Never were truer words spoken of a mathematical formula.
Out on Friday — some über-Geeky weekend reading for those (still?) interested in financial risk management. It comes courtesy of The Joint Forum, the taskforce set up by the Basel Committee on Banking Supervision, IOSCO and the International Association of Insurance Supervisors, to examine topics of mutual interest.
This particular document looks at ways of improving risk aggregation models post financial crisis — think VaR, VarCovar and the like. So there’s plenty of talk of better capturing tail risk, or extreme losses that have a low probability. Copulas — in particular the Gaussian or normal distribution kind — get some specific treatment.
These were made infamous in the recent crisis, for their inability to adequately capture correlation or tail risk, and it appears they have not come out of it unscathed.
From the report:
To get a better view on and understanding of the tail dependencies and tail risks some interviewed firms are moving or moved away from the classical aggregation methods often based on Gaussian or log-normal distribution assumptions to aggregate their risks. For instance, interviewed firms are considering or implemented methods based on non-Gaussian copulas and distributions. These developments are considered also to be aided by the fact that there is now more tail data available. Though, this trend is clearly more observed among the insurance-based groups. The main motivation of firms to move away from Gaussian copula is its inability to quantify any tail dependence. A limited number of interviewed firms use copulas chosen from the more general class of elliptical copulas, which have the advantage that tail dependence can be explicitly incorporated into the aggregation to, for instance, capture the fact that during times when losses occur, more severe losses are observed in different risk categories simultaneously than during good times. Deriving and employing the appropriate copula was said to be a great challenge. Several interviewed firms also noted that “copulas have an image problem with senior management” as they are mathematically moderately complex and require expert judgment.
Lots of firms have tried to tweak their copulas accordingly, including the use of “stressed correlations” to help address tail risk. For instance, the report quotes one unnamed firm that that replaced Basel II internal ratings-based correlations with its own versions. But just tweaking the variables may not solve all-things copulous.
There’s the issue of tail dependence — that’s the tendency of severe losses to cluster together — which Gaussian copulas aren’t really able to capture. Elliptical copulas — which enable you to specify different correlations between variables — may do this better, but even then getting the underlying assumptions right is technically difficult.
There’s an interesting (related) aside in the report too — about what seems to be a sometimes futile attempt to escape the dominance of copulas in risk management.
Finally, note that some of the trends discussed cannot be considered to be entirely “new”. For instance, after the stock market crash of 1987, market participants started to talk about and consider the necessity of “non-Gaussian” distribution assumptions in their risk measurement, as the crash showed that market prices do not follow Gaussian processes. However, the classical distribution assumptions (such as Gaussian, lognormal, or Pareto distributions) still seem to be the most commonly used distributional assumptions in many areas of risk management practice.
Copulas and finance. It’s a copu-cated relationship. Get it?
Related links:
Building a better Gaussian copula – FT Alphaville
The formula that felled Wall Street- Sam Jones, FT
Fat tails, tail dependence, and micro-correlations - Resources for the future paper
Why didn’t people in finance pay attention to Mandelbrot? – Justin Fox