Elliptical distribution: Definition & Examples

An elliptical distribution is the probability law of an affine transformation of a spherically symmetric random vector: stretch, rotate and shift a law whose directions are uniform. It lets quants model joint risk factors with a common geometry while choosing light or heavy tails. The multivariate normal distribution and multivariate t-distribution are members of this family. Knowing that a vector is elliptical does not determine its tail probabilities.

Geometry plus a radial law

A useful representation is

\mathbf{X}\overset{d}{=}\boldsymbol\mu+RA\mathbf{U}, \qquad S=AA^\top.

Here

A

is a fixed

d\times k

matrix of rank

k

\mathbf{U}

is uniform on the unit sphere in

\mathbb{R}^k

, and the radius

R\geq0

is independent of

\mathbf{U}

. The vector

\boldsymbol\mu

is the location and

S

is a scatter or dispersion matrix. The law of

R

controls how often observations lie far from the centre. Changing the scale of

A

and compensating in

R

leaves the law unchanged, so a scatter convention must be stated. This representation includes singular laws supported on a lower dimensional subspace. Hult and Lindskog, §2.3 give the general rank-based construction.

For positive definite

S

, if a joint density exists, it has the form

f_{\mathbf{X}}(\mathbf{x}) =|S|^{-1/2}g\!\left((\mathbf{x}-\boldsymbol\mu)^\top S^{-1}(\mathbf{x}-\boldsymbol\mu)\right).

It is constant along ellipsoids of squared radius

Q=(\mathbf{x}-\boldsymbol\mu)^\top S^{-1}(\mathbf{x}-\boldsymbol\mu)

. A nonmonotone

g

can give the same density on several ellipsoids. Positive definite scatter alone does not guarantee a density: setting

R=1

and

A=I_2

places all mass on the unit circle, which has zero area. This is still an elliptical law. The ETH treatment, §3.3, explicitly conditions its density formula on density existence.

Scatter is not automatically covariance

If $\mathbb{E}[R^2]<\infty$ , then

\mathbb{E}[\mathbf{X}]=\boldsymbol\mu, \qquad \text{Cov}(\mathbf{X})=\frac{\mathbb{E}[R^2]}{k}S.

Without the required moments, location need not be a mean and scatter need not be a covariance matrix. When component variances are finite and nonzero, the common multiplier cancels in their correlation. Hult and Lindskog, §2.3 distinguish this geometric parameter from moment-based correlation.

For a concrete synthetic risk-factor model in arbitrary model units, take a bivariate

t

with

\nu=5

, location zero, and scale matrix

S=\begin{pmatrix}1&0.5\\0.5&2\end{pmatrix}.

The parametrisation is

\mathbf{X}=\sqrt{5/V}\,A\mathbf{Z}

, where

V\sim\chi^2_5

\mathbf{Z}\sim\mathcal{N}_2(\mathbf{0},I_2)

V

and

\mathbf{Z}

are independent, and

AA^\top=S

. This is the scale/shape convention used by SciPy's multivariate t documentation. Then

\text{Cov}(\mathbf{X})=\frac{5}{3}S =\begin{pmatrix}5/3&5/6\\5/6&10/3\end{pmatrix}.

For a nondegenerate $t_\nu$ law, the mean exists only for $\nu>1$ and covariance is finite only for $\nu>2$ . In our example, $Y=(X_1+X_2)/2$ has $t_5$ scale parameter $\sqrt{\mathbf{w}^\top S\mathbf{w}}=1$ but variance $5/3$ , where $\mathbf{w}=(1/2,1/2)^\top$ . Treating its scale squared as its variance would miss the factor $5/3$ .

Normal and t5 laws with zero location and the same scatter have identical elliptical boundaries Q equals 1, 4 and 9. Outside Q equals 9, the t5 probability is 7.62 percent and the normal probability is 1.11 percent. The survival-probability axis is logarithmic.

Original synthetic comparison. The boundaries are squared-radius levels, not confidence levels. Both laws use the displayed scatter S; their covariances differ. The right panel compares probabilities outside the ellipses, not one-sided portfolio losses.

For this bivariate

t_5

Q/2\sim F_{2,5}

, so

\mathbb{P}(Q>9)=(1+9/5)^{-5/2}=7.62\%

. For a normal law with covariance

S

Q\sim\chi^2_2

and the probability is

e^{-9/2}=1.11\%

. These are deliberately same-scatter comparisons. A comparison holding covariance fixed would rescale the

t

model.

What survives portfolio aggregation

For any fixed matrix $B$ and vector $\mathbf{b}$ , $B\mathbf{X}+\mathbf{b}$ remains elliptical, with location $B\boldsymbol\mu+\mathbf{b}$ and scatter $BSB^\top$ . Thus linear portfolio risk reduces to a one-dimensional distribution. For a return vector $\mathbf{X}$ and loss $L=-\mathbf{w}^\top\mathbf{X}$ ,

\text{VaR}_{\alpha}(L) =-\mathbf{w}^\top\boldsymbol\mu +\sqrt{\mathbf{w}^\top S\mathbf{w}}\,q_\alpha,

when the scale is positive. Here

q_\alpha

is the quantile of the unit-scale univariate projection implied by the specified elliptical law. It is a normal quantile for the normal model and a standard

t_\nu

quantile for the specified

t

model. Elliptical geometry alone supplies no numerical value at risk; the radial law is essential. Nonlinear option P&L need not inherit elliptical closure.

Symmetry also does not imply independence. With diagonal scatter, a multivariate

t_5

has zero correlations yet shares the random multiplier

\sqrt{5/V}

across components: unusually large magnitudes can arrive together. Heavy-tailed elliptical laws can have tail dependence; Gaussian laws with correlations strictly between minus one and one do not have asymptotic pairwise tail dependence. Elliptical symmetry also pairs upper and lower extremes, which limits its ability to represent different crash and rally behaviour. Hult and Lindskog, §§2.3–2.4 study these radial-tail distinctions.

For the Gaussian special case, the Multivariate Normal Distribution lesson develops density contours, linear portfolios and conditional distributions. Its public preview introduces the lesson; the full teaching requires membership.

Sources

Embrechts, McNeil and Straumann — Correlation and Dependency in Risk Management: Properties and Pitfalls, §§3.2–3.4: spherical/elliptical definitions, density conditions, moments and portfolio structure.
Hult and Lindskog — Multivariate extremes, aggregation and dependence in elliptical distributions, reproduced as Chapter 2 of Hult's 2003 thesis, §§2.3–2.4: rank-based radial representation, scatter normalisation, normal/ $t$ radial laws and tail dependence.
SciPy — scipy.stats.multivariate_t, Parameters and Notes: explicit location, shape and degrees-of-freedom convention and multivariate $t$ density.

Geometry plus a radial law

Scatter is not automatically covariance

What survives portfolio aggregation

Sources

Related terms

Correlation

Independence

Multivariate normal distribution

Probability density function

Value at risk