Improved Risk Reporting with Factor-Based Diversification Measures

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

2. Portfolio Diversification Measures

In this section, we present a comparative analysis of various measures of portfolio diversification/concentration, with a discussion of their respective merits and shortcomings. 2.1 Weight-Based Measures of Portfolio Diversification A key distinction exists between weight- based measures of portfolio concentration, which are based on the analysis of the portfolio weight distribution independently of the risk characteristics of the constituents of the portfolio, and risk-based measures of portfolio concentration, which incorporate information about the correlation and volatility structure of the return on the portfolio constituents. In a nutshell, weight-based measures can be regarded as measures of naive diversification, while risk-based measures can be regarded as measures of scientific diversification. Most weight-based measures of portfolio concentration provide a quantitative estimate of the effective number of constituents (ENC) in a portfolio, in an attempt to alleviate the problems related to the use of the nominal number of constituents in a portfolio which can be very misleading, in particular in case of a very ill-balanced allocation of the portfolio to the various constituents (e.g., one security makes up for 99% of the portfolio while other securities make up collectively for the remaining 1%). We denote with w the weight vector representing the percentage invested in each asset of a given portfolio, and we define the following class of diversification/ concentration measures:

,

α ≥ 0, α ≠ 1. (2.1)

Taking α = 2 leads to a diversification measure defined as the inverse of the Herfindahl Index, which is itself a well-known measure of portfolio concentration, or . Portfolio diversification (respectively, concentration) is increasing (respectively, decreasing) in the ENC measure. Note that this measure is directly proportional to the inverse of the variance of the portfolio weights, as can be seen from the following lemma. Lemma 1 If a portfolio contains N constituents, then the ENC 2 measure of the portfolio can be expressed in terms of the variance of the weight distribution as:

2 - This result is known in information theory under the following statement: the Rényi entropy converges to the Shannon entropy.

(2.2)

Proof. See Appendix B.1. It can be shown that when α converges to 1, then ENC α converges to the entropy of the distribution of the portfolio weights: 2

(2.3)

It is straightforward to check that, for positive weights, ENC α reaches a minimum equal to 1 if the portfolio is fully concentrated in a single constituent, and a maximum equal to N , the nominal number of constituents, achieved for the equally-weighted portfolio. These properties justify using this family of measures to compute the effective number

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An EDHEC-Risk Institute Publication