Improved Risk Reporting with Factor-Based Diversification Measures

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

Executive Summary

Introducing an Improved Measure of Diversification Risk reporting is increasingly regarded by sophisticated investors as an important ingredient in their decision-making process, and a large number of indicators are now available to help them assess the risks of their portfolio. The 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, the probability of a severe outcome in the future, or the reward that an investor can expect in exchange for bearing those risks. In this context, it appears to be of critical importance for investors and asset managers to also be able to rely on forward-looking risk indicators for their portfolios. Common intuition and portfolio theory both suggest that the degree of diversification of a portfolio is a key indicator when assessing its ability to generate attractive risk-adjusted performance across various market conditions. The benefits of diversification are intuitively clear: efficient diversification generates a reduction of unrewarded risks that leads to an enhancement of the portfolio risk-adjusted performance. On the other hand, in the absence of a formal definition for diversification, it is not as straightforward a task as it might seem to provide a quantitative measure of how well or poorly diversified a portfolio is. The usual definition of diversification is that it is the practice of not “putting all your eggs in one basket”. Having eggs (dollars)

spread across many baskets is, however, a rather loose prescription in the absence of a formal definition for the true meaning of “many” and “baskets”. An initial approach to measuring portfolio diversification would consist of a simple count of the number of constituents the portfolio is invested in. One key problem with this approach is that what matters from a risk perspective is not the nominal number of constituents in a portfolio, but instead its effective number of constituents (ENC). To understand the nuance, let us consider the example of a fictitious equity portfolio that would allocate 99% of the wealth to one stock and spread the remaining 1% of the wealth to the 499 remaining stocks within the S&P 500 index universe. While the nominal number of stocks in that portfolio (defined as the number of stocks that receive some non zero allocation) is 500, it is clear that the effective number of stocks in the portfolio is hardly greater than one, and that this poorly diversified portfolio will behave essentially like a highly concentrated one-stock portfolio from a risk perspective. In this context, it appears that a natural and meaningful measure of the effective number of constituents (ENC) in a portfolio is given by the entropy of the portfolio weight distribution. This quantity, a dispersion measure for probability distributions commonly used in statistics and information theory, is indeed equal to the nominal number N for a well-balanced equally- weighted portfolio, but would converge to 1 if the allocation to all assets but one converges to zero as in the example above, thus confirming the extreme concentration in this portfolio.

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An EDHEC-Risk Institute Publication