A Convex-Risk-Measure Based Model and Genetic Algorithm for Portfolio Selection

A convex risk measure called weighted expected shortfall (briefly denoted as WES (Chen and Yang, 2011)) is adopted as the risk measure. This measure can reflect the reasonable risk in the stock markets. Then a portfolio optimization model based on this risk measure is set up. Furthermore, a genetic algorithm is proposed for this portfolio optimizationmodel. At last, simulations are made on randomly chosen ten stocks for 60 days (during January 2, 2014 to April 2, 2014) fromWind database (CFD) in Shenzhen Stock Exchange, and the results indicate that the proposed model is reasonable and the proposed algorithm is effective.


Introduction
In 1952, Markowitz proposed the first quantitative risk measure (i.e., variance) for portfolio selection [1].Because variance as the risk measure has essential drawbacks such as only being applicable to the cases in which the return obeys the normal distribution or elliptical distribution, it is rarely used as the risk measure currently.After that, value at risk (VaR) was proposed by Baumol [2] as a risk measure and it has been a widely used risk measure to manage or control risk for portfolio selection.For example, the Basle Committee on Banking Supervision allows banks to use VaR when determining their asset-adequacy requirements arising from their exposure to market risk [3].The reason for VaR to be a popular risk measure is that it is easily understood and can exactly answer the following question: under the normal market environment and given confidence level, what is the maximal potential loss of an investor in a certain period?Just as VaR definition mentioned, given some confidence level  ∈ (0, 1), the VaR of the portfolio at the confidence level  is given by the smallest number  such that the probability of the loss  exceeding  is not larger than (1 − ); that is, Since VaR appears, many researchers have paid attention to use VaR in their portfolio models.For example, authors in [4] integrate GARCH model and VaR model and got the better results.However, VaR model has some serious drawbacks: (1) it usually gives preserved estimation on the risk of the markets [5]; (2) it does not satisfy subadditivity (i.e., the total risk of the sum of two investments is not larger than that of the sum of their risks [6]); and (3) it is not a coherent risk measure [7].Thus, some researchers thought that VaR was not a good risk measure and even some researchers thought that it was seductive but dangerous (e.g., [8]).To get more reasonable risk measures, researchers discussed the conditions which a reasonable risk measure should satisfy (e.g., [6,7,9,10]).The widely accepted and representative condition for a reasonable risk measure is coherent condition, which defines a risk measure () to be a coherent measure if it satisfies the following properties: (I) monotonity: if  ≤ ,  () ≥  (), for any two portfolios  and ; (II) subadditivity:  ( + ) ≤  () +  (), for any two portfolios  and ; (III) positive homogeneity:  () =  (), for any portfolio , and  ≥ 0; (IV) translational invariance: If  is a deterministic portfolio with guaranteed return , then  ( + ) =  ()− for any portfolio .
Monotonity illustrates that if portfolio y always has better values than portfolio x under almost all scenarios, then the risk of y should be less than the risk of x.Subadditivity implies that the risk of two portfolios together cannot get any worse than adding the two risks separately; this is the diversification principle.Loosely speaking, positive homogeneity represents that if you double your portfolio then you will double your risk.Translational invariance indicates that after adding an amount of money  to the original portfolio , the risk will be decreased by amount .
Based on concept of coherent risk measure, many new risk measures were proposed.For examples, Artzner et al. proposed a coherent risk measure WEC [7].WEC is a good risk measure in theory, but it seriously depends on the distribution of random variable .To get easily computed coherent risk measure, Acerbi et al. [11] proposed the expected shortfall as the risk measure (ES) and Rockafellar and Uryasev [12] proposed a coherent risk measure: conditional value at risk (CVaR).Although the definitions of ES and CVaR are different, their key ideas are same.ES has an important property [13]: any coherent risk measure can be described by spectral risk measures and any coherent spectral risk measure can be represented by the linear combination of ES with different confidence levels.That means that ES is the basic components of coherent risk measures.CVaR also has attractive properties such as its formula can be easily obtained and thus can be easily used for portfolio selection.Recently, with the further deep study of coherent risk measures, some new coherent risk measures with good performance were proposed; for example, Rosazza Gianin [14] proposed a coherent risk measure based on g-expectation operator which can be applicable not only to the estimation of financial asserts, but also to the derivatives.Also, by choosing different g-expectation operator, this risk measure can satisfy the different prefers of investors to the risk.To handle the problems with nonnormal distribution and leptokurtosis (i.e., fat tails), Chen and Wang [15] proposed a new class of coherent risk measures based on -norms.However, aforementioned coherent risk measures are only one side coherent risk measures [16]; that is, the distribution information on only one side of supply side and demand side is considered.From the point of view of competition, when an investor is going to buy a stock, he or she should also pay attention to the behavior of the seller.That is to say, to measure the risk more exactly, the behavior of investors to supply side and demand side should be simultaneously considered.In other words, two-side risk measures are needed.For example, Chen and Wang [16] proposed a two-side coherent risk measure which is easily applied.
Although there have been a lot of progresses in the research of coherent risk measures, there exist two serious problems in coherent risk measures.(1) Positive homogeneity is not reasonable in many cases and the condition is too strong [17,18]; (2) translational invariance is not reasonable in many cases and more and more researchers do not adopt this condition in their risk measures (e.g., [19][20][21]).To relax the condition in positive homogeneity and get more reasonable risk measures, some authors suggested replacing subadditivity and positive homogeneity by the following convexity: and proposed convex risk measures (e.g., [22,23]).From above analysis, one can see that a good and reasonable risk measure should satisfy monotonity and convexity.For this purpose, some risk measures satisfying monotonity and convexity were proposed (e.g., [24,25]) and experiments indicate that the measures are reasonable.
In this paper, we adopt a convex risk measure called weighted expected shortfall (WES) as the risk measure [24] and propose a portfolio optimization model based on this risk measure.Then we design a genetic algorithm for this portfolio optimization model.At last, simulations are made on real data in the financial markets and the results indicate that the proposed model is reasonable and the proposed algorithm is effective.
The remaining parts are organized as follows.In Section 2, the portfolio optimization model is set up.The proposed genetic algorithm is proposed in Section 3. The computer simulations are made on real data in financial markets in Section 4, and the conclusions are made in Section 5.

