An application of the law of small numbers

What is the link between the law of small numbers and a responsible investment strategy? It is far from being obvious. The law of small numbers is a bizarre mathematical rule describing the asymptotic properties of the counts of rare events. A responsible investment strategy is, according to many, a strategy in which the investor rules out investment in companies with poor environmental, social or governance track record, thus cleaning up his consciousness at the expense of less diversification and thus more risk in his portfolio. See the instructive article by the Nobel prize of Economics winner H. Markowitz on this topic. So, not very much in common, but let us dig a little bit deeper.

Most mathematical methods applicable in finance stem from the law of large numbers. With respect to quantitative investment decision making, a consequence of the law of large numbers is the fact that, for a sufficiently large and diversified portfolio, the specific risk from the single assets disappears. This motivates the passive investment approach and was one of the reasons behind the rapid development of collective (e.g. mutual fund) investment schemes during the last century. To apply the law of large numbers or the related central limit theorem, one needs to estimate the asset’s average value as a proxy for the estimated return, the asset’s volatility as a proxy for the risk and the correlations between the assets in the portfolio as a proxy for their dependency. These quantities are typically extracted from market prices, and rarely deliver any information about how well the governments, the corporations or the institutions behind the asset score on environmental, social or governance issues.

Now here is the problem: all common mathematical techniques applicable in finance do not consider any environmental, social or governance factors. On the other hand, examples like Enron in 2001 and Tepco-Fukushima in 2011 clearly show that rare events closely related to these factors do happen and do have the potential to destroy investor’s value up and even beyond the specific ‘black swan’ asset he is holding. The law of large numbers is a helpless tool when it comes up to such cases. Instead, the law of small numbers has to come in place.

Probably the first practical application of the law of small numbers was done by the Russian economist and statistician of Polish origin Ladislaus Bortkiewicz (1868 – 1931) (who did most of his research while living and working in Germany). In his book he investigated the Prussian horse-kick data. The data give the number of soldiers killed by being kicked by a horse each year in each of 14 cavalry corps over a 20-year period. It was found to follow a Poisson distribution. Since that time, this statistical distribution and its generalizations have found numerous applications in fields like biology, astronomy and insurance mathematics, whenever the topic is about counting of rare events in a large population. It is therefore natural to use these mathematical techniques when it comes up to counting the rare events which burst whole companies, regions or industry sectors in a large, well diversified financial asset portfolio. The question to be answered should be: does the 21st century investor need to worry about rare events more than he worries about horse-kick accidents?

The topics around the law of small numbers are much more delicate to handle compared to law of large numbers. In some cases, the investor would be in the position of the army general whose soldiers are from time to time getting killed by horse-kicks. It is surely unpleasant but not epidemic, happens from time to time, and the impact of it can be reduced through standard techniques. It is surely unpleasant to invest in a corporation known for its corrupting practices and low environmental standards. However, as long as the rare event of this company bursting due to these low governance standards is not contagious, there is no need to worry. The impact on portfolio level can be mitigated via standard techniques like diversification. For example, using correlations as a sole dependence measurement technique would always leave the investor with a sense of security. The asymptotic distribution in this case is of Poisson type, and there are no clusters of rare events. In contrast, a prudent risk management strategy would investigate for additional sources of non-linear dependence (and I believe it is not hard to see such sources). The asymptotic distribution could be more like the negative binomial one, allowing for whole clusters of rare events to happen over short time periods. It is not possible to hedge or diversify to limit the harm in this situation.

The paper I have written about in the previous post is a small attempt to formulate these ideas a little bit more accurately. I feel like the idea is good but I am facing timing limitations to implement it. I have even created a contest on prizes.org for improving the paper so if anybody wants to get 50$ please go here, register and take part!