I’ve been doing a lot of thinking and a little bit of work on Ole Peters’ concept of “Time averaging”. This was flagged up in Mark Buchanan’s excellent blog (http://physicsoffinance.blogspot.co.uk/2012/11/ergodicity-biggest-mistake-in-economics.html). The non technical papers (e.g. http://www.towerswatson.com/united-kingdom/research/8271) and TEDx video (http://www.youtube.com/watch?feature=player_embedded&v=LGqOH3sYmQA) are excellent introductions for laymen such as me.

I agree with Peters and Buchanan that this is fundamentally important. It has serious implications for distribution of wealth and equity, but most importantly in terms of risk tolerance. I have had a hunch for a while (after looking at what was the expected 95% confidence limit in a portfolio over say 30 years) that something was wrong in the conventional, risk-neutral, view of the world. Peters’ work hits it fair and square. It is clear that the arithmetic average (ensemble average) is not a relevant statistic for decision making. Far more interesting are either median or percentile outcomes which tell you far more about where you will likely end up, as opposed to the occasional lucky fool.

The St. Petersburg paradox Peters uses to examine this is an extreme example, but not the only one. Nevertheless, I decided to run a simple spreadsheet looking at the distribution of results you get if you play the following game:

Start with $1.

Toss a coin.

Heads you gain 50%

Tails you lose 40%

Repeat 100 times.

If you ask Excel to run this for 8,000 players, the log distribution, as you could expect, is roughly “bell shaped” (see below). But the big surprise is what the average is (or ensemble average to be precise). In the figure below, the amazing result is that the average is 1.76, even though the peak of the distribution is around minus 2.

 

Image

This may not seem that dramatic in log space, but the implication is that the “normal” or “common” result is that after 100 coin tosses, most people will end up with between $0.1 and $1, whereas a few extreme winners mean that the average outcome is $5,800 (!)

Extend this analogy to the GDP of a country and you can see why median earnings are far more important to society than GDP, which can be skewed dramatically by disproportionate returns to the global elite (Top 1% anyone???).

 

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