Thanks, Khalid. OK, since you asked for it...
The Kelly criterion and the conceptually-similar optimal F are designed to find the ""optimal"" risk level for a particular trading system. I put quotes around ""optimal"" because it is certainly not optimal in terms of safety. The so-called ""optimal"" point is the point at which you maximize your returns. I will call it the "max returns" point so it doesn't sound quite so much like the recommended risk level.
Increasing your risk level increases your returns AND increases your drawdowns, until you reach the "max returns" risk level. Increasing beyond that point continues to increase your drawdowns, but your returns DECREASE. Furthermore the drawdowns at the "max returns" risk level are horrific, generally 90-95% or so. No one with any sense of self-preservation should trade anywhere NEAR the "max returns" level.
That means you need to know where the "max returns" level is. That's what Kelly is for.
The Kelly "max returns" risk level can be calculated in several ways. The original research applied to betting situations, where wins were always the same size and losses were always the same size. There are also equations to calculate Kelly for trading situations. These equations take the mathematical variance of the wins/losses into account, and are quite accurate for trading applications. (There are plenty of references for this online, see e.g. the link below.) However you can get a good approximation of the Kelly level with a simple equation. Kelly is approximately: AverageTrade / AverageWin. This is not an exact calculation, but the market conditions will certainly change in the future anyway. You really don't need to calculate your risk level to six decimals. A good approximation is far better than the blind guess most traders use, and is "good enough" for our needs.
So if you look at all your trades and the average profit of all trades is $50, and the average profit of all WINNING trades is $100, your Kelly value is 50 / 100 = 0.50. That means that if you wanted to trade at the "max returns" risk level, you would risk 0.50 or 50% of your entire account on each trade. (!!!!!) Over the long run, you will maximize your returns with this system if you risk 50% of your account each time. The drawdowns will likely drive you insane but the returns will be maximized.
But the intent of Kelly calculations is not to find and then trade at the "max returns" point. As I said, that's suicidal. The object of finding the Kelly level is to determine where the "max returns" point is, so you can choose a safe risk level well below it. You normally select some fraction of Kelly, and trade at that level. So if you have a Kelly value of 0.50, and you chose to trade at 5% of Kelly, then you would risk 5% * 0.50 = 2.5% of your account on each trade.
In fact, you can get a very good feel for your likely future drawdown level, given your chosen fraction of Kelly. For any Kelly value, if K is your fraction of Kelly, and D is a particular drawdown level (D = 0.8 is a 20% drawdown), then the probability of a drawdown D is:
P(D) = D^(2/K-1)
(See e.g. this paper. As I said, these formulae don't apply precisely to the variable-sized wins/losses in trading, but they're a good approximation.)
For a 20% drawdown, D = 1-20% = 0.8. So if you trade at full Kelly, the probability of hitting a 20% drawdown is 0.8^(2/1.0-1) = 80%. But if you trade at 10% of Kelly, the chance of that same 20% drawdown is only 0.8^(2/0.1-1) = 1.4%. If you trade at 2x Kelly, your chances of that (or any other) drawdown are 0.8^(2/2.0-1) = 1. At 2*Kelly you're guaranteed to hit any and all drawdown levels, including 100%.
That risk of a particular drawdown looks like this, for varying levels of K (fraction of Kelly):
So your risk of drawdown increases rapidly as you increase your size, but then the rate of increase actually drops off and the curve flattens out. But that's no-man's-land, and you don't want to be there anyway.
So that's a calculation of risk of drawdown, which can be extended to a risk-of-ruin calculation. It doesn't directly address your expected return. For that I have a spreadsheet that I use to backtest one of my systems, and I can specify the Kelly fraction I want to trade at. Here is the shape of the "return vs. Kelly fraction" from those actual calculations on real trades:
So the returns increase smoothly to the ""optimum"" at full Kelly, then drop off to zero at 1.51 * Kelly. (These numbers will vary somewhat depending on the particular backtest history, but the concepts remain the same.)
So in theory you get your maximal returns at full Kelly, and it drops off fairly smoothly from there.
But remember: no one in his right mind would trade anywhere near full Kelly!! Notice the red Drawdown line. At full Kelly it's at 95%. Can YOU survive a 95% drawdown?? If you decide you can only tolerate a 20% drawdown, you should be trading down around 10-15% of full Kelly.
Furthermore this is the idealized Kelly, based on history. In actual forward trading, you are likely to encounter different market conditions, and the actual Kelly for those future trades will probably be different. In actual forward trading of this particular system, the returns peak at about 72% of full Kelly. That's because the forward-traded period included a worse drawdown than what I saw in the Kelly-calculation period. That's the kind of thing you have to expect and plan for, and that's one reason why you have to assume your historic Kelly values are optimistic.
So that's a "quick" summary of Kelly. Khalid does not share my enthusiasm for Kelly position sizing, to put it mildly.
He is very careful and conservative on his position sizing and goes by general rules of thumb, such as "risk 1% of account." Those rules are generally very safe and are appropriate for most traders. I take a different approach; I apply some additional science and math to the question to find out the appropriate risk levels for a particular trading system. That lets me comfortably and safely risk a larger amount than Khalid might. I know the likely behavior of the system in the future, I know the probability of hitting a particular drawdown level, and I accept the risk of known higher drawdowns for the higher returns I want.For example, let's look at the system I mentioned above. If we follow the 1% rule of thumb, we make 18% CAGR and we see only 4% drawdowns in this backtest.
If I choose to trade at 10% of Kelly, for this system that means risking about 4% per trade. The returns increase to 86% CAGR. In 2 years and over 500 trades, you never saw worse than a 12% drawdown.
I said above that trading at 10% of Kelly means you have a 1.4% chance of a 20% drawdown. I'm very comfortable with that level of risk. I'm very willing to accept that level of risk to increase my returns from 18% to 86%.
By the way, even my more aggressive risk levels are lower than where a lot of people trade, because they don't know where their system's "max returns" point is. They just look at the return they can get with their bigger betsizes and they're oblivious to the much higher risks they're taking. That's why so many people blow up their accounts with a winning system -- they're just trading too large! If you don't know exactly how your system performs and where your "max returns" point is, you're playing with fire to take on larger position sizes.
Khalid's approach is MUCH safer if you haven't carefully analyzed your system's behavior. Unless you understand how to do the above analysis, and you understand the implications of that analysis, you should stick with Khalid's approach and play it safe.
I prefer to understand and characterize my particular system, so I can project my likely risk of drawdown, and empirically choose a risk level that matches my risk tolerance. THIS is the real reason to use Kelly, IMHO -- so you can knowledgeably choose your risk level and have a good idea of what drawdown to expect. The fact that it generally lets you collect a much higher return is a terrific side benefit.