## 8. Model 4: Correlated Sectors, Inter-Sector Covariance = 0.2

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---

### Introduction and purpose
- Presents theoretical background and implementation of Credit Risk+ (actuarial approach).
- Objective: determine probability distribution of portfolio losses to compute credit Value at Risk (VaR) and economic capital.
- Reasons for Credit Risk+ popularity:
  - Limited input data and assumptions.
  - Uses same basic inputs as Basel II IRB approach.
  - Provides analytical solution for loss distribution.
  - Exposes concentration as a key credit risk driver.

### Model taxonomy (progression of models)
- Model 1 (Basic):
  - Bernoulli default events; non-random default probabilities; independence across obligors.
  - PGF computed by convolution and FFT.
- Model 2 (Fixed, Poisson approximation):
  - Bernoulli → Poisson approximation for small n p; closed-form PGF under Poisson.
- Model 3 (Credit Risk+):
  - Random default probabilities driven by mutually independent Gamma-distributed factors; Poisson approximation used; analytical PGF obtained.
- Model 4 (Extension with correlated factors):
  - Introduces common Gamma factor Γ so that k γ are independent only conditional on Γ, producing positive unconditional covariance among sector factors; yields PGF for correlated-sectors model.

### Poisson approximation (key derivation and usage)
- Bernoulli PGF formula preserved as in source.
- For small n p and |z|≤1, Taylor expansion ln(1+w)=w−w2/2+...; neglecting higher-order terms yields approximation ln(1+ w ) ≈ w.
- Poisson PGF approximation (as given):
  - ()()(  )(  ),0 exp1exp x x n Dnnn x p Gzpzpz x ∞ = ≈−=−⎡⎤⎣⎦∑ !
- Individual normalized loss PGF under Poisson:
  - ()() ,exp1 n n n Gzpz ν λ =−⎡⎤⎣⎦.
- Poisson approximation enables analytical solutions when default probabilities are random (Credit Risk+).

### Model 4 specifics (correlated sectors)
- Introduces common Gamma-distributed variable Γ with shape parameter 2 1/ασ= and scale parameter 2 βσ=; means and variances given by ()()22 1E Var αβ αβσ=⋅=Γ=⋅=.
- k γ remains Gamma-distributed conditional on Γ with k α = Γ⋅k α (k β constant).
- Unconditional covariance between any two sector factors k γ and l γ becomes positive:
  - ()2 ,0 kl Covγγ σ=> (unconditional covariance >0 and equals variance of Γ effect).
- Joint MGF Mγ(z) derived by integrating conditional MGFs over Γ.
- Final PGF for total normalized loss in correlated-sectors model (preserved expression from source):
  - ()() 2 , 2 1 11 expln 1ln 1 K kk CORR k k Gz Pz Gz λ σβ σβ =⎧⎫⎡⎤⎪⎪=− +−⋅ ≡ ⎨⎬⎢⎥⎪⎪⎣⎦⎩⎭∑
  - where ()() 1 1 n N n knk n Pz pz ν ω = =− ∑.

### Numerical implementation and algorithms
- Two main computational challenges for recursive Panjer recursion:
  - Overflow when expected defaults large.
  - Numerical instability accumulating errors in recursion leading to inaccurate upper-tail/VaR.
- Alternative recursive scheme (Giese (2003) + Haaf, Reis and Schoenmakers (2003)):
  - Numerically stable; unified implementation for Poisson-based models; can fail for very large numbers of obligors.
- FFT-based algorithm (Melchiori (2004)):
  - Uses characteristic function and FFT/IFFT to compute loss distribution efficiently; successfully applied to a portfolio containing 679,000 obligors.
  - For model with Poisson defaults and fixed probabilities: define l PGF and use Π = exp(1IFFT FFT μ μπ ) (notation as in source).
- Recursive polynomial transformations:
  - Exponential and logarithmic transforms of polynomials computed by recursive coefficient formulas (given in source).
- Implementation limitation condition (exact source expression preserved):
  - max 1 750        1,, N n kn k n pkKωμ = ⋅<  ≈= ∑ K.  

