| This thesis studies the application of Sharpe rule and Value-at-Risk (VaR) in dealing with the portfolio improvement problem. It proposes that a portion of portfolio value should be invested in some other assets 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 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. 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.;Moreover, we suppose to use VaR instead of variance as a risk measure in the formulation. 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. |
