A stochastic programming model for multiperiod portfolio selection with new constraints.

Author:Seyedhosseini, S.M.

    Portfolio selection, as originally articulated by Markowitz (Markowitz (1952, 1956, 1959) ), has been one of the important fields of research in modern finance. In these types of decision problems, the future condition of stock market should be predicted by historical data due to the high volatility of market environments, so the amount of uncertainty in estimating stock rate of return is unavoidable. To overcome this uncertainty many researches have been accomplished in two directions. In one direction, portfolio selection was treated in fuzzy environment; in another direction, the problem was dealt with in stochastic environment. There are a variety of models developed in both lines.

    When chance constrained programming (CCP) was introduced by Charnes and Cooper in 1959 as a tool to deal with uncertain decisions, it is believed that it can play important role in financial environment decisions. Charnes and Cooper 1959 are defined chance constrained programming as: "select certain random variables as functions of random variables with known distributions in such a manner as (a) to maximize a functional of both classes of random variables subject to (b) constraints on these variables which must be maintained at prescribed levels of probability". After that, Liu (1999) generalized CCP to the case with not only stochastic constraints but also stochastic objectives.

    Using chance constrained programming in financial decisions and portfolio analysis was initialed with Brockett et al. (1992), Charnes et al. (1993), Li (1995) and Williams (1997). Aouni et al. (2004), use Chance constrained programming to model the portfolio selection problem by converting the stochastic compromise program into a deterministic one. Thereafter they can develop their former chance constrained compromise programming model with considering conflicting in decision maker multi objectives that you can see it in (Abdelaziz, 2005). Huang (2006) proposes two types of credibility-based portfolio selection model, according to two types of chance criteria: the objective is to maximize the investor's return at a given threshold confidence level and the objective is to maximize the credibility of achieving a specified return level subject to the constraints. Yan (2009) represents security returns as birandom variables then he can solve portfolio problem according to birandom theory.

    However, it is a single period model which makes a one-off decision at the beginning of the period and holds on until the end of the period so single-period framework suffers from an important deficiency. It is impracticable and difficult to apply to long-term investors having goals at particular dates in the future, for which the investment decisions should be made with regard to temporal issues besides static risk-reward trade-offs. To satisfy this necessity, many researchers develop this models toward formulate from the beginning the allocation problem over a horizon composed of multiple periods some of hem are: Mossin (1968), Chryssikou (1998), Morey and Morey (1999), Leippold et al. (2004), Bertsimas and Pachamanova (2008) and Briec and Kerstens (2009).

    The multi-period portfolio optimization proposed by Bertsimas and Pachamanova could be developed to incorporate realistic features such as borrowing and lending rates. The proposed method of this paper considers borrowing and lending rates as part of multi-period investment planning. In this paper it is considered that the rates are stochastic variables with normal distributions and the results are discussed using a practical example. We believe this features make our proposed method more realistic since most of the brokerage houses provide the opportunity to make an acquisition on different assets by borrowing the money from the brokerage.

    This paper is organized as follows. First some preliminaries about chance constrained programming are presented in section 2. In section 3 the proposed model is developed and described. The model is reformulated in chance constraint form in section 4. In Section 5 the characteristics of proposed GA is explained. Numerical results which research the performance of proposed model and also the GA are discussed in Section 6. Finally, in Section 7 conclusions are given to summarize the contribution...

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