Dependence and Copulas

Dependence cases

The one-factor model associates each credit with a latent asset value

${𝐴}_{𝑖}={𝛽}_{𝑖}𝑍+\sqrt{1-{𝛽}_{𝑖}^2}{𝜀}_{𝑖}$,

where $𝑍$ is the common market factor and each ${𝜀}_{𝑖}$ is an independent idiosyncratic factor. The coefficients ${𝛽}_{𝑖}$ determine how much each credit is exposed to the common factor; the asset correlation of two credits is ${𝜌}_{𝑖,𝑗}={𝛽}_{𝑖}{𝛽}_{𝑗}$. The Lecture 9 notes introduce this model on PDF page 1.

Let ${𝑝}_{𝑖\left(𝑇\right)}=𝑃\left({𝜏}_{𝑖}\le 𝑇\right)$ be credit $𝑖$’s default probability by time $𝑇$. Changing the signs and sizes of the market-factor loadings gives three limiting dependence cases.

Minimum dependence

The smallest joint-default probability comes from opposite exposures to the same market factor. In the limit, set ${𝛽}_{𝑖}=1$ and ${𝛽}_{𝑗}=-1$. Both idiosyncratic components disappear and the asset correlation is ${𝜌}_{𝑖,𝑗}=-1$. The joint-default probability is

$𝑃\left({𝜏}_{𝑖}\le 𝑇,{𝜏}_{𝑗}\le 𝑇\right)=max\left({𝑝}_{𝑖\left(𝑇\right)}+{𝑝}_{𝑗\left(𝑇\right)}-1,0\right)$.

It is zero whenever the two default probabilities sum to at most one. If their sum exceeds one, the joint probability is the excess over one.

Independence

Set ${𝛽}_{𝑖}={𝛽}_{𝑗}=0$. Each asset value is then only its own idiosyncratic factor, the correlation is zero, and the default events are independent:

$𝑃\left({𝜏}_{𝑖}\le 𝑇,{𝜏}_{𝑗}\le 𝑇\right)={𝑝}_{𝑖\left(𝑇\right)}{𝑝}_{𝑗\left(𝑇\right)}$.

Maximum dependence

Set ${𝛽}_{𝑖}={𝛽}_{𝑗}=1$. Both credits have the same asset value $𝑍$ and their asset correlation is $1$. The joint-default probability is the smaller of their marginal default probabilities:

$𝑃\left({𝜏}_{𝑖}\le 𝑇,{𝜏}_{𝑗}\le 𝑇\right)=min\left({𝑝}_{𝑖\left(𝑇\right)},{𝑝}_{𝑗\left(𝑇\right)}\right)$.

Lecture 9, slide 5 (PDF p. 5), captions its graph “The joint default probability as a function of the asset correlation (4 cases).”

Conditional hazard and portfolio loss

The April 17 handwritten lecture notes then condition on a realized common factor $𝑍=𝑧$. With a flat deterministic conditional hazard rate, they write conditional survival as

${𝑆}_{𝑖\left(𝑇|𝑧\right)}={𝑒}^{-{𝜆}_{𝑖\left(𝑇|𝑧\right)}𝑇}=\mathrm{\Phi }\left(\frac{{𝛽}_{𝑖}𝑧-{𝐶}_{𝑖\left(𝑇\right)}}{\sqrt{1-{𝛽}_{𝑖}^2}}\right)$,

so that

${𝜆}_{𝑖\left(𝑇|𝑧\right)}=-\frac{1}{𝑇}ln\mathrm{\Phi }\left(\frac{{𝛽}_{𝑖}𝑧-{𝐶}_{𝑖\left(𝑇\right)}}{\sqrt{1-{𝛽}_{𝑖}^2}}\right)$.

Here ${𝐶}_{𝑖\left(𝑇\right)}$ is the default threshold calibrated from the single-name survival curve. Conditioning on $𝑧$ makes the credits independent: compute the portfolio-loss distribution at that factor value, then integrate over the common factor to obtain the unconditional distribution. The notes state this procedure on PDF page 4.

Lecture 9, slide 6 (PDF p. 6), captions the conditional hazard-rate plot: “Plot of conditional hazard rate distribution for $𝛽=0,0.2,0.4$,” assuming an unconditional hazard rate of 2% and a one-year horizon. Slide 3 (PDF p. 3) shows the time dependence of survival probabilities and default thresholds for issuer curves with flat deterministic hazard rates.