Correlation-Adjusted Position Size Calculator

Q: What correlation should I enter if I have not measured it?

There is no default that is right, and the calculator does not supply one beyond a starting figure. What can be said is which way the error costs more: correlations drift upwards under stress, so a figure that is too low understates combined risk at the moment the limit is being tested. Running the same positions at two or three correlations and reading the spread is more informative than picking one.

Q: How is this different from the portfolio heat calculator?

Heat is the straight sum of the risks — what the book costs if every stop is hit together — and it makes no assumption about how the positions relate. This calculator takes the same risks and combines them at a correlation you supply, so overlapping positions contribute more than unrelated ones. Both figures are reported here, because the gap between them is the point.

Q: Combined risk is lower than the sum. Does that mean I have spare capacity?

It means the arithmetic of combining correlated risks gives a smaller number than adding them, under the correlation you entered. The straight sum has not stopped being possible: every stop hitting at once is exactly what happens in a broad sell-off, and that is also when correlations are highest. The combined figure is a description of the model, not headroom the account has been granted.

Q: What does 'effective independent positions' actually mean?

It is the number of equal, uncorrelated positions that would carry the same combined risk as your book. Four positions at a correlation of zero give four; the same four at a correlation of one give one, because they are functionally a single trade in four pieces. It converts an abstract correlation into a count, which is usually easier to react to.

Q: Why can I not enter a negative correlation?

Because a single average correlation is only a valid matrix over a limited negative range. Below −1 ÷ (n − 1) the implied matrix is not positive semi-definite and the expression under the root goes negative, meaning the question has no answer rather than a negative one. Genuine hedges need pairwise correlations, which this calculator does not accept.

Q: Does the scaling factor mean I should resize my positions?

It reports arithmetic, not an instruction. The factor is simply limit ÷ combined risk, and multiplying every position by it lands the combined figure exactly on the limit you typed in — upwards when there is room, downwards when there is not. Whether the limit itself is the right one, and whether every position deserves the same treatment, are decisions the calculation does not make.

Size correlated positions so their combined risk matches your limit, rather than adding up as if they were independent.

Adding up what four open positions risk assumes all four stop out on the same day. Ignoring the overlap entirely assumes none of them do. Neither describes a real book, and the distance between the two answers is where most accounts actually sit. This correlation-adjusted position size calculator puts a figure on that distance by combining the individual risks with the standard portfolio variance expression instead of a plain sum.

Formula: combined risk = sqrt(sum of variances + 2 x sum of pairwise covariances)

Your figures

Currency

Account balance

Total risk limit

The most you are prepared to lose across every open position at once.

Average correlation

How closely these positions move together, from 0 (independent) to 1 (they move as one). Your own estimate — nothing is fetched — and a pessimistic figure is the safer entry, because correlations rise towards 1 in exactly the conditions that make them matter.

Open positions

For each position, either the cash you lose if it stops out or the same risk as a percentage of the account. Cash wins if both are filled in, and blank rows are skipped.

Risk if stopped out 1

Risk % 1

Risk if stopped out 2

Risk % 2

Risk if stopped out 3

Risk % 3

Risk if stopped out 4

Risk % 4

Combined risk

US$837.85

3.35% of the account across 4 positions, against a 6.00% limit

Room left under your limitYour limit is US$1,500.00.: US$662.15 · 2.65%
Naive total riskThe straight sum of the risks, which is what portfolio heat measures and what every position stopping out together would cost.: US$1,000.00 · 4.00%
Diversification benefitHow much of the straight sum the correlation entered removes. At a correlation of 1 it removes nothing at all.: US$162.15 · 16.21% of the sum
Effective independent positionsHow many equal, uncorrelated positions would carry this combined risk. 4 positions at a correlation of 1 behave like one.: 1.42
Scale every position byMultiplying every risk below by this factor puts the combined risk exactly on your limit. Above 1 the limit has not been reached; below 1 it has been passed.: 1.790x
Positions countedAt an average correlation of 0.60: 4
Position 1 — US$300.00 entered1.20% of the account · 30.00% of the total risk · shown at 1.790x: US$537.09
Position 2 — US$250.00 entered1.00% of the account · 25.00% of the total risk · shown at 1.790x: US$447.57
Position 3 — US$250.00 entered1.00% of the account · 25.00% of the total risk · shown at 1.790x: US$447.57
Position 4 — US$200.00 entered0.80% of the account · 20.00% of the total risk · shown at 1.790x: US$358.06

What this means

Adding the risks up gives US$1,000.00, which is what these 4 positions cost if every stop is hit. Combining them at a correlation of 0.60 instead — each risk treated as a standard deviation, so the total is the square root of the summed variances plus the pairwise covariances — gives US$837.85, or 3.35% of the account. The difference of US$162.15 is what the correlation you entered is worth, and it is why this set behaves like 1.42 independent positions rather than 4. Multiplying every position by 1.790x lands the combined figure exactly on your 6.00% limit. As entered, the combined risk is within that limit.

