The Multivariate Normal Distribution

The multivariate normal is what the The Normal Distribution becomes when you have several quantities that vary together, say the daily returns of ten stocks. Instead of one bell curve on a line, you get a smooth mound over a whole space, and its level sets are tilted ellipses whose slant encodes how the variables move in concert. It's the default model whenever you need many correlated Gaussians, which in finance is almost always: portfolio risk, factor models, Kalman filters and pricing engines all lean on it.

What makes it so workable is that just two ingredients pin it down completely. A mean vector $\boldsymbol{\mu}$ says where each variable centres, and a covariance matrix $\Sigma$ says how wide each one is and how every pair co-moves. The density for a $d$-dimensional vector $\mathbf{x}$ is

$$f(\mathbf{x}) = \frac{1}{(2\pi)^{d/2}\,|\Sigma|^{1/2}} \exp\!\left(-\tfrac{1}{2}(\mathbf{x} - \boldsymbol{\mu})^\top \Sigma^{-1} (\mathbf{x} - \boldsymbol{\mu})\right) .$$

Don't be scared of it. The piece that matters is the quadratic $(\mathbf{x} - \boldsymbol{\mu})^\top \Sigma^{-1} (\mathbf{x} - \boldsymbol{\mu})$ inside the exponent: it's a "distance from the centre" that has been re-scaled by the covariance, so it measures how many standard deviations away a point is accounting for correlation. Everything else is a normaliser. Here $\boldsymbol{\mu}$ is the vector of means, $\Sigma$ is the covariance matrix, $\Sigma^{-1}$ is its inverse, and $|\Sigma|$ its determinant.

The clearest way to feel it is in two dimensions. Below is a cloud of draws from a bivariate normal; drag the correlation $\rho$ and watch the round blob tilt into an ellipse. At $\rho = 0$ it's a circular cloud (the two variables ignore each other); push toward $\pm 1$ and it collapses toward a line (knowing one nearly tells you the other). The green line is the conditional mean, where $Y$ is expected to sit for each value of $X$.

Three properties that make it special

The multivariate normal is beloved because it's closed under the operations quants actually do:

• Linear combinations stay normal. Any weighted sum of jointly normal variables is itself normal. A portfolio $\mathbf{w}^\top \mathbf{r}$ of Gaussian returns is Gaussian, with variance $\mathbf{w}^\top \Sigma \mathbf{w}$, which is precisely the formula every risk model computes.
• Marginals and conditionals stay normal. Slice out any subset of variables and you get another (smaller) multivariate normal. Condition on some of them and the rest stay normal too, with a mean that shifts linearly, the mathematical root of linear regression.
• Uncorrelated implies independent. For jointly normal variables, zero correlation is the same as full independence, a coincidence that holds for essentially no other distribution.

Worked example: a two-asset portfolio

Two assets have returns that are jointly normal, each with mean 0 and standard deviation $1\%$, and a correlation of $\rho = 0.5$. Their covariance matrix is

$$\Sigma = \begin{pmatrix} 1 & 0.5 \\ 0.5 & 1 \end{pmatrix} .$$

Hold an equal-weight portfolio, $\mathbf{w} = (0.5,\, 0.5)$. Its variance is $\mathbf{w}^\top \Sigma \mathbf{w}$, which for two assets works out to

$$\sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2 w_1 w_2 \rho\, \sigma_1 \sigma_2 = 0.25 + 0.25 + 2(0.25)(0.5) = 0.75 .$$

So the portfolio's standard deviation is $\sqrt{0.75} \approx 0.87\%$, less than either asset's $1\%$ because the correlation is below 1, that's diversification, quantified in one matrix multiply. And the conditional expectation of asset 2 given asset 1 moved by $x$ is $E[R_2 \mid R_1 = x] = \rho\, x = 0.5x$: a $+2\%$ day for asset 1 nudges your best guess for asset 2 to $+1\%$.

Where it misleads

• Marginal normality is not joint normality. Each variable can be perfectly bell-shaped on its own while the pair is not jointly normal, and then "uncorrelated implies independent" fails. Joint normality is a stronger claim than a normal histogram per column.
• $\Sigma$ must be a valid covariance matrix. It has to be symmetric and positive semidefinite, no negative variances allowed for any portfolio. Estimated covariance matrices on noisy data often come out ill-conditioned, which is why shrinkage exists.
• Thin tails, again. Like its one-dimensional parent, the multivariate normal badly underweights joint extremes. Real markets crash together far more often than a Gaussian says, so correlation-based risk models understate the danger exactly when everything moves at once.

The multivariate normal is the natural home of Joint and Marginal Distributions with Gaussian pieces, the source of the linear form of Conditional Expectation, and the object whose covariance you spend so much effort estimating well (Covariance Matrix Estimation). Master the mean-vector-and-covariance-matrix picture and most of Gaussian quant finance opens up.