  Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) are popular risk
measures from academic, industrial and regulatory perspectives. The problem of
minimizing CVaR is theoretically known to be of Neyman-Pearson type binary
solution. We add a constraint on expected return to investigate the Mean-CVaR
portfolio selection problem in a dynamic setting: the investor is faced with a
Markowitz type of risk reward problem at final horizon where variance as a
measure of risk is replaced by CVaR. Based on the complete market assumption,
we give an analytical solution in general. The novelty of our solution is that
it is no longer Neyman-Pearson type where the final optimal portfolio takes
only two values. Instead, in the case where the portfolio value is required to
be bounded from above, the optimal solution takes three values; while in the
case where there is no upper bound, the optimal investment portfolio does not
exist, though a three-level portfolio still provides a sub-optimal solution.
