// LINEAR ALGEBRA · COVARIANCE
PSD Matrix and Covariance Questions in Quant Interviews
'Is this a valid covariance matrix?' and 'how low can the third correlation go?' are standard quant research and trading questions. They appear live in interviews and as items in timed quizzes, and they all reduce to one fact: a covariance matrix must be symmetric positive semidefinite.
The format at a glance
| Where it appears | Live interviews and timed quant quizzes (G-Research, Goldman Strats style) |
|---|---|
| Typical time budget | Aim for about 90 seconds an item on a timed quiz |
| Core test (2x2) | Diagonal entries ≥ 0 and determinant ≥ 0 |
| Covariance bound | |Cov(X, Y)| ≤ σx σy |
| Three-asset bound | ρAC ∈ [ab - √((1 - a²)(1 - b²)), ab + √((1 - a²)(1 - b²))] |
| Calculator | Rarely allowed; keep the arithmetic small |
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Why covariance matrices must be PSD
For any weights w, the variance of the portfolio wᵀX is wᵀΣw, and a variance cannot be negative. So wᵀΣw ≥ 0 for every w, which is the definition of positive semidefinite. A matrix that fails this would let you build a portfolio with negative variance, which is impossible.
For a 2x2 symmetric matrix, PSD is equivalent to both diagonal entries being non-negative and the determinant being non-negative. The determinant condition is the Cauchy-Schwarz bound: Cov(X, Y)² ≤ Var(X)Var(Y).
Worked example: the third correlation
Assets A and B have correlation 0.9, and B and C have correlation 0.9. How low can the correlation between A and C be? The 3x3 correlation matrix is PSD exactly when 1 - a² - b² - c² + 2abc ≥ 0. Solving for c gives c between ab - √((1 - a²)(1 - b²)) and ab + √((1 - a²)(1 - b²)).
Here ab = 0.81 and (1 - a²)(1 - b²) = 0.19 × 0.19, whose root is 0.19. So c lies between 0.62 and 1. Two highly correlated pairs force the third pair to be fairly correlated too.
Question mechanics
- Classify a 2x2 matrix as definite, semidefinite or indefinite from its trace and determinant.
- Find the largest possible covariance given two variances.
- Check whether a proposed 3x3 correlation matrix is valid.
- Var(aX + bY) = a²Var(X) + b²Var(Y) + 2ab Cov(X, Y), where dropping the 2 is the usual slip.
Frequently asked questions
How do you check if a 2x2 matrix is a valid covariance matrix?
It must be symmetric, with non-negative diagonal entries and a non-negative determinant.
If corr(A, B) = corr(B, C) = 0.9, how low can corr(A, C) be?
0.62. The bound is ab - √((1 - a²)(1 - b²)) = 0.81 - 0.19.
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