Model Parameters
Corporate Bond Terms
Duffie-Singleton Credit Visualizer
Term Structure of Defaultable Zero Bonds & Credit Spreads
| Maturity | RMV Zero ($100) | RP Zero ($100) | Survival $Q(T)$ | Cum PD ($1-Q$) | Zero Yield | RMV Spread | RP Spread |
|---|
Matrix 1: Intensity Volatility $\sigma$ vs. Mean-Reversion $\kappa$
10-Year Credit Spread (bps) across hazard rate volatility and reversion speed.
| $\sigma_\lambda \backslash \kappa_\lambda$ | 0.05 | 0.15 | 0.25 | 0.40 | 0.60 |
|---|
Matrix 2: Hazard Rate $\lambda_0$ vs. Loss Given Default $L$
Corporate Bond Price ($) across initial default intensity and loss severity.
| $\lambda_0 \backslash L$ | 30% | 45% | 60% | 75% | 90% |
|---|
Theoretical Foundations: Duffie-Singleton (1999) Credit Risk
1. The Default-Adjusted Short Rate
In Darrell Duffie and Kenneth J. Singleton's (1999) landmark paper, default is modeled as an exogenous jump process with conditional intensity $\lambda(t)$. Debtholders suffer a fractional loss of market value $L(t)$ upon default (Recovery of Market Value, or RMV). Over any infinitesimal interval $dt$:
$$R(t) = r(t) + \lambda(t) L(t) = r(t) + s(t)$$
Because loss is proportional to market value just prior to default ($V_{t-}$), defaultable contingent claims can be priced as if they were default-free discount bonds by replacing the risk-free rate $r(t)$ with the default-adjusted rate $R(t)$.
2. Affine Term Structure Solution
When hazard rate intensity $\lambda(t)$ follows an affine square-root diffusion (CIR process) $d\lambda_t = \kappa(\theta - \lambda_t) dt + \sigma \sqrt{\lambda_t} dW_t$, the expectation $\mathbb{E}[e^{-L \int_0^\tau \lambda_u du}]$ has an exact closed-form analytical solution:
$$P_{\text{def}}(0, \tau) = e^{-r \tau} \exp\left( A(\tau) - B(\tau) \lambda_0 \right)$$
where $B(\tau)$ and $A(\tau)$ are analytical functions of $\kappa, \theta, \sigma$, and effective jump size $L$. This guarantees non-negative intensity and preserves mathematical tractability across multi-asset bond portfolios.