New approach for multiperiod portfolio optimization with different rates for borrowing and lending.

Author:Seyedhosseini, S.M.

    Portfolio selection is defined as selecting a combination of assets among portfolios to reach the investment goal. In a typical portfolio management, one is responsible to allocate funding to different assets by buying and selling them. Modern portfolio theory (MPT) was introduced in 1952 by Harry Markowitz. Markowitz's MPT has led to a new paradigm in portfolio selecting for investors in order to construct a portfolio with the highest expected return at a given level of risk (the lowest level of risk at a given expected return). Markowitz presents three nonlinear models and explained that the unique optimal solution for all three models is equal (Markowitz, 1952, 1956, 1959).There are many researches have been performed by experts in order to solve and develop Markowitz's seminal model. Because of the limitations of a factual market, lots of these attempts have tried to make his model more practical.

    The portfolio selection strategy was extended for a planning horizon in stochastic form by Samuelson (Samuelson, 1969) and Merton (Merton, 1969).

    In spite of comprehensive success of Markowitz' model, the 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, one may formulate from the beginning the allocation problem over a horizon composed of multiple periods (T >1 periods).

    Merton presents a mathematical model for the optimum consumption and the portfolio rules in a continuous time horizon (Merton, 1971, 1996). Merton in his work shows how to construct and analyze optimal continuous-time allocation problems under uncertainty. His objective is to maximize the net expected utility of consumption plus the expected utility of terminal wealth. Mossin also presents a multi-period optimization technique (Mossin, 1968). Chryssikou uses approximate dynamic programming algorithms to provide a near-optimal dynamic trading strategy for special types of utility functions when a closed form solution to the discrete-time multi period problem with quadratic transaction costs is not attainable (Chryssikou, 1998). Hakansson uses mean-variance and quadratic approximations in implementing dynamic investment strategies (Hakansson, 1971, 1993). Techniques from approximate dynamic programming have been successfully employed for efficient optimal policy computations: for example, Sadjadi et al. propose a dynamic programming approach to solve efficient frontier with the consideration of transaction cost (Sadjadi et al., 2004). Their approach led to a closed form solution of the mean variance portfolio selection is presented by.

    Li et al. consider a two-step method where a dynamic programming is employed to solve an auxiliary problem in the first phase and the solution to the auxiliary problem is then manipulated to obtain the optimal mean-variance portfolio policy and the corresponding efficient frontier (Li et al., 1998, 2000). Leippold et al. introduce a geometric approach to multi period mean variance optimization of assets and liabilities (Leippold et al., 2004). Morey and Morey introduce the same idea in a multi-period or temporal setting (Morey and Morey, 1999). They propose two types of efficiency measures: The first efficiency measure attempts to contract all risk dimensions proportionally where the second one focuses on augmenting all return dimensions as much as possible in a proportional way. Yan and Miao present the multi-period semi-variance model where variance is substituted by semi-variance in Markowitz's portfolio selection model. They point out that for this class of portfolio model, that the hybrid GA with PSO is effective and feasible (Yan and Miao, 2007).

    Briec and Kerstens develop multi-horizon mean-variance portfolio analysis in the (Morey and Morey, 1999) in several ways. First...

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