Expectancy: what a system earns per trade

After this lesson you can compute what any trading approach earns per trade, and explain why a 70% win rate can still lose money.

A coin game

Van K. Tharp famously taught this subject with bags of marbles before showing anyone a chart, and the tradition is worth keeping.

Here's a coin: heads you win €2, tails you lose €1. Fair coin. Should you play? Work it out per flip: half the time +€2, half the time −€1, so 0.5 × 2 − 0.5 × 1 = +€0.50 per flip, on average. Play for as long as they'll let you.

That number is the game's expectancy: the average result per attempt. In trading we compute it in R, using the numbers your journal has been collecting since course 2:

expectancy = (win rate × average win in R) − (loss rate × average loss in R)

Two systems

System A wins 40% of the time. Its average winner is +2.5R and its average loser is −1R, because its trader keeps the stop contract.

0.40 × 2.5 − 0.60 × 1.0 = 1.0 − 0.6 = +0.4R per trade

System B wins 70% of the time, and it feels wonderful to trade. Its average winner is +0.5R, because it takes profits quickly. Its average loser is −1.5R, because it gives losing trades "room". Both of those habits are what push the win rate up to 70%.

0.70 × 0.5 − 0.30 × 1.5 = 0.35 − 0.45 = −0.10R per trade

System B loses money. Its trader is right seven times out of ten and still finishes behind, because the wins are small and the losses are allowed to run half again past 1R. System A's trader is wrong more often than not and earns +0.4R a trade, before costs. The win rate on its own didn't settle either case. The formula needs all four numbers before it says anything.

One consumer note while the arithmetic is fresh: an advertisement leading with a "93% win rate" has told you nothing about expectancy. School VIII comes back to the services that leave the other numbers out on purpose.

Where the streaks fit

Course 1's losing streaks live inside these numbers. System A loses 60% of the time, so eight or nine losses in a row will show up somewhere in a few hundred trades. Expectancy says what the system earns per trade over time, and it says nothing about the order the results arrive in. Holding both facts at once is what lets you sit through a streak without abandoning a system that still works. It's also what makes the harder situation hard, because a broken system and a bad run look identical for a long while. Telling them apart is a sample-size question, and School VIII treats it properly; the honest short version is that ten trades tell you almost nothing about which system you're holding.

Check yourself

1. Win rate 50%, average win +1.8R, average loss −1R. Expectancy? (0.9 − 0.5 = +0.4R per trade.)
2. Win rate 80%, average win +0.4R, average loss −2R. Expectancy? (0.32 − 0.40 = −0.08R, so it loses money despite winning four trades in five.)
3. A signal service advertises its win rate and nothing else. Which two numbers are missing? (Average win and average loss, in R. Without them the win rate is unusable.)

The habit this lesson installs

Judge a system by what it earns per trade, never by how often it wins.

Next: Course 5 — "Money stops: deciding the exit while you're calm."