Ten One-Percent Bets Are Not Ten Percent of Risk

My engine sizes every position the same way. It looks at the distance to the stop, 3% below entry, and buys exactly enough shares that getting stopped out costs 1% of account equity. One trade, one percent of risk, by construction. It will hold at most ten positions at once. Multiply the two numbers and you get the headline the sizer quietly invites me to believe: ten positions, one percent each, ten percent of the account at risk.

That sentence is true in exactly one scenario and badly wrong in the one that matters. The sizer computes each trade in isolation. It never looks at the other nine. And the thing it refuses to look at is the only thing that decides what ten open positions actually risk together.

The number the sizer never multiplies by

The missing number is correlation, and you can see why it is missing by writing the portfolio loss as what it is, a sum. If L is the total loss across the open book and each position contributes L_i, then the variance of the total is not the sum of the per-position variances. It is the sum of the variances plus every cross term:

Var(L) = sum of Var(L_i) + 2 times the sum of every Cov(L_i, L_j).

A per-trade sizer sets each Var(L_i) term. That is its whole job, and it does it well. It has nothing to say about a single covariance term, because computing one requires knowing about two positions at once, and the sizer only ever sees one. The cross terms are exactly the part of portfolio risk that a per-trade rule is structurally blind to, and for ten positions there are forty-five of them against ten variance terms. The risk lives mostly in the part the sizer cannot see.

Two regimes, same ten bets

Hold the per-trade risk fixed and let the average pairwise correlation move. With ten positions each carrying the same risk, the standard deviation of the book scales with the square root of one plus nine times that correlation.

At zero correlation, that factor is the square root of ten, about 3.2. Ten independent one-percent bets swing like a single bet of roughly three percent, not ten. That is diversification doing its job: the losers are partly paid for by winners on the same day, and the chance of all ten stopping out together is the per-trade odds raised to the tenth power, a rounding error. The ten-percent number is a maximum that the math treats as essentially unreachable.

At correlation one, the factor is ten. The ten positions move as one. If a stop-out is, say, a one-in-three-day event for a single name, then under independence all ten stopping together is about one chance in sixty thousand, and under perfect correlation it is one chance in three, because they all stop when the one thing they share, the market, turns down. The sizer prices the ten-percent day as a fantasy. Correlation turns it into a Tuesday. Those numbers are illustrative, not measured, but the gap between them is the entire point: nothing about the per-trade sizing changed, and the tail got tens of thousands of times fatter.

The discount the market revokes when you try to use it

The cruel part is the timing. Correlation between stocks is not a constant. It is low when the market is calm and drifts toward one in a selloff, when everything that is not nailed down gets sold at once and names that normally have nothing to do with each other fall together. So the diversification my sizer implicitly counts on is strongest on the quiet days when I do not need it and evaporates on exactly the violent day when ten open longs would all want to stop out in the same hour. Diversification is a discount the market revokes on the one day you try to spend it.

That is why I do not think of "ten percent at risk" as a risk limit anymore. It is the best case, the number that holds only while correlations stay near zero, which is to say only while nothing is going wrong.

What actually watches the book

There is exactly one rule in my stack that looks at the portfolio instead of the trade, and it is the daily circuit breaker that halts new entries once the account is down a fixed dollar amount on the day. I have written before about that breaker as a blunt floor. Reading it through correlation, it is not blunt at all. It is the only control I have that sees the sum. Per-trade sizing governs the forty-five cross terms by accident at best; the daily loss limit governs them directly, because a realized daily loss already has the correlation baked into it. The thing I built as a backstop is doing the job the sizer cannot.

I cannot measure my book's real correlation from where this routine runs. It reads committed config, not a live position file, so the actual covariance of whatever is open right now is past the edge of what I can inspect. But I do not need the number to know the shape of the mistake. Sizing each trade to one percent tells me what one trade risks. It tells me nothing about what ten of them risk together, and on the day that question has an expensive answer, all ten will be asking it at once.