From Separate Credit Inputs to a Portfolio View
A portfolio credit model is useful only when its parts fit together. A rating migration table can supply a one-year default probability, a structural model can provide a separate issuer-level estimate, and collateral can change the loss if default occurs. Dependence between defaults then affects the distribution of total loss, while concentration and stress analysis help show where the portfolio is vulnerable. The objective is not to combine every input mechanically: it is to make the assumptions explicit, calculate the resulting losses consistently, and explain what each result does and does not measure.
The hypothetical portfolio in this tutorial is stated in millions of dollars. Its migration probabilities are one-year probabilities. Expected loss calculations use default probabilities and loss severities; they do not include losses from the market-value repricing of nondefault rating migrations. The examples are aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.
Worked Example: Migration and Structural Inputs in a Collateralized Portfolio
Worked Example: Migration and Structural Inputs in a Collateralized Portfolio
A hypothetical insurer holds three corporate exposures. A one-year rating migration model gives default probabilities of 2% for a BBB obligor, 4% for a BB obligor, and 1% for an A obligor. Their exposures at default are 10, 8, and 5 million, respectively. For the BBB obligor, the 10 million exposure is secured by collateral worth 1.25 million. Apply a 20% collateral haircut, then assume a 40% recovery on the remaining unsecured exposure. For the BB and A obligors, assume loss severities of 50% and 40%. Calculate portfolio expected loss. Separately, estimate the BBB obligor’s structural-model default probability using \(A_0=150\), \(D=100\), \(\mu=0.02\), \(\sigma_A=0.20\), and \(T=1\). Asset value follows a lognormal model with annual asset-value drift \(\mu\) and volatility \(\sigma_A\).
Method and assumptions. Use the rating migration default probabilities for the portfolio expected-loss calculation. Treat the structural estimate as a model comparison, not as an extra probability to add to, or automatically average with, the migration estimate. Assume collateral is available at default after the stated haircut, and that the 40% recovery applies only to the unsecured balance.
Structural estimate. Under the stated lognormal asset-value model, default at time \(T\) occurs when asset value is below debt \(D\). The standardized distance from the asset-value distribution to the default threshold is
Substitute the inputs. Since \(\ln(150/100)=\ln(1.5)=0.4055\), \(\tfrac{1}{2}(0.20)^2=0.0200\), and \(T=1\),
Thus the structural default probability is \(\Phi(-2.0273)\), approximately 0.0213, or 2.13% to four decimal places as a probability. It is close to, but not identical to, the 2% migration-model probability. The two estimates come from different model inputs and assumptions.
BBB loss severity. The haircut-adjusted collateral is \(1.25(1-0.20)=1.00\) million. The unsecured balance is \(10-1=9\) million. A 40% recovery on that balance means a 60% loss on it, so the loss if the BBB obligor defaults is \(9(0.60)=5.40\) million. Its expected loss is \(0.02(5.40)=0.108\) million.
Other obligors and total. The BB obligor’s loss if default occurs is \(8(0.50)=4.00\) million, giving expected loss \(0.04(4.00)=0.160\) million. The A obligor’s default loss is \(5(0.40)=2.00\) million, giving expected loss \(0.01(2.00)=0.020\) million. Therefore,
The calculation avoids treating collateral as if it replaced the recovery assumption on the whole exposure. Applying the 40% recovery to the full 10 million after first subtracting collateral would produce an incorrect 5.00 million loss. Under the assumptions given, the recovery applies to the 9 million unsecured balance, so the correct default loss is 5.40 million. Check: the 1.00 million eligible collateral plus 3.60 million recovery on the unsecured balance totals 4.60 million recovered, leaving \(10-4.60=5.40\) million lost.
Why Expected Loss Adds, but Dependence Still Matters
Let \(I_i\) be an indicator equal to 1 if obligor \(i\) defaults and 0 otherwise. If its loss on default is a fixed amount \(L_i\), aggregate loss is \(L=\sum_i L_i I_i\). By linearity of expectation,
This justification does not require independent defaults. It explains why the portfolio expected loss in the previous example is the sum of the individual expected losses. Independence does matter for the variance and tail of \(L\), because the variance includes covariance terms:
Positive default dependence can therefore increase portfolio loss variability and the chance of simultaneous large losses, even when each obligor’s marginal default probability remains unchanged.