Portfolio Optimization Model
In static state, risk can be seen as a random variable  on a probability space (Ω, F, ), where  represents uncertain rate of return of portfolio.Then, for a given confidence level , a new risk measure is defined in [24,26] as follows.
Also, in [24,26], a computable formula (or an estimation) of WES  () was given.Suppose that there are  risky assets and one risk-free asset and the portfolio for these assets is denoted by  = ( 1 ,  2 , . . .,   ,  +1 ), where ( 1 ,  2 , . . .,   ) is the portfolio for  risky assets, respectively, and  +1 is the portfolio for the risk-free asset.Let   represent the rate of return of the th risky asset for  = 1 ∼  and the rate of return of the risk-free asset for  =  + 1, respectively, in the period  for  = 1 ∼ .Let   represent the rate of return of the th risky asset for  = 1 ∼  and the rate of return of the risk-free asset for  =  + 1, respectively.It can be estimated by Let   represent the dividend yield of the th risky asset for  = 1 ∼  in the period  for  = 1 ∼ .Let   represent the dividend yield of the th risky asset for  = 1 ∼ .It can be estimated by Let  0 = ( 0 1 ,  0 2 , . . .,  0  ,  0 +1 ) be the initial portfolio, let   be the per unit transaction cost of the th risky asset, let   be the asset income marginal tax rate, let  0 be ordinary marginal income tax rate, and let   be the given target rate of return.Let If we choose the proper confidence level  and parameter  such that (1 − )  is an integer, then, according to [26], the risk based on the new measure WES  can be calculated by where  () (1) is the smallest element of { ()  |  = 1 ∼ } and  () () is the th smallest element of { ()  |  = 1 ∼ }.
Also, according to [26], an optimization portfolio model can be formulated as ) Also, the condition       = 0 is required to be satisfied in optimal solution in [26].This is a nonlinear constraint, but this condition was not put in the model in order to make the model easily solved (although this is not reasonable).Thus the optimal solution obtained for the model ( 14)-( 20) may not satisfy the condition       = 0.This will result in the obtained solution being not a true optimal solution.
Note that   ( = 1 ∼  + 1),    , and    ( = 1 ∼ ) are variables in the above model; thus, there are total (3 + 1) variables and the problem dimension is (3 + 1).But when we carefully check this model, we can find that these variables are not independent.In fact, from formulas (7) and (8), it can be seen that    and    are not independent to   and can be completely presented by   .Thus, variables    and    can be deleted for  = 1 ∼ .Also note that when we delete variables    and    ,  = 1 ∼ , constraints ( 17) and ( 18) as well as       = 0 will be automatically satisfied.Furthermore, in order to estimate   in formula (10) more precisely, we can use a large number of historic data (i.e., a large number ) in formulas ( 5), (6), and (10), but too large  will result in the increasing of computation of risk function WES  ().In order to reduce the computation load, we have to choose a proper .In this way, we can simplify the model ( 14)- (20).
In summary, we can modify the above model by deleting variables    and    ,  = 1 ∼  and constraints ( 17) and ( 18) as well as using proper value of  to set up a new simplified optimization portfolio model as follows: This optimization model has only ( + 1) dimensions which are much lower than those of the original model ( 14)- (20).
The optimization portfolio model in previous section is a nonlinear optimization problem.It is very hard to get its global optimal solution using the traditional optimization methods.Genetic algorithms (briefly, GAs) are a new kind of intelligent optimization methods which are designed for these difficult optimization problems [27][28][29].They exploit a set of potential solutions, named a population, and detect the optimal solution through cooperation and competition among individuals of the population.However, for GAs in global optimization, the major challenges are that an algorithm may be trapped in the local optima of the objective function and the convergent speed may be slow.These issues are particularly challenging when the dimension of the problem is high and there are numerous local optima.In order to improve the GAs, researchers have incorporated other techniques to enhance their performance.One important technique is to design more efficient crossover operators to enhance the local search ability of GAs [28,29].In this section, we first design an efficient crossover operator which can explore the search space efficiently.Then we design a mutation operator which can adaptively exploit the search space.Based on these, a new genetic algorithm is proposed.

