A Generalized Measure for the Optimal Portfolio Selection Problem and its Explicit Solution

In this paper, we offer a novel class of utility functions applied to optimal portfolio selection. This class incorporates as special cases important measures such as the mean-variance, Sharpe ratio, mean-standard deviation and others. We provide an explicit solution to the problem of optimal portfolio selection based on this class. Furthermore, we show that each measure in this class generally reduces to the efficient frontier that coincides or belongs to the classical mean-variance efficient frontier. In addition, a condition is provided for the existence of the a one-to-one correspondence between the parameter of this class of utility functions and the trade-off parameter λ in the mean-variance utility function. This correspondence essentially provides insight into the choice of this parameter. We illustrate our results by taking a portfolio of stocks from National Association of Securities Dealers Automated Quotation (NASDAQ).


Introduction
The portfolio selection problem is of both theoretical and of practical interest (Castellano and Cerqueti 2014;Li and Hoi 2014;Shen et al. 2014;Fletcher 2015;Fulga 2016;Ray and Jenamani 2016).The pioneering work of Markowitz (1952) (see Elton and Gruber. 1987;Steinbach 2001) on optimal portfolio selection (OPS) introduced the classical mean variance (MV) approach which has been studied extensively in the financial literature ( Landsman and Valdez 2003;Li and Hoi 2014;Bera et al. 2015;Graham et al. 2015;Qin 2015;Markowitz 2014).Here R = π T X is the portfolio return, π and X are n × 1 vectors of weights and excess returns, respectively, and 1 T π = 1, where 1 is vector of n ones.E (R) = π T µ, where µ T = (µ 1 , µ 2 , ..., µ n ) is the vector of expected returns and Var (R) = π T Σπ, where Σ is the n × n covariance matrix of X, the vector of portfolio excess returns.Let us notice that the mean-variance model is the balanced difference between the expectation of returns, which measures an average asset profit, and the variance of returns, which measures the associated risk.Maximization of MV (R) is motivated, therefore, by maximization of average asset profit and minimization of asset risk.Let us also notice that in the case when short selling is possible, the mean-variance OPS problem has an analytic explicit-form solution, because it reduces to a classical quadratic programming problem.
Another key approach in the OPS theory is the maximization of the Sharpe ratio (see, for instance, Sharpe 1998), which was introduced in Sharpe (1966), and has the form where R f is the risk free rate.Here we have a ratio between expectation of returns and their standard deviation.Again, the maximization of the Sharpe ratio is motivated by maximization of expected profit and minimization of risk.We observe that there is no analytic solution to this maximization problem even for the situation when short selling is permitted.
In this paper, we offer a novel utility measure, which generalizes both the mean-variance, Sharpe measures and many other celebrated risk measures, and represents a much more flexible measure for incorporating E(R) and Var(R), Here t(x), p(x) and v(x) are differentiable functions, where t(x) is a monotonic increasing function, i.e., t (x) > 0, p(x) is a positive convex or concave function, and v(x) is defined on [0, ∞) and is positive on (0, ∞).
We found an analytical explicit-form solution of the optimization of the general functional F (π) subject to a system of affine equality constraints, when short selling is permitted.The solution takes the following explicit form where both π 0 ∈ R n and z ∈ R n have explicit closed-forms and they do not depend on the functionals t, p, and v, which means that these vectors are the same for different OPS models (e.g., the MV and the Sharpe ratio models), while the parameter w * ∈ R + depends on the models that are used.The paper is organized as follows: In Section 2 we provide an explicit solution to the maximization of (3) subject to (5).In Section 3 we give a numerical illustration of the main results for a special case of the general model.Section 4 provides a conclusion to the paper.

