Improved Risk Reporting with Factor-Based Diversification Measures

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

Appendices

A. Robustness Checks In this paper, we chose as a base case to compute the different diversification measures for the period starting in 1959 and ending in 2012 using a 1-year rolling window without overlap and using the entropy metric, see Equations (2.3) and (2.6). However, we could have used a rolling window with overlap, as well as other rolling window periods, and other kinds of metrics to compute the dispersion of the weights in the ENC, or of the factor contributions in the ENB. In order to test the robustness of our analysis, we present in this section some results using different time-periods and another type of dispersion measure than the one used in the base case to compute the diversification measures. We first present the ENC and the ENB computed with a PCA, an MLT and the Fama-French factors using the inverse Herfindhal metric. In a second step, we present the diversification measures we obtain using a 1-year rolling window with a 1-week lag overlap, and we present 2-year and 5-year rolling windows with a 1-week lag overlap as well. As a base case, we use the entropy metric to compute the ENC and the ENB. We propose now to look at the inverse Herfindahl metric (also called the “ L 2 -norm”) calculated as follows:

The ENC and ENB measures computed with the inverse Herfindhal metric can be seen in Figure 16. On this figure we notice that the diversification measures obtained with the entropy metric (Figure 1) have similar fluctuations as the ones obtained using the inverse Herfindahl metric. We also use the inverse Herfindhal metric with a 1-year window without overlap in order to compute the ENC and ENB (using the MLT approach) of the equally-weighted version of the S&P500 (see Figure 17). As for the cap-weighted S&P500, we notice that the diversification measures computed with both metrics have similar fluctuations. However, we notice that the values obtained with the inverse Herfindahl metric are in general lower than the ones obtained with the entropy metric. Finally, we present the results of diversification measures computed with a 1-year rolling window on a weekly basis (with overlap) and the results obtained using a 2-year and a 5-year rolling window with overlap (respectively in Figures 18, 19 and 20). We notice that the 2-year and 5-year rolling window periods with overlap lead to similar shapes for the diversification measures than the 1-year rolling window period with overlap. However, the longer the length of the rolling-window, the smoother the curves of the three diversification measures. Therefore, from these robustness checks we can conclude that the findings we obtained with our diversification measures do not depend on our choice of the metric, nor on the length of the time period used to estimate the covariance matrix.

,

where each p k is a function of w 1

, ..., w N .

56

An EDHEC-Risk Institute Publication