FINC6023 · Financial Risk Management
Multivariate Models, Correlations & Copulas
Multivariate Models, Correlations & Copulas is how the unit builds dependence between many risk factors without estimating every pairwise covariance. A factor model collapses the covariance structure onto a few common drivers (σᵢ² = βᵢ²σ_m² + σ_εi²); the Gaussian copula separates each variable's own marginal distribution from the dependence by mapping every percentile to a standard-normal score and imposing multivariate-normal dependence on the scores; and Cholesky factorisation generates correlated samples. The exam treats copulas as understand-and-interpret, not compute.
What this chapter covers
- 01Single-factor model Rᵢ = αᵢ + βᵢR_m + εᵢ → σᵢ² = βᵢ²σ_m² + σ_εi²
- 02Covariance from a factor model σᵢⱼ = βᵢβⱼσ_m²
- 03Correlation ρ = cov(V₁,V₂)/(SD(V₁)SD(V₂))
- 04Independence ≠ zero correlation (the triangular-distribution example)
- 05Bivariate-normal conditionals μ_{x|y} and σ_{x|y}
- 06Gaussian copula: percentile-to-percentile mapping onto standard normals; Sklar's theorem
- 07Cholesky correlated sampling ε₂ = ρz₁ + z₂√(1−ρ²)
Correlation implied by a single-factor model
- 2 marksTotal variance of Stock 1: σ₁² = β₁²σ_m² + σ_ε1² = 1.2²(0.20)² + 0.15² = 0.0576 + 0.0225 = 0.0801, so σ₁ = √0.0801 = 0.2830 (28.30%).
- 2 marksTotal variance of Stock 2: σ₂² = 0.8²(0.20)² + 0.10² = 0.0256 + 0.0100 = 0.0356, so σ₂ = √0.0356 = 0.1887 (18.87%).
- 1 markCovariance via the common factor: σ₁₂ = β₁β₂σ_m² = 1.2 × 0.8 × 0.04 = 0.0384.
- 1 markCorrelation ρ = σ₁₂/(σ₁σ₂) = 0.0384/(0.2830 × 0.1887) = 0.0384/0.05341.
- 1 markCompute: ρ = 0.719.
Key terms
- Factor model
- A model expressing each return as loadings on a few common factors plus idiosyncratic noise: Rᵢ = αᵢ + βᵢR_m + εᵢ. It collapses the N(N−1)/2 pairwise covariances onto a few factor parameters.
- Correlation vs independence
- Correlation measures only LINEAR dependence; two variables can be perfectly dependent yet have zero correlation (the V ∈ {−1,0,1} triangular example). Zero correlation does not imply independence.
- Gaussian copula
- A construction that separates each variable's own marginal distribution from the dependence: map each marginal percentile to a standard-normal score and impose multivariate-normal dependence on the scores. Sklar's theorem guarantees the decomposition.
- Cholesky factorisation
- A method to generate correlated random draws from independent normals; for two variables ε₁ = z₁ and ε₂ = ρz₁ + z₂√(1−ρ²) produces a pair with correlation ρ.
Multivariate Models, Correlations & Copulas FAQ
Do I have to compute copulas in the exam?
No — the Week-13 Review states copulas are understand-and-interpret only. You should be able to explain the percentile-to-standard-normal mapping, why it separates marginals from dependence, and the role of the Gaussian copula in the crisis, but the calculations focus on factor models, correlation and bivariate normals.
Why does a factor model save so much work?
A full covariance matrix needs N(N−1)/2 distinct covariances. A single-factor model only needs each asset's beta and idiosyncratic variance plus the factor variance, because every covariance is forced through the common factor: σᵢⱼ = βᵢβⱼσ_m². For large portfolios this is the difference between estimable and not.
If two variables have zero correlation, are they independent?
Not necessarily. Correlation captures only linear association. The classic example takes a variable equal to −1, 0 or 1 and a second variable that is a deterministic function of it, giving zero correlation but obvious dependence. Independence is strictly stronger than zero correlation.
Exam move
Be fluent with the single-factor variance and covariance formulas (most computation lives here) and rehearse the 'zero correlation ≠ independence' one-liner. For copulas, learn the percentile-mapping story well enough to explain, not compute.
