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Multivariate Normal Distribution

The multivariate normal

\mathbf{X} \sim \mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})

Every marginal and every conditional of it is normal too, which is what makes it the tractable case.

Fully specified by a mean vector and a covariance matrix. For $n$ variables that is $n$ means and $\frac{n(n+1)}{2}$ covariances, and nothing else is needed, which is both its power and its limitation.

Why it dominates portfolio mathematics

Closed under linear combinations. Any weighted sum $\mathbf{a}^\top\mathbf{X}$ is univariate normal, with

\mathbf{a}^\top\mathbf{X} \sim \mathcal{N}\left(\mathbf{a}^\top\boldsymbol{\mu},\; \mathbf{a}^\top\boldsymbol{\Sigma}\mathbf{a}\right)

A portfolio is a linear combination of assets, so a portfolio of jointly normal assets is itself normal. Its whole distribution follows from two numbers, which is why mean-variance optimisation works so cleanly.

Conditionals are normal too, with a mean that shifts linearly with the conditioning variable. That linear relationship is exactly a regression coefficient, which is why regression and multivariate normality fit together so neatly.

Uncorrelated implies independent, uniquely for this distribution. In general zero correlation does not imply independence; under joint normality it does. This is a special property and it is routinely over-generalised.

Key takeaway

Under joint normality, mean and covariance are the complete description. That is why the entire classical portfolio theory framework needs only two moments, and why relaxing normality complicates everything at once.

Where it fails

The failure is specific and worth being precise about: the multivariate normal has thin joint tails.

Under joint normality, extreme moves in several assets simultaneously are astronomically unlikely. Real markets deliver exactly that pattern regularly, because a common shock hits everything at once.

Formally, the Gaussian copula has zero tail dependence: conditional on one variable being extreme, the probability another is extreme goes to zero at the limit. Empirically, tail dependence in financial assets is strongly positive.

This is the mathematical statement behind "correlations go to 1 in a crisis". The correlation parameter may be correctly estimated at 0.3 for ordinary days, and the model still assigns essentially no probability to the joint crash, because the shape of the distribution forbids it rather than the parameter being wrong.

Tip

The problem is not the correlation estimate; it is the distribution's shape. Raising correlations in a stress test does not fix a model whose structure rules out simultaneous extremes.

What is used instead

Multivariate t-distributions have fatter tails and non-zero tail dependence, with one extra parameter.

Copula approaches separate the marginals from the dependence structure, so tail behaviour can be modelled deliberately.

Historical or scenario methods avoid a parametric joint distribution altogether, at the cost of being limited to what has already happened.

Test your knowledge

Let $ X = (X_1, X_2)^\top \sim \mathcal{N}(\mu, \Sigma) $ with $ \mu = \begin{bmatrix} 1 \\ 2 \end{bmatrix} $ and $ \Sigma = \begin{bmatrix} 4 & 2 \\ 2 & 3 \end{bmatrix} $. Which statement is true?

The covariance matrix implies $ X_1 $ and $ X_2 $ are uncorrelated

The linear combination $ Y = X_1 + X_2 $ is normally distributed

$ X_1 $ and $ X_2 $ are independent because their variances are nonzero

The distribution of $ X $ is symmetric about the origin

A risk model built on the multivariate normal assigns almost no probability to several assets crashing together, even though the correlation parameters were estimated correctly. What is the problem, and does raising correlations in a stress test fix it?

The correlation estimates are stale, and refreshing them on recent data resolves the understatement

The model needs more assets to capture the joint behaviour, and the problem disappears as the dimension grows

The means are misspecified, and centring the distribution correctly restores the joint tail probability

The shape has zero tail dependence, so simultaneous extremes are ruled out structurally

PreviousJoint, Marginal, and Conditional Distributions

NextCovariance Matrices and Linear Combinations

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