Main Results
The generalized measure we introduce is based on the maximization of (3) subject to the linear constraints where B = (b ij ) m,n i,j=1 is m × n, m < n, rectangular matrix of full rank, c is some m × 1 vector and 0 is a vector-column of m zeros.The motivation of such a system of linear constraints arises from models, such as the MV, where the linear constraint 1 T π = 1 ascertains that the sum of the weights of the portfolio is equal to one, with positive and negative values corresponding to long and short positions, respectively.More elaborate systems of constraints associated with optimal portfolio selection can be found in Best and Grauer (1990); Landsman (2008b), and in Landsman and Makov (2016).For example, in the case where c = (1, m 1 , m 2 ) T and B is of dimension 3 × n and is equal to Equation (5) represents the case when the expected return of a sub-portfolio of size j (the first j returns) is equal to m 1 and the expected return of a remaining sub-portfolio of size n − j (the remaining n − j returns) is equal to m 2 .
We show that π * , the solution of the maximization of (3) subject to (5), is taking an explicit form, which coincides with the solution of the mean-variance problem ( 16), where the trade-off parameter is λ * = 1/(2w * ), and w * is a solution of an algebraic equation to be presented later.
Before we formulate the main theorem, we begin by introducing certain notations and partitions, which will be helpful in the sequel.Consider the natural partitions of vector π T = (π T 1 , π T 2 ) where π 1 = (π 1 , ..., π n−m ) T and π 2 = (π n−m+1 , ..., π n ) T , and consider the respective partitions of and Notice that as Σ is positive definite, Q is also positive definite (see Landsman 2008a).
Denote by and where v(x) and p(x) are functions in (3).We note that function v(x) must satisfies the following condition: (C) The function h(x) = v(ax 2 + 2qx + r) is strictly convex on R for any numbers a > 0, q, and r such that q 2 − ar < 0.
This condition is essentially weaker than a condition of convexity of v.For example, v(x) = √ x is concave but condition (C) still holds.
Theorem 1. Suppose that u 1 (x) > 0 and u 2 (x) > 0. If the following equation with respect to w, has a positive solution w * , this solution is unique and the problem of maximization subject to (5) coincides with the solution of the mean-variance optimal problem, where the trade-off parameter λ takes the value In other words the solution to ( 11) is given by Here The proof of the Theorem is given in the Appendix A. This Theorem essentially generalizes Theorem 1 of Landsman and Makov (2016), where a special case was considered when t(x) = log x, p(x) = exp(x), v(x) = exp(λs(x)).Notice that in Landsman and Makov (2016) vector z was presented in a less convenient form than that of (A19).
Theorem 1 may have the following economic interpretation: it expresses the competition between the minimum variance portfolio (π 0 ) and vector z; these vectors appear in the classical mean variance portfolio and do not depend on the choice of the utility function and tuning parameter w * , which reflects by investor's preference.From expression of vector z, (A19), immediately follows that Bz = 0.
Then, for the special case, when m = 1, matrix B = 1 T and c = 1 we have 1 T z = 0 and the vector z can be interpreted as a self-financing portfolio in the sense that long-positions are financed by corresponding short positions (see, for instance, Panjer and Boyle (1998), sct.8.2.1).This case is considered in more details below, in Remark 1.
We now state a property related to the efficient frontier of the class of utility measures generated by (3).First of all we notice that for our utility function ( 3) These inequalities are very natural and important.They say that utility function F is increased with respect to E(R) and decreased with respect to Var(R).
Corollary 1.The efficient frontier of the optimization problem (3) subject to (5) coincides with or belongs to the efficient frontier of the classical mean-variance utility function and has a parabolic form (see below (A22) and (A23)).
The proof of the Corollary is given in the Appendix A.
Remark 1.In the special case when m = 1, matrix B = 1 T and c = 1, we can conclude that which conform with Panjer and Boyle (1998, (8.2.7)-(8.2.9)).See also (Steinbach 2001, Theorem 1.4).From (A22) follows that elements of the efficient frontier are The orthogonality property (A20) for this case was proved by Best and Best and Grauer (1990).

