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The investment portfolio selection is an example of the general problem of deciding between random variable when larger outcomes are preferred. Decisions are required on the amount of capital to be invested in each of a number of available assets such that at the end of investment period the return is as high as possible.

Consider a set of n assets, with asset giving a return at the end of the investment period. is a random variable since the future price of the asset is not known. Let be the proportion of capital invested in asset, where is the capital invested in asset and w is the total amount of the capital to be invested and let x = () be the portfolio resulting from this choice. This portfolio's return is the random variable with distribution function that depends on the choice of . Further the random variables are treated as discrete and described by realizations under T states of world generated using scenario generation or finite sampling of historical data. Let state occur with probability. Thus the random returns are defined on a discrete probability space {, with a field and P(. Let be the return of asset i under scenario. Thus the random variable representing the return of asset i is finitely distributed over with probabilities The random variable representing the return of portfolio x = () is finitely distributed over where

The portfolio selection problem is thus a choice of a feasible set of decision vectors i.e. = () with some constraints like , (Under no short selling). The problem of choosing between two feasible sets of weights say x and y hence becomes a choice between random variables and. The criteria by which one random variable is considered better than the other need to be specified and the models for choosing between random variables are required.

Since the seminal work of Markowitz, the research on quantitative investment portfolio selection and management has focused mainly on three extensions (i) Developing alternative portfolio selection models for adequately capturing investors' preferences (ii) Capturing practical features of portfolio planning in the models (iii) Developing computational algorithms for solving the problems in an efficient way.

The biggest drawback of mean-risk models is that the most appropriate risk measure to use is still a matter of debate and an active area of research. Further the portfolio selection models do not adequately capture all the attributes of stocks for decision making. A few later works have tried the idea of restricting the risk of a distribution from two different perspectives. In [48], a mean-absolute deviation-skewness portfolio optimization model is proposed, in which the lower semi-third moment of the portfolio return is maximized subject to constraints on the mean and on the absolute deviation of the portfolio return. An optimization approach is proposed in which the objective function is approximated by a linear one and thus leading to a LP model. They further claim that this approach generates a portfolio with a “large ...
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