iv)“” stands for of the solution of the minimum-variance portfolio.(v)“” stands for the solution of the no-short-sale-constrained minimum-variance model.

Proposition 1. *For a given in -RMVP, there exists a in -NCMVP such that -RMVP and -NCMVP have the same optimal solution.*

*Proof. *The Lagrangian corresponding to the optimization problem stated in (8) isThe KKT conditions (necessary and sufficient ones) of the Lagrangian (9) are as follows:The Lagrangian corresponding to the optimization problem stated in (7) isThe KKT conditions (necessary and sufficient ones) of the Lagrangian (14) are as follows:Let be the optimal solution of -NCMVP and () be the corresponding Lagrange multiplier. In optimization, condition (19) is known as the complementary condition. We can express the complementary condition asor, equivalently,Roughly speaking, this means the optimal Lagrange multiplier is zero unless the constraint is active at the optimum.

Comparing the KKT conditions (10)–(13) and (15)–(19), we can obtain the following from (20):(i)For a given in -RMVP, we have an optimal solution for -RMVP. Then, there is such that . That is, .(ii)For a given in -NCMVP, we have an optimal solution for -NCMVP. Then, there is such that -RMVP and -NCMVP have the same optimal solution.This implies the conclusion as follows:(1)The solution () of systems (10)–(13) is considered for a given .(2)Considering , , , and , conditions (15)–(19) are satisfied.(3)Since these conditions are necessary and sufficient, then is the optimal solution of problem -NCMVP for .

Proposition 2. *(1) The active and feasible range of the parameter in -NCMVP is . (2) The active and feasible range of the parameter τ in -RMVP is , where . (3) Given, there exists such that -NCMVP and -RMVP have the same solution.*

*Proof. *(1)From the condition (21), we can obtain from the Lagrangian (9) and the Lagrangian (14) that the upper bound of the parameter is corresponding to the Lagrange multiplier . From the above conclusion, it is obtained that if we set , the -regularization minimum-variance portfolio model degenerates into the minimum-variance portfolio. Therefore, the maximum active value of is equal to . That is, if , then the constraint in -NCMVPis not active. Moreover, it is obvious that(2)Suppose that the two weight vectors and are minimizers for the objective function in -RMVP, corresponding to the values and , respectively, and both satisfy the constraint . By using the respective minimization properties of and , we can obtain which implies that If , we have from (25) that . If all the are nonnegative, but some of the are negative, then we have . It implies that the optimal portfolio with nonnegative entries obtained by the minimization procedure corresponds to the upper bound of and thus typically to the sparsest solution. If , then the optimal portfolio obtained by -RMVP is nonnegative. Hence, the KKT conditions (necessary and sufficient ones) of the Lagrangian (9) are as follows: We firstly solve the no-short-sale-constrained minimum-variance model to obtain the optimal portfolio . Then, we select any . Substituting it into (25), we can obtain It implies is a constant for any . Substituting (28) into (26), we can obtain Hence, we can set where the value is calculated with any such that .(3)Given , we solve -NCMVP to obtain the optimal portfolio . It is obvious that is also an optimal solution to -RMVP for some . Then, we select any and from the optimal portfolio in -NCMVP. From (10) and (11), we can obtain

*Remark. *For an unconstrained minimizer of the -penalized least-squares objective functionif , then . But, for a constrained minimizer of the -penalized least-squares optimization problem, this case does not occur.

From (2), the -regularization minimum-variance portfolio model also has the following equivalent multivariate regression form:The Lagrangian corresponding to the optimization problem stated in (34) isWhen , we have and . If , then the Lagrange multiplier . Moreover, . The reason for this is .

#### 4. Numerical Experiments

In this section, we apply the models described above to construct optimal portfolios and test the proposed propositions by employing real market data.

