160. Quantitative analysis for beginners. Dumb Gaussian approximation

Back in Munich, when we were discussing the crisis (apart from so many other things ) with Serge Winitzki, he eventually stated that the crisis’ takeoff  is ultimately related to the ignorance of financial analysts   – namely, they ignore the fact that rare (in the sense of Gaussian statistics) events are not that rare when one considers real dynamics of stock prices.

Since I strongly doubted that Wallstreet financial analysts are so much ignorant (in this particular sense ) but was unable to prove at that time  why Sergey’s statement is wrong, I decided to go quickly through the literature. Of course, I immediately discovered that even in the end of 19th century no single financial analyst was paid an equivalent of present 100000 USD/year for attempts to approximate dynamics of stock prices by Gaussian distributions.

But before turning to my amazing discoveries , let me first explain why Gaussian distribution for the behavior of stocks  is actually not that bad What is actually meant by the Gaussian distribution?

Let be the price of a stock. We will call the quantity return (per unit time) of the stock. The claim is actually that return satisfies the following stochastic (Langevin) equation

, (1)

where the noise has correlation properties

, , (2)

of the Gaussian random field. The quantity determines  the volatility of the stock.

If the growth rate is contant, the solution of the equation (1) can be immediately found. It has the form

(3)

Quick exercise. Check it

Trivially, it descibes the exponentially growing price of the stock You might be surprised, but this is actually what is typically observed on markets in the long run. See for example this behavior of the Dow Jones Industrial for the last 30 years:

If one ignores an unpleasant 2000-2010 decade, DJIA behavior is nearly exponential. Also, the spectrum of the noise is typically nearly flat – especially, for markets with huge volume like FOREX. The basic recommendation that follows from the plot above (very well known to people who invest into mutual funds) – just follow the market. Invest in some index mutual fund and enjoy watching how your money grow together with the market. This strategy should work especially well for rich people like Worren Buffet, whose assets are bold enough to sustain strong fluctuations on the market – ultimately, markets grow exponentially.

But is it actually write? Take a look on the plot above now taking into account the 2000-2010 decade – it seems that exponentially growth was very effectively stopped (like it was in the middle 60s for example, when stagnation of DJIA continued for 20 years – till 1980). What is wrong with common lore about mutual funds we presented above and our naive formula (1).

Actually, many things we did not take into account are simultaneously important. first of all, volatility itself is a function of time, as well as the growth rate . Second, rare events have much higher probability than the one determined by the exponential tail of the Gaussian distribution, as was observed already in the end of the 19th century by Pareto.

But this is probably a subject for another post.