Improved Risk Reporting with Factor-Based Diversification Measures

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

2. Portfolio Diversification Measures

matrix, Ω smp

, and the diagonal matrix of

between the whole portfolio and the (weighted) sum of the components are meaningful diversification measures, they still provide very little information about the effective number of bets in a portfolio, which is what we turn to next. 2.2.1 Measures of the Effective Number of Correlated Bets (ENCB) To try and identify a meaningful measure of the number of bets (baskets) to which investors’ dollars (eggs) are allocated, one can first define the contribution of each constituent to the overall variance of the portfolio as Roncalli (2013)

constituents’ volatilities, D ; 2. Then, we diagonalise the correlation matrix as is the diagonal matrix of eigenvalues sorted by decreasing order and R is a matrix of normalised eigenvectors; 3. Then, we identify k systematic factors 4 , which correspond to the largest k eigenvalues of Ω smp . The remaining eigenvalues are set to 0, leading to a new diagonal matrix of eigenvalues E 2 ; 4. The resulting robustified correlation Ω matrix is obtained by computing RE 2 R' , and replacing the diagonal elements with 1 (otherwise it may not be a true correlation matrix because its diagonal elements may be different from 1); 5. Finally, the robustified covariance matrix is equal to: Σ = D Ω D . It is therefore the robustified covariance matrix Σ that is used to derive risk-based measures of portfolio diversification. In order to take into account the covariance matrix, Goetzmann et al. (2005) use the ratio of the variance of the portfolio to the weighted average variance of the portfolio constituents: 5 This measure takes into account not only the weights of each constituents, but also the correlation structure. More specifically, a portfolio that concentrates weights in assets with high correlation will tend to have portfolio risk higher than the average standalone risk of each of its constituents. Thus it will have a high Goetzmann-Li-Rouwenhorst measure, that is, high correlation-adjusted concentration. While such risk-based measures of distance

3 - We apply the PCA on the correlation matrix because the volatility may vary from one constituent to another which stresses the need for returns’ normalisation. 4 - The rule we use to determine k comes from “random matrix theory”, and states that any eigenvalue e 2 that is below the threshold , where N represents the number of constituents and T the number of observations is considered as statistical noise and should not be counted as a factor. 5 - See Fernholz (1999) or Choueifaty and Coignard (2008) for related diversification/concentration measures.

where [ X ] k denotes the kth element of vector X . This leads to the following scaled contributions:

, where

Note the portfolios such that the contribution q k of each constituent to the variance are all equal is named risk parity portfolio (see Roncalli (2013) for conditions of existence and unicity of the risk parity portfolio). To account for the presence of cross-sectional dispersion in the correlation matrix, one can apply the naive measure of concentration ENC introduced above to the contributions to portfolio risk. This allows us to define the effective number of correlated bets in a portfolio as the dispersion of the variance contributions of its constituents:

24

An EDHEC-Risk Institute Publication