One average correlation stands in for every pair. Real matrices are not uniform, and a pair moving together at 0.9 inside an average of 0.4 is invisible here.
Correlations are not stable. They rise towards 1 in precisely the sell-offs that test a limit, so a figure measured in calm conditions understates the risk of the day that matters.
Negative correlations are not accepted. A single average below −1 ÷ (n − 1) describes a correlation matrix that cannot exist, and a genuine hedge needs position-by-position correlations rather than one average.
Each risk is treated as a standard deviation, which is the standard portfolio variance formula applied to stop distances. It describes how the risks combine, not how likely any of them is to be reached.
Assumes each stop fills at its price. A gap takes the loss beyond the amount entered, and gaps are exactly when correlated positions gap together.

The link reopens this calculator with the same numbers. Nothing you enter is sent to a server.

Continue with these figures

About the Correlation-Adjusted Position Size calculation

Beyond the risks themselves it needs one input: the average pairwise correlation between the positions, from 0 for independent to 1 for moving as a single trade. Nothing is measured for you — the figure is your own estimate, and the calculator applies it to every pair. At a correlation of 1 combined risk equals the plain sum exactly, and at 0 it collapses to the root sum of squares, so the two ends of the range behave the way intuition says they should.

The output is the combined risk of the correlated positions in cash and as a percentage of the account, the difference between that and the naive sum, how many genuinely independent positions the set behaves like, and the multiplier that would put combined risk exactly on the total risk limit entered. The multiplier is a complete answer rather than an approximation, because scaling every position by the same factor scales the combined figure by precisely that factor.

This correlated trades risk calculator is the one that follows portfolio heat. Heat measures what a book costs if every stop is hit at once; this measures what the same book is worth treating as risk once the positions are allowed to be related, and reports both so the gap is visible.

Correlation-Adjusted Position Size formula

combined risk = √( Σ risk_i² + 2ρ Σ_{i<j} risk_i × risk_j )

computed from two running sums, with
  S = Σ risk_i        and        Q = Σ risk_i²

combined risk = √( Q + ρ × (S² − Q) )

naive total risk        = S
diversification benefit = S − combined risk

risk limit = account balance × limit % ÷ 100
scale by   = risk limit ÷ combined risk

effective independent positions = (S ÷ combined risk)²

for n equal positions this reduces to:
  combined risk = risk × √( n + n(n − 1)ρ )

risk_i: The cash lost if position i stops out — not the position's value, and not the margin posted against it.
ρ: The average pairwise correlation, 0 to 1. It is your estimate of how closely these particular positions move together, applied uniformly to every pair.
S: The naive total: the straight sum of the risks, which is what the portfolio heat calculator reports and what every stop hitting together would cost.
combined risk: The risks added as standard deviations rather than as cash, so overlapping positions count more than once and unrelated ones count less.
scale by: The single multiplier that lands combined risk exactly on the limit. Above 1 the limit has not been reached; below 1 it has been passed.

Each position's risk is treated as a standard deviation and combined with the portfolio variance formula. That is a modelling choice and it is worth stating plainly: the expression says nothing about how likely any individual stop is to be hit, only about how the losses aggregate when they are. What it buys is correct behaviour at both ends of the correlation range, which no simpler haircut on the total achieves.

Those two ends are the sanity check on the whole thing. At ρ = 1 every pair moves together, everything under the root collapses to S², and combined risk is exactly the naive sum — a correlation of one means diversification buys nothing whatsoever. At ρ = 0 it becomes the root sum of squares, which for four equal positions is twice one position's risk, not four times it and not a quarter of it. Anything that fails either of those checks is not a correlation adjustment.

The second form in the block is the one computed. The pairwise sum Σ_{i<j} risk_i risk_j needs a loop over every pair, whereas (S² − Q) gives twice the same total from two running sums, and the two are exactly equal rather than approximately so. The general form is used rather than the equal-position shortcut because real books hold unequal positions, and replacing them with their average understates the sum of squares — which understates the combined risk, in the direction that flatters the account.

