Using Copulas to Calculate Mixed Default and Survival Probabilities

Summary

The note considers a basket of three credits with specified marginal default probabilities and a known copula for their joint default times. It asks how to express the event that the first name survives to time t while the other two default by t.

The answer obtains this mixed event by subtracting the probability that all three names default from the joint default probability of the second and third names. The latter is evaluated by applying the copula with the first marginal set to one. This is a direct inclusion-exclusion identity and does not require constructing a separate survival copula. The result assumes the stated copula describes the joint distribution and that its marginal inputs are correctly specified; the note gives no calibration method, empirical evidence, or discussion of model fit.

Key ideas

• A mixed default and survival probability can be expressed as a difference of joint default probabilities.
• The probability that names two and three default is given by the copula evaluated with the first marginal equal to one.
• Subtracting the probability of joint default across all three names enforces survival of the first name.
• The calculation depends on the chosen copula and marginal default probabilities.

Tags

# Copulas and default probability


# Copulas and default probability












Assume a basket of 3 credits, each with some unconditional default probability ${q_i}(t) = \Pr [{\tau _i} \le t]$.

Consider the joint CDF $H$ of the default times is given by $H(t,t,t) = \Pr [{\tau _1} \le t,{\tau _2} \le t,{\tau _3} \le t] = C({q_1}(t),{q_2}(t),{q_3}(t))$, where $C$ is a known copula function (e.g. Archimedan).

My question is: is there some (possibly Copula-based) representation of a function $G$ defined as $G(t,t,t) = \Pr [{\tau _1} > t,{\tau _2} \le t,{\tau _3} \le t]$ ?

I know a survival copula ${\bar C}$ can be constructed from $C$ but this is not entirely what I want as I want a joint probability of the last two names to default and the first name to survive.

Thanks

## Answer by M. Jeunesse (score 3, accepted)

https://quant.stackexchange.com/a/26326

$$\text{Pr}[\tau_1>t,\tau_2\leq t,\tau_3\leq t]=\text{Pr}[\tau_2\leq t,\tau_3\leq t] - \text{Pr}[\tau_1\leq t,\tau_2\leq t,\tau_3\leq t]$$

$$\text{Pr}[\tau_2\leq t,\tau_3\leq t]=C(1,q_2(t),q_3(t))$$

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