### Numerical examples and empirical results (sample portfolio)
- Sample portfolio: CSFP reference portfolio used (Table 1); computations and experiments use that 25-obligor sample in examples.
- Monte Carlo benchmark:
  - Monte Carlo for Bernoulli defaults with random Gamma-driven probabilities: 5000 draws of k γ, and for each, 5000 draws of n D → overall 25 million random combinations used.
- Effects of Poisson approximation (non-random probabilities):
  - Comparing Model 1 (Bernoulli) vs Model 2 (Poisson) on 25-obligor CSFP portfolio (default probabilities ranging from 3 percent to 30 percent):
    - VaR is 8.67 percent larger in the Poisson-defaults model than in the Bernoulli-defaults model for this portfolio.
  - Experiment varying common default probability 1 to 30 percent:
    - Ratio bp VaR p VaR decreases from 1 to 0.86 as common default probability increases from 1 to 30 percent (i.e., Poisson approximation error increases with default probability).
  - General tendency: Poisson approximation tends to overestimate VaR.
- Random default probabilities, uncorrelated factors (Models 3a and 3b vs Monte Carlo):
  - Model 3a: Poisson defaults, random probabilities, CSFP weighted-average estimation of 2 k σ.
  - Model 3b: same but 2 k σ estimated by least squares (LS).
  - Results:
    - VaR for model 3b is 9 percent higher than Monte Carlo benchmark.
    - VaR for model 3a is only 1 percent higher than Monte Carlo benchmark.
  - Explanation: CSFP weighted-average underestimates k σ, which partly offsets Poisson overestimation.
  - Conclusion: Credit Risk+ with CSFP sigma estimation can yield VaR close to Monte Carlo for the sample.
- Random default probabilities, correlated factors (Model 4 experiments):
  - Var(Γ) set to 0.1 and 0.2 in experiments.
  - Results:
    - VaR is higher in Model 4 than in Model 3a.
    - VaR increases with the variance of Γ (inter-sector covariance), reflecting increased systemic risk and uncertainty in default probabilities.

### Practical guidance and model selection
- Model 1 (Bernoulli with known probabilities):
  - Use when default probabilities are known with little uncertainty, probabilities are relatively high, and portfolio size not more than a few thousand obligors.
- Credit Risk+ and extensions (Models 3 and 4):
  - Provide quasi-instantaneous solutions even for very large portfolios at the cost of approximations (Poisson approximation; latent-factor linear model).
  - Suitable when default probabilities are influenced by random latent factors and computational speed / scalability is required.
- Estimation of k σ (sector factor standard deviations):
  - CSFP weighted-average approach is simple, tends to understate k σ (but can offset Poisson bias).
  - Least squares yields theoretically consistent estimates via solving linear system for 2 k σ, but may require constrained least squares and may not guarantee strictly positive solutions in all cases.
- Algorithms:
  - Use numerically stable recursive schemes (Giese-type) for moderate-size portfolios where recursion limits hold.
  - Use FFT-based algorithm (Melchiori) for very large portfolios (hundreds of thousands of obligors).

### Key numeric highlights
- VaR is 8.67 percent larger in the model with Poisson defaults than in the model with Bernoulli defaults (CSFP 25-obligor portfolio).
- In Poisson vs Bernoulli experiment varying common default probability from 1 to 30 percent, bp VaR p VaR decreases steadily from 1 to 0.86 as the common default probability increases from 1 to 30 percent.
- VaR for model 3b is 9 percent higher than the VaR computed by Monte Carlo simulation.
- VaR in model 3a is only 1 percent higher than the VaR computed by Monte Carlo simulation.
- FFT-based algorithm successfully applied to a portfolio containing 679,000 obligors.
- Monte Carlo benchmark used 5000 draws of k γ and 5000 draws of n D per draw → 25 million random combinations.
- Correlated-factor experiments presented for inter-sector covariance (variance of Γ) = 0.1 and = 0.2 (Model 4 results show VaR increases with this variance).

