Ron Leisching on the search for better models
Value-at-risk (VAR) models grossly underestimated losses in many portfolios last October. VAR models assume normally distributed returns. The assumption made the mathematics simple, but the model unrealistic. Extreme value theory can provide more accurate worst-case loss estimates for funds using risk budgeting.
Normal curves came from the central limit theorem: add up enough short-run random returns, and long-term you should get a normal distribution. But this is only true if returns are random. And it does not specify what “long term” is. After a five standard deviation loss, there may be no long term for the portfolio.
Recently there has been widespread use of insurance statistical theory in finance. Insurers are not concerned with “normal” outcomes. They are only interested in the frequency and severity of extreme events – for example, insuring an excess loss layer of £100m (E150m) above an initial property loss of £30m. Whether it is the San Andreas fault, or floods in the Netherlands, the statistics are the same. This area of statistics is known as “extreme value theory”.
Extreme value theory uses distributions that are specifically designed to fit the “fat tails” – the very large moves. The normal curve is thin-tailed – the probability of a large move drops to zero at an exponential rate. For example, the intraday yen/dollar move that occurred on October 7 last year was a 7.6 standard deviation event – impossible if the return distribution was normal.
Most papers on extreme value theory are highly technical. However, a recent unpublished memorandum by Mark Robson and Ian Bond of the Bank of England provides a useful introduction to the topic. The paper analyses daily returns in the FT Allshare Index. By counting the number of extreme moves, it is easy to show that the normal curve is not a realistic model for daily UK equity returns. The authors show how one can derive robust estimates of the frequency with which very large moves occur in practice.
So what are the better models? There are three classes of extreme value distributions – Fréchet, Gumbel and Weibull; financial distributions are usually Fréchet. By using extreme value curves, we can calculate the chance of a larger loss than has historically occurred. This tells the fund the implicit capital and risk budget it is allocating to this activity. Most short-run financial returns show intermittent “extreme moves”. As an example, look at the monthly swings in the foreign currency value of an international equity portfolio in the chart above (80 months’ data from 1992).
The largest monthly loss was over 7%. Losing money at an 84% annualised rate gets board attention. And this live example is on a portfolio of over £2bn. Usually long-term investors ignore short-run losses. But currency is an exposure, not an investment. There can be no expectation of any currency losses “coming back”. (If losses did “come back”, then every foreign exchange trading bank could make money easily after each large move.) Hence the fund decided to manage down this unrewarded risk.
Long-term investors are less concerned about large monthly equity investment losses. This is an investment, not an exposure: the capital is still invested in a productive equity assets, losses can be expected to come back.
However for European funds considering moving out of bonds into equities, the situation is very different. Analyse exposure to the equity/bond exchange rate and you will find the long-term equity outperformance is concentrated in under 7% of the time periods. All the long-term equity outperformance is in the extreme tail of the distribution. This is why market timing is so dangerous – get it wrong and the losses (or opportunity losses) can be dramatic.
Euroland funds are revisiting their strategic asset allocation. This shows the need for an aggressive move out of domestic bonds, and into equities and into non-euro-denominated assets. The equity/bond exchange rate risk and foreign currency/euro exchange rate risk are key issues faced by funds.
This dramatic allocation change will improve the fund surplus long term. However the short-run risks are not normally distributed. Extreme value theory can help measure the worst case outcome – in advance.
Ron Liesching is managing director of research at Pareto Partners in London