The multivariate normal

The multivariate normal (MVN) is the most-used joint distribution in quantitative finance. It is closed under marginalisation, conditioning, and linear combinations; its dependence structure is fully captured by a covariance matrix; and its tails are zero-tail-dependent. The last property is the catch — but for most short-horizon risk modelling, MVN is the appropriate first model.

Definition and density

X ~ N(μ, Σ),    X ∈ Rⁿ,    Σ positive definite
f(x) = (2π)^(-n/2) |Σ|^(-1/2) exp(-(1/2)(x - μ)ᵀ Σ⁻¹ (x - μ))

μ is the mean vector. Σ is the n×n covariance matrix. The exponent contains the Mahalanobis distance squared: (x - μ)ᵀΣ⁻¹(x - μ).

Mahalanobis distance

A unit-free distance between a point and a distribution. Replaces 'how many standard deviations away' for the multivariate case: it standardises by the full covariance, not just the marginal scales.

• Mahalanobis-squared distance for X ~ N(μ, Σ) is χ²_n distributed.
• Used for multivariate outlier detection, statistical arbitrage signals, model anomaly detection.
• Sensitive to Σ-estimation error: garbage Σ gives garbage distances.

Marginal and conditional distributions

Partition X = (X₁, X₂)ᵀ with corresponding partitions of μ and Σ. Then:

• Marginal: X₁ ~ N(μ₁, Σ₁₁). Just drop the rows and columns.
• Conditional: X₁ | X₂ = x₂ ~ N(μ₁ + Σ₁₂Σ₂₂⁻¹(x₂ - μ₂), Σ₁₁ - Σ₁₂Σ₂₂⁻¹Σ₂₁).

Linear transformations

X ~ N(μ, Σ),   Y = AX + b ⟹ Y ~ N(Aμ + b, AΣAᵀ)

Every linear combination of MVN components is normal. This is why portfolio returns are normal under the MVN assumption — and why so much of risk modelling collapses to elegant closed-form expressions.

Sampling from MVN

Generate Z ~ N(0, I) (i.i.d. standard normals), then X = μ + L Z where L = chol(Σ). The Cholesky factor L is much cheaper than the eigen-square-root and is the production-standard approach.

import numpy as np

def sample_mvn(mu, Sigma, n_draws):
    L = np.linalg.cholesky(Sigma)
    Z = np.random.randn(len(mu), n_draws)
    return mu[:, None] + L @ Z

Empirical adequacy

Daily equity log returns are usually well-approximated as MVN within sample but show clear non-normality in joint tails (cluster crashes, correlation breakdowns). Weekly and monthly returns get closer to MVN. For risk modelling beyond a one-day horizon: stress tests, regime models, or t-copulas with normal marginals.