Common risk measures provide little information about causes of risk in portfolios. Lionel Martellini and Romain Deguest, of the Edhec-Risk Institute, highlight academic work on the relationship between diversification and risk.
Risk reporting is increasingly regarded by sophisticated investors as an important ingredient in their decision-making process. Most commonly used risk measures such as volatility (a measure of average risk), value-at-risk (a measure of extreme risk) or tracking error (a measure of relative risk), however, are typically backward-looking risk measures computed over one historical scenario.
As a result, they provide very little information, if any, regarding the possible causes of the portfolio riskiness and the probability of a severe outcome in the future, and their usefulness in a decision-making context remains limited. For example, research has found that an extremely risky portfolio such as a leveraged long position in far out-of-the-money put options may well appear extremely safe in terms of the historical values of these risk measures, that is until a severe market correction takes place.
In this context, it is of critical importance for investors and asset managers to be able to rely on more forward-looking estimates of loss potential for their portfolios. The main focus of recent Edhec research – contained within Improved Risk Reporting with Factor-Based Diversification Measures and conducted with the support of Caceis as part of the New Frontiers in Risk Assessment and Performance Reporting research chair – is on analysing meaningful measures of how well, or poorly diversified, a portfolio is, exploring the implication in terms of advanced risk reporting techniques, and assessing whether a relationship exists between a suitable measure of the degree of diversification of a portfolio and its performance in various market conditions.
BETTER UNDERSTANDING OF DIVERSIFICATION
While the benefits of diversification are intuitively clear, the proverbial recommendation of “spreading eggs across many different baskets” is relatively vague, and what exactly a well-diversified portfolio is remains somewhat ambiguous in the absence of a formal quantitative framework for analysing such questions. Fortunately, recent advances in financial engineering have paved the way for a better understanding of the true meaning of diversification.
In particular, academic research by Ang et al. (Evaluation of Active Management of the Norwegian Government Pension Fund – Global published in 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, convertible bond returns are subject to equity risk, volatility risk, interest rate risk and credit risk. As a consequence, analysing the optimal allocation to such hybrid securities as part of a broad bond portfolio is not likely to lead to particularly useful insights.
Conversely, a seemingly well-diversified allocation to many asset classes that essentially load on the same risk factor (for example, equity risk) can eventually generate a portfolio with very concentrated risk exposure. More generally, given that security and asset class returns can be explained by their exposure to pervasive systematic risk factors, looking through the asset class decomposition level to focus on the underlying factor decomposition level appears to be a perfectly legitimate approach, which is also supported by standard asset pricing models, such as the intertemporal Capital Asset Pricing Model or the arbitrage pricing theory.
Two main benefits can be expected from shifting to a representation expressed in terms of risk factors, as opposed to asset classes. On the one hand, allocating to risk factors may provide a cheaper, as well as more liquid and transparent, access to underlying sources of returns in markets where the value added by existing active investment vehicles has been put in question. For example, Ang et al argue in favour of replicating mutual fund returns with suitably designed portfolios of factor exposures such as the value, small-cap and momentum factors. (In the same vein, Hasanhodzic and Lo (2007) argue for the passive replication of hedge fund vehicles, even though Amenc et al. (2010) found that the ability of linear factor models to replicate hedge fund performance is modest at best.)
Similar arguments have been made for private equity and real estate funds, for example. On the other hand, allocating to risk factors should provide a better risk management mechanism, in that it allows investors to achieve ex-ante control of the factor exposure of their portfolios, as opposed to merely relying on ex-post measures of such exposures.
In our research, we first review a number of weight-based measures of (naive) diversification as well as risk-based measures of (scientific) diversification that have been introduced in the academic and practitioner literatures, and analyse the shortcomings associated with these measures. We then argue that the effective number of (uncorrelated) bets (ENB), formally defined in Meucci A’s 2009 paper, Managing Diversification Risk, as the dispersion of the factor exposure distribution, provides a more meaningful assessment of how well-balanced is an investor’s dollar (egg) allocation to various baskets (factors). We also provide an empirical illustration of the usefulness of this measure for intra-class and inter-class diversification. For intra-class diversification, we cast the empirical analysis in the context of various popular equity indices, with a particular emphasis on the S&P 500 index. For inter-class diversification, we analyse policy portfolios for two sets of pension funds, the first set being a large sample of the 1,000 largest US pension funds and the second set being a small sample of the world 10 largest pension funds.
In a first application to international equity indices, we use the minimal linear torsion approach (Meucci A, Santangelo A, Deguest R, Measuring portfolio diversification based on optimized uncorrelated factors, 2013) to turn correlated constituents into uncorrelated factors, and find statistical evidence of a positive (negative) time-series and cross-sectional relationship between the ENB risk diversification measure and performance in bear (bull) markets. We find a weaker relationship when using other diversification measures such as the effective number of constituents, thus confirming the relevance of the effective number of bets on uncorrelated risk factors as a meaningful measure of diversification.
Finally, we find the predictive power of the effective number of bet diversification measure for equity market performance to be statistically and economically significant, comparable to the predictive power of the dividend yield for example (see Cochrane JH Where the market is going – 1997), with an explanatory power that increases with the holding period. In a second application to US pension fund policy portfolios, we find that better diversified policy portfolios in the sense of a higher number of uncorrelated bets tend to perform better on average in bear markets, even though top performers are, as expected, policy portfolios highly concentrated in the best performing asset class for the sample period under consideration. Overall, our results suggest that the effective number of (uncorrelated) bets could be a useful risk indicator to be added to risk reports for equity and policy portfolios.
Further research
Our work can be extended in several dimensions. In particular, the analysis developed in this paper could be used not only to measure, but also to manage, diversification within an equity or policy portfolio.
We encourage interested readers to look at Meucci A, et al 2013, and Risk parity and beyond – from asset allocation to risk allocation decisions (Deguest R, Martellini L, Meucci A – 2013) for a thorough empirical and theoretical analysis of the properties of portfolios designed to maximise the effective number of bets.
Regarding the empirical analysis of diversification for equity portfolios, another natural extension of our work would consist in comparing the degree of diversification for various weighting schemes based on the same investment universe.
While we have looked at cap-weighted and equally weighted equity index portfolios only, it would be useful to also assess the degree of diversification achieved by minimum variance portfolios or risk parity portfolios, among other examples of so-called smart beta indices. Yet another useful extension of the research would consist in repeating for endowments the analysis we have conducted for pension funds.
We leave this extension for further research.
Lionel Martellini is professor of finance, Edhec Business School and scientific director, Edhec-Risk Institute, and Romain Deguest is senior research engineer, Edhec-Risk Institute
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