Diversification+Optimization=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, due to the difficulty of obtaining good estimates for the parameters involved in the function to be optimized, several authors have proposed alternative approaches where they look for the best way of diversifying the risk of the portfolio. A straightforward approach to diversify the risk of a portfolio seems to be that of using an Equally Weighted (EW) portfolio. However, if the market contains assets with very different intrinsic risks, then this leads to a portfolio with limited total risk diversification (Maillard et al, 2010). Nonetheless, the EW portfolio seems to have very good theoretical properties (Pflug et al, 2012) and often shows undominated performance in real markets (DeMiguel et al, 2009). Another naive approach, frequently used in practice, to achieve approximately equal risk contribution of all assets, is to take weights proportional to 1/(\sigma_i), where \sigma_i is the volatility of asset i. However, a more thorough approach to risk diversification requires to formalize the notion of “risk contribution” of each asset and then to manage it by a model. The term “Risk Parity” was first introduced by Qian (2005). He developed the idea of Risk Parity Portfolios, where an equal amount of risk is allocated to stocks and bonds. Risk Parity has later been formalized in a model which aims at making the total risk contributions of all assets equal among them (Maillard et al, 2010). This model can be solved by solving a nonlinear convex optimization problem or a system of equations and inequalities (Maillard et al, 2010; Spinu, 2013). The risk measure commonly used in the RP approach is volatility. However alternative risk measures can also be considered (see, e.g., Boudt et al (2013); Cesarone and Colucci (2015) and the comprehensive recent monograph by Roncalli (2014) on Risk Parity and Risk Budgeting). After the global financial crisis of 2008, the interest for the defensive strategy of the RP approach has continuously grown over the years, both among academics and among practitioners (Anderson et al, 2012; Asness et al, 2012; Bai et al, 2015; Bertrand and Lapointe, 2015; Boudt et al, 2013; Cesarone and Colucci, 2015; Chan-Lau, 2014; Chaves et al, 2011; Clarke et al, 2013; Corkery et al, 2013; Dagher, 2012; Lee, 2011; Lohre et al, 2014; Maillard et al, 2010; Qian, 2005, 2011; Roncalli, 2014; Sorensen and Alonso, 2015; Spinu, 2013), thus becoming a popular asset allocation strategy1. Furthermore, several alternative approaches to diversify risk have been recently proposed in the literature (see, e.g., Cesarone and Tardella (2016); Choueifaty et al (2013); Meucci et al (2015)). Due to its formulation, all assets are selected in a RP portfolio, since, by construction, each asset gives a positive (equal) contribution to total risk and must hence have a positive weight. We propose here a new approach that tries 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