We address an optimal portfolio
selection problem of maximizing so we call CVaR (Conditional
Value-at-Risk) based Sharpe ratio of return rate of portfolio,
which is defined as the ratio of the expected excess return to
CVaR. The Sharpe ratio defined as the ratio of expected excess
return to standard deviation, the most common traditional
performance measure, takes standard deviation as a risk measure,
however, its has been received a lot of criticisms. In our CVaR
based Sharpe ratio, the standard deviation is replaced by CVaR,
which is a remarkable coherent risk measure which overcomes
essential defects of standard deviation. Although our new
performance measure is expected to enlarge the applicable area of
practical investment problems for which the original Sharpe ratio
is not suitable, however, we should device effective computational
methods to solve optimal portfolio selection problems with very
large number of investment opportunities.

In order to deal with rather complicated non-concave objective
function, which comes from the introduction of CVaR, in this
paper, we propose a Genetic Algorithm (GA) approach. The paper
briefly reviews the literature in the area of application of Genetic
Algorithms to financial problems, and then details the development of
Genetic Algorithm for portfolio selection. In order to evaluate CVaR for
each portfolio, by utilizing the results of Rockafellar and
Uryasev (2000), we introduce an auxiliary decision variable to
obtain a tractable concave maximization problem. Furthermore, if
we estimate or approximate required expected values by sampling
methods or historical data, we can reduce this concave
maximization problem to an LP (Linear Programming) problem.
Therefore, our problem could be solved by GA which incorporates LP
for evaluating values of CVaR. Numerical experiments from real
Japanese financial data are conducted to test our approach to
conclude that, by suitable choice of probability parameter of three
evolution operation, we could effectively solve optimal selection
problems with practical sizes.