In "Is the Potential for International Diversification Disappearing? A Dynamic Copula Approach" by Christoffersen et al. (2012), the authors present the concept of Conditional Diversification Benefit of a portfolio, based on the expected shortfall, defined as (confidence level $\alpha$ is omitted for legibility reasons):
$$CDB_t = \frac{\overline{ES}_t-ES_t}{\overline{ES}_t-\underline{ES}_t}$$
where $\overline{ES}_t$ is the upper bound of portfolio's ES when there is no diversification, i.e. $\overline{ES}_t = \sum_{i=1}^n w_i ES_{i,t}$ (where $i$ stands for asset $i$), $\underline{ES}_t$ is the lower bound of portfolio's expected shortfall, which by definition coincides with the respective VaR, and $ES_t$ is the actual estimate. In their paper (where each asset is an index representing the economy of a country) they find that $CDB_t$ has declined over the last 15-20 years as a result of increasing dependence between different countries' economies.
In my project I'm using sector-specific indexes rather than country-specific indexes. My findings differ from Christoffersen et al. (2012): my $CDB_t$ stays relatively constant over time, with some minor fluctuations (the biggest upward fluctuation during the financial crisis), and a plot of the average correlations (average correlation of asset $i$ with assets $(1, ..., i-1, i+1, ...n))$ confirms that these have actually declined, leading to increasing $CDB_t$ during the crisis.
Is this not plausible, or may this be given by the fact that different sectors are diversely hit by the crisis, with some sectors that lost tons of money while others maybe profited from it, or at least are not harmed by it (for example health care companies vs. financial institutions), whereas country-related indexes (which in developed countries include stocks from all industries) are constructed in a fairly similar way, and therefore tend to correlate more during volatile times? Maybe you have an intuitive answer, or a reference to industry correlations during the crisis?