### Conclusion (implementation takeaways)
- Choice of model depends on data availability, portfolio size, computational constraints, and acceptable approximation error.
- Credit Risk+ and its correlated-factor extension (Model 4) extend tractable analytical PGF solutions to settings with random default probabilities and inter-sector correlation, enabling large-scale, fast risk estimation for supervisory and monitoring tasks.
- For the sample portfolio, Credit Risk+ results were close to Monte Carlo despite Poisson and latent-factor approximations; inter-sector correlation (Model 4) materially increases VaR as inter-sector covariance rises.

*Source: IMF staff paper chapter "8. Model 4: Correlated Sectors, Inter-Sector Covariance = 0.2" (from the provided PDF content).*

### References..............................................................................................................

### _wp06134 - References

### Tables
- 1. Credit Swiss Financial Product Reference Portfolio ...........................................................24
- 2. Summary of the Assumptions Used in the Different Models ..............................................25

### Figures
- 1. Model 1: Fixed Probabilities, Bernoulli Defaults ................................................................26
- 2. Model 2: Fixed Probabilities, Poisson Defaults...................................................................26
- 3. Ratio of Bernoulli VaR to Poisson VaR for CSFP Portfolio ...............................................27
- 4. Bernoulli Defaults, Random Probabilities, Monte Carlo Simulation ..................................28
- 5. Model 3a: Poisson Defaults, Uncorrelated Factors, Weighted Average .............................28
- 6. Model 3b: Poisson Defaults, Uncorrelated Factors, Least Squares .....................................29
- 7. Model 4: Correlated Sectors, Inter-Sector Covariance = 0.1...............................................30

*Source: _wp06134 - References.*

### 8. Model 4: Correlated Sectors, Inter-Sector Covariance = 0.2...............................................30

### 8. Model 4: Correlated Sectors, Inter-Sector Covariance = 0.2

### Introduction and purpose
- Presents theoretical background and implementation of Credit Risk+ (actuarial approach).
- Objective: determine probability distribution of portfolio losses to compute credit Value at Risk (VaR) and economic capital.
- Highlights reasons for Credit Risk+ popularity:
  - Limited input data and assumptions.
  - Uses same basic inputs as Basel II IRB approach.
  - Provides analytical solution for loss distribution.
  - Exposes concentration as a key credit risk driver.

### Model taxonomy (progression of models)
- Model 1 (Basic): Bernoulli default events; non-random default probabilities; independence across obligors; PGF computed by convolution and FFT.
- Model 2 (Fixed, Poisson approximation): Bernoulli → Poisson approximation for small n p; closed-form PGF under Poisson.
- Model 3 (Credit Risk+): Random default probabilities driven by mutually independent Gamma-distributed factors; Poisson approximation used; analytical PGF obtained.
- Model 4 (Extension with correlated factors): Introduces common Gamma factor Γ so that k γ are independent only conditional on Γ, producing positive unconditional covariance among sector factors; yields PGF for correlated-sectors model.

### Poisson approximation (key derivation and usage)
- Bernoulli PGF: ()()() ,111 Dnnnn Gzp pz pz=−+⋅=+ −.
- For small n p and |z|≤1, define w()1 n pz=− ; Taylor expansion ln(1+w)=w−w2/2+... ; neglecting higher-order terms yields approximation ln(1+ w ) ≈ w.
- Using expansions results in Poisson PGF approximation:
  - ()()(  )(  ),0 exp1exp x x n Dnnn x p Gzpzpz x ∞ = ≈−=−⎡⎤⎣⎦∑ !
  - Individual normalized loss PGF under Poisson: ()() ,exp1 n n n Gzpz ν λ =−⎡⎤⎣⎦.
- Poisson approximation enables analytical solutions when default probabilities are random (Credit Risk+).

### Model 4 specifics (correlated sectors)
- Introduces common Gamma-distributed variable Γ with shape parameter 2 1/ασ= and scale parameter 2 βσ=; means and variances given by ()()22 1E Var αβ αβσ=⋅=Γ=⋅=.
- k γ remains Gamma-distributed conditional on Γ with k α = Γ⋅k α (k β constant).
- Unconditional covariance between any two sector factors k γ and l γ becomes positive:
  - ()2 ,0 kl Covγγ σ=> (text shows unconditional covariance >0 and equals variance of Γ effect).
- Joint MGF Mγ(z) derived by integrating conditional MGFs over Γ; final PGF for total normalized loss in correlated-sectors model:
  - ()() 2 , 2 1 11 expln 1ln 1 K kk CORR k k Gz Pz Gz λ σβ σβ =⎧⎫⎡⎤⎪⎪=− +−⋅ ≡ ⎨⎬⎢⎥⎪⎪⎣⎦⎩⎭∑
  - where ()() 1 1 n N n knk n Pz pz ν ω = =− ∑.

