Finding Optimal Portfolios with Constraints on Value at Risk

Value at risk is an important measure of extent to which a given portfolio is subject to different kinds of risk present in financial markets. Considerable amount of research was dedicated during recent years to development of acceptable methods for evaluation of this risk measure. In this paper we go beyond estimation of value at risk and address the following questions. Suppose that a boundary for acceptable value at risk is fixed. How to find a portfolio among given set of securities which would provide the maximal yield and at the same time satisfy the constraints on value at risk; Suppose that the market conditions are changing. How to obtain a portfolio rebalancing strategy which would keep portfolio within given boundary on value at risk and maximize in some sense the yield of resulting sequence of portfolios. In order to solve these problems we adapt and further develop algorithmic tools which have their origin in stochastic optimization and were considered recently in machine learning.