Lesson 23 of 25 · 7 min

Joint-distribution modelling, conceptually, and what it was used for

Two holdings both falling past two standard deviations is a 1-in-509-month event at 0.29 and a 1-in-99-month event at 0.81.

A correlation is one number summarising an entire joint distribution, and it is lossy in one specific place: the corner where both holdings fall a long way at once. That corner is where a collateral outcome is decided.

Joint-distribution modelling means specifying the whole shape of how two or more variables move together rather than one summary statistic. The approach that made this tractable splits the problem in two: describe each holding's own distribution however you like, then describe the linkage between them separately. The linking function is called a copula, and its appeal is that the two halves can be chosen independently — fat-tailed marginals with a mild linkage, or well-behaved marginals with a linkage that binds hard in the tail.

Put numbers on the corner. Under a joint-normal assumption each holding falls more than two standard deviations in 2.275 percent of months. If the two were independent, both would do so in 0.0518 percent of months — once in 1,932. At a correlation of 0.29 the probability is 0.1965 percent, once in 509 months. At 0.81 it is 1.0118 percent, once in 99 months. Same two holdings, same marginal behaviour, and the joint event goes from once in forty-two years to once in eight.

Through the 2000s this machinery was used to value structured-credit instruments, where the question was the joint default behaviour of large pools of obligations and the answer was governed by a few linkage parameters. In 2007 and 2008 the realised joint behaviour sat far into the corner those models treated as nearly empty. The instructive part is where the failure was not: the descriptions of individual obligations. The linkage in the tail was the problem, and lesson 16 showed why that is the parameter least amenable to estimation.

The general lesson survives the specific episode. A single correlation is a linear summary, so it treats the corner as an extrapolation from the middle of the distribution, where all the data is. Every conclusion about simultaneous severe declines is therefore an extrapolation, whatever notation it is written in.

Key terms

Joint distribution: The full description of how two or more variables move together, including how often they reach extremes at the same time. Correlation is one summary of it.
Copula: A function describing the linkage between variables separately from each variable's own distribution. It makes the dependence structure an explicit choice rather than an assumption.
Tail dependence: How strongly two variables reach their extremes together. Not captured by a correlation, and the property a simultaneous-decline calculation actually rests on.

Worked example

The probability of the corner

Two holdings, joint-normal, each falling more than two standard deviations in 2.275% of months. Only the correlation between them changes. Illustrative figures.

Each holding alone, below -2 standard deviations: 2.275% of months
Both, if independent: 0.0518%
Both, at a correlation of 0.29: 0.1965%
Both, at a correlation of 0.60: 0.5500%
Both, at a correlation of 0.81: 1.0118%
Ratio between the 0.81 and 0.29 cases: 5.15

Every row describes the same two holdings with the same individual behaviour. Only the linkage changed, and the joint event became five times more frequent.

Check yourself

Two holdings each fall more than two standard deviations in 2.275% of months. Independently, both would do so once in 1,932 months. At a correlation of 0.60 the joint probability is 0.55%. How much more frequent is that, and why is the multiple so large?

Saved in this browser only. No account, nothing sent anywhere.

Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.