An Analysis Of Optimal Portfolio With A Beta Constraint

Since Markowitz proposed the classic mean- variance portfolio theory in 1952, portfolio theory have emerged a number of important research results, such as the new risk measurement methods, the corrected mean-variance model after easing the hypothesis and so on. This article is based on the Markowitz mean-variance model, and adds the Beta constraint to that model, our article is aimed at investigating the properties of the optimal portfolio and the varieties of the efficient frontier, in addition, we also take an empirical test for the optimal portfolio performance with Beta constraint. This study and previous studies’ biggest difference is that previous studies either considered only the total risk of the portfolio(revenue’s variance) or only the systematic risk portfolio(revenue’s Beta coefficient), but this paper puts Both properly together and combines to form a model which has a smallest total risk with a given systemic risk.After reviewing and summarizing the past results of portfolio theory, the paper proposes the optimal mean-variance model with Beta constraint. Then we analyze and test the model from the perspective of both theoretical and empirical separately, the theoretical part investigates the properties of the optimal portfolio and the varieties of the efficient frontier and the empirical part test the performance of optimal portfolio.In the theoretical analysis, based on the analysis of the optimal results with a Beta constrained model, we come to four properties of the optimal portfolio:(1) the optimal portfolio satisfies the three- fund separation theorem;(2) the optimal portfolio’s construction process takes three steps;(3) Beta constraint can hedge the portfolio’s systematic risk;(4) The optimal portfolio with Beta constraint is non-efficient. This paper also uses a numerical example to examine the changes of efficient frontier with beta constraint, we found: Efficient Frontier is always located to the right of Markowitz’s, and that led to a non-efficient portfolio, and the size of the no-efficiency is also inextricably linked with the Beta.In the empirical analysis, this paper takes two stages of bull and bear markets to examine the optimal portfolio performance with beta constraint. The empirical results show that: the expected return in a bull market with beta constraint increases as beta increases, but the Sharpe ratio is decreasing as beta increases; while the expected return in a bear market with Beta constraint increases as beta decreases, but Sharpe ratio decreases as beta decreases. According to the empirical results we can conclude that: an ideal investment portfolio with beta constraints should be such that, its expected return is higher than the MV model and its Sharpe ratio should be consistent with the MV model.