You did not open six positions. You opened two.

The average pairwise correlation between them. Six positions sized at 1% of capital each do not carry six times the risk of one, and they do not carry the 6% that the sum implies; what they carry is decided almost entirely by that single quantity. At an average correlation of 0.385 — the worked matrix in section 03 — six positions carry the dispersion of 2.05 independent positions. At 0.80 they carry the dispersion of 1.20. The formula takes a line of arithmetic, it is exact for any set of equally-risked positions whatever the individual pairs look like, and it has a ceiling that no amount of position-adding can pass.

The second half of the article is about what correlation does not do. It does not change what you expect to make: expectation is linear, so the average total is identical at every correlation. What it changes is how often everything happens at once. In the worked six-position example the expected total holds at +2.1% of capital throughout, while the probability that all six positions stop out on the same set of moves — the one case in which portfolio heat becomes a real loss — rises from 2.8% under independence to 29.2% at a driver correlation of 0.75.

Somewhere between the largest single position and the sum of all of them — and exactly where is decided by correlation. If you hold n positions each sized to lose R of your capital, the sum n × R is what traders call portfolio heat. It is the amount you lose if every one of those positions is stopped out. That is a real and useful number, but it is not the risk you are carrying. It is the loss in exactly one of the possible combinations of outcomes, and an overstatement of the loss in all the others.

The quantity that describes the whole set is the dispersion of the combined outcome, and stating it correctly needs a second symbol. R is the money a position can lose. s is the standard deviation of what that position actually returns. Those are different numbers, and they are equal only for a symmetric bet that wins R as often as it loses R. For the positions worked through on this page — a 45% chance of a 2R target against a 55% chance of the stop — s is 1.49 times R. Keeping the two apart is the difference between an identity that is true and one that is only nearly true.

The denominator is the part to hold onto: that ratio is dispersion against the summed dispersions, not against heat, and the two coincide only in the symmetric case where s equals R. Read correctly it says the combined spread is the naive sum of the individual spreads divided by the square root of the effective bet count — and that naive sum is reached only at a correlation of exactly 1.

Turning that into money needs the ratio of s to R, and this is where the two quantities visibly separate. Each 1% position on this page carries a dispersion of 1.49% of capital, so six of them sum to 8.95% of dispersion against a 6.00% heat:

Both readings are worth having, and they answer different questions. Against the summed dispersions, correlation takes six positions from 40.8% to 91.3%: the square-root benefit of diversification being cancelled as the bet count collapses. Against heat, the dispersion passes 100% before the correlation reaches 0.5, which looks alarming and is not — heat bounds the loss, while dispersion measures spread in both directions, and a position paying 2R on a win contributes more upside spread than downside. Comparing a spread against a loss bound is exactly the confusion this section exists to remove, and the 3.66% figure at zero correlation is the same number section 06 derives from the full distribution.

A trader who has internalised "diversification cuts risk by the square root" is carrying the 40.8% number in their head. At a correlation of 0.5 — unremarkable for positions taken in the same session on related instruments — the honest number is 76.4%. The mistake is not in the square root. The square root is right. It is being applied to the position count instead of to the bet count.

That sentence is the practical content of this section. A sizing rule of the form "never risk more than 1% on a trade" is a constraint on R. It says nothing at all about n, and n is chosen by mood, opportunity and boredom rather than by arithmetic. A trader can obey their sizing rule with complete discipline on every individual entry and still be carrying five times the exposure they believe they have, because the rule they were disciplined about was never the rule that mattered. That is the same shape as the argument in why evaluations actually fail: what ends an account is usually a rule rather than a bad trade — and a drawdown rule is measured against the whole account, which is every open position at once. Correlation is why that total is larger than the number the trader added up.

It tells you how much of two instruments' movement was shared, on average, in the window you measured, in the second moment only. It is exactly the product of the two instruments' sensitivities to whatever they have in common. It does not tell you they will move together on any particular day, it does not say which drives which, it does not describe the joint behaviour of the tails, and — the part almost nobody accounts for — it is not the correlation between your two trade results.

Correlation is the product of loadings. Suppose two instruments are each partly driven by one common thing and otherwise independent: instrument X responds to driver Z with sensitivity a, instrument Y with sensitivity b, and each carries its own unrelated noise. Standardise everything to unit variance and the covariance is a × b, with unit variances top and bottom, so the correlation is simply a × b.

That product is more instructive than it looks. Two instruments that are each 80% "about" the same driver correlate at 0.64, not 0.80 — the shared part gets multiplied, not averaged. It also runs the other way: an instrument only 30% about a driver still correlates 0.27 with one that is 90% about it, which is how a position taken specifically to be different ends up carrying a quarter of the same bet.

Your trade result is not the instrument's return. A position with a stop and a target does not return the instrument's move; it returns one of two numbers depending on which threshold the move crossed first. Thresholds throw information away, and the correlation between two trade results is therefore lower than the correlation between the two instruments. For symmetric thresholds — an even chance of stop or target — the relationship is exact and classical, an orthant-probability identity for the bivariate normal:

The identity is exact when the stop and target are equally likely. Away from that, it drifts: computed numerically, a stop probability anywhere between 0.45 and 0.55 stays within 0.002 of the identity, and at a stop probability of 0.70 it is still within 0.017. For the range a trading system actually occupies it can be treated as exact, and it is worth knowing which direction it errs in: measuring correlation on your trade log will make a portfolio look better diversified than the instruments underneath it are. Whether a number computed from your own trade list can carry that much weight in the first place is a separate question, and which of your journal statistics are real is where it belongs.

