History of Asset Prices and the Rare Event
March 12, 2007
N-alpha Capital, a small hedge fund managing around $100 million in assets recently blew up. It lost around 25% of its capital in a span of a few days when the markets tumbled two weeks ago. The manager had put in stop losses of around 500 points on a couple of index futures (thinking that they will never be hit) with large bets and all stops got hit. In a rare moment of lucidity he confided to a friend over beer that there was nothing in the history (of asset markets) for the last one year to suggest that such a fall was coming. This was an amazing statement. Rarely do you see a hedge fund manager who plays the statistician! An unfortunate fact was also that the manager had gone long on most of the equity markets and he became a victim of a strong positive correlation, a rare event.
It is an oddity and a paradox of life, and the financial markets, that a rare event isn't all that rare. It occurs with a great degree of certainty.
Read today's South China Morning Post ( Hong Kong ) where the US Federal Reserve Chairman, Bernanke's testimony to the Congress on Feb 28 is quoted. Here is what the paper wrote: Testifying before the Congress on February 28, the day after the market's big fall, Mr Bernanke said that the markets had been functioning well and he had not seen anything in the recent economic data to alter his view for "moderate growth going forward".
To paraphrase Nassim Taleb* it is like saying that we examined the last 50 or 60 years of data on Ben Bernanke and realized that he did not die and therefore we can reject the hypothesis of him being mortal at a high confidence level. Can we say that? If an insurance company had only one client and if that client, say a middle aged man, was healthy and had no history of any major medical illness will it then write a life insurance policy on this individual for say, a hundred million dollars?
The essence of skew - the one that we talk about day in and day out in all contexts- is the rare event (once again quoting Taleb). And no time series of data can expose the rare event, nor can any time series help us in forecasting a rare event. Conventional statistics abhors all outliers ( remember your thesis advisor who asked you to throw out a couple of data points because they were not in line with the observations? ) and the rare event is an outlier. It could be a ten standard deviation event, something that is likely to happen in once in more than a trillion years and yet it happens once in every five years or so. Skew tells us that we will be making money (profits) more often and lose money (losses) less often, but when we lose we lose big time. On the average, we keep making money for a while, maybe even for quite long time, and then we take a big hit; we blow up. That's what happens to many hedge funds; they make money quite regularly, show handsome returns for a few years and then one fine morning they blow up. LTCM, Amaranth and others, all the same story. They are all the victims of the skew, the rare event!
If you really have to look at a time series of past data (out of habit or peer pressure) then try to see how often your position generated positive returns and how often you got negative returns. This may tell you, in a historical context, which side of the skew you are on and may help you to design a better macro strategy. Whether you want to bleed for a long time on this asset and then make a big positive payoff or else you want to make small gains for a long time and then blow off. In other words, in a macro sense, whether you want to be long options or short options. But if you ask us, this is a futile exercise.
To get a glimpse of the rare event - you can never uncover it - you need to look at how a time series will behave in the future, and not how it has behaved in the past. And to do so you need to simulate a stochastic path of the asset price or your P & L with a wide variety of input parameters. Out of the million paths that you generate, one of them may contain a rare event - say, a market fall of 8% in a day or a complete busting of your P & L.
The fact that the correlation between equity markets across the world came close to one on a particular day, which ravaged the fund manager's portfolio above, may seem obvious in a post mortem analysis but in fact it was also a rare event!
In the end, like Nuclear Physics, in finance there is no choice but to resort to Monte Carlo simulation.
* Interview of Nassim Table with Derivativesstragey.com, November 1999.
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