A mathematical function modeling the dependence structure between several risks independently of their individual laws, a central and treacherous tool for accumulation.
A copula is a mathematical function that links the individual distributions of several random variables to their joint distribution, isolating the dependence structure that connects them independently of their own laws. Sklar's theorem guarantees that such a decomposition always exists, which makes copulas a highly flexible tool for modeling how risks move together. Their value is decisive for insurance and reinsurance, because the dependence between risks, and even more so dependence in extreme events, is precisely what determines the scale of an accumulation. A well-chosen copula captures the fact that certain risks, independent in normal times, become strongly correlated in crisis scenarios. But the tool is also formidably treacherous. The massive use of the Gaussian copula in 2000s finance, which seriously underestimated tail dependence, contributed to the mispricing of risk at the heart of the 2008 crisis, illustrating how a poor choice of dependence structure can conceal a real accumulation. In cyber, where the dependencies between insureds are numerous, hidden and non-stationary, the choice of copula is a critical and unresolved issue.
A cyber model that assumes, through a poorly suited copula, a low correlation between insureds may conclude that accumulation is modest. If the true tail dependence is strong, a systemic attack then reveals a loss far greater than the model had forecast.
copula, fonction copule, copule gaussienne