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Kelly Value and Equity Curve of a trading system
Do you remember the moment when you had started to read your first book on how to win in the markets? I don't know which one was yours, but I'm sure the bestseller I was reading had something in common with your book. I think those are the widely accepted market wisdoms like "Trend is your friend" or "Let your winners run and cut your losers soon". Am I right?
That sounds good. That sounds easy to repeat to other traders. And everyone nods. What about "Don't put all your eggs in one basket!" maxim? But what does it mean? Do any of these industry slogans give you any edge at all? The answer is :"Yes and No" at the same time. Yes if you have developed a trading system, back-tested it on the historical data and strictly follow its signals in your real-life trading. Without overriding any single signal. Just taking it religiously! And the answer is definitely NO if you trade without any plan or statistically tested strategy. You heard on CNBC Maria talking about "over-all up trend in the stock market" and Goldman Sachs's Abby speaking about "best opportunity to buy stocks ever" and you push the green button, reasoning that indeed the stock market is in a great secular bull market (so the trend is up) so that's why I buy. But the market continues to decline and at the bottom you receive the margin calls from your broker even if you diversified the capital among the leaders. Yeah... Market crash. Sounds familiar?
So the first assumption for further discussion is that you have developed yourself/leased/bought a trading system and back-tested it on history big enough to produce at least 50 to 60 trades. It can be a trend-following system, range system or whatever else. You see that it wins in 60% of trades and an average winning trade is 1.5 times greater than a losing one,e.g. So the win probability equals 0.60 and win/loss ratio is 1.5 Can you still lose your money trading this winning on average system?
We have simulated 100 traders accounts and their equity curves during more than 450 trades, assuming that the process of trading is absolutely random with the only known and determined parameters : win/loss ratio and win probability. You can play yourself with other parameters and see the possible future development of your equity curve! Click here to see our Java applet.
As you can see it's a very efficient system in the long run. You start with initial capital equal to 100 points and end up with 300 points in your account on average! Though there's one trader who was not lucky enough and whose account experienced a temporal drawdown to 93. This example represent a system with the positive mathematical expectation of profit. If you tried our applet then you can see that it's equal to 0.50 In other words, every buck put in a trade returns 50 cents of profit on average.
If you try a system with win/loss equal to 1.00 and win probability equal 0.50 than you will get something like this. On average the system is neutral and an average account should have a horizontal straight equity curve. But in reality as in the markets there are traders who are just more lucky than other ones. So money flows from unlucky hands to lucky ones in this zero-sum game. I think you understand that in such case your playing the markets is nothing more than trivial coin tossing.
So before committing any dollar to the market check and see the possible future of your account. And don't count on your past being just lucky!
The next and most important question is this one. How many eggs should I put in one basket in order to achieve the best results for my account as a whole in the long run? Or how much percentage of the capital should I put in the trade when the system rings to buy? If I risk too much eggs then I can ruin all my account if a losing streak got my winning on average strategy. If I commit too small amount of money the risks are low and so are the returns. The right value is somewhere between and the Kelly's formula gives an exact answer.
The Kelly Criterion arose from the work of John Kelly at AT&T's Bell Labs in 1956. His original formulas dealt with long-distance telephone transmission signal noise. But the gambling community quickly understood that the same approach may help them to calculate the optimal amount to bet on a horse and the best way to take advantage of overlays and underlays, maximizing the growth of your bankroll over the long term. Nowadays, Kelly Criterion is a recognized money management system and whenever the question of optimal betting size pops up in handicapping or money management books you always see Kelly formula mentioned.
The Kelly's formula is : Kelly % = W - (1-W)/R
The math behind the system is pretty complicated. Kelly's original paper is all but unreadable to non math majors.
For more in-depth information about using Kelly's value in stock trading and long term investing please read the following literature (free ebooks):
THE MATHEMATICS OF GAMBLING
THE KELLY CRITERION IN BLACKJACK, SPORTS BETTING, AND THE STOCK MARKET
The Winners and Losers of the Zero Sum Game
Also it might be interesting for you to see our Java applet
P.S. Our personal experience tells us that even you have a perfect trading system and calculations above show that you should invest up to 50% or more in a single trade, just DO NOT DO THAT! Don't tie more than 25-30% of your capital to any single trade despite any calculations! Don't be too greedy and then Mr. Market will reward you for your passion.