### Numerical implementation and algorithms
- Two main computational challenges for recursive Panjer recursion: overflow when expected defaults large; numerical instability accumulating errors in recursion leading to inaccurate upper-tail/VaR.
- Alternative recursive scheme (Giese (2003) + Haaf, Reis and Schoenmakers (2003)):
  - Numerically stable; unified implementation for Poisson-based models; can fail for very large numbers of obligors.
- FFT-based algorithm (Melchiori (2004)):
  - Uses characteristic function and FFT/IFFT to compute loss distribution efficiently; successfully applied to a portfolio containing 679,000 obligors.
  - For model with Poisson defaults and fixed probabilities:
    - Define l PGF and use Π = exp(1IFFT FFT μ μπ ) (notation as in source).
- Recursive polynomial transformations:
  - Exponential and logarithmic transforms of polynomials computed by recursive coefficient formulas (given in source).
- Implementation limitation condition (exact source expression preserved):
  - max 1 750        1,, N n kn k n pkKωμ = ⋅<  ≈= ∑ K.  

### Numerical examples and empirical results (sample portfolio)
- Sample portfolio: CSFP reference portfolio used (Table 1); computations and experiments use that 25-obligor sample in examples.
- Monte Carlo benchmark:
  - Monte Carlo for Bernoulli defaults with random Gamma-driven probabilities: 5000 draws of k γ, and for each, 5000 draws of n D → overall 25 million random combinations used.
- Effects of Poisson approximation (non-random probabilities):
  - Comparing Model 1 (Bernoulli) vs Model 2 (Poisson) on 25-obligor CSFP portfolio (default probabilities ranging from 3 percent to 30 percent):
    - VaR is 8.67 percent larger in the Poisson-defaults model than in the Bernoulli-defaults model for this portfolio.
  - Experiment varying common default probability 1 to 30 percent:
    - Ratio bp VaR p VaR decreases from 1 to 0.86 as common default probability increases from 1 to 30 percent (i.e., Poisson approximation error increases with default probability).
  - General tendency: Poisson approximation tends to overestimate VaR.
- Random default probabilities, uncorrelated factors (Models 3a and 3b vs Monte Carlo):
  - Model 3a: Poisson defaults, random probabilities, CSFP weighted-average estimation of 2 k σ.
  - Model 3b: same but 2 k σ estimated by least squares (LS).
  - Results:
    - VaR for model 3b is 9 percent higher than Monte Carlo benchmark (overestimation due to Poisson approximation).
    - VaR for model 3a is only 1 percent higher than Monte Carlo. Explanation: CSFP weighted-average underestimates k σ, which partly offsets Poisson overestimation.
  - Conclusion: despite approximations, Credit Risk+ with CSFP sigma estimation can yield VaR close to Monte Carlo for the sample.
- Random default probabilities, correlated factors (Model 4 experiments):
  - Model 4 introduces common Γ producing inter-sector covariance (variance of Γ equals covariance between any two factors).
  - Portfolio loss distributions computed for two values of Var(Γ): 0.1 and 0.2.
  - Results:
    - VaR is higher in Model 4 than in Model 3a.
    - VaR increases with the variance of Γ (inter-sector covariance), reflecting increased systemic risk and uncertainty in default probabilities.
  - Model 4 figures include the case "Inter-Sector Covariance = 0.2".

