One Type of Optimal Portfolio Selection in Birandom Environments

In order to solve the portfolio problem when security returns are birandom variables, firstly we propose a new definition of risk, then one type of portfolio selection based on expected value and risk is provided according to birandom theory. Furthermore, A hybrid intelligent algorithm by integrating birandom simulation and genetic algorithm is designed. Finally, one numerical experiment is provided to illustrate the effectiveness of the hybrid intelligent algorithm.


Introduction
The theory of portfolio selection was initially provided by Markowitz (1952, p.77) and has been greatly developed since then.It is concerned with selecting a combination of securities among portfolios containing large number of securities to reach the goal of obtaining satisfactory investment return.In his path-break work, Markowitz proposed a principle that when making investment decision, an investor should always strike a balance between maximizing the return and minimizing the risk, i.e., the investor maximize return for a given level of risk, or one should minimize risk for a predetermined return level.More importantly, Morkowitz initially quantified investment return as the expected value of returns of securities, and risk as variance from the expected value.After Maokowitz's work, scholars have been showing great enthusiasm in portfolio management, trying different mathematical approaches to develop the theory of portfolio selection.Traditionally, returns of individual securities were assumed to be stochastic variables, and many researchers were focused on extending Markowitz's mean-variance models and on developing new mathematical approaches to solve the problems of computation.Peng (2007,p.433)proposed concept of birandom variable and the framework of birandom programming.However, investors may come across birandom returns in portfolio selection situations.For example, security returns are usually regarded to be normally distributed random variables, but the expected value may be still random variable, thus investors have to face random returns with random parameters, to deal with this type of uncertainty, we propose the security returns could be regarded as birandom variables.As a general mathematical description for this kind of stochastic phenomenon with incomplete statistical information, birandom variable is defined as a mapping with some kind of measurability from a probability space to a collection of random variables.
In general, there are three types of risk definitions in portfolio selection problems.Variance is the earliest and most commonly accepted definition of risk for portfolio selection initially proposed by Markowitz (1952, p.77).A variety of extensions to Markowitz's mean-variance models has been proposed.Semivariance is the second type of risk definitions, and was also proposed by Markowitz (1959).Semivariance is an improvement of variance because semivariance only measures portfolio return below the expected value.Many models have been built to minimize semivariance in different cases.The third popular definition of risk is a probability of a bad outcome initially by Roy (1952, p.431).Much research has been undertaken to find ways of minimizing the probability of the bad outcome.Recently, Huang (2007, p.5404) proposed another new definition of risk for portfolio selection in fuzzy and random fuzzy environments.The detailed exposition on the definition of risk had been recorded in the literature, the interested readers may consult it.We can regard it as the fourth type of risk.Her work has enriched the risk theory for portfolio selection.We try to do something for portfolio selection in birandom environments, and give a new risk definition and a model for portfolio selection according to the proposed risk.
The rest of this paper is arranged as follows.After reviewing some necessary knowledge about birandom variable in section 2, in section 3, one type of risk for portfolio selection model under birandom environment is proposed.In section 4, we give a model for portfolio selection from the point of the new definition of risk.To provide a general method for solving the new models, in section 5, a hybrid intelligent algorithm integrating genetic algorithm and birandom simulation is designed.To better illustrate the modeling idea and demonstrate the effectiveness of the proposed algorithm, one numerical example is provided in section 6.

Preliminaries
Birandom variable theory was introduced by Peng (2007, p.433).To better understand the proposed model for portfolio selection, let us briefly review some necessary knowledge about birandom variable.
Definition 1 A birandom variable ξ is a mapping from a probability space Pr) , , ( Α Ω to a collection of random variables such that for any Borel subset B of the real line R , the induced function , and . Assume that ξ is a function on Pr) , , ( Α Ω as follows.
and 2 ξ is a normally distributed random variable with mean 0 and variance 1 , i.e., ] . Then ξ is a birandom variable according to the definition.
The following are the definitions of the expected value operator and variance of birandom variance and the primitive chance of birandom event.
Definition 2 (Peng (2007, p.4330)).Let ξ be a birandom variable defined on the probability space Pr) , , ( Α Ω . Then the expected value of birandom variable ξ is defined as dt provided that at least one of the above two integrals is finite., and be a vector-valued Borel measurable function.Then the primitive chance of birandom event characterized by 0 , and

New definition of risk
In reality, some investors are only sensitive to one preset bad case.They regard as safe those securities whose chance of this bad case occurring is lower than the investors' tolerance level.Other investors consider all the possible unfavorable cases, and only those securities whose chance of every unfavorable case occurring is lower than the investors' tolerance level are regarded as safe.We will define the risk from this perspective.
Definition 4 Let ξ be a birandom variable on the probability space gives the chance of the occurrence of all events when the birandom return ξ is r less than the target return b .
From theorem 1 and theorem 2 we can derive that the risk curve ) , ( r f δ is a decreasing function with respect to δ and r , that is, the greater the δ , the smaller the Let ξ be a birandom return of a portfolio A , and , Which is exactly the ordinary chance measure of birandom variable.

