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.

1. Kelly criterion is the sure way to ruin. Lars assumes he knows the win rate. In reality, nobody knows the win rate. As a result, the Kelly criterion cannot be applied. Your knowledge of probability theory is very weak and so is Lars' because the win rate is already derived from a limiting process, i.e. you cannot go back in time and use the win rate to start with. Who and what can guarantee a win rate of 60% in investment field?

"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."

There is no such a thing as an "expected return of a single round". This is a common misconception of first year math students because of the units of the expected value. The average used in the Towers and Watson paper converged to the expected value only at the limit of infinite trials.

"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."

This is expected, as at the limit the time-average return should converge to the expected value which is also 0. But you insist on defending the Towers and Watson paper in which they assume that in two coin tosses of equal probability one will be a head and the other will be a tail, a gross misconception some high school kids have. In that experiment there are four possible outcomes: (heads, heads), (heads, tails), (tails, tails), and (tail, heads). The average of all those possible time-averages returns is 0 as expected. Is this so hard to understand?

The problem with most economists is that they do not study probability and statistics and lack understanding of the subject.

Lars' method is the sure way to ruin. One loser and he is down 20%. Another one and he is down 36%. Another one and he is out of business. It will take only three losers for the average fund to go out of business when using Kelly.

2. Hi Mark, thanks for pointing to this work. Immediately I myself wrote an article about Peters and ergodicity for the danish newspaper Ingeniøren - ing.dk/artikel/134205 - and also talked with Alex Adamou. Interesting stuff. Sadly, it's in danish.

1. You guys need to understand probability theory before dealing with this stuff. Apparently, you do not. The notion of risk of ruin is as old as casinos. What you bet is what you can afford to lose. Averages converge to expected values only at the limit, weak or strong. Nobody in the investment industry has ever sized bets based on expected value and this is just a straw man invented by those who do not understand the notion of risk.

http://www.quantwolf.com/doc/cointoss/cointoss.html

Risking your entire wealth makes sense only if the probability of win is 1. What you do not understand from this simple result? Anything else is a strawman argument.

However, even the Kelly criterion is not appropriate when drawdown is used as a performance measure or drawdown adjusted returns are used to rank performance. In that case, you risk what you can afford to risk at every run.

3. Regarding Lars Syll's post I think his equation (7)
(7) 0.6 log (1.2) + 0.4 log (0.98) ≈ 0.11.
(7) 0.6 log (1.2) + 0.4 log (0.8)
??

1. You're right and it has now been corrected (I inadvertently must have read 0.2 as 0.02)!

2. This comment has been removed by the author.

4. So what? The growth rate was overestimated by a factor of about 5.50. This is finance, right. :)

This is the least of Lars problems. In general, optimal growth rate is path dependent. His example involved a stationary win rate and equal R:R. These are never true in the investment field.

5. Thanks for give us information about Physics of Finance. Thank you very much.

6. In regards to 2), a very simple way of thinking about it is to multiply your starting capital (1) by 0.5. If now you multiply your result (0.5) by 1.5, you get "only" 0.75: after a 50% decrease of capital you need a 100% appreciation to get back to your starting point.
The same holds if first you have a certain % increase of your capital, and then the same % decrease of the increased capital. Your final capital will be lower than the starting one.
Can be counterintuitive, but if you think about it, it makes perfect sense.
Andrea

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