What the coefficient does not carry is the tail. Two sets of positions can report exactly the same correlation and behave completely differently in the corner where everything loses at once, because one number cannot describe a whole dependence structure. That is worth showing rather than asserting. Draw 800,000 pairs twice — once with normal dependence, once with a heavy-tailed elliptical structure — and tune both to the same measured correlation of 0.60. Then ask the only question that matters at the end of a bad day: given that one position is having one of its worst sessions, how often is the other?

The measured correlation is 0.60 in both columns. Under normal dependence the co-movement drains away the further out you look, and tends to zero in the limit. Under the heavy-tailed structure it barely moves, and at the 1% threshold the second position is nearly twice as likely to be down there with the first. The laboratory below uses the normal structure, so its joint-loss figures sit at the lower end of that range by construction. They are a floor on how often everything goes wrong together rather than a forecast of it, and the demonstration above is roughly the size of the gap.

When they load on the same driver. Because correlation is the product of loadings, positions can look entirely different — different instrument, different thesis, different chart — and still be one bet, provided each of them is mostly about the same underlying thing. There is no requirement that the positions resemble each other. There is only a requirement that they each resemble the driver.

Shared drivers come from ordinary places: an index and a component weighted heavily inside it, two positions carried into the same scheduled release, two instruments that can only be exited into the same book, one directional view expressed in two instruments, or two systems fitted on the same features and the same history. Enumerating them is not the useful step, though, because the arithmetic does not care which mechanism supplied the driver. It cares about one number only: the loading.

Section 02 gives the correlation between two positions loading equally on a shared driver as a × a. Substituting that into the effective-bet formula collapses the entire question to that single input:

The last column is the result worth carrying away, because it simplifies exactly to 1 divided by the loading squared. The number of bets a book of same-driver positions can ever reach depends on the loading and on nothing else — not on how many positions there are, not on which instruments they are in, not on why each was taken. And because the loading enters squared, halving it quadruples the ceiling: positions 0.80 loaded on one driver are capped at 1.56 effective bets and the same set at 0.40 is capped at 6.25. Checked at three loading pairs, the ratio is 4.00 every time.

The matrix below is the shape this produces. It was generated from two orthogonal factor loadings rather than typed, which is worth saying plainly: a matrix of plausible-looking numbers typed by hand is very often not a valid correlation matrix at all, and a portfolio calculation fed an invalid matrix returns a confident number that corresponds to no possible world. Generating from loadings guarantees validity by construction, and the eigenvalues are printed beside the figure so the claim can be checked rather than believed.

Position F is the interesting one in the other direction. It correlates 0.105 to 0.150 with everything else, it was sized identically to the rest, and it carries 9.2% of the portfolio's risk against the 16.7% of the money at risk it was given. It is the only genuine diversifier in the set and it is the smallest contributor to the outcome — which is the subject of section 08.

N = n / (1 + (n - 1) × C), where n is the number of positions and C is the average pairwise correlation. That is the whole calculation. Six positions at an average correlation of 0.385 give 6 / (1 + 5 × 0.385) = 2.05 effective bets. The definition behind it is the square of the ratio between the naive sum of the individual risks and the portfolio's actual dispersion — the number of genuinely independent, equally-risked positions that would produce the dispersion you have.

The step worth doing slowly is why only the average correlation survives. Write the definition out for positions carrying equal money at risk R. The numerator is the naive sum squared, which is (nR) squared. The denominator is the portfolio variance, which is R squared multiplied by the sum of every entry in the correlation matrix — n ones on the diagonal plus n(n-1) off-diagonal entries whose sum is, by definition, n(n-1) times their average. The R squared terms cancel:

So a portfolio with one violently correlated pair and four loose ones has the same effective bet count as a portfolio where every pair sits at the average — as far as dispersion is concerned. That is a genuinely useful simplification and it is also a genuine limitation, and both should be carried forward: N is a second-moment quantity. It tells you how variance scales. It says nothing about which corner the losses arrive in, which is why the laboratory computes the full distribution and not only this number.

Then there is the ceiling. Let n grow without limit in the formula and the numerator and denominator both grow linearly, so N approaches 1 / C. That is not an approximation or a rule of thumb; it is the limit of the expression. At an average pairwise correlation of 0.5 you cannot hold more than two effective bets no matter how many tickets you write.

This reframes what adding a position is for. Below the ceiling, an extra position buys diversification. Near it, an extra position buys nothing but adds a term to the heat, a commission, an execution and something else to watch. The arithmetic does not say that holding many correlated positions is wrong — it says that if the reason for holding them was diversification, the diversification is not being delivered, and the decision should be made on some other ground.

The same collapse applies to evidence as it does to exposure, and by the same formula. A run of correlated trades — or a set of correlated backtests — holds fewer independent observations than its count suggests, which is what makes “how many things did you try” a harder question than counting them.