Friday, November 16, 2012

Why time matters

I'm still thinking about the ideas of Ole Peters and the importance difference between time and ensemble averages. A few comments suggest that some people I have "lost the plot," but I'm convinced this issue is indeed extremely important and generally underappreciated. A few things to add for now:

1. This post by economist Lars Syll from earlier this year does an excellent job of laying out the main issues and linking them to the Kelly criterion: a practical criterion for playing risky gambles that is based explicitly on time averages. Lars couldn't have explained the basic ideas more clearly.

2. From some comments on other blogs, similar to some I've seen here, many people familiar with probability theory find it hard to accept that a time average expected return of a random multiplicative process is just not equal to the (usual) expected return of a single round. It isn't. Start with any number you like, multiply it by a long sequence of numbers, each either 0.9 or 1.1 drawn with equal probability, and you will find that the number tends to get smaller. In the limit of an infinite sequence, the result heads to 0. And the result quoted in the Towers and Watson paper, a 1% decline on average per period, is correct.

 3. I came across an interesting comment from Tim Johnson, writing on Rick Bookstaber's blog:
The model Peters develops appears to be remarkably similar to the one Durand proposed in 1957 (The Journal of Finance, 12, 348–363) and is discussed by Szezkely and Richards (The American Statistician, 2004, Vol. 58, No. 3).

I do not disagree with your assessment that there has been an error in economics for the past 77 years (just one?) but mathematicians working in finance have generally ignored Samuelson's attacks on logarithmic utility. Poundstone's book on the Kelly Criterion is a good description of the battle in the 1960s and I there is a rich contemporary literature that develops Kelly's ideas...
The paper by Szezkely and Richards is indeed worth a read, although I'm convinced that Peters has gone considerably further than Durand.