FINC6023 Chap.6 Multivariate Models, Correlations & Copulas
Multivariate Models, Correlations & Copulas
Multivariate Models, Correlations and Copulas explains how dependence enters portfolio and credit models. A full covariance matrix grows quickly with the number of risk factors, so a factor model represents returns through common drivers plus idiosyncratic shocks.
Under a one-factor specification, each asset's variance includes systematic and idiosyncratic pieces, while covariance between assets comes from the common factor when idiosyncratic terms are uncorrelated.
Covariance and correlation are related but not interchangeable. Covariance carries the units and scale of the variables. Correlation standardises covariance into a unit-free number.
Zero correlation removes linear association but does not generally imply independence. Under joint normality the distinction narrows, which is one reason normal models are convenient and also one reason their assumptions need scrutiny.
A copula separates marginal distributions from dependence.
Each observed value is mapped through its own marginal distribution to a percentile, then through the inverse normal distribution to a standard-normal score. A Gaussian dependence structure is imposed on those scores while the original marginals remain intact. This construction is useful but can understate simultaneous extreme moves when tail dependence is stronger than the Gaussian model permits.
Cholesky factorisation performs a related computational task by converting independent normal draws into correlated shocks for simulation.
The exam treatment is mainly interpretive for copulas and computational for factor-model variance, covariance, conditional normal moments, and correlated sampling. Explain what is held fixed and what is changed at each stage.
A strong answer never says that mapping values to normal scores makes the original data normally distributed; it changes the representation used for dependence, not the marginal facts.
What this chapter covers
- 01
Single-factor model Rᵢ = αᵢ + βᵢR_m + εᵢ → σᵢ² = βᵢ²σ_m² + σ_εi²
- 02
Covariance from a factor model σᵢⱼ = βᵢβⱼσ_m²
- 03
Correlation ρ = cov(V₁,V₂)/(SD(V₁)SD(V₂))
- 04
Independence ≠ zero correlation (the triangular-distribution example)
- 05
Bivariate-normal conditionals μ_{x|y} and σ_{x|y}
- 06
Gaussian copula: percentile-to-percentile mapping onto standard normals; Sklar's theorem
- 07
Cholesky correlated sampling ε₂ = ρz₁ + z₂√(1−ρ²)
Worked example · free
Correlation implied by a single-factor model
- 2Total 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%).
- 2Total 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%).
- 1Covariance via the common factor: σ₁₂ = β₁β₂σ_m² = 1.2 × 0.8 × 0.04 = 0.0384.
- 1Correlation ρ = σ₁₂/(σ₁σ₂) = 0.0384/(0.2830 × 0.1887) = 0.0384/0.05341.
- 1Compute: 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.
- Marginal distribution
- A marginal distribution describes one risk factor on its own, separate from the dependence structure that determines how it moves jointly with other factors.
- Tail dependence
- Tail dependence is the tendency of variables to experience extreme outcomes together, a feature that can be understated by a Gaussian copula even when ordinary correlation appears reasonable.
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.
What does a Gaussian copula preserve?
It preserves each variable's marginal distribution while using percentile-to-normal-score mappings to impose a Gaussian dependence structure. The transformed scores are normal by construction, but the original returns do not become normal. This distinction is essential when explaining why a copula can combine non-normal marginals with a chosen joint model.
Where does Cholesky factorisation enter risk measurement?
Monte Carlo simulation begins with independent random draws. Multiplying them by a Cholesky factor produces shocks with the desired covariance matrix. In the two-variable case, verify the variance of each constructed shock and their covariance before using the draws, because a sign or square-root error changes every simulated portfolio result.
Exam move
Organise revision around three contrasts. First separate total variance from covariance in a factor model: idiosyncratic volatility increases an asset's variance but does not create covariance when the idiosyncratic shocks are independent. Second separate correlation from independence by keeping one nonlinear counterexample you can explain without calculation.
Third separate marginals from dependence in a copula: the percentile mapping preserves each variable's marginal ranking while the copula supplies the joint structure. Practise a factor-model calculation that ends by recovering correlation, then check that the result lies between minus one and one.
For conditional-normal questions, write how the observed value shifts the conditional mean and how correlation reduces conditional variance. For Cholesky, verify the constructed covariance algebraically before using simulated draws. In conceptual practice, explain the Gaussian copula's thin joint tails and why a fixed dependence assumption can be most misleading during stress.
Avoid the inaccurate shortcut that transformed scores make the underlying returns normal.
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