Optimal Portfolios for Special Cases
The investigated generalized measure, includes, as special cases, key measures employed in risk management.
We now show some special cases of the generalized measure F (π) : Special case 1: Sharpe ratio (SR).For t (x) = x, p (x) = x − R f and v(x) = √ x, (3) becomes the Sharpe ratio measure ( 2) Special case 2: Generalized Sharpe ratio (GSR).We introduce a generalized Sharpe ratio which is a special case of (3), for which t (x) = x, p (x) = x − R f and v(x) = x β , and found an analytic solution of the corresponding optimization problem for β ≥ 1/2.The economic justification for introducing this utility function can be explained by investor's wish to allow faster decrease in the utility when the variance increases.In this section, we show that as the investors are more risk-averse they should consider larger value of β, the power function parameter, so β is subjective for each investor.The β parameter can be interpreted as a new risk aversion parameter, which was absent in the classical Sharpe ratio.This risk measure is much more flexible than the classical Sharpe ratio for which β = 1/2.In Section 2.1.4we explain this in more details.Special case 3: Mean Variance (MV).In this case, t (x) = ln (x) , p (x) = e x and v(x) = e λx , λ > 0, and the goal function (3) takes the form Special case 4: Mean Standard Deviation (MSD).In the case t (x) = ln (x) , p (x) = e x and v(x) = e λ √ x , λ > 0, (3) has the form This functional arises in risk management and actuarial theory.For instance, the value at risk and the expected shortfall takes the form (17) for returns having elliptical distribution (see, for example, Landsman and Valdez (2003)).

Optimal Portfolio Selection With MV Measure
This case was implicitly considered in the proof of Theorem 1. Recall that for MV t (x) = ln (x) , p(x) = e x and v(x) = e λx , λ > 0, and the functions u 1 and u 2 take the forms u 1 (x) = λ, u 2 (x) = 1.Therefore, Equation (10) simply reduces to It is clear that w * is positive and unique, and the point of maximum π * is (12) with w * = 1/(2λ).Clearly, this optimal solution is only meaningful if λ is specified.Notice that the MV measure can be also computed explicitly from the main Theorem of Landsman and Makov (2016), but, the solution in Landsman and Makov (2016) is much more complex in its form than Theorem 1.This proposed solution well conforms with Panjer and Boyle (1998, (8.2.7)-(8.2.9)).

Optimal Portfolio Selection With MSD
As was discussed in the introduction, for the MSD utility function, t (x) = ln (x) , p(x) = e x and v(x) = e β √ x , β > 0 and the main functional reduces to the form which has a form of a combination of linear functional and a square root of a quadratic functional.
The optimal problem with this functional was considered in details in Landsman (2008aLandsman ( , 2008b).
Here we only show how the solution of this problem can be obtained from Theorem 1.We note that the functions u 1 and u 2 have the form and Equation (10), which reduces to has the positive solution, Now, substituting w * into (12), we finally get the maximization solution x * , which conforms with Theorem 1 in Landsman (2008b).Applying Theorem 1 we can say that the optimal portfolio coincides with mean-variance portfolio when the trade off parameter is Note that although the optimal solution can be derived, in principle, from the mean-variance utility function, the choice of λ remains a guess, unless calculated (λ * ) using the main theorem of this paper.

Optimal Portfolio Selection With Sharpe Ratio
Recall that the Sharpe ratio functional has the form It is clear that for the solution of the maximization problem it is enough to consider π T µ > R f .Then t(x) = x, v(x) = x 1/2 , x > 0 and p(x) = x−R f , x > R f .Consequently functions u 1 and u 2 simply have the form Thus Equation ( 10) reduces to algebraic equation of the first degree and the unique solution w * is the solution of ( 18) is positive and then we obtain the explicit maximum solution π * which has the form given in ( 12).Hitherto, an optimal portfolio selection for the Sharpe ratio utility function has been unavailable.We establish that the optimal solution is a special case of the mean-variance utility function, where, using (18), λ * can now be calculated to be λ * = 1 2w * .

