Copulas are functions that link univariate marginal distributions to a multivariate distribution. Stated in another way a multivariate distribution can be decomposed into two contributions, the marginal distributions and the dependence structure as given by the copula model. This allows for the separate specification and examination of marginal distributions and the dependence structure. Marginal distributions may be chosen without constraint and independently of the chosen copula function. The marginal distributions can be determined by fitting available data on each risk factor to statistical methods and estimating the parameters of these methods, whereas the copula function can be chosen independently of the marginal distributions, from among a number of families of copulas available, based on the dependency structure that the user would want imposed within a risk model. For example, a user may choose a copula function having tail dependence to capture the simultaneous occurrence of large losses from various risk factors in extreme adverse or stress situations.
Applications have extended from the hard sciences, such as mathematics, to the softer social sciences in recent years particular in the fields of insurance and finance. In general copulas are tools for understanding relationships among multivariate outcomes, examining the simultaneous interaction or interdependency of several variables. In insurance they have been used to determine joint life mortality distributions; in credit risk applications they have been used in the determination of joint probability of default for companies within a bank’s credit portfolio and in pricing collateralized debt obligations (CDOs); in market risk applications they have been used to value investment portfolios, to determine the profit and loss distribution to use in calculating a Value-at-Risk (VAR) measure; in capital allocation and risk assessment applications they have been used to aggregate the various sources of risk (market, credit, operational, etc) that an entity may be exposed to; in Monte Carlo simulations of multivariate outcomes as copula constructions allows for easy simulation of them; etc.
Copulas allow for the definition of a dependency structure between the marginal distributions of each risk factor that can vary under different circumstances, i.e. a non-linear dependency structure unlike the traditional linear correlation coefficient measure which implies constant dependency. Linear correlation fails for extreme scenarios because, as we have witnessed in the recent financial crisis, in these scenarios dependency tends to change often becoming greater whereas linear correlation assumes dependence remains unchanged for all scenarios. The use of copulas can allow the user to focus on the unknown elements of dependence of extreme and uncommon events while maintaining the known information on the marginal distributions and the dependence during common events.
The Gaussian copula is one of the most widely used copula functions because users find it easy to implement when generating Monte Carlo simulations of multivariate outcomes. However the model has limitations in that it is not able to capture any tail dependence between variables. It should be a choice for risk factors that exhibit linear correlation but it has been used inappropriately for valuing instruments (e.g. CDOs), risk factors, etc that do not have this characteristic and which show greater dependence in the tail end of the distributions or alternatively in extreme events. The result has been an underestimation of the risk that contributed toward the most recent financial crisis. Better choices for such distributions would be Archimedean copulas or the Student –t copula (with low degrees of freedom) that capture extreme events.
The problem with using most copula approaches is that they can be mathematically complex. Their specification can be very abstract and difficult to interpret. Estimating parameters for a given copula function is also a difficult task requiring significant statistical and technical expertise not only for the immediate practitioner but also for the management who have to make decisions based on the output from these methods.
Bibliography, References & Additional reading
- Coping with Copulas, Thorsten Schmidt, December 2006
- Tails of Copulas, Gary G. Venter
- Modeling Copulas, Marten Dorey, Phil Joubert
- ERM in Insurance Groups. Modeling risk concentration and default risk
- Optimal Capital Budgeting. Presentation.