Survival probability
S(t) = exp(-integral_0^t lambda(s) ds)Integrated intensity accumulates conditional default arrival risk over time.
Research & Systems
Credit Intensity Models and Default Correlation
Intensity models treat default as a random time with a hazard rate, which can be linked across names via factors.
Mental model
Default can be modelled as a random arrival whose likelihood and dependence vary across names.
In reduced-form models, default time is generated from an intensity (hazard rate) process, not a structural threshold.
Common factors in intensities generate dependent default times and cluster risk in portfolios.
Explore the mechanism
How do time-varying default arrival rates and common factors shape single-name and joint default risk?
Intensity models treat default as a random time driven by a hazard rate. When a common factor drives part of that intensity, different names share stress periods and default times become clustered rather than independent.
Numbers
λ base ≈ 5.0% per year
λᵢ idiosyncratic ≈ 3.0% per year
λ_f base factor ≈ 2.0% per year
λ_calm ≈ 4.4% per year
λ_stress ≈ 6.6% per year
P(default A by T) ≈ 21.4%
P(both defaults) ≈ 4.7%
P(at least one default) ≈ 38.1%
Pairwise default correlation ≈ 0.01
Interpretation
Increasing ρ moves more of λ into the common factor: the purple segments grow relative to the grey ones, especially in the stress scenario. Single-name default probabilities change only moderately, but the joint default and “at least one default” bars grow faster.
The correlation number summarises clustering: with ρ near zero, defaults are almost independent; with large ρ, stress periods dominate and defaults become much more likely to occur together. This is exactly the effect that matters for portfolio credit, tranches, and systemic stress testing.
UNDERSTAND
What to notice
Default correlation matters for CDO tranches, tail risk, and systemic stress testing.
The quantitative structure
Notation
lambda(t)- default intensity at time t
S(t)- survival probability to time t
This model assumes
- Default is modeled as an inaccessible random time with intensity under the selected measure.
- Recovery and dependence structures are specified separately.
A small example
Constant-intensity survival
Assume a constant annual default intensity of 2% for five years.
- Integrated intensity is 0.02 * 5 = 0.10.
- Survival probability is exp(-0.10) = 90.48%.
- Cumulative default probability is 1 - 90.48% = 9.52%.
ResultUnder the constant-intensity model, five-year default probability is approximately 9.52%.
Where it appears
- pricing credit derivatives
- simulating portfolio defaults
- estimating counterparty exposure adjustments
Where the intuition breaks
What this view leaves out
- Intensity models may fit market spreads while masking recovery, liquidity, contagion, wrong-way risk, and structural regime changes.
Common mistakes
“A hazard rate is the probability of default.”
Not quite.It is an instantaneous conditional default rate; a finite-horizon probability requires integration and survival conditioning.
Where this concept sits
Follow the strongest prerequisites, applications, extensions, and contrasts.
Connections for Credit Intensity Models and Default Correlation
Credit Intensity Models and Default CorrelationYOU ARE HERE
Used byXVA (CVA/FVA) MechanicsCredit intensities and dependence assumptions feed counterparty default and exposure-adjustment calculations.
Used byScenario and Stress EngineCredit stress scenarios use default intensities and dependence assumptions to translate shocks into portfolio losses.
Supported byCovariance Estimation and CorrelationDefault correlation is a dependence problem, so the general ideas of covariance and correlation provide useful statistical grounding.
Used by · XVA (CVA/FVA) Mechanics
Credit intensities and dependence assumptions feed counterparty default and exposure-adjustment calculations.
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