Improved Risk Reporting with Factor-Based Diversification Measures

Improved Risk Reporting with Factor-Based Diversification Measures — February 2014

Executive Summary

On the other hand, if one is indeed entitled to considering that a well-balanced allocation of dollars (eggs) to identical securities (baskets) may be regarded as a well-diversified allocation, the existence of differences in risks across securities would require some adjustment to the proposed measure of sound diversification. In other words, what needs to be well-balanced is not the number of eggs in each basket per se, but rather the risk contribution of each basket. In this context, a well-diversified portfolio would seek to have more eggs in more robust baskets, and fewer eggs in frailer baskets. At this stage, the need remains for a critical assessment of what should be the proper interpretation for the "baskets" in this proverbial definition of diversification. The straightforward approach, which suggests that baskets are asset classes in an asset allocation context, or securities for a portfolio constructed within a given asset class, is in fact misleading or at least severely incomplete. Indeed recent research (e.g. Ang et al. (2009)) has highlighted that risk and allocation decisions could be best expressed in terms of rewarded risk factors, as opposed to standard asset class decompositions, which can be somewhat arbitrary. For example, a seemingly well-diversified allocation to many asset classes that essentially load on the same risk factor (e.g., equity risk) can eventually generate a portfolio with a very concentrated set of risk exposures. Going back to the eggs-and- baskets analogy, having a well-balanced allocation of eggs across many different baskets that would be tied together can hardly be regarded as an astute way to ensure a proper diversification of the risks involved in carrying eggs to the market.

In other words, baskets should be interpreted as uncorrelated risk factors , as opposed to correlated asset classes , and it is only if the distribution of the contributions of various factors to the risk of the portfolio is well-balanced that the investor's portfolio can truly be regarded as well-diversified. Putting all these elements together, we propose using the effective number of bets (ENB) in our empirical analysis, which would serve as a meaningful measure of diversification for investors' portfolios (see Meucci (2009) and Deguest, Martellini and Meucci (2013) for more details). 1 One natural way to turn correlated asset returns into uncorrelated factor returns is to use principal component analysis (PCA). While useful in other contexts, the PCA approach suffers from a number of shortcomings when estimating the effective number of bets. The first shortcoming is the difficulty in interpreting the factors, which are pure statistical artefacts. The second shortcoming, particularly severe in the context of the design of a diversification measure, is that by construction, principal components are defined so as to achieve the highest possible explanatory power. As a result, the contribution of the first few factors is often overwhelmingly large with respect to the contribution of other factors, and the portfolio diversification measure empirically tends to be biased towards low values. 2 A competing approach to extracting uncorrelated factors from a basket of correlated constituents, which we use in the analysis that follows, is the minimal linear torsion (MLT) approach, which focuses on extracted uncorrelated factors that are as close as possible to the original constituents, in the sense that they have the same volatility as the original

1 - ENB is formally defined as the entropy of the distribution of contributions of uncorrelated factors to the risk of the portfolio. 2 - For example, the ENB measure is equal to 1 for an equally-weighted portfolio based on a universe of assets with equal volatility and pairwise correlation values, regardless of the correlation value, while the intuition would suggest that for a vanishing constant correlation value such a portfolio should have a number of bets equal to the number of assets. This counter-intuitive full-concentration effect follows because the equally- weighted portfolio is in this case fully exposed to the first principal component and not exposed to any other principal component (see Meucci, Santangelo and Deguest (2013)).

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An EDHEC-Risk Institute Publication