What is your strategy?
- Thread starter kut2k2
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Help kut2k2 and myself to come up with a generalized approach to rank trading strategies (not necessarily the ones with a binary outcome) . This is where I'd like to see the direction of this discussion.
I think I am onto something. I found a confirmation of (r^2) / (s^2) = Sharpe^2 as a "special quantity" in here (see section 7, "Wall Street: the biggest game", equations 7.4):
http://www.bjmath.com/bjmath/thorp/paper.htm
close, reduces variance by 1 divided by root 2 = 71%
Reduces variance by 2, and reduces STD by 1 divided by root 2 = 71%
You need to include the covariance matrix of the three choices, because the three outcomes are not independent. The results would be a strategy containing more than one choice, as this will allow you to reduce the overall variance without decreasing the arithmetic return which in turn will allow you to increase the geometric return as a result of the reduction in variance ( it would be as if you had a extra option which is closer to a sure gain which will allow the kelly criterion to allocate more of the bankroll to invest in each successive bet).
Continuous Kelly says that the leverage should be R/S^2. When you time that by R, you get R^2/S^2, or square of sharp.
(This is assuming that risk free rate is zero).
Right. And furthermore, I propose that (R^2 / S^2) can be used as a universal performance metric for strategy selection, including any trading strategy, and even more generally, including any betting strategy, such as the one stated in the beginning.
You guys keep arguing about candidate performance metrics which are monotone increasing transforms... none of these can be any better of a metric than the base figure.
Think of it like this, if a strategy maximizes some metric, x, relative to other strategies, then it also maximizes x^2, ln(x) and ln(x^2).
In economics they use a log utility function to account for risk aversion. It has two problems: 1) it implies 'upside risk' is bad (mentioned earlier in this thread), and 2) it gives minus infinite utility if money goes to zero, so it's only really meaningful for life-bankroll type money, not just one bet.
If you are interested in using this type of approach to get a best risk-adjusted return, then you have to calculate the logged value of income in every terminal state of nature, before averaging them/taking the expectation. In the case of this thread, that means you would take the log of all the terminal payoff values and then average the results. This is of course problematic if you go broke ever, and I believe this is one motivation that led to the Kelly criterion being expressed as a percentage rather than a fixed amount...
Right, but what is the "base" metric for a betting strategy? I am arguing that it should be r/s, or better yet, (r/s)^2, since it captures the relative value of a strategy better.
I propose the System Achievement Score.
SAS == 4*k*max[ 0, E ]*PF*min[ 1, N/mant ] ,
where
SAS is the System Achievement Score,
k is the solution to the Kelly equation (see below),
E is the expectation (see below),
PF is the profit factor (see below),
N is the number of trades in the SAS evaluation,
mant is the minimum acceptable number of trades.
E == sum[ Ri ]_i=1toN / N ,
where
Ri is the return (%) of the i'th trade.
The Kelly equation is
0 == sum[ Ri/(1+k*Ri) ]_i=1toN
PF == sum[ max[ 0, Ri ] ]_i=1toN / sum[ max[ 0, -Ri ] ]_i=1toN
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