I propose that the Kelly fraction is a far better measure of risk than standard deviation. At least the risk reflected in Kelly is pure downside, whereas the 'risk' reflected in standard deviation is only partially downside.
You're mixing apples and oranges. Of course the Kelly fraction doesn't give asset allocation, its calculation is totally dependent on asset allocation. BUT it does reflect the quality of any given asset allocation. For all your complaining, you've yet to show a better method of determining how to bet on a finite-spin game.
The BEST strategy! (... so far) : Bet 13% of bankroll on Red ; Bet 7.8% of bankroll on R-16 ; Bet 5.2% of bankroll on R-14. Kelly == 0.2593 , Expectation == 1.37297 , k*E == 0.356
Interesting discussion - and probably a good one to refer anyone too regards the Kelly formula. I can see both sides of the argument, and ultimately it boils down to what you are trying to get to (risk v return) and the limitations of reality (10 spins v infinite) Kelly has two advantages - to minimise the risk of ruin and maximise the return over the long run, neither is particularly applicable here in the constraints of the question. $1000 is not going to ruin you, and you only have 10 spins. While I think Kelly gives the best formula - and yes it can be applied to one spin. In reality you should be prepared to spend your whole bank roll on a game where by the odds are more in your favour than normal. (Hence I also see where Dom was coming from - there is reality and there is theory) Personally..... Due to the limited number of spins, luck plays a much bigger part hence you should go for it....in which case simply betting a fixed amount on the greatest payout and expectation - in the hope that you get a couple of lucky spins and being prepared to blow the lot. (Keep a scotch in hand just in case) When will you get another chance to have a roulette wheel like this again!
Regarding root 2 increase in variance, assume a random variable X has a normal distribution with mean y and variance z. What's the mean and variance of 2X? Use basic to intermediate statistics to figure it out. As for the second part, the optimal " risk adjusted kelly fraction", a combination of differential equations and simultaneous equations will give you the answer.
The answer you arrive at depends on how you approach the situation. 1) Is this a real-life event where you're playing with your own money and you're free to gamble it away with no other consequence? 2) Is this an interview question where your prospective employers are looking to see how well you handle THEIR money? I have the very intense feeling that exposing yourself as a gambler means you flunked the interview. I also have the strong feeling that nonlinear5's 1%-on-Red is also a failing grade. When the interviewers ask why such a timid approach and all you can say is that your made-up performance metric told you to bet that way, oh boy! If ever there was rock-solid evidence that standard deviation is a perfectly lousy measure of risk, it's the recommendation of 1% on Red in this scenario.
agree - its more likely to see if you are sensible and have an idea about sizing. However I have seen it before where the employers want someone who is a gambler - then they simply control what they gamble in....but they need folks to have a go at things and not be timid when the odds are stacked favorably. Hence rather than simply asking - how would you position size something, the question is more stacked to see how they respond. Interesting either way, and its a shame your original poster of the question never really came back.
The "continuous" version of Kelly also uses standard deviation: CK = r / s^2 That happens to be directly related to Sharpe's ratio: CK = r / s^2 = S / s CK = continuous Kelly r = return s = standard deviation of returns S = Sharpe's ratio Now, I know that you fight a campaign against the standard deviation, and that you question the continuous Kelly, but it has been proven as rigorously as the discrete Kelly. The continuous Kelly is a generalized version of the discrete Kelly. The continuous Kelly is designed to account for the non-binary outcomes, such as the ones occurring in a trading.
OK let's try a different question. Which of the three strategies below is best according to MCS? A) bet 10% on Red ; B) bet 5% on Red and 5% on R-16 ; C) bet 5% on Red, 3% on R-16 and 2% on R-14. Now we can make an apples-to-apples comparison looking only at asset allocation and with no preordained performance metric.