Center with a radius

Two correlations with a common third variable cannot combine arbitrarily: the 3 by 3 correlation matrix must stay positive semi-definite. If assets A and B each correlate 0.6 with the market, their mutual correlation can sit anywhere from 0.36 - 0.64 = -0.28 up to 0.36 + 0.64 = 1.00, and nothing outside that band is consistent.

Words you need

Correlation: Covariance scaled to lie between −1 and 1. +1 means they move in lockstep.
R²: The share of a quantity's variance that a regression explains, from 0 to 1.

Why it works

A correlation matrix is a Gram matrix of standardized variables, so its determinant cannot go negative. Setting that determinant to zero traces the exact lower and upper curves for the third correlation.

r1 r2 - sqrt((1 - r1^2)(1 - r2^2)) <= rho_AB <= r1 r2 + sqrt((1 - r1^2)(1 - r2^2))

Spot it: An interviewer gives two pairwise correlations and asks what the third can be.

Technique reference and sources

Read the band as a center with a radius. The center is the product r1 * r2; the radius is the product of the two residual roots sqrt(1 - r1^2) * sqrt(1 - r2^2). The feasible band is center minus radius to center plus radius.

When this method fits

An interviewer gives two pairwise correlations and asks what the third can be.
You must reject an impossible correlation matrix before pricing a basket.
Two assets share a strong common factor and you need their worst-case co-move.
You see Pythagorean values like 0.6 and 0.8, whose residual roots come out exact.

Multiply for the center. Center = r1 * r2. For 0.6 and 0.6: 0.6 * 0.6 = 0.36.
Root each residual. 1 - 0.36 = 0.64, and sqrt(0.64) = 0.8. Do the same for the second correlation.
Multiply the roots for the radius. Radius = 0.8 * 0.8 = 0.64.
Add and subtract. Band is 0.36 - 0.64 = -0.28 to 0.36 + 0.64 = 1.00.

Pythagorean pairs keep roots exact: With 0.6 the residual root is 0.8, and with 0.8 it is 0.6, because 1 - 0.36 = 0.64 and 1 - 0.64 = 0.36. Radii then multiply cleanly.

0.6 and 0.6: center 0.36, radius 0.64, band -0.28 to 1.00
0.6 and 0.8: center 0.48, radius 0.48, band 0 to 0.96

Strong shared factors force nonnegativity: When both correlations reach 0.8, the band is 0.64 - 0.36 = 0.28 to 1.00: the pair cannot go negative at all.

0.8 and 0.8: band 0.28 to 1.00
0.6 and 0.8: band 0 to 0.96

Common traps

Claiming rho_AB = -0.5 is feasible when both market correlations are 0.6 and 0.8. The band there is 0 to 0.96, since 0.48 - 0.48 = 0. Negative values are impossible.

Multiplying the correlations and stopping: rho_AB = 0.36 exactly. The product is only the center. The radius 0.64 opens the band from -0.28 to 1.00.

Forgetting to root the residuals: radius = 0.64 * 0.64. Root first, then multiply: sqrt(0.64) = 0.8, so the radius is 0.8 * 0.8 = 0.64.

Another example: Lowest consistent correlation at 0.8 and 0.8

Center: 0.8 * 0.8 = 0.64
Residuals: 1 - 0.64 = 0.36 on each side, sqrt(0.36) = 0.6
Radius: 0.6 * 0.6 = 0.36
Lower edge: 0.64 - 0.36 = 0.28

Source notes

These explanations and questions are written for QuantArc. Links show the underlying mathematics and learning design.

Harvard Stat 110: Probability (Joe Blitzstein)
Free course site with videos and the open companion book Introduction to Probability; the topic spine for counting, Bayes, and linearity. Lesson examples and questions are original.

Grinstead and Snell, Introduction to Probability (free PDF)
Free textbook PDF confirmed live: counting methods, poker-style dealing models, and the random-walk chapter behind gambler's ruin and first-step equations. Lesson examples and questions are original.

IES: Organizing Instruction and Study to Improve Student Learning
Supports worked examples, retrieval, explanations and spaced study. Does not establish QuantArc's exact lesson length, mastery threshold or review intervals.