Expectancy

An edge is a number, not a feeling about a chart: E = p·b − (1−p), measured in units of what you risk. If you cannot write yours down you do not have an edge, you have a hope about one, and the two behave very differently when the money is real. A setup winning seven times in ten can lose 0.125R on every trade while one winning four times in ten makes 1.40R, and the whole of the difference is one line of arithmetic.

Prerequisites: Lesson 10, which defined R and the cost this number has to clear, and lesson 16, where you last computed a breakeven win rate without knowing it was a special case.

The win rate you actually need

Set E to zero and solve for p, and you get the win rate at which a setup exactly pays for itself:

breakeven p = (1 + s) ÷ (b + 1)

Take the two setups from the top. The first has b = 0.25, so it breaks even at 1 ÷ 1.25, or 80 per cent, and it wins 70 per cent of the time: E = 0.70 × 0.25 − 0.30 = −0.125R a trade. The second has b = 5, so it breaks even at 1 ÷ 6, or 16.7 per cent, and it wins 40 per cent: E = 0.40 × 5 − 0.60 = +1.40R a trade. Seventy per cent was not enough and forty per cent was more than twice what was needed.

Now the whole relationship, with and without the cost of trading. The right-hand column uses s = 0.17, the figure lesson 16 built from lesson 12’s mid-cap:

Reward to risk, b	Breakeven win rate, no costs	With s = 0.17
0.25	80.0%	93.6%
0.5	66.7%	78.0%
1	50.0%	58.5%
2	33.3%	39.0%
5	16.7%	19.5%

The middle row is worth pausing on, because you have seen both of its numbers before. At b = 1 with no costs the breakeven is 50 per cent, which is the base lesson 10 started from before it added its charges; at b = 1 with s = 0.17 it is 58.5 per cent, which is exactly the figure lesson 16 arrived at from the other direction. Three lessons, one formula, and the agreement is not a coincidence — the earlier two were this one with b fixed at 1.

Read down the first column and the trade-off is plain. Every step you take towards a higher win rate is bought by accepting a smaller b, and the table prices the exchange rate. What it will not do is tell you which row to stand on, because that is a question about which of the two you can actually deliver, and only your own record answers it.

What this does not settle

That a positive E is a promise. It is an average, and an average says nothing whatever about the order in which the trades arrive. A method with a genuinely positive expectancy will still hand you losing runs long enough to make you doubt it, and how long is lesson 18.

Nor is an E computed from a handful of trades an E. Both p and b are estimates from a sample, and a small sample gives you a number with a wide error around it that looks exactly as precise as a good one. How many trades it takes before the figure means anything is lesson 19, and it is a larger number than almost anyone expects.

And a positive E does not tell you how much to bet. It tells you the game is worth playing; the size of the bet is a separate calculation with its own failure mode, and that is lesson 20. Getting the first right and the second wrong is a well-populated way to lose money with a real edge.

Finally, none of this can be computed from memory. p and b are facts about your last fifty trades, and if those were not written down at the time then you do not have the inputs, only an impression of them. Lesson 23 is about the ten fields that fix that, and it is duller than this lesson and worth more.

And p and b are not two dials you can turn separately, which is what the table quietly implies and the third problem below asks you to do. They are joined at the exit: move your target further away and b rises while p falls, because fewer trades reach it. The exchange rate the table prices is real, but the trade is forced rather than offered, and the only honest version of the third problem is to measure how much p actually fell the last time you moved a target, which is a number your record has and this page does not.

Two numbers, one line of arithmetic, and an answer that does not care how the chart looked. Most people never write it down, which is why most people are arguing about the wrong variable.

Problems

1. Compute yours. From your last fifty trades, take the fraction that won and call it p, then divide the average winner by the average loser and call it b. Put them into p·b − (1−p). You now have one number that describes your trading, in units that will still mean something when your account is a different size. Write it where you will see it.
2. Find the row you are standing on. Look your b up in the table and read across to the win rate you need. Compare it with the p you just computed. The distance between those two numbers is your entire edge, expressed as the thing you would have to stop being wrong about — and if it is negative, you have learned something that fifty more trades would only have confirmed more expensively.
3. Move one variable. Recompute E with b multiplied by 1.5 and p left alone, then again with p raised by five percentage points and b left alone. One of the two will move your number further than the other, and which one it is depends on where you sit in the table. That is the variable to work on, and you now have a reason for choosing it rather than a preference.

Sources. Van K. Tharp, Trade Your Way to Financial Freedom, which is where the practice of quoting every result as a multiple of the amount risked comes from, and the reason this lesson measures in R rather than in currency. J. L. Kelly Jr., “A New Interpretation of Information Rate” (Bell System Technical Journal 35, 1956), which establishes that a positive expectation is the precondition for any sizing rule at all — and which lesson 20 takes up. Edward Thorp, Beat the Dealer (1962), still the clearest demonstration that a small positive expectation, sized correctly and repeated, beats a game that most players are certain cannot be beaten.

You can now say what your edge is worth per trade. The next lesson asks what that average feels like on the way to being an average, because the answer is what makes people abandon a method that was working.

Educational only. Trading involves substantial risk of loss. Not financial advice. Past performance does not guarantee future results.