AN OPTIMAL TRADE-OFF MODEL FOR PORTFOLIO SELECTION WITH SENSITIVITY OF PARAMETERS

. In this paper, we propose an optimal trade-oﬀ model for portfolio selection with sensitivity of parameters, which are estimated from historical data. Mathematically, the model is a quadratic programming problem, whose objective function contains three terms. The ﬁrst term is a measurement of risk. And the later two are the maximum and minimum sensitivity, which are non-convex and non-smooth functions and lead to the whole model to be an intractable problem. Then we transform this quadratic programming problem into an unconstrained composite problem equivalently. Furthermore, we develop a modiﬁed accelerated gradient (AG) algorithm to solve the unconstrained composite problem. The convergence and the convergence rate of our algorithm are derived. Finally, we perform both the empirical analysis and the numerical experiments. The empirical analysis indicates that the opti- mal trade-oﬀ model results in a stable return with lower risk under the stress test. The numerical experiments demonstrate that the modiﬁed AG algorithm outperforms the existed AG algorithm for both CPU time and the iterations, respectively.


1.
Introduction. In 1952 Markowitz [17] proposed the mean-variance model for portfolio selection and opened a new era of quantitative analysis in the portfolio selection. The mean-variance model seeks to reduce the total variance of the portfolio with the certain level of the expected return. For mean-variance model, the return is measured by the mean value and the risk is quantified by the variance of the portfolio return, respectively. Based on Markowitz's work, different extensions of the mean-variance model have been proposed, such as considering the maximum individual risk [25], marginal risk [27], probabilistic risk [24], cardinality constraint [23,26] and so on [5].
Since the mean and the variance in the mean-variance model are measured with lacking and missing historical data, it may cause the uncertainty of estimation errors. The accuracy of the parameters in the mean-variance model directly affects both measurements of return and risk. Chopra et al. [4] investigated the effectiveness of estimation error of the mean, variance and covariance and indicated that even a small error of parameters might cause a large deviation for portfolio selection. To overcome the uncertainty of parameters, the popular and useful method is robust optimization, whose main idea is to consider the worst-case portfolio given an uncertainty interval of parameters. The seminal work was done by Goldfarb et al. [9] who proposed a robust mean-variance model for portfolio selection and reformulated it into a second-order cone programming problem. However, Scherer [22] pointed out that the robust mean-variance model was over conservation and could not bring the extra return. Oppositely, it might increase the computational cost. Recently, Cui [6] proposed a mean-variance model with sensitivity of parameters as a class of constraints for portfolio selection. In her model both upper and lower bound of parameters are given as constants to control the error of parameters. However, it is difficult to choose the upper and lower bounds for the parameters and it is still an optimization problem. Similarly, Li et al. [15] presented a tradeoff mean-variance model between total risk and maximum relative marginal risk to avoid choosing parameters. Motivated by Cui and Li's researches, we use the sensitivity of parameters defined in [6] to formulate a mean-variance model, in which we balance the risk and the sensitivity of parameters by a scalar to replace the choice of the upper and lower bounds of parameters.
