Portfolio optimization by multi-stage stochastic mode

Portfolio selection problem is a classical financial problem introduced by Markowitz and includes two parts consist of risk and return. This model established a fundamental base for single-period portfolio selection. In the real world the portfolio strategies are usually multi-period because the investor is able to re balance his portfolio in each time period. Re balancing the portfolio, the investor incurs transaction costs. Also, multi- period models can be defined as stochastic. Scenario tree generation can improve the performance of multi-period stochastic programming. At the beginning variance was considered as a risk measure. However, both theories and practices indicate that variance is not a good measure of risk and has some disadvantage. Therefore, downside risk measures, such as semivariance, downside beta coefficient, Value-at-Risk (VaR), or Conditional Value-at-Risk (CVaR) should be replaced with variance. In this paper, a multi objective model is presented for portfolio selection, characterized on the basis of three parameters: the expected value, downside coefficient and Conditional value at risk (CVaR) at a specified confidence level.