Optimization of value-at-risk portfolios in uncertain lognormal models

A value-at-risk portfolio selection model to maximize not only the expected daily geometric return but also value-at-risk is discussed. The analytical solutions of the value-at-risk portfolio problem are derived. From the analytical results, this paper gives formulae to show the explicit relations among the following important parameters in portfolio: Value-at-risk, the expected daily geometric return, the risk probability of falling and bankruptcy and the falling rate of 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.