Usually, the mean-variance models presented above are non-convex quadratically constrained quadratic programming (QCQP) problem. The methods to solve nonconvex QCQP problems are mainly branch-and-bound algorithms. Basic references include Horst et al. [11], Al-Khayyal et al. [1], Raber [21], Audet et al. [2], and Linderoth [16]. Recently, Deng et al. [7] proposed a branch-and-cut algorithm to speed up the computational effort for a portfolio selection which is formulated into a semidefinate programming (SDP) problem. It is well-known that the branchand-bound algorithm performances badly with large-scale optimization problems. Nesterov's accelerated gradient (AG) algorithm [18] has attracted much attention recently partly due to the increasing demand solving large-scale convex programming (CP) problems by using the fast first-order algorithms. Then Nesterov [20] used AG algorithm and a new approach for constructing efficient schemes to solve a simple non-smooth CP program. Lan [14] further showed that the AG algorithm, when employed with proper stepsize policies, is optimal for solving not only smooth CP problems, but also general non-smooth and stochastic CP problems. More recently, Ghadimi et al. [8] developed a class of AG algorithms to solve the non-convex and stochastic optimal problems. Inspired by the AG algorithm, in this paper, we propose an optimal trade-off model for portfolio selection with sensitivity of parameters, which are estimated from historical data. The feature of model is to balance the risk and the error caused by parameters. The sensitivity of parameters are measured by maximum and minimum sensitivity for all assets. Mathematically, the model is a quadratic programming problem, whose objective function contains three terms. The first term is a measurement of risk. And the later two are the maximum and minimum sensitivity, which are non-convex and non-smooth functions, and lead to the whole model to be an intractable problem. Then we transform this quadratic programming problem into an unconstrained composite problem equivalently. Furthermore, we develop a modified AG algorithm to solve the unconstrained composite problem. The convergence and the convergence rate of our algorithm are derived. Finally, we perform both the empirical analysis and the numerical experiments. The empirical analysis indicates that the optimal trade-off model results in a stable return with lower risk under the stress test. The numerical experiments demonstrate that the modified AG algorithm outperforms the existed AG algorithm for both CPU time and the iterations, respectively.
The rest of this paper is organized as follows. In Section 2, we recall some preliminaries of this paper. In Section 3, we propose an optimal trade-off model for portfolio selection with sensitivity of parameters and transform it to an unconstrained composite problem. In Section 4, we develop a modified AG algorithm to solve the unconstrained composite problem. The empirical analysis is derived in Section 5. The numerical experiments are shown in Section 6. Finally, some concluding remarks are made in Section 7.
2. Preliminaries. In this section, we briefly recall some basic definitions and basic theorems that will be used below. We first recall the mean-variance model, the concept of the sensitivity of parameters and the portfolio selection model with the sensitivity of parameter constraints in [6]. Then we recall the definition of Lipschitz continuous, subgradient of a convex function in [19] and the AG algorithm in [8], respectively.
In the mean-variance model, consider n risky assets with random rates of return r = (r 1 , r 2 , . . . , r n ) for an investor. Let x = (x 1 , x 2 , . . . , x n ) be the amount of the portfolio to be invested in the n risky assets. Then the random return of portfolio x is R(x) = r, x , and the expected value and the variance are where µ and Σ are the mean vector and the covariance matrix of r. The Markowitz's mean-variance model can be formulated as follows: where ρ is the expected return of the investor and e = (1, 1, . . . , 1). The variance of portfolio selection can be rewritten as where ρ ij is the correlation coefficient among assets and σ = (σ 1 , . . . , σ n ) is the vector of standard deviation. Cui [6] introduced the concept of sensitivity of parameters.
Definition 2.1. [6] The sensitivity of the parameters for asset i is defined as the partial derivative of variance of portfolio x with respect to the standard deviation of individual asset σ i :

YANQIN BAI, YUDAN WEI AND QIAN LI
Moreover, as ∂µx T ∂µi = x i , i = 1, . . . , n, restricting the upper and lower bounds of x is also necessary. Given the estimated parameters µ and Σ, Cui [6] developed a portfolio selection model with the sensitivity of parameters constraints, i.e., where m ≤ n denotes the number of assets that the sensitivity of parameters should be restricted, k, s are the lower and upper bounds of sensitivity of parameters, and l, u are the lower and upper bounds of x. Since Σ i is an indefinite matrix, (MV sc ) is a non-convex quadratically constrained quadratic program.
where the norm · denotes the Euclidean norm. We denote f (x) ∈ C 1,1 L f (R n ).
The subgradient set of function f at point x 0 is denoted as ∂f (x 0 ).
Consider a class of composite problems given by is convex, and X is a simple convex (possibly non-smooth) function with bounded domain.

