**Abstract:**

We consider the problem of optimal portfolio choice using the
CVaR and VaR measures for a market consisting of $n$ risky assets and a risk-less asset and
where short positions are allowed. When the mean return vector and variance/covariance matrix of
the risky assets are specified without specifying a return distribution, we derive the distributionally
robust portfolio rules. We also address uncertainty
(ambiguity) in the mean return vector in addition to distribution ambiguity, and derive the portfolio
rules when the uncertainty in the return vector is modelled via an ellipsoidal uncertainty set.
In the presence of
a risk-less asset, the optimal portfolio rule exhibits an extreme behaviour
while in a market without the riskless asset we obtain a closed-form portfolio rule that
generalizes earlier results.
Key words : Robust portfolio choice, ellipsoidal uncertainty, Conditional Value-at-Risk,
Value-at-Risk , distributional robustness.