Optimal Portfolio With Generalized Sharpe Ratio
The functional of the GSR was introduced previously, as follows: In this functional, v(x) = x β , where β ≥ 1 2 is a power function.Recall t(x) = x, and p(x) = x − R f , x > R f .Now we consider the case where β > 1 2 , (the case β = 1/2 was considered in the previous subsection).Then functions u 1 and u 2 take the form and Equation ( 10) reduces to a quadratic equation with respect to w b where Notice that (20) has the unique positive solution.
and when δ > 0 the maximum solution x * takes the form of ( 12).
We now return to the economic interpretation of this functional, comparing it with Sharpe ratio.In fact, returning to the arguments given after Theorem 1 we obtain that For β > 1/2, the speed of decrease of the generalized Sharpe ratio S β in Var is higher than that of decrease of the Sharpe ratio S β=1/2 .
As for the efficient frontier, we conclude from ( 21) that the efficient frontier corresponding to the generalized and classical Sharpe ratios belongs to the efficient frontier corresponding to the mean-variance model (See proof of Corollary 1), with the following set showing the relation between the MV risk aversion parameter, λ, and the generalized Sharpe ratio risk aversion parameter, β, Applying Theorem 1 we can say that the optimal portfolio coincides with mean-variance portfolio when the trade-off parameter is The following  Note that acknowledging the fact that the optimal solution corresponds to a solution derived from the mean-variance utility function is not enough since the value of the trade of λ remains unknown.The result provided here allows us to calculate its value.

Numerical Illustrations
We illustrate the results of the OPS problem for the mean standard deviation utility function and for special cases of the generalized Sharpe ratio with β = 0.5, 1, 2, and free rate R f = 0.00016 for a portfolio of 10 stocks from NASDAQ: Facebook, Intel, Frontier, Micron, Apple, Qualcomm, Sirius, Applied Materials, Cisco and Yahoo, for a three month period of time, from 4 January 15 to 7 January 15-daily returns (For the data, see http: //www.nasdaq.com/).Here µ is the vector of expected returns and Σ is the covariance matrix or returns presented in Tables 1 and 2, respectively.Furthermore, we assume the traditional linear constraint 1 T π = 1.Then, using formula (24) we can calculate trade off parameters λ * and then obtain the vectors of solutions, for each model, are given in Table 3.In the following Figure we present the spectrum of optimal portfolios, given in Table 3.
From Figure 2 it is evident that the amplitude of the weights decreases with increasing values of β.This well conforms with the graph of λ * , presented in Figure 1.As a result it can be observed that the portfolio become more robust when β increases.This agrees with the expression of portfolio variance, given in the second formula of (A21), associated with the risk of the portfolio.According to Figure 2, the stocks: Intel, Apple, Qualcomm, App.Mat. and Cisco should be invested in a long position, while the other stocks should be invested in a short position.Furthermore, all of the models are in agreement that the stocks with the largest impact on the portfolio are Intel and Sirius.Table 3, presents the solutions π * for the different models: MSD, SR and GSR, and together with the graphical illustration (Figure 2) show that all of the mean-variance-based risk measures share the same patterns in terms of higher/lower gains, so the main difference between them is the amplitude.This numerical illustration shows that investors with a more risk-averse character should consider larger values of β than the Sharpe ratio measure.Therefore, the flexibility of the generalized Sharpe ratio is governed by the different values of β, the subjective parameter corresponding to the level of risk aversion.The generalized Sharpe ratio shows how to implement, naturally, a risk aversion parameter into the framework of the standard Sharpe ratio (2).

Conclusions
In this paper, we suggested a novel generalized optimal portfolio selection utility measure which incorporates many popular portfolio selection utility measures as special cases.We further provided an explicit solution to this new utility measure, thus reestablishing existing optimal solutions to portfolio selection and offered closed-form solutions to optimal portfolio selections based on new utility measures which hitherto have not been investigated.We note that while the optimal solution for each of these utility functions can be obtained using the mean-variance utility function, the corresponding λ is simply unavailable without the methodology presented here.Furthermore, we demonstrated our results using a portfolio consisting ten stocks from NASDAQ in a period of three months and analyzed the results for the mean standard deviation utility function and for the generalized Sharpe ratio.
Figure shows the relation between the λ * and β.As we can see from Figure 1, λ * increased almost proportionally when β increases.
21 and B 22 are of dimensions m × (n − m) and m × m, respectively.As matrix B is full rank, suppose, without loss of generality, that matrix B 22 is non singular.Furthermore, we define m × (n − m) and (n − m) × m matrices

Table 2 .
Covariance matrix of the returns.

Table 3 .
The solutions π * for different models: Mean standard deviation (MSD), Sharpe ratio (SR) and generalized SR.