Lemma 2.4. [8]
If Ψ(·) is defined in (CP), then for any x, y ∈ R n , we have is a proper closed convex function with bounded domain, then there exists a constant M such that P(x, y, c) ≤ M for any c ∈ (0, +∞) and x, y ∈ R n , where P(x, y, c) is given by Ghadimi et al. [8] also introduced an important quantity that will be a termination criterion in the AG algorithm, i.e., Ghadimi et al. [8] developed the AG algorithm for solving problem (CP). The AG algorithm is described as follows.

3.
Optimal trade-off model and equivalent transformation. In this section, we first propose an optimal trade-off model for portfolio selection with sensitivity of parameters. Then we analyze the feature of model. Based on the properties of model, we finally transform it to an unconstrained composite problem equivalently.

3.1.
Optimal trade-off model. In our model, we use the maximum and minimum sensitivity of parameters of all assets as a measure of sensitivity. The set of assets whose the sensitivity of parameters should be restricted is denoted as S = {1, . . . , m}.
The objective function contains two parts in our model, the first one is the classical risk term denoted by xΣx T , and the second one is the sensitivity of parameters denoted by max i∈S {xΣ i x T } − min i∈S {xΣ i x T }. Our goal is to minimize the risk, and also to make the the sensitivity of parameters small. This is naturally described as an optimization problem with two objectives. To solve this bi-criterion problem, we minimize the weight sum of the objectives by introducing a τ > 0, as a trade-off factor. We propose an optimal trade-off model for portfolio selection with sensitivity of parameters, i.e., where τ ∈ (0, ∞) is a trade-off factor. Since Σ i is an indefinite matrix, xΣ i x T is a class of non-convex quadratic functions. Due to the nonconvexity and nonsmoothness of the objective function, it is hard to solve (TMV sc ).
can be rewritten as, Thus, the objective function of (TMV sc1 ) is a composite function that contains a convex quadratic function and two non-convex maximum functions. Then, we transform it equivalently to an unconstrained composite problem in Subsection 3.2.

Unconstrained optimization problem.
To transform (TMV sc1 ) to an unconstrained composite equivalently, we first introduce an indicator function of X = {x| µ, x ≥ ρ, e, x = 1, l ≤ x ≤ u} given by, Then (TMV sc1 ) can be equivalently expressed as an unconstrained optimization, It is obviously that both terms of max i∈S {xΣ i x T } and max i∈S {−xΣ i x T } are non-convex and non-smooth. Thus (UTMV sc ) is still difficult to solve.
To convexify max i∈S Therefore, (UTMV sc ) is equivalent to the following problem.
Thus, Ψ(x) is a non-convex differentiable function which consists of a convex function and a concave function. X (x) is a proper convex with bounded domain x ∈ X but non-smooth function.
[12] Given a matrix A ∈ R n×n , its spectral norm is defined as the largest singular value of A, i.e., where A H denotes the conjugate transpose of A.
For any x, y ∈ X, there exits a constant L Ψ such that Therefore, ∇Ψ(x) is Lipschitz continuous.
Obviously, (UTMV sc1 ) has the same form of (CP). Therefore, (UTMV sc1 ) can be solved by Algorithm 2.1. However, if m, the number of the assets for restricting the sensitivity of parameters, is chosen large, then solving (2) and (3) of Algorithm 2.1 may cause the computational difficulty. To overcome it, we develop a modified AG algorithm in the following section. 4. The modified AG algorithm. In this section, we first develop a modified AG algorithm to solve the problem (UTMV sc1 ), which is described as Algorithm 4.1. Then we derive the convergence and the convergence rate of Algorithm 4.1. Denote The key step in Algorithm 2.1 is to solve problem (P), which can be reformulated equivalently into the following quadratic convex programming problem.
