Dependence and Copulas

Dependence cases

Relevant variables for these credits

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Let’s look at, let’s say, minimum dependence:

In what situation might we have such minimum dependence?

The k where we have the lowest probability that we have both names defaulting.

Both of these credits have an idiosyncratic component and a market exposure z.

So one way we can achieve minimum dependence is to have an opposite exposure to z.

If we have a large negative exposure to z, we’ll say beta is large. For credit j, we can change the exposure to market variable z, and instead of having a large negative number, it becomes a large positive numbers, and we lower the probability of default.

For minimum dependence, if we take it in the limit, for β_i = -β_j

In the limit, we can take β_i = 1, β_j = -1

One has 100% exposure to z, the other has negative exposure to z, and both of them have zero idiosyncratic component.

So correlation is inverse, ρ = -1.

In this case, such probability is dependent on the survival probability of the names, \((1 - Q_i(T) - Q_j(T))_+\)

The joint probability of default is zero as long as Q_i(T) + Q_j(T) > 1

Independence means correlation is 0. And to do that, we have to remove market exposure. So in the limit, we take

β_i = β_j = 0, which means the correlation is zero, it’s entirely idiosyncratic.

Maximum Dependence is when β_i = β_j = 1, maximum market exposure. It’s the minimum of either 1 - Q_i(T), 1 - Q_j(T).

If you want to illustrate this,

Slide 5