Expectancy & the Edge

Risk of ruin has two inputs: how big you bet, and how good your edge is. Last lesson we saw that bet size can sink even a winning system. This lesson nails down the other input — the edge itself — with the single number that summarizes whether a system makes money at all: expectancy. We’ll build it from win rate and payoff, recast every trade into clean R-multiples so a $50 loss and a$5,000 loss live on the same ruler, and uncover why a system that loses most of its trades can still be wildly profitable.

Expectancy: the average profit per trade

Analogy. Imagine a slightly biased coin-flip game at a fair. Each play, you either win some amount or lose some amount, with some probability of each. Expectancy is what you’d pocket per play if you played forever — the long-run average. It folds together how often you win and how much you win or lose into one verdict: per trade, does this system feed you or bleed you?

Definition. For a system with win probability $p$, average win $W$, and average loss $L$ (both as positive amounts), the expectancy per trade is $E = p \cdot W - (1 - p) \cdot L.$ The first term is what your wins contribute on average; the second is what your losses cost on average. A system has an edge precisely when $E > 0$.

Worked example 1 — the trend follower. A trend-following system wins only $p = 0.40$ of the time, but its average win is $W = 1\,200$ and its average loss is $L = 500$ dollars: $E = (0.40)(1\,200) - (0.60)(500) = 480 - 300 = +220 \text{ dollars per trade}.$ It loses 6 trades out of 10, yet earns $220 on average every single trade. Over 1,000 trades that's about$220,000 of expected profit — from a system that’s wrong most of the time.

Worked example 2 — the premium seller. A different system wins $p = 0.80$ of the time (it feels great), but its wins are small, $W = 300$, and its rare losses are brutal, $L = 1\,600$ (dollars): $E = (0.80)(300) - (0.20)(1\,600) = 240 - 320 = -80 \text{ dollars per trade}.$ It wins four times out of five and still loses money — the picture-of-the-account-going-up-most-days hides a negative edge that the occasional disaster more than erases. High win rate, negative expectancy.

R-multiples: one ruler for every trade

Dollar amounts make trades hard to compare — a $50 loss on a tiny account and a$5,000 loss on a large one might be the same risk. The fix is to measure everything in units of the amount you risked. That unit is R.

Definition. R is your initial risk on a trade — typically the distance from entry to your stop-loss, times position size — i.e. the amount you’d lose if the trade goes against you and you exit at the stop. Every trade’s outcome is then an R-multiple: profit (or loss) divided by R.

• A trade that hits its stop loses exactly −1R.
• A trade that makes three times your initial risk is +3R.
• A trade closed early for half the risk is −0.5R.

Why this is powerful. Once every trade is an R-multiple, position size drops out and you can compare and combine trades across instruments, account sizes, and time. Expectancy in R becomes the system’s edge per unit of risk: $E_R = p \cdot (\text{avg win in R}) - (1 - p) \cdot (\text{avg loss in R}).$ If you always exit losers at the stop, the average loss is exactly 1R and the formula simplifies to $E_R = p \cdot (\text{avg win in R}) - (1 - p)$.

Worked example. A system over 100 trades: 40 winners averaging +2.5R, 60 losers averaging −1R (they all hit the stop). Expectancy in R: $E_R = (0.40)(2.5) - (0.60)(1) = 1.0 - 0.6 = +0.40\text{R per trade}.$ Read that as: every trade earns, on average, 0.4 times whatever you chose to risk. Risk $100 per trade → +$40 expected per trade. Risk $1,000 → +$400. The R-expectancy is the same; only your chosen R scales the dollars. This is the cleanest possible statement of an edge, and it’s exactly the quantity the risk-of-ruin formula will consume next lesson.

Win rate versus payoff ratio: the great trade-off

Expectancy has two knobs, and they trade against each other. Understanding their relationship is what frees you from the tyranny of the win rate.

Definition. The payoff ratio (or reward-to-risk) is the average win divided by the average loss: $b = \frac{W}{L} \quad(\text{or in R: } b = \text{avg win in R, when avg loss} = 1\text{R}).$ Expectancy in R is then $E_R = p \cdot b - (1 - p)$. Setting $E_R = 0$ gives the breakeven win rate for a given payoff ratio: $p_{\text{breakeven}} = \frac{1}{1 + b}.$ Any win rate above this line is profitable; below it, you bleed.

The trade-off in numbers. This single formula explains why wildly different systems all work:

Worked example. A trend follower wins only 30% of the time with a payoff ratio of $b = 4$. Is it profitable? Its breakeven win rate is $1/(1+4) = 1/5 = 20\%$. It wins 30%, comfortably above 20%, so $E_R = 0.30 \times 4 - 0.70 = 1.2 - 0.7 = +0.5\text{R}$. Profitable — because the payoff ratio bought it a low breakeven bar. The same 30% win rate at a payoff ratio of 1 would give $0.30 - 0.70 = -0.4\text{R}$, a disaster. The win rate didn’t change; the payoff ratio decided everything.

When to use which lens

When to use expectancy in dollars

• Reporting actual P&L and comparing to costs, salaries, or capital requirements — dollars are what pay the bills.
• Sanity-checking whether a real edge is large enough to matter after commissions and slippage (subtract costs from W and add them to L before computing E).

When to use R-multiples

• Comparing systems across instruments and account sizes — R strips out scale.
• Feeding the risk-of-ruin formula (next lesson), which wants edge per unit of risk, not raw dollars.
• Tracking a track record trade-by-trade without your changing account size distorting the picture.

Pitfall — averages hide the distribution

Expectancy is a mean. Two systems with identical +0.4R expectancy can have utterly different risk of ruin if one has a few enormous winners (most trades are small losers) and the other has steady modest wins. The same average can wrap a smooth ride or a white-knuckle one — which is exactly why the next lessons move from the average edge to the distribution of outcomes and the probability of ruin.

Putting it together

Expectancy is the per-trade verdict on an edge: $E = p \cdot W - (1-p) \cdot L$, positive when the system makes money on average. A high win rate alone proves nothing — an 80%-win system can have negative expectancy if its rare losses are large enough. R-multiples put every trade on one ruler by measuring outcomes in units of risk taken (a stopped-out loser is −1R, a triple-risk winner is +3R), and expectancy in R is the pure per-unit-of-risk edge that scales with whatever you choose to risk. The payoff ratio $b = W/L$ trades off against win rate through the breakeven line $p_{\text{breakeven}} = 1/(1+b)$: trend followers clear a low bar with rare huge wins, scalpers clear a high bar with constant small wins, and both can carry identical edges. Judge every system against its own breakeven, net of costs — and remember that expectancy is only the mean, which is why survival (the distribution and the risk of ruin) is the story still to come.

Next up — the risk-of-ruin formula — we feed this edge into the classic survival equation: how win probability, payoff odds, and the number of units of risk in your bankroll combine into a probability of going broke, why ruin falls exponentially as you hold more units of capital, and several derivations worked end to end.