Because the expression is homogeneous of degree one in the risks, multiplying every position by k multiplies combined risk by exactly k. That is what makes one scaling factor a complete answer: limit ÷ combined risk, applied across the board, lands the combined figure precisely on the limit while leaving the relative sizes of the positions untouched. The per-position rows show what each becomes at that factor.

The effective independent position count is (S ÷ combined risk)², and it answers the question the correlation figure is really being asked. Four positions at ρ = 0 count as four; the same four at ρ = 1 count as one; in between the number is usually a good deal smaller than the position count suggests, and it is the most direct way of seeing what a book of related trades is really doing.

What no portfolio risk correlation calculator can do is know the real correlation matrix. One average stands in for every pair, and correlations are neither stable nor symmetric in their behaviour — they rise towards one in precisely the conditions that test a risk limit, which is an argument for entering a stressed figure rather than a measured one.

Correlation-Adjusted Position Size: a worked example

A $25,000 account holding four positions risking $300, $250, $250 and $200, at an average correlation of 0.6, against a 6% total risk limit.

Naive total risk, S300 + 250 + 250 + 200 = $1,000, or 4.00%
Sum of squares, Q300² + 250² + 250² + 200² = 255,000
Pairwise termS² − Q = 1,000,000 − 255,000 = 745,000
Variance255,000 + 0.6 × 745,000 = 702,000
Combined risk√702,000 = $837.85, or 3.35% of the account
Diversification benefit1,000 − 837.85 = $162.15, 16.21% of the sum
Effective independent positions(1,000 ÷ 837.85)² = 1.42
Risk limit in cash25,000 × 6% = $1,500
Room left under the limit1,500 − 837.85 = $662.15, or 2.65%
Scale every position by1,500 ÷ 837.85 = 1.790x

Combined risk is $837.85 — 3.35% of the account rather than the 4.00% the straight sum reports — so the correlation entered is worth $162.15. Four positions at 0.6 behave like 1.42 independent ones, not four. Drop the correlation to 0 and the same four combine to $504.98 and behave like 3.92 independent positions; raise it to 1 and combined risk is the full $1,000. The multiplier of 1.790 would put the combined figure exactly on the $1,500 limit.

How to use the Correlation-Adjusted Position Size calculator

Enter the account balance and the total risk limit you hold yourself to across every open position at once. The limit is yours to set; the calculator reports against it and does not propose one.
Add a row per open position, entering the cash lost if that position's stop is hit, or the same risk as a percentage of the account. Cash takes precedence when both are filled in, and blank rows are skipped.
Enter the average correlation. The correlation calculator will measure it from two return series, but the figure that belongs here is what it becomes under stress rather than what it has been in quiet conditions.
Read the combined risk against the limit first, then the multiplier underneath it. Above 1 there is room left; below 1 the positions sit above the limit as entered.
Compare the effective independent position count against the number of positions. A book of eight trades behaving like two is telling you something about the effective risk of multiple positions that the individual sizes cannot.
Recalculate as the book changes. Adding a position correlated with the existing ones raises combined risk by more than its own size implies.

Limitations of the Correlation-Adjusted Position Size calculator

Every calculation makes assumptions. These are the ones that matter here.

One average correlation stands in for every pair. A pair moving together at 0.9 inside an average of 0.4 is completely invisible here, and that pair is the one that does the damage.
Correlations are not stable, and they rise towards one in exactly the sell-offs that test a limit. A figure measured in calm conditions understates the risk of the day it matters, which is the strongest argument for entering a pessimistic number.
Correlation is itself estimated from a sample. Over sixty daily observations the standard error is wide enough that 0.4 and 0.6 are hard to tell apart, and the combined figure inherits every bit of that uncertainty without displaying it.
Treating each risk as a standard deviation describes how losses aggregate, not how likely any of them is to occur. Nothing here estimates the probability that a stop is reached.
Every stop is assumed to fill at its price. A gap takes the loss past the amount entered, and correlated positions are precisely the ones that gap together.
Negative correlations are rejected. A single average below −1 ÷ (n − 1) implies a correlation matrix that cannot exist, and a genuine hedge needs position-by-position correlations rather than one average across the book.

Frequently asked questions about Correlation-Adjusted Position Size

What correlation should I enter if I have not measured it?