{"categories":["Credit"],"contentHtml":"

Dependence cases

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

𝐴𝑖=𝛽𝑖𝑍+1βˆ’π›½π‘–2πœ€π‘–,

where 𝑍 is the common market factor and each πœ€π‘– is an independent idiosyncratic factor. The coefficients 𝛽𝑖 determine how much each credit is exposed to the common factor; the asset correlation of two credits is πœŒπ‘–,𝑗=𝛽𝑖𝛽𝑗. The Lecture 9 notes introduce this model on PDF page 1.

Let 𝑝𝑖(𝑇)=𝑃(πœπ‘–β‰€π‘‡) 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 𝛽𝑖=1 and 𝛽𝑗=βˆ’1. Both idiosyncratic components disappear and the asset correlation is πœŒπ‘–,𝑗=βˆ’1. The joint-default probability is

𝑃(πœπ‘–β‰€π‘‡,πœπ‘—β‰€π‘‡)=max(𝑝𝑖(𝑇)+𝑝𝑗(𝑇)βˆ’1,0).

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 𝛽𝑖=𝛽𝑗=0. Each asset value is then only its own idiosyncratic factor, the correlation is zero, and the default events are independent:

𝑃(πœπ‘–β‰€π‘‡,πœπ‘—β‰€π‘‡)=𝑝𝑖(𝑇)𝑝𝑗(𝑇).

Maximum dependence

Set 𝛽𝑖=𝛽𝑗=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(πœπ‘–β‰€π‘‡,πœπ‘—β‰€π‘‡)=min(𝑝𝑖(𝑇),𝑝𝑗(𝑇)).

Lecture 9, slide 5 (PDF p. 5), captions its graph β€œThe 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

𝑆𝑖(𝑇|𝑧)=π‘’βˆ’πœ†π‘–(𝑇|𝑧)𝑇=Ξ¦(π›½π‘–π‘§βˆ’πΆπ‘–(𝑇)1βˆ’π›½π‘–2),

so that

πœ†π‘–(𝑇|𝑧)=βˆ’1𝑇lnΞ¦(π›½π‘–π‘§βˆ’πΆπ‘–(𝑇)1βˆ’π›½π‘–2).

Here 𝐢𝑖(𝑇) 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: β€œPlot 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.

","contentSourceFormat":"typst","contentTypst":"= Dependence cases\n\nThe one-factor model associates each credit with a latent asset value\n\n$A_i = beta_i Z + sqrt(1-beta_i^2) epsilon_i$,\n\nwhere $Z$ is the common market factor and each $epsilon_i$ is an independent idiosyncratic factor. The coefficients $beta_i$ determine how much each credit is exposed to the common factor; the asset correlation of two credits is $rho_(i,j)=beta_i beta_j$. The Lecture 9 notes introduce this model on PDF page 1.\n\nLet $p_i(T)=P(tau_i<=T)$ be credit $i$'s default probability by time $T$. Changing the signs and sizes of the market-factor loadings gives three limiting dependence cases.\n\n== Minimum dependence\n\nThe smallest joint-default probability comes from opposite exposures to the same market factor. In the limit, set $beta_i=1$ and $beta_j=-1$. Both idiosyncratic components disappear and the asset correlation is $rho_(i,j)=-1$. The joint-default probability is\n\n$P(tau_i<=T, tau_j<=T)=max(p_i(T)+p_j(T)-1, 0)$.\n\nIt 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.\n\n== Independence\n\nSet $beta_i=beta_j=0$. Each asset value is then only its own idiosyncratic factor, the correlation is zero, and the default events are independent:\n\n$P(tau_i<=T, tau_j<=T)=p_i(T)p_j(T)$.\n\n== Maximum dependence\n\nSet $beta_i=beta_j=1$. 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:\n\n$P(tau_i<=T, tau_j<=T)=min(p_i(T),p_j(T))$.\n\nLecture 9, slide 5 (PDF p. 5), captions its graph β€œThe joint default probability as a function of the asset correlation (4 cases).”\n\n== Conditional hazard and portfolio loss\n\nThe 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\n\n$S_i(T|z)=e^(-lambda_i(T|z)T)=Phi(frac(beta_i z-C_i(T), sqrt(1-beta_i^2)))$,\n\nso that\n\n$lambda_i(T|z)=-frac(1,T) ln Phi(frac(beta_i z-C_i(T), sqrt(1-beta_i^2)))$.\n\nHere $C_i(T)$ 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.\n\nLecture 9, slide 6 (PDF p. 6), captions the conditional hazard-rate plot: β€œPlot of conditional hazard rate distribution for $beta=0, 0.2, 0.4$,” 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.","dataUrl":"https://sharifhsn.dev/api/posts/advanced-derivatives-week-12.json","date":"2025-04-17","datePublished":"2025-04-17","description":"Lecture notes on minimum, independent, and maximum dependence in a one-factor credit model.","site":"https://sharifhsn.dev","slug":"advanced-derivatives-week-12","source":"Advanced Derivatives","sourceUrl":null,"tags":["Credit","Dependence","Copulas"],"title":"Dependence and Copulas","url":"https://sharifhsn.dev/blog/advanced-derivatives-week-12/","version":"1","wordCount":591}