##### 4.1. Data and Models

In our numerical experiments, the tested portfolio models have the following meanings:(i)“MVP” stands for the minimum-variance portfolio model.(ii)“RMVP” stands for the minimum-variance portfolio model with a multivariate regression form.(iii)“CMVP” stands for the no-short-sales-constrained minimum-variance portfolio model (Jagannathan and Ma, 2003).(iv)“RCMVP” stands for the no-short-sales-constrained minimum-variance portfolio model with a multivariate regression form.(v)“-NCMVP” stands for the -norm-constrained minimum-variance portfolio model (DeMiguel et al., 2009).(vi)“-RMVP” stands for the -regularization minimum-variance portfolio model.

Note that because of the convexity of the norm , solving the above models is a easy task for which the standard software solution exists. We compute the optimal solutions of the above models by using the optimization package CVX (Grant and Boyd [31]). All the codes were run on Matlab 2015a.

In our numerical experiments, we select 10 assets from S&P 500 with 500 historical daily stock return data for the illustration purpose. We use these data to obtain the following covariance matrix:

##### 4.2. Test Results of the Proposed Portfolio Models

The optimal portfolios of tested models vary with different values of the parameters. In this section, we will give optimal portfolios of tested models for different values of the parameters.

Optimal portfolios for MVP and RMVP are given in Table 1, from which we can find the minimum-variance portfolio model and the minimum-variance portfolio model with a multivariate regression form have the same optimal solution, and .

Optimal portfolios for CMVP and RCMVP are given in Table 2, from which we can find the no-short-sales-constrained minimum-variance portfolio model and the no-short-sales-constrained minimum-variance portfolio model with a multivariate regression form have the same optimal solution, and . The number of positive weight assets () is five. Moreover, we can obtain that is equal to

Hence, we have from (40) that

That is, the upper bound of in -RMVP is .

The test results of the -NCMVP are given in Table 3. From Table 3, the following conclusions are drawn:(1)For , the optimal portfolios by -NCMVP are the same as those by CMVP.(2)For , the optimal portfolios by -NCMVP are the same as those by MVP.(3)For , the optimal portfolios by -NCMVP are the same as those by MVP.

For , we can obtain that is equal to

From (32), we have

The test results of the -regularization minimum-variance portfolio model are given in Table 4. From Table 4, the following conclusions are drawn:(1)If , the -regularization minimum-variance portfolio model degenerates into the minimum-variance portfolio model and has the same optimal solution as the -NCMVP with .(2)-RMVP with has the same optimal portfolios as the -norm-constrained minimum-variance portfolio model with .(3)For , the optimal portfolio obtained by -RMVP is the same as that obtained by -NCMVP with , and also as the CMVP. From the tested results, we can find the no-short-sales-constrained minimum-variance portfolio model in the study of Jagannathan and Ma [5] has the sparest portfolios.

#### 5. Conclusion

In this paper, by analyzing the KKT conditions (necessary and sufficient ones) of Lagrangian functions, we investigate the relation between the portfolio weight -norm-constrained method and the objective function -regularization method in minimum-variance portfolio selection problems. We give the range of parameters for the two models and the corresponding relationship of parameters.

From the above discussion, we can find that, for appropriate choices of and , the solutions of -NCMVP and -RMVP coincide. From an application perspective, -NCMVP is often preferred because the parameter is easy to determine.

If in -NCMVP, we have . This implies -NCMVP with is equivalent to the no-short-sale-constrained minimum-variance minimization problem since -NCMVP with has the sparsest solution. Hence, if we need the sparsest solution, we can solve the no-short-sale-constrained minimum-variance model to obtain it. The no-short-sale-constrained minimum-variance model is a traditional quadratic programming problem which is easy to solve.

The proposed propositions for minimum-variance portfolio selection problems with -norm constraints or regularization can be easily extended to the Markowitz mean-variance portfolio selection model.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The author declares that there are no conflicts of interest.

#### Acknowledgments

The authors would like to thank Professors Grant and Boyd for providing the optimization package CVX for numerical experiments. This research was supported by the NSF of China (71771030 and 11301041) and by the Scientific Research Fund of Hunan Provincial Education Department (16B005).