I disagree. We don't know the inputs into the formula. Let's take a simple continous kelly formula, that uses the first two moments of the return distribution, mean and standard deviation. Let's collapse those into a Sharpe ratio. If I have a 40 year backtest with a true SR of 0.5, then statistically all I can say is that there is a 95% chance that my Sharpe ratio is between 0.1 and 0.9. With a SR of 1.0 things are a bit easier; the 95% confidence is 0.6 to 1.3. Or in Kelly terms, I should be running my strategy (full Kelly) at an annualised volatility target of between 10% and 90% in one case; and 60% and 130% with the higher SR. Or half Kelly, the ranges are between 5% and 45% for SR 0.5, and 30% and 65% for SR 1.0. Of course this assumes the expected SR of my strategy remains the same in the future as it was in my backtest. Okay it's the assumption every systematic trader makes, but it's still an assumption. This adds a further cloud of uncertainty. I run my system at 25% annualised volatility; my backtest has a SR of around 1.0. So if my backtest is correct (and I have reason to think it's conservative at least to a degree, due to a lack of instrument breadth in the past, and an ultra robust fitting method) then even at the lower bound of confidence this is less than full Kelly. If you use higher moments (skew, kurtosis), as you did on the other thread, that doesn't help. Just more parameters you don't know the precise value of. For me personally my own return distribution is sufficiently tame that I would only see a few percentage points change in the correct Kelly. Sure it's better to use a more precise formula. I'm heartily in favour of being reminded that if your returns are non Gaussian you need to be running at a lot less risk (i.e read here from ''the most overlooked characteristic of a strategy is the skew of it's returns"). But let's not fool ourselves that in the real world we can work out the right fraction to 4 decimal places. GAT
Sorry but that is bollocks. First of all, we know exactly the inputs into the new Kelly formula : R1, R2, ... RN , the trade returns. Second, we don't need the right fraction to 4 decimal places, we only need it to 2 decimal places. Third, MEGO whenever I see the words 'Sharpe Ratio'. Perfectly useless statistic; I always avoid it. What we absolutely don't need are any stupid summary statistics like winrate, average win, average loss, drawdown or standard deviation. Here's the info you're missing: OP puzzle: Exact Kelly fraction = 0.778 New Kelly formula = 0.843 Proprietary Kelly formula. = 0.782 Second puzzle: Exact Kelly fraction = 0.994 New Kelly formula = 1.255 Proprietary Kelly formula = 0.995 If I drop the third decimal in the proprietary answers, I match the exact Kelly fractions to 2 decimals. And a formula is a direct one-step calculation. No iterations necessary. What's the main difference between the new formula and the proprietary formula? S5 and S6. S1, S2, S3, S4, S5 and S6 can all be instantly updated with each newly completed trade. Therefore the Kelly formulae can likewise be instantly updated. Good luck doing that with an iterative solution. The right inputs in the right combinations can work miracles.
Let me try putting my point across a different way, as you seem to have had the red mist descend on seeing the words 'sharpe ratio' without the main point of the post coming across. In reality: "There is a 35% chance that you win 20% of your bet ; There is a 25% chance that you lose 15% of your bet ;; There is a 20% chance that you win 12% of your bet ; There is a 15% chance that you lose 6% of your bet ; There is a 4% chance that you win 50% of your bet ; There is a 1% chance that you lose 100% of your bet." .... we don't have this kind of information. This isn't a casino. We're not card counting blackjack with an edge that we can quantity. It's the stock / futures / FX market. We can estimate the figures above from a backtest, if we're trading systematically (I wouldn't even presume to guess how a discretionary trader would do this). But this will give us a point estimate, and does not reflect the uncertainty of the fact that our backtest is just a sample from an unknown data set. It ignores overfitting bias. It ignores the fact that the future won't be like the past. This estimation error massively exceeds the 10% difference between these estimates: Exact Kelly fraction = 0.778 New Kelly formula = 0.843 Proprietary Kelly formula. = 0.782 As I pointed out just looking at the first two moments of the distribution alone the uncertainty error is at best a factor of 2. The overfitting bias could easily be another factor of 2, and sometimes more. The future not being like the past; let's not even go there. Using more moments of the distribution, or a different formula, does not deal with any of these problems. Don't get me wrong, it makes sense to use a model that is theoretically correct, rather than one that is wrong. But we shouldn't let the fact that it's a model which will be accurate if we know the inputs perfectly - which we are a long way from doing. GAT
Where suddenly do S5, and S6 come from? I think they weren't present in your past postings? This is really getting funnier and funnier.
