Copula Time-To-Default for Credit Sensitive Assets

Computing the time of state of default of credit sensitive assets

Risksvr incorporates different copula based methodologies. The first and foremost is to compute the time of a state of default from a cash-flow. The other is it's opposite state : survival. Finally long term FX rates can also be simulated according to a Copula based approach.

All these methodologies are based on the same underlying principle of time-to-default, with their own implementation details, as specification and parameterization change according to context.

Time-To-Default Correlated Asset Approach

Simulating Time To Default From Asset Correlations and Credit Curves

Module Breakdown:

This document assumes you are familiar with the notion of Credit-Curves

Credit Curves are covered in detail in the Related Topics section above.

The Time to Default modules uses the cumulative probability of each Asset's Rating expressed in it's Credit Curve to extract the time of occurrence of this event.

	=Is the cumulative default probability of an asset or obligor with a Rating R up to a given time period t.
	=Is the default time of the asset.

We can therefore seek the probability of an event occurring at a given time t prior to a known event T or we can seek the time by knowing the probability!

The cumulative default probability is often computed by taking the opposite probability of Survival (No survival), Either way, Default or Survival or actually be it any other way (hazards, cumulative hazards, forward no-default, marginal conditional default). They all lead to the exact same result, provided they are continuous or dicrete ceteris paribus. (See Credit-Curve interpolation).

	= the correlation matrix between assets / obligors.
	= the multivariate (normal if Gaussian Copula) distribution function.
	= the univariate standard (normal if Gaussian Copula) cumulative distribution function.
	= the univariate standard (normal if Gaussian Copula) cumulative distribution function with uniform random ui.

Default time of a Given asset is computed by creating a dependency between the Copula function C(u,1,u2,u3,Un) of a series of individual univariate functions and the standard multivariate distribution, function accordingly:

Correlated default events and default times are then simulated by first simulating the multivariate distribution from the normalized asset or obligor correlation y(i) with i=1,...,n

Compute u(I)=

Once we have obtained u(1),u(2),...u(n) we can obtain default times from the credit curve by seeking the corresponding inverse default CDF [Cumulative Density Function].

Note: To activate Time-To-Default, you must enable Time-To-Default is active in your analysis.

For any of these models, you must also supply:

Obligor / Asset/Country/FX Correlation
A credit-[default]-curve (Hazards, Survival or EDF) or a curve that expresses the event we are seeking over one or more time-steps (often referred as timeline).
A Name, an identifier to bind a curve and a row/column in the matrix. (usually the rating rank, the rating system, obligor name, credit curve name, etc)