  In this paper, we propose a general bi-objective model for portfolio
selection, aiming to maximize both a diversification measure and the portfolio
expected return. Within this general framework, we focus on maximizing a
diversification measure recently proposed by Choueifaty and Coignard for the
case of volatility as a risk measure. We first show that the maximum
diversification approach is actually equivalent to the Risk Parity approach
using volatility under the assumption of equicorrelated assets. Then, we extend
the maximum diversification approach formulated for general risk measures.
Finally, we provide explicit formulations of our bi-objective model for
different risk measures, such as volatility, Mean Absolute Deviation,
Conditional Value-at-Risk, and Expectiles, and we present extensive
out-of-sample performance results for the portfolios obtained with our model.
The empirical analysis, based on five real-world data sets, shows that the
return-diversification approach provides portfolios that tend to outperform the
strategies based only on a diversification method or on the classical
risk-return approach.
