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Investment is a monetary asset purchased with the idea that the asset will provide income in the future or will later be sold at a higher price for a profit. To get the best investment decision, we shall do a portfolio establishment. By doing portfolio establishment, investor can identify the securities and determine the allocation of asset to obtain an efficient portfolio. Efficient portfolio is a portfolio that provides the greatest expected return for a given level of risk, or equivalently, the lowest risk for a given expected return. This paper provides the greatest expected return for a given level of risk. An efficient portfolio most preferred by an investor because its risk/reward characteristics approximate the investor's utility function is called optimal portfolio.

 

This paper studies about the optimal portfolio selection problem subject to a maximum Value-at-Risk (denoted by MVaR) constraint. Portfolio selection problem has both theoritical and practical significance in finance. The problem is to search the best allocation of wealth among some kinds of securities and consumption are also considered. Our idea is generated on basis of Yiu (2010). The model contains regime switching market modes. The model parameters switch over time according to a continuous-time Markov chain, whose states are interpreted as the states of an economy. We constrain a VaR value for the portfolio in a short time duration over different states of the chain and MVaR is defined as the maximum value of the VaRs in all economy states. We suppose that the price dynamics of the risky asset is governed by a Markov-modulated geometric Brownian motion (GBM).

We formulate the problem by maximizing the discounted utility of consumption over a finite period of time. We obtain a regime switching HJB equation and then derive a system of coupled HJB equation corresponding to the economy states by using the dynamic programming principle. We also present the method of the Lagrange multiplier to deal with the MVaR constraint.

 

The following system of coupled HJB equation to solve the problem is the following form:

and

Dynamic programming, Hamilton-Jacobi-Bellman (HJB) equation, maximum value-at-risk, optimal portfolio selection, regime switching, utility maximization