Copula
A mathematical function modeling the dependence structure between several risks independently of their individual laws, a central and treacherous tool for accumulation.
David X. Li, 2000, Journal of Fixed Income: the Gaussian copula is applied to default correlation, and banks take it up to price CDOs. It seriously underestimated tail dependence, which contributed to the mispricing of risk that fed the 2008 crisis. A copula is the function that links the individual distributions of several random variables to their joint distribution, isolating the dependence structure from their own laws. Sklar's theorem (1959) guarantees that such a decomposition always exists, hence the flexibility of the tool. In insurance and reinsurance, dependence between risks, especially in the extremes, sets the scale of an accumulation. A well-chosen copula captures risks that are independent in normal times and become strongly correlated in a crisis; a poorly chosen one hides a real accumulation. In cyber, where dependencies between insureds are numerous, hidden and non-stationary, the choice of copula remains an open question.
Textbook case: a cyber model that assumes, through a poorly suited copula, a low correlation between insureds may conclude that accumulation is modest. Let a systemic attack hit much of the portfolio at once and the tail dependence the copula flattened comes due: the loss far exceeds what the model had forecast.
copula, copula function, dependency modelling
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