is a convex programming problem that can be solved by CVX software package developed by Grant et al. [10]. However, choosing m assets from n assets, (P1) has to be solved twice for two terms of sensitive of parameters at each iteration in Algorithm 2.1, which obviously increases the computational cost. To alleviate expensive computational cost associated with the different paraments, our idea is to only solve (P1) once to obtain x ag k . Then the iteration x k is obtained in terms of combination of x ag k−1 and x ag k . In other words, our algorithm only requires the solution of one subproblem (P1) and the computational cost is decreased. Our algorithm is described as follows. Step 0. Input a feasible solution x 0 and the accuracy parameter . Let x ag 0 = x 0 , k = 1, α k = 2 k+1 , β k = 1 2LΨ .
Note that the subproblem (5) guarantees the feasibility of solution and its solution x ag k is used to compute the search direction x ag k − x ag k−1 . From the termination criterion (1) and (5), the new termination criterion is as follows.
The following Lemma 4.1 shows that x ag k approaches to a stationary point of (UTMV sc1 ) as (7) decreasing.
Proof. By (5) and the first-order optimal condition, we have Thus, we obtain By Lemma 3.2, and β k = 1 2LΨ , we have If it satisfies −∇Ψ(x ag k ) ∈ ∂X (x ag k ) + B( 3 2 ), x ag k is called an -stationary point of (UTMV sc1 ). In the following part, we prove the convergence and convergence rate of Algorithm 4.1.
Proof. By Lemma 3.2 and Lemma 2.4, we have the following inequality, Specially, for the right hand side of inequality, we have and for the left hand side of inequality, we have Combining (8) and (9), we have to compute ∇Ψ(x md k ), x ag k − α k x − (1 − α k )x ag k−1 . Since x ag k is the optimal solution of (5), we have Multiplying (10) by α k and (11) by (1 − α k ), respectively, we obtain Then denoting Φ(x) = Ψ(x) + X (x) and combining (8), (9) and (12), we have where the last equality follows form the equation (4). Substracting Φ(x) from both side of the above inequality, re-arranging the terms, we obtain where Observing Lemma 2.5, there exists a constant M 1 such that x ag k ≤ M 1 and x ag k ∈ X. Moreover, by the presume that x k is bounded, there also exists a constant M 2 such that x k ≤ M 2 . Therefore, letting M = max{M 1 , M 2 }, x = x * , and using Jensen's inequality for · , we have Replacing the above result in (13), we obtain where β k = 1 2LΨ and the last inequality follows that From α k = 2 k+1 and (14), we have .
Therefore, substituting Γ N in the above equation, we have min k=1,...,N Therefore, in view of Lemma 4.1, x ag N is an -stationary point of (UTMV sc1 ) at most 5. Empirical analysis. In this section, we take the empirical analysis for our proposed optimal trade-off model compared with the MV model. We first describe a stress test problem using data from Standard & Poor's 500 (S&P 500) and compare the efficient frontiers and the ratios of mean to standard deviation of the portfolios between our model and the MV model. Then we analyse the out-of-sample performances of the portfolios using Monte Carlo simulation. Finally, we explore the region of the parameter τ in our model. Let l i = 0, u i = 0.3, i = 1, . . . , 33, m = n = 33, and = 10 −4 . The proposed model is solved by the modified AG algorithm which is implemented in Matlab R2012(a) win64-bit on a PC with 2.30GHZ CPU processor.

Sample analysis.
We consider a portfolio selection problem with S&P 500 historical data from the period of July 24, 2006 to February 24, 2012. Weekly rates of return are used to estimate their mean µ, standard deviation σ and the covariant matrix Σ. We select 33 stocks according to the statistic analysis in [13,3], listed in Table 1.
