Portfolio improvement and asset allocation

Abstract

This thesis studies the application of the Sharpe rule and Value-at-Risk in dealing with the portfolio improvement problem. It proposes that a portion of the portfolio value should be invested in some other assets for portfolio improvement. The generalized Sharpe rule is first used to assess the performances of assets or portfolios. Analytic results are derived to show that some assets with better performance are selected for portfolio improvement. By applying the Sharpe rule, it can be determined that new stocks are worthy of adding to the old portfolio if the average return rate of these stocks is greater than the return rate of the old portfolio multiplied by the sum of the elasticity of the Value-at-Risk (VaR) and 1. One attraction of our approach is diversification. Consideration is also given to the 'optimal' number of new assets to be added in two specific cases (i.e., arithmetic series and geometric series regarding the sequences of expected returns and standard deviations). Some interesting simulation results show that a new portfolio with the 'highest' Sharpe ratio can be obtained by adding only a few new assets. Motivated from the simulations that a few new assets need to be added for portfolio improvement, we also formulate the portfolio improvement problem using the mean-variance approach with equality cardinality constraint. In the formulation, variance is regarded as the risk. The equality cardinality constraint restricts that a given number of new stocks are selected for portfolio improvement. Under the assumptions that all the stocks are uncorrelated, analytical solutions to the formulated problem are derived for two specific cases: the expected returns of stocks are all equal to the desired return, and the expected returns of stocks are not all equal. The problem is also formulated with inequality cardinality constraint. Comparison is conducted to the problems formulated with equality cardinality constraint and with inequality cardinality constraint. Though the inequality cardinality constraint is set, numerical results show that in most of our simulated cases, the inequality cardinality constraint becomes equality at the optimal solution. The need of innovation and progress in risk management leads to the popularity of VaR. In another formulation of the portfolio improvement problem, we propose to use VaR instead of variance as a risk measure. Due to some desirable properties of Conditional VaR (CVaR), it makes CVaR much easier to be handled than VaR. The portfolio improvement problem is formulated into a mean-CVaR problem. The problem is then solved under the normality and non-normality assumptions about the portfolio returns. Experimental results show that as the number of scenarios increases, the loss random variable approaches normality under the former assumption; however, such convergence is not observed under the latter assumption.