What is your strategy?

What's the K*E for {R-16: 8%, R-14: 5%, Red: 11%} ?

 

Sorry, too complicated. My computing has been reduced for the time being to an android tablet which really sucks. I'll tackle the Kelly eq when it reduces to a quadratic but not when it's a cubic or higher.

I'll never complain about Windows again (at least not until I get back on a Windows box ). Man, I miss Excel.

 

Haha, yeah, you get used to familiar things.

Okay, I think we nailed this problem, with a combined effort. What makes it very interesting is that it is related to the real-world trading strategy selection problem, which is this:

Given N different trading strategies, which criteria (or performance metric) should one use to rank these strategies from "best" to "worst"? The K*E seems to work well for the problem stated in this thread, where the outcome is binary, and both K and E can be calculated precisely. What about the generalized case with the trading strategies, where the outcome is more "analog", or "continuous", such as in this case of the series of trading returns:

{+1%, -2%, +3%, -1%, +10%, -8%, ... }

Here is my first shot at it. In the generalized case, the equivalent of K*E is CK*R,
where
CK = Continuous Kelly
R = strategy return over a given period

The product of the two makes sense, because R indicates the expectancy, and CK indicates the optimal leverage (which adjusts for risk), so CK*R is the "optimally-leveraged return". So, it appears to me that CK*R would be a good metric to use.

Now,
CK = r / s^2,
where r is the mean return, and s is the standard deviation of returns.

So,
Performance Metric = CK * R = (R * r) / (s^2)

What do you guys think?

 

If I followed your notation correctly, that would be the per-bet risk-sized return on bankroll for the strategy, which IMO is a very good figure of merit and would have to be the basis for anything better.

What might be better:
- Some additional indication of how many opportunities of said type per unit time you get. Lots of bets that get sized down for risk but which occur in series can collectively be far better than one better bet. Example metric: expected annual return (log space) assuming 1/5 Kelly.
- The degree to which a given strategy is worth keeping in your portfolio in light of the other strategies available (this is the classic "horses" Kelly at work assuming independence)

 

I'm not convinced about the formula for CK. It seems too simple.

 

Thank you for your feedback, SplawnDarts. With regards to the "number of opportunities", I think it's already (implicitly) incorporated in the proposed metric:

Performance Metric = CK * R = (R * r) / (s^2)

That's because R is (in an arithmetic return sense) equivalent to r*N, where N is the number of trades.

To put in in other words, let's consider two strategies A and B. Let's say they both made the same overall return over the same time period, and the standard deviation of the trade returns is also the same for A and B. The only difference is that A made 10 trades, while B made 100 trades. So, while B had 10 times more opportunities, both strategies have an identical risk-reward profile, and therefore should be scored the same. Do you agree?

 

Well, I am trying to convince myself, too. To this end, I am going to apply the metric (R * r) / (s^2) to the original problem, and see what I get. If I come up with {R-16: 8%, R-14: 5%, Red: 11%} as the top strategy using this metric, that would confirm its validity.

 

My initial result of the Monte-Carlo came out nonsensical. I think that's because the quantity (R * r) / (s^2) does not make sense when r (average return) is negative. I need to figure out how to penalize it correctly, so that the average calculated metric over many trials makes sense.

 

You guys will not believe me, but i worked out r16, r14, red yesterday v simply.

Consider a bet on red alone, the kelly fraction comes to 24.32%.

We know the Kelly fraction for r16 and r14 is 8.26% and 5.48%, respectively.

If these three bets were mutually exclusive, the problem is solved, you just bet the respective percentage of the bankroll per spin.

However, we have the complication that the bet on r14 and r16 is duplicated or overlaps with RED. So all i did was to subtract the Kelly fraction for r14 and r16 from the Kelly fraction of red! So now the red Kelly fraction is 24.32-8.26-5.48 = 10.58%

So we bet 8.26% on R16, 5.48% on R14 and 10.58% on Red.

QED

 

Yeah, that was the simplest thing to do, and it is a good approximation of the solution, which is
8.108% on R16, 5.405% on R14, 10.811% on Red.