To compare the efficient frontier of the (TMV sc ) model and the MV model, we first observe the efficient frontiers without stress testing. Set the trade-off factor τ = 1, 5, 10, 20, and the expected return ρ ∈ [7 × 10 −3 , 11 × 10 −3 ]. For each value of ρ, we obtain the optimal risk by performing the (TMV sc ) model and the MV model. The efficient frontier is composed of investments with the highest return and the   where i denotes seven assets with largest sensitivity of parameters. Moreover, we reconstruct the MV model with new µ and σ. The portfolios of the reconstructed MV model are regarded as the optimal portfolios. Similar to Figure 1, the efficient frontiers under stress scenario are obtained. As shown in Figure 2, the efficient frontiers of the (TMV sc ) model are above the efficient frontier of the MV model. We also observe that the efficient frontier of the (TMV sc ) model is closed to the optimal one for τ = 5. We further compute the ratios of mean to standard deviation of the portfolios. As the numerical results in Table 2, for the case of τ = 5 and τ = 10, the ratios of mean to standard deviation of the (TMV sc ) model are greater than the ratios of the MV model. Moreover, the ratio of mean to standard deviation of the (TMV sc ) model are even greater than the optimal one when ρ = 9 × 10 −3 and τ = 5. As to the criterion of ratio of mean to standard deviation, the (TMV sc ) model performs better than the MV model.

5.2.
Out-of-sample analysis. We now compare the out-of-sample performances of the portfolios generated by the (TMV sc ) model and the MV model.
In particular, the weekly rates of return are generated by Monte Carlo simulation with 10000 samples from the normal distribution N ( µ i , σ i 2 ). Using the weekly data, we compute the return and the risk of portfolios in Subsection 5.1. The ratios of the mean to the standard deviation are obtained and listed in Table 3. Table 3. Out-of-sample performance related to the ratio of mean to standard deviation   Table 3, the ratios of mean to standard deviation of the (TMV sc ) model are greater than the ratios of the MV model for τ = 5. Moreover, the ratios of mean to standard deviation of the (TMV sc ) model are greater than the optimal one for the case of ρ = 8 × 10 −3 and ρ = 9 × 10 −3 . Fixed ρ = 8 × 10 −3 , we compare the accumulated returns of our model and the MV model in Figure 3.
Obviously, the accumulated return of our model is upward that of the MV model. The empirical analysis illustrates that the (TMV sc ) model makes sense for a stable return and a low risk. 5.3. Discussion of τ . The parameter τ balances the risk and sensitivity of parameters in the (TMV sc ) model. To explore the region of τ , we preform the (TMV sc ) model and compare both optimal value of risk and that of the sensitivity of parameters, respectively, based on data from S&P 500 and Hongkong (HK) stocks. Obviously, the larger the value of τ is, the smaller the value of the risk xΣx T is. Moreover, we compare the rate of the risk, defined by Rate R = Risk(T M V sc)/Risk(M V ) − 1, and the rate of the sensitivity of parameters defined by Rate S = |Sensitivity(T M V sc)/Sensitivity(M V ) − 1|, respectively. The numerical results are listed and illustrated in Table 4. Fixed ρ = 7 × 10 −3 and τ = 5, the risk of the (TMV sc ) model is 0.4902 × 10 −3 . In contrast to the risk of the MV model, there is almost 20% increasing. Similarly, the sensitivity of the (TMV sc ) model is 1.7136 × 10 −3 . It is almost 57% decreasing, compared with that of the MV model. Let the value of τ increase as τ = 10, τ = 15 and τ = 20, respectively. We observe that the values of Rate R and Rate S go to less, respectively. Summarily, we conclude that τ makes sense to balance the risk and the sensitivity of parameters in the (TMV sc ) model with a range τ ∈ [5,20].
