The Minimization of the Risk of Falling in Portfolios under Uncertainty

A portfolio model to minimize the risk of falling under uncertainty is discussed. The risk of falling is represented by the value-at-risk of rate of return. Introducing the perception-based extension of the value-at-risk, this paper formulates a portfolio problem to minimize the risk of falling with fuzzy random variables. In the proposed model, randomness and fuzziness are evaluated respectively by the probabilistic expectation and the mean with evaluation weights and λ-mean functions. The analytical solutions of the portfolio problem regarding the risk of falling are derived. This paper gives formulae to show the explicit relations among the following important parameters in portfolio: The expected rate of return, the risk probability of falling and bankruptcy, and the rate of falling regarding the asset prices. A numerical example is given to explain how to obtain the optimal portfolio and these parameters from the asset prices in the stock market. Several figures are shown to observe the relation among these parameters at the optimal portfolios. Keywords— Value-at-risk (VaR), risk-sensitive portfolio, fuzzy random variable, perception-based extension, the risk probability, the rate of falling.