| In the case of minimizing risk with a given level of expected return, we discuss the portfolio selection problem when the asset returns are characterized by a Gaussian distribution and heavy tailed distribution.; More specifically, under the Gaussian assumption, we give the explicit solutions to the problems of minimizing risk variance, CaR and EaR respectively. When a compound Poisson process is assumed, we derive explicit solutions to the variance, CaR and EaR. Furthermore, we give the explicit solution for the CaR when a Levy distribution is considered.; For the more realistic process---normal inverse process, we are able to obtain the analytical solution for the EaR with the help of the explicit form of its probability density function.; Moreover, we give numerical results using Monte Carlo simulation for each risk measure discussed above by assuming that the stock returns follow Gaussian and Compound Poisson models, respectively. Finally, we give a comparison of the risk curves between these two processes and characterize the sensitivity of the risk curves for various values of the model parameters. |