### Practical guidance and model selection
- Model 1 (Bernoulli with known probabilities):
  - Use when default probabilities are known with little uncertainty, probabilities are relatively high, and portfolio size not more than a few thousand obligors.
- Credit Risk+ and extensions (Models 3 and 4):
  - Provide quasi-instantaneous solutions even for very large portfolios at the cost of approximations (Poisson approximation; latent-factor linear model).
  - Suitable when default probabilities are influenced by random latent factors and computational speed / scalability is required.
- Estimation of k σ (sector factor standard deviations):
  - CSFP weighted-average approach is simple, tends to understate k σ (but can offset Poisson bias).
  - Least squares yields theoretically consistent estimates via solving linear system for 2 k σ, but may require constrained least squares and may not guarantee strictly positive solutions in all cases.
- Algorithms:
  - Use numerically stable recursive schemes (Giese-type) for moderate-size portfolios where recursion limits hold.
  - Use FFT-based algorithm (Melchiori) for very large portfolios (hundreds of thousands of obligors).

### Key numeric highlights (preserved exactly as in source)
- VaR is 8.67 percent larger in the model with Poisson defaults than in the model with Bernoulli defaults (CSFP 25-obligor portfolio).
- In Poisson vs Bernoulli experiment varying common default probability from 1 to 30 percent, bp VaR p VaR decreases steadily from 1 to 0.86 as the common default probability increases from 1 to 30 percent.
- VaR for model 3b is 9 percent higher than the VaR computed by Monte Carlo simulation.
- VaR in model 3a is only 1 percent higher than the VaR computed by Monte Carlo simulation.
- FFT-based algorithm successfully applied to a portfolio containing 679,000 obligors.
- Monte Carlo benchmark used 5000 draws of k γ and 5000 draws of n D per draw → 25 million random combinations.
- Correlated-factor experiments presented for inter-sector covariance (variance of Γ) = 0.1 and = 0.2 (Model 4 results show VaR increases with this variance).

### Conclusion (implementation takeaways)
- Choice of model depends on data availability, portfolio size, computational constraints, and acceptable approximation error.
- Credit Risk+ and its correlated-factor extension (Model 4) extend tractable analytical PGF solutions to settings with random default probabilities and inter-sector correlation, enabling large-scale, fast risk estimation for supervisory and monitoring tasks.
- For the sample portfolio, Credit Risk+ results were close to Monte Carlo despite Poisson and latent-factor approximations; inter-sector correlation (Model 4) materially increases VaR as inter-sector covariance rises.

*Source: IMF staff paper chapter "8. Model 4: Correlated Sectors, Inter-Sector Covariance = 0.2" (from the provided PDF content).*

### References

### References

### Key works on Credit Risk+ and credit risk models
- Crouhy, Michel, Dan Galai and Robert Mark, 2000, “A Comparative Analysis of Current Credit Risk Models,” Journal of Banking & Finance, Vol. 24, pp. 59–117.
- CSFP, 1997, “Credit Risk+: A Credit Risk Management Framework,” Credit Suisse First Boston.
- Giese, Gotz, 2003, “Enhancing Credit Risk+,” Risk, Vol. 16, No. 4, pp. 73–77.
- Gordy, Michael B., 2002, “Saddlepoint Approximation of Credit Risk+,” Journal of Banking and Finance, 26, pp. 1335–1353.
- Melchiori, Mario R., 2004, “Credit Risk+ by FFT,” working paper, Universidad Nacional del Litoral, Santa Fe, Argentina.

### Computational methods and numerical stability
- Haaf, Hermann, Oliver Reiss and John Schoenmakers, 2003, “Numerically Stable Computation of Credit Risk+,” working paper, Weierstrass Institute, Berlin.
- Gordy, Michael B., 2002, “Saddlepoint Approximation of Credit Risk+,” Journal of Banking and Finance, 26, pp. 1335–1353.
- Melchiori, Mario R., 2004, “Credit Risk+ by FFT,” working paper, Universidad Nacional del Litoral, Santa Fe, Argentina.

### Supervisory and framework sources
- Austrian Financial Market Authority and Oesterreichische Nationalbank, 2004, “New Quantitative Models of Banking Supervision,” Vienna.
- CSFP, 1997, “Credit Risk+: A Credit Risk Management Framework,” Credit Suisse First Boston.
- Koyluoglu, H. Ugur and Andrew Hickman, 1998, “A Generalized Framework for Credit Risk Portfolio Models,” unpublished working paper.

*Source: _wp06134 - References*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2006/_wp06134.pdf_