Crossover Operator.
In this section, the uniform design method [28][29][30] is used to design a new crossover operator.The main objective of uniform design is to sample a small set of points from a given set of points, such that the sampled points are uniformly scattered on the interested region.The crossover operator based on the uniform design is similar to a local search scheme; thus, it can effectively explore the search space.The detail is as follows.

Denote
Algorithm 2 (crossover operator).( 1) Generate  approximately uniformly distributed points in   by aforementioned formulas, and denote the set of these points by In simulations, the parameter value is taken as  = 2 + 3 and  = 5.
(2) Generate  uniformly distributed points in set [, ] by Then, the points in  are offspring of  and .

Experimental Results for Real Stock Market
We use the proposed algorithm to the optimization portfolio model (21).The results are given in Table 2.The rates of return of the 10 selected stocks for 60 days are given in Tables 3 and 4.
It can be seen from Table 2 that, for four of five cases with different risk aversion coefficients, the risk values are very small (smaller than 0.05).Only for case  = 10, the risk value is relatively large (0.0933).Thus, in general speaking, we got very good portfolios.

Table 1 :
Stock names and their stock symbols randomly chosen.

Table 3 :
The rates of return of 10 stocks for the first 30 days.
To evaluate the performance of the proposed optimization portfolio model based on a new risk measure called WES and the genetic algorithm, we conducted the experiments on randomly chosen ten stocks for 60 days (during January 2, 2014 to April 2, 2014) from Wind database (CFD) in Shenzhen Stock Exchange and the names and numbers of these ten stocks are shown in Table 1, where VKA represents A-shares of China Vanke Company Limited, PAB represents

Table 4 :
The rates of return of 10 stocks for the last 30 days.PingAn Bank Company Limited, BLL represents stock of Baolilai Investment Company Limited, SZPRDA represents A-shares of Shenzhen Properties and Resources Development (Group) Limited, CSGA represents A-shares of CSG Holding Company Limited, SH represents stock of Shahe Industry, SZHA represents A-shares of Shenzhen Zhongheng Huafa Company Limited, SVOTA represents Ashares of Shenzhen Victor Onward Textile Industrial Company Limited, Konka A represents A-shares of Konka Group Company Limited, and SSIA represents A-shares of Shenzhen Shenbao Industrial Company Limited.We collect the daily returns and daily return rates from Shenzhen Stock Exchange in Wind database (CFD) in this period in the experiments.