Risk Management, PRMIA

Hi,

Read your concerns and it's an interesting area which I happen to look very closely at. Basically, the assumption on normality holds very well for 95% of real returns when looking at monthly returns in the global equity market. The problem starts on the left side beyond that where it's quite clear that there are a higher degree of extreme losses than a normal distribution would suggest translated in a bit of excess kurtosis. The way I've chosen to deal with this is to take into account the higher moments of the distribution and penalise high kurtosis and negative skew by introducing a Modified VaR methodology that uses the Cornish-Fisher expansion. Nothing is perfect however, but to systematically overestimate risk must be better that systematically underestimate risk using a pure Gaussian frame work. If you wish to contact me for a discussion my number is: +44 20 7956 2846 email j.e@hythesecs.com

You've picked a seriously contentious issue and one that has been doing the rounds for at least 20-years.

In 1993 I discussed Operational Metrics with some of the great and good then, long before Risk/Credit Metrics appeared and certain much of the Credit markets. Even then data was being discussed as the main problem we face and the improper use of rules to create it.

If you wish I can share some of this as it will be part of my PhD in Management Cybernetics over the next 2-years and I'd welcome the interaction and view point you have. It's a mature program PhD as I'm now 56 and have been through the risk side for both insurance and banking.

Warm regards

Stefan

Referring to your sentence "Nassim Taleb and Benoit Mandelbrot have overstated the reliance of financial risk managers on the Gaussian distribution" a little syntatical correction is needed: if anything, Nassim stated that risk managers rely too much on Gaussians.

What the 'true' distribution can only be known if one has a lot of data, and if these are sampled in an environment where everything else remains stationary. This is rarely the case in real life. Sensitivities to risk factors and correlations change over time and risk management and transfer techniques are developed. There is nothing wrong about Gaussians per se. If one is worried about extreme tail events, one can model them with normal mixture distributions, or one can keep the normal regime Gaussian and model the extremes using other distributions.

I'm just an educator not a practitioner (sorry) but I have an opinion:

I think Taleb setup a straw man. I rarely read an expert who suggests either (i) normal is the default distribution or (ii) that it ought to be used without help; e.g., stress test.

What is special about the normal distribution? I can think of only two things. 1. It's easy (elegant) b/c it only needs two parameters (the same reason that should make it suspect) and, the more important special feature 2. the central limit theorem. A beautiful thing that says sampling means converge to normal; that's really why we see the normal so much. But, that is about the center (the central tendency), not the edges where it totally breaks down. In short, normal for returns (e.g., averages) but not for risk (edges).

Except for CLT (which is profound at the center), what is possibly special about a Gaussian distribution except it is suspiciously easy to use?

I would agree with all of these comments, the normal distribution is ideal for ``normal`` cases, how does a business unit normally operate? If however, you are looking at reserving capital for a one in a hundred year event, a rare place that is NOT normal, then the normal distribution doesn`t depict the event at all well as Nassim puts it.

When it comes to modeling normal events, sometimes we don`t even use that Gaussian distribution but curve fit the loss data via QQ plots / Kolmogorov-Smirnov / Anderson-Darling goodness of fit test to another distribution (Weibull, Gamma, Beta, Erlang, there are so many distributions). The largest issue for operational risk is the definition of loss and the associated collection of data which to date has huge paucity. A lot of analysts say Peak Over Thresholds is great way to estimate the potential maximum loss on an Independent Identically Distributed iid variable, but we have to choose a suitably high threshold for the excess function (usually through mean excess) to work in a manner that it represents the iid appropriately. So look at another method such as Block Maxima where we separate the sample into non-overlapping subsamples of fixed length of time. Then there are also the GEV functions which have already been mentioned however we have to define the location, scale parameter and tail index parameter for a good estimation. Brilliant, easy now do this with 3 data points, really, what does one do? These Extreme Value Approaches work great in the fields of engineering which is where they have been applied often because we can create a controlled experiment environment and repeat the test varying specific conditions that simulate real life. Lock down against a control and run the engine again, now turn it up and run it harder, change the pressure, run it harder, oh dear a turbine blade snapped off and went through the wall. Let’s run the experiment again with some other adjustments and so on. Operational risk in a bank isn’t an experiment, its thousands of them squared and already in play not considering the people factor.

That is why it is so difficult to dimension, I believe.

Beaumont, have a look at the book `Operational Risk` – Measurement and Modelling by Jack L King, published by Wiley & Sons ISBN: 978-0-471-85209-4 it is one of the best outlines I have seen for a long time for risk, it was written a while back but none the less really covers the space … Jack King talks about Delta – EVT models and the differences between Normal and the Extreme Position but he also shows it with examples.

On the 19th to the 25th of January this month Andreas Jobst from the International Monetary Fund (IMF) is going to be discussing a research paper “Operational Risk – The Sting is still in the tail but the Poison Depends on the Dose” you might want to go up to the Oprisk Austega site on http://oprisk.austega.com and download and have a look at what he has written, good piece of work, really tests the EVT theory out against Basel.

Hi, Beaumont

I agree with most of the replies above. The assumption of normality and indeed all assumptions in finance make the proverbial ass out of U and ME unless tested. Some assets are close to normally distributed but in all cases you should check using either a JB or KS test first.

In the field of Hedge Funds less than 10% of the 12000 or so funds we have tested have 'normally' distributed returns. (We test 13 different distribution types including the Normal, Mixture of Normals, 3-parameter log-nomal, Skew T, Cornish Fisher modified Normal, Uniform, Gumbel, Pearson IV and Johnson family of distributions to arrive at this result ). This should not be that surprising given that Hedge funds are attempting to generate an asymmetric payoff by definition but it does have a material bearing on their relative 'riskyness' and on the allocations arrising from 90%= of optimisation software out there that continues to 'assume' normality.

It could be argued that asymmetry is a measure of the lack of efficiency of a market in pricing all relevant information instantaneously due to various frictions such as illiquidity (Hedge funds in general), opacity or the arcaneness of the price (eg CDO's). Thus from this perspective the CLT holds and markets tend towards normality / efficiency but 'only for the most part' as De Moivre observed or if you prefer 'in the long run' about which Keynes said 'yes but in the long run we are all dead'.

This perfectly describes the way markets are and explains why so called alpha exists which it should not in a perfect world.

I believe that the Gaussian assumption is still very widely used for risk modeling, but that is beginning to change.

Use of the Cornish Fisher expansion is now much more common than it was even a few years ago: it has the advantage of being almost as "elegant" as the Gaussian in that it utilizes the first four moments of the distribution and thus models the deviations from normality that are most easily discussed with members of the business (skewness, kurtosis). as with any model, it has points where it breaks down, but the sensitivities of Cornish Fisher are easy to calculate and visualize so that you know when it is breaking down.

I am very concerned about model risk issues in any model that requires parameter estimation: if parameters need to be estimated via MLE or any other method then the often very short historical return periods are clearly inadequate for any suitable confidence level. While I use copulas in my work, I am concerned with the model risk inherent in fitting these very tricky distributions.