Portfolio Parameters
CreditRisk+ EngineEach exposure is mapped to integer units: $\nu_i = \text{round}(\text{EAD}_i \times \text{LGD}_i / L_0)$.
35%
0% (Pure Poisson)
35% (Typical Cycle)
100% (Severe Crisis)
| Obligor | Sector | EAD ($M) | LGD (%) | PD (%) | Del |
|---|
CreditRisk+ Actuarial Mathematical Architecture
1. Stochastic Poisson Intensity & Gamma Mixture:
$$N_i \sim \text{Poisson}(\lambda_i), \quad \lambda_i = \text{PD}_i \cdot S$$ $$S \sim \text{Gamma}(\alpha, \beta), \quad \mathbb{E}[S] = 1, \quad \text{Var}(S) = \sigma_S^2$$Compounding Poisson with Gamma yields a Negative Binomial distribution for defaults.
2. Discretization & Unit Exposure Bands:
$$\nu_i = \left\lfloor \frac{\text{EAD}_i \cdot \text{LGD}_i}{L_0} + 0.5 \right\rfloor, \quad \mu_j = \sum_{i: \nu_i = j} \text{PD}_i$$3. Probability Generating Function (PGF):
$$G_L(z) = \left(1 - \sigma_S^2 \sum_{j=1}^M \mu_j (z^j - 1)\right)^{-1/\sigma_S^2}$$4. Exact Unexpected Loss Variance:
$$\text{Var}(L) = \sum_{i} \text{PD}_i E_i^2 + \sigma_S^2 \left(\sum_i \text{PD}_i E_i\right)^2$$ $$\text{CreditVaR}_\alpha = q_\alpha(L) - \mathbb{E}[L]$$
Total Exposure at Default
$110.0M
Portfolio Notional
Expected Loss (EL)
$1.32M (1.20%)
$\sum \text{PD}_i \times \text{LGD}_i \times \text{EAD}_i$
Credit Volatility ($\sigma_L$)
$2.41M (2.19%)
Unexpected Loss (UL)
99% CreditVaR
$7.68M (6.98%)
1-Year Tail Loss
99% Expected Shortfall
$10.45M (9.50%)
Tail CVaR
99.9% Economic Capital
$13.18M (11.98%)
Solvency II / Basel Extreme
Exact discrete probability mass function derived via characteristic function inversion without Monte Carlo noise. Red bars denote tail losses beyond 99% CreditVaR.
Notice how systematic factor volatility ($\sigma_S$) fattens the tail of default counts, transforming independent Poisson arrivals into clustered Negative Binomial distributions.
Decomposition of Expected Loss, Credit Volatility, and 99% Economic Capital contributions across sectors.
Obligor Exposure Discretization & Loss Breakdown
Actuarial Unit Allocation| Obligor | Sector | EAD | LGD | Net Loss ($M) | Unit Band ($\nu_i$) | PD (%) | Expected Loss |
|---|
CreditRisk+ Sensitivity Analysis (Dual 5x5 Heatmaps)
Factor Vol ($\sigma_S$) vs Grid ($L_0$) on 99% CreditVaR ($M)
| $\sigma_S \downarrow$ \ $L_0 \rightarrow$ | $0.1M | $0.25M | $0.5M | $1.0M | $2.0M |
|---|
Baseline PD Multiplier vs Factor Vol on 99.9% Economic Capital ($M)
| PD Scale $\downarrow$ \ $\sigma_S \rightarrow$ | 10% | 30% | 50% | 70% | 90% |
|---|
CreditRisk+ Quantitative Mastery Quiz
Frequently Asked Questions: CreditRisk+ & Actuarial Credit Risk
CreditRisk+ is an actuarial portfolio credit risk management framework developed by Credit Suisse Financial Products (CSFP) in 1997. Unlike mark-to-market structural models that simulate latent asset returns, CreditRisk+ focuses exclusively on discrete default events governed by Poisson point processes with stochastic default intensities.
CreditRisk+ models correlation through systematic background risk factors representing macroeconomic or industry sectors. Default intensities are conditioned on Gamma-distributed factors. Compounding a Poisson arrival process with a Gamma-distributed intensity yields a Negative Binomial distribution for defaults, producing realistic default clustering and fat-tailed credit losses.
Discretizing each obligor's net loss at default into an integer multiple of a standard unit exposure amount transforms the continuous portfolio loss into a discrete polynomial probability generating function. This enables exact analytical distribution solutions via Panjer recursion or Fast Fourier Transform (FFT) in milliseconds.
The probability generating function (PGF) of portfolio loss is inverted either by Panjer's recursive formulation or by discrete Fourier transform inversion of the characteristic function. This produces the complete, exact discrete probability mass function P(L = n) without any Monte Carlo simulation noise.
CreditMetrics is a mark-to-market framework modeling rating migrations via Gaussian copula asset returns and Monte Carlo simulation. Vasicek's ASRF model is an asymptotic structural model for infinitely granular portfolios under Basel IRB. CreditRisk+ is an actuarial default-only model for finite portfolios that yields exact analytical loss distributions via generating functions.