Birandom portfolio selection
Let us select a portfolio according to the definition of risk in the preceding content.Let i x denotes the investment proportions in security i , i ξ the birandom return for the th , respectively.Let r denote the loss severity indicator, and ) (r α the confidence curve preset by the investor.To obtain the maximum investment return and avoid risk, the investor should select an optimal combination of securities from the portfolio safe point.We use the expected value of the securities to express the investment return.Thus we should set a goal of maximizing the expected return of a portfolio, and require that the risk curve ) , ( r f δ is not larger than the confidence curve ) (r α .Let b be the target return and δ the preset credibility level.Then the model is formulated as follows: When the birandom returns degenerate to random, the chance constraint becomes thus the model is the following Furthermore, if the investor only concerns one preset loss severity level 0 r , then the model ( 2) can be converted into the formulation:

Hybrid intelligent algorithm
Since the two-fold uncertainty of birandom variable, it is difficult to analytically solve the models (1), ( 2) and (3).To provide a general solution to the models, we design a hybrid intelligent algorithm integrating genetic algorithm (GA) and birandom simulation.Roughly speaking, in the proposed hybrid intelligent algorithm, the technique of birandom simulation is applied to compute the expected value and the chance measure, then birandom simulation and GA are integrated for solving the birandom models.

Birandom simulation
In this section, we first discuss the calculation of the expected value and the chance measure of birandom variables.
Let i ξ be birandom variables and i x decision variables, . Let b be the target return and δ the preset credibility level.The number r denotes all possible loss severity indicator.In order to solve the proposed models, we must handle the following two types of uncertain function.
may be estimated by the following procedure.
Step 2. Generate ω from Ω according to the probability measure Pr.
Step 4. Repeat the second to the third steps N times.
Step 2. randomly generate a real number r according to the confidence curve given by the investor. Step3.
from Ω according to the probability measure Pr.
Step 4. Compute the probability , respectively, by stochastic simulation.
Step 5. Set ' N as the integer part of N δ .
Step 6.Return the th Step 7. If β is no larger than , else 0 * l l = .
Step 8. repeat the second to the fifth steps for a given number times.
Step 9.If 1 = l , then return YES, else return NO.
Remark: here YES means that the investment proportion x is feasible; NO means that x is infeasible.

Genetic algorithm
Representation structure: A solution ) , , , ( , where the genes , and the relation between x and V are formulated as follows: , based on the relation between x and V , the feasibility of the randomly generated chromosomes is obvious.
Step 3 Given the rank order of the chromosomes according to the objective values, and the values of the rank-based evaluation function of the chromosomes.
Step 4 Compute the fitness of each chromosome according to the rank-based evaluation function.
Step 5 Select the chromosomes by spinning the roulette wheel.
Step 6 Update the chromosomes by crossover and mutation operations.
Step 7 Repeat the second step to the sixth step for a given number of cycles.
Step 8 Take the best chromosome as the solution of portfolio selection.

Numerical example
To illustrate the modeling idea and to test the effectiveness of the designed hybrid intelligent algorithm, let us consider one numerical example.The example is performed on a personal computer by using C++ programming language.The parameters in the HIA are set as follows: the probability of crossover and left-continuous function of α .Theorem 2(Peng (2007, p.4330)).Let ξ be a birandom variable and α a given number in and left-continuous function of x .
, and δ the preset confidence level and b the target return.Then the curve risk curve of an investment in the portfolio, and r the loss severity indicator.The greater the indicator r , the more severe the loss a portfolio selection is risky, an investor must first decide what is his or her maximum tolerance level of each bad event occurring.Usually, the worse the event, the lower the tolerance level.Then for every where b is the target return and δ the preset credibility level.The number r denotes all possible loss severity indicator.If the investor is only concern with one special loss severity indicator 0 r , then the risk becomes the chance )

Example 2
Assume that there are 5 securities, the returns of securities are all birandom variables. holds.