To further verify the property of τ and its region, we preform the (TMV sc ) model again by new data of HK stocks to compare the risk, the sensitivity of parameters, Rate R and Rate S , respectively. The corresponding numerical results are recorded in Table 5. As showed in Table 5, fixed ρ = 3 × 10 −3 and τ = 10, the risk of the (TMV sc ) model is 0.2819 × 10 −3 , greater almost 14% than that of the MV model. The sensitivity of the (TMV sc ) model is 4.8713 × 10 −3 , less almost 38% than that of the MV model. We observe the similar conclusion to the previously noted numerical results in Table 4 with a range τ ∈ [20,40]. Moreover, we find that the region of τ depends on the different data. 6. Numerical experiments. In this section, we first compare the CPU time and iterations between the modified algorithm and Algorithm 2.1, respectively, based on the data from S&P 500. We then present the computational results of Algorithm 4.1, based on randomly generated data. The parameters in Algorithm 2.1 are set as α k = 2 k+1 , β k = 1 2LΨ , and λ k = kβ k 2 . The algorithms are implemented in Matlab R2012(a) win64-bit on a PC with 2.30GHZ CPU processor.
6.1. Numerical experiments for S&P 500. We first perform the (TMV sc ) model by Algorithm 2.1 and Algorithm 4.1 and compare the CPU time and iterations of algorithms, respectively, based on the data from S&P 500.
For each instance, the optimal value of the model (TMV sc ) solved by Algorithm 4.1 is denoted as Opt . Meanwhile, Algorithm 2.1 is terminated as the objective value is not more than Opt. The CPU time and iterations are listed in Table 6, respectively. Where the iteration is denoted as N iter . It is obviously that the CPU time in Algorithm 2.1 are even more than 1000 seconds and iterations are more than 350, while all the CPU time of Algorithm 4.1 are less than 200 seconds and iterations are less than 110. Therefore, Algorithm 4.1 solves the model (TMV sc ) more efficiently than Algorithm 2.1. To further analysis the effectiveness of Algorithm 4.1 on (TMV sc ), we test it with different m and n.
In the following implementation, τ = 5, ρ = 4 × 10 −3 and the termination criterion = 10 −4 . For each pair of m and n, we generate five instances by selecting n stocks randomly from S&P 500 to obtain the computational results in Table 7. For the rest parts of numerical experiments, we mainly report the minimum, maximum, average CPU time and average iterations. According to the numerical results in Table 7, when n ≤ 100 and m is fixed, we find that the CPU time are relative stable. However, the CPU time is almost double when m doubled. It means that the CPU time of Algorithm 4.1 is more sensitive to m than n.

6.2.
Numerical experiments for randomly data. We further present the numerical results of Algorithm 4.1 in the instances generated randomly by the same method mentioned in [27].
In particular, the instances are generated by the single factor model as followed.
where r i is the return of asset i, r m is the return of market index and e i is the residual return of asset i. Therefore, we have µ i = α i +β i E(r m ), σ ii = β 2 i V ar(r m )+V ar(e i ) and σ ij = β i β j V ar(r m ), where the parameters are set as followed.
• E(r m ) = 4×10 −3 , V ar(r m ) = 0.03, and V ar(e i ) ∈ [0, 0.2×10 −3 ], i = 1, · · · , n is randomly generated by the uniform distribution. • τ = 5, ρ = 4 × 10 −3 . For each instance generated, the computational results are listed in Table 8. It has been seen that Algorithm 4.1 is sensitive to m. Moreover, given fixed m, if n increases, then the CPU time decreases. This observation shows that the CPU time depends on the different value of m/n. Given m ∈ [5, 50], we illustrate the behavior    Figure 4, where the lines with •, ,× denote the CPU time with m/n = 1, 1/2, 1/3, respectively. 7. Conclusions. In this paper, we have proposed an optimal trade-off model for portfolio selection with sensitivity of parameters. Furthermore, we have developed a modified AG algorithm to solve the optimal trade-off model. The convergence and the convergence rate of our algorithm have been derived. The empirical analysis has indicated that the optimal trade-off model results in a stable return with lower risk under the stress test. The numerical experiments have been demonstrated that the modified AG algorithm outperforms the existed AG algorithm in both the CPU time and the iterations, respectively.