A high win rate is not an edge

The most persuasive statistic in trading is also the least informative. A strategy that wins nine times in ten can lose money steadily. One that wins three times in ten can be excellent. What separates them fits on a single line, and it is almost never quoted next to the win rate.

The break-even win rate

The expected value of a repeated bet is the chance of winning times what you win, minus the chance of losing times what you lose. Setting that to zero and solving gives the only number that makes a win rate interpretable:

break-even win rate = amount at risk ÷ (amount at risk + amount collected)

Until that number is known, a win rate describes the strategy's shape rather than the strategy itself, and shape can be manufactured. Any strategy at all can be given an arbitrarily high win rate by taking profits early and letting losses run, which converts frequent small gains into infrequent large losses without adding one unit of skill; the resulting equity curve has a well-known and unflattering name in the industry.

Three structures

Consider three ways of committing one hundred units of capital. The first is a defined-risk structure that collects a modest premium and keeps it unless the market moves against the position by a meaningful amount — the archetype of a high-win-rate design. The second is the same shape with a thinner premium. The third is the opposite archetype: a trend-following pattern that is wrong most of the time and occasionally right by a lot.

One 86 percent structure earns eight and the other loses four, while the 30 percent structure earns twenty — more than twice the best of the high-win-rate designs, and wrong seven times out of ten. Anybody selecting on win rate alone would have ranked these three in exactly the wrong order.

A win rate quoted without its break-even is a mood, not a statistic.

Hazards specific to high win rates

The first row of that table is genuinely profitable, so high-win-rate structures are not a trick. They do, however, carry three specific hazards that the low-win-rate archetype does not, and all three are hard to see from the equity curve.

1. Margin above break-even

Eighty-six percent sounds like an enormous distance from breaking even, but the margin is eight points, and a deterioration of eight points in the win rate — the sort of thing a modest change in market conditions, a slightly worse fill, or a slightly less generous premium can produce — takes the whole edge to zero. The trend structure breaks even at 25 percent and is running at 30, so it has five points of margin on a much smaller base, but its edge does not evaporate from a small drift in the win rate, because its payoff ratio is doing the work rather than its hit rate.

2. The losses are not independent

Of the three hazards this is the most damaging, because in high-win-rate structures the loss usually happens for a common reason — the market fell, volatility spiked, the same thing went wrong everywhere at once — so the losses arrive together rather than one at a time in random order.

That matters statistically, because the arithmetic above quietly assumes each trade is an independent draw. If your losses come in clusters, the number of independent observations in your record is closer to the number of distinct stress episodes the record contains than to the number of trades: over two decades perhaps a dozen, and over a two-year track record often zero, which is why a two-year record of a high-win-rate structure tells you almost nothing about the thing that will eventually determine its outcome.

3. Time to detect a deterioration

Suppose the structure's true win rate has slipped from the profitable 86 percent to the break-even 78. The measurement error on a proportion puts the number of trades needed to detect that in two parts: at about seventy-five trades the two rates are far enough apart to be formally distinguishable, though that is the coin-flip version, the point at which you would notice the deterioration roughly half the time; for a four-in-five chance of catching it you need about a hundred and sixty-five. Both figures assume every trade is an independent draw, which clustered losses rule out.

A strategy taking a couple of positions a month needs the better part of seven years to accumulate a hundred and sixty-five trades. The equity curve will look smooth and reassuring for most of that time, because that is what a high win rate does. The general form of this problem — a test that cannot see the thing it was built to find — is the subject of a separate article.

The breadth constraint

There is a further arithmetic that hits small accounts specifically, and it has nothing to do with the quality of the signal.

The number of positions you can hold at once is a division rather than a decision. If your risk budget is two hundred units and the narrowest structure available in the listed market puts one hundred units at risk, you hold two positions — not because a model chose two, but because the market's contract sizes and strike increments do not offer anything smaller and two is what fits. Neither a better signal nor a larger opportunity set changes that, because it is a floor division and it binds absolutely.

This constraint matters more than the per-trade edge, because the quality of a portfolio's return per unit of risk depends on two things: the edge per bet, and how many roughly independent bets you can hold at once. Richard Grinold formalised this in 1989, and the shape of the result is that the achievable return-to-risk ratio scales with the square root of the number of independent bets, so with two bets the multiplier is the square root of two and rescues nothing. A genuinely positive edge per trade can therefore coexist with a portfolio that is mostly noise, because the edge cannot be expressed often enough or in enough independent places.

This is why so many small accounts running a validated, positive-expectancy design produce underwhelming amounts of money: not because the design is fake, but because two positions at a time, a handful of times a year, at a modest edge per position, multiplies out to a small number, from which the friction discussed in the companion article then takes a share.

What an edge requires

Three terms, all of which have to be present:

1. Positive expected value per unit of risk, to which the win rate contributes only jointly with the payoff ratio; neither number means anything alone.
2. Enough independent bets to express it, which is governed by capital, by instrument granularity, by how often the opportunity appears, and by how correlated the positions are with one another.
3. Enough of it left after friction: costs are per trade and the edge is per trade too, so this is a direct subtraction, and at small size the fixed components of cost do not shrink.

A strategy quoted on the first term alone is quoted on a third of the question; quoted on the win rate alone, it is quoted on half of a third.

Five questions to ask about any win rate

• What is the break-even win rate? If the answer is not immediately available, the person quoting the win rate has not computed the edge.
• What does a loss look like, and is one in the sample? A record that contains no stress episode has not been tested against the thing that produces the losses.
• Are the losses correlated with each other? If yes, the effective sample size is the number of episodes, not the number of trades.
• How many positions can be held at once? And is that a modelling choice or a floor division forced by contract sizes?
• What is the friction as an annual number? Not per trade, but per year against the annual edge.

Scope and limitations

• This article is not an argument that high-win-rate structures are bad. Several of them are well documented and genuinely positive. It is an argument that the win rate is not the evidence, and that a design whose margin is a few points of hit rate needs to be monitored on the payoff ratio rather than on the hit rate.
• Low-win-rate structures are not thereby good; most are not. The point is symmetric, in that neither hit rate carries information without its counterpart.
• Expected value is not a complete account of risk, because it says nothing about the path. A structure with positive expectation and a fat left tail can ruin an account before the expectation has time to assert itself, which is a separate calculation and no less important.
• None of this describes a particular strategy, product or account, and none of it is a recommendation to trade.

Further reading

• Richard C. Grinold, “The fundamental law of active management”, Journal of Portfolio Management, 1989 — where the breadth term comes from.
• Edward O. Thorp, “The Kelly criterion in blackjack, sports betting and the stock market”, 2006 — on why how much you stake matters as much as whether the stake has a positive expectation. The criterion in the title is the classical rule for sizing a repeated favourable bet.
• Roni Israelov and Lars Nielsen, “Still not cheap: portfolio protection in calm markets”, Journal of Portfolio Management, 2015 — a careful worked example of a structure whose intuitive appeal and actual expectation point in different directions.