Diversified Optimal Portfolios: a new approach to portfolio selection

The classical approach to portfolio selection calls for finding a feasible portfolio that optimizes one of the several proposed risk measures, or (expected) utility functions, or performance indexes. However, the optimization approach might be misleading due to the difficulty of obtaining good estimates for the parameters involved in the function to be optimized and to the high sensitivity of the optimal solutions to the input data. This observation has led some researchers to claim that a straightforward capital diversification, i.e., the Equally Weighted portfolio can hardly be beaten by an optimized portfolio [3]. However, if the market contains assets with very different intrinsic risks, then this leads to a portfolio with limited total risk diversification. Therefore, alternative risk diversification approaches to portfolio selection have been proposed, such as the practitioners’ approach of taking weights proportional to 1/\sigma_i, where \sigma_i is the volatility of asset i. A more thorough approach to risk diversification requires to formalize the notion of risk contribution of each asset, and then to manage itby a model. For example the Risk Parity approach (see [5], and references therein) aims at a portfolio where the total risk contributions of all assets are equal among them [4]. The original risk parity approach was applied to volatility. However alternative risk measures can also be considered (see, e.g., [1]). It can also be shown that the Risk Parity approach is actually dominated by Equal Risk Bounding [2], where the total risk contributions of all assets are bounded by a common threshold which is then minimized. Furthermore, several alternative approaches to diversify risk have recently appeared in the literature. We propose here a new approach that tries to reduce the impact of data estimation errors and to join the benefits of the optimization and of the diversification approaches by choosing the portfolio that is best diversified (e.g., Equally Weighted or Risk Parity) on a subset of assets of the market, and that optimizes an appropriate risk, or utility, or performance measure among all portfolios of this type. We show that this approach yields portfolios that are only slightly suboptimal in-sample, and generally show improved out-of-sample performance with respect to their purely diversified or purely optimized counterparts.