Sorry I misunderstood. Of course the puzzles are total artifices. In the real world we'd just have a long string of trade returns. And that is precisely what the new Kelly formula and proprietary Kelly formula are designed to deal with, although they obviously can deal with the artificial trading scenarios as well. My main point is that the formulae are update-able, so no need to try to figure out probabilities and so forth. The formula will take whatever string of trade returns you feed it and compute the current best trading fraction. For the new formula, I would use it for half-Kelly or perhaps ⅔-Kelly trading. I use the proprietary formula (2 decimals as stated) for full Kelly trading. Always updating as stated. The 'experts' concerned about Kelly blow-ups have never seen anything like my formulae so their fears have no foundation. They're still doing the winrate/rr-ratio kindergarten stuff.
S5 and S6 are inputs into the proprietary Kelly formula. Same format as S1, S2, S3 and S4 in the new Kelly formula.
So essentially you are adding new information (i.e. new trades) to your existing trading stats and recalculating Kelly? Would that be right?
Yes, using a formula that accounts for higher moments is important (whether 3, 4, 5 or 6). But I still think that Kelly blowups usually happen not because of the wrong formula, but because of a poor appreciation of the uncertainty of returns. I'm going to labour the point, in case there are people reading this thread without the same appreciation of the nuances that you obviously have. The first order problem is that if are over confident about your mean versus your standard deviation (there I won't use the 's' word again); as I've already discussed. It doesn't matter how you update your estimate of optimal Kelly, you're going to need thousands of data points just to know that there is a 95% chance your optimal Kelly is somewhere between 2% and 4% (using more realistic numbers). The second order problem is you have a negative skew / evil kurtosis, but don't realise you do, because you haven't yet had your LTCM moment. So you get a long string of small positive returns. Even the fanciest Kelly formula will be taking you to fill your boots. Or if you prefer There is a 95% chance that you win 1% of your bet ; There is a 5% chance that you lose 10% of your bet There is a 0.01% chance that you lose 100% of your bet ... comes out to around 97% (using excel solver, with rounding, for the avoidance of any ambiguity) But if the true distribution is actually: There is a 95% chance that you win 1% of your bet ; There is a 4.8% chance that you lose 10% of your bet There is a 0.2% chance that you lose 100% of your bet ... optimal comes in around 55% But then is why we use half kelly, right? We'd just about get away with it. But if the true distribution is actually: There is a 95% chance that you win 1% of your bet ; There is a 4.5% chance that you lose 10% of your bet There is a 0.4% chance that you lose 100% of your bet Then you should be using something more like 17%. My question is does anyone really have a well calibrated idea of whether something has a 0.01%, 0.2% or 0.4% chance of happening? I refer you to the behavioural finance work on this, plus everything Taleb has written. Even if the underlying distribution of returns is stable (a massively unrealistic assumption as I've already said), you'd need tens of thousands of trades to narrow down a sufficiently narrow window for that figure. (this is a more long winded way of saying that having to estimate higher moments doesn't make your life any easier) And in real life, as I've already said we don't have a stable distribution or enough data; nine times out of 10 we're LTCM in 1998 or the CDO guys in 2006 and the -100% currently has a realised probability of zero. So you have to guess what the odds of -100% are. If you're tied into a gaussian thought process you'd not bother assiging any probability, since even on the last set of numbers -100% is a 15 sigma event. Even if you manage to get yourself to confront the remote possibility of a total loss how do you know whether that possibility is 0.2% or 0.4%? The only way out of this conundrum is to avoid trading anything with such evil skew (or anything that's likely to have such nasty skew in the future), or if you absolutely must do so then use an extremely conservative fraction of optimal Kelly. One Kelly formula might tell you that optimal is 90%, another might say 80%, but in certain situations you'd be insane to use more than 5%. GAT
This is trivial stuff. Just use a black swan factor (BSF): BSF == 0.9/max[ .0001 , X*max[ -Ri ]_i=1toN ] BSF prevents your next losing trade from wiping you out even if it is X times as large as your current worst loss. Choose the smallest value of X that lets you sleep at night. Default: X = 2. Use min[ BSF, your (fractional) Kelly estimate ] as your trading fraction.