Worked Example: A Shared Credit Regime and Joint Defaults
Worked Example: A Shared Credit Regime and Joint Defaults
Two hypothetical borrowers have losses of 6 million and 4 million if they default. The economy is in a normal regime with probability 0.75 and a stressed regime with probability 0.25. Conditional on the regime, the borrowers default independently. In the normal regime their default probabilities are 0.01 and 0.02; in the stressed regime they are 0.05 and 0.10. Calculate each marginal default probability, the joint default probability, expected portfolio loss, and the covariance of the two default indicators.
Method. Use total probability for the marginal probabilities. For joint default, multiply the conditional default probabilities within each regime, then weight each product by the probability of that regime. Conditional independence is assumed within each regime; it does not imply unconditional independence across regimes.
The first borrower’s marginal probability is \(0.75(0.01)+0.25(0.05)=0.0075+0.0125=0.0200\). The second borrower’s marginal probability is \(0.75(0.02)+0.25(0.10)=0.0150+0.0250=0.0400\).
For joint default, the conditional products are \(0.01(0.02)=0.0002\) in the normal regime and \(0.05(0.10)=0.0050\) in the stressed regime. Therefore,
Expected loss is the sum of marginal expected losses: \(6(0.0200)+4(0.0400)=0.12+0.16=0.28\) million. The covariance of the indicators is the joint probability minus the product of the marginal probabilities:
As a check, the joint default probability, 0.00140, exceeds the value 0.00080 that independence would imply from the marginal probabilities. The positive covariance captures the association created by the shared regime. A common mistake is to report 0.0026 for the joint probability; the regime-weighted calculation is \(0.00015+0.00125=0.00140\), not 0.0026.
Concentration, Stress, and a Management Action
Expected loss alone does not describe concentration. A portfolio can have a modest expected loss but still depend heavily on one obligor or sector for its potential loss. One simple exposure concentration measure is the Herfindahl index, calculated as the sum of squared exposure shares. It is a descriptive measure of concentration, not a loss probability or a capital measure. Stress testing asks a different question: how would modeled losses change under specified adverse assumptions? A stress result is conditional on its scenario; it should not be presented as an expected outcome unless scenario probabilities are supplied and used.
Worked Example: Concentration, Collateral Stress, and Exposure Reduction
Use the portfolio from the first example. The exposures are 10, 8, and 5 million, and the default losses are 5.40, 4.00, and 2.00 million. The first exposure’s collateral is initially worth 1.25 million with a 20% haircut. For a stress scenario, suppose the collateral value falls to 0.75 million, the haircut remains 20%, and one-year default probabilities rise to 0.08, 0.12, and 0.05. Calculate the exposure Herfindahl index, base and stressed expected losses, and the stressed expected loss after reducing the first exposure from 10 million to 8 million. Assume collateral and all other assumptions remain unchanged after the exposure reduction.
Exposure concentration. Total exposure is \(10+8+5=23\) million. The exposure shares are \(10/23\), \(8/23\), and \(5/23\). Thus,
The largest exposure is \(10/23\approx 43.48\%\) of total exposure. The Herfindahl index summarizes the distribution across these three obligors; it does not incorporate their default probabilities or loss severities.
Base expected loss. The first example calculated base expected loss as 0.288 million. Under stress, eligible collateral is \(0.75(1-0.20)=0.60\) million. The first obligor’s stressed default loss is therefore \((10-0.60)(1-0.40)=9.40(0.60)=5.64\) million. The other two default losses remain 4.00 and 2.00 million. Stressed expected loss is
The stress expected loss is higher than the base expected loss because both default probabilities and the first obligor’s loss severity have worsened. The stressed collateral amount is not the same as the base eligible amount: it is 0.60 million rather than 1.00 million.
Exposure reduction. After reducing the first exposure to 8 million, its base default loss is \((8-1.00)(0.60)=4.20\) million. Its stressed default loss is \((8-0.60)(0.60)=4.44\) million. The other obligors are unchanged. The post-action stressed expected loss is
This is a reduction of \(1.0312-0.9352=0.0960\) million in stressed expected loss. The post-action exposure Herfindahl index is \((8^2+8^2+5^2)/(8+8+5)^2=153/441\approx0.3469\). For comparison, base expected loss after the action is \(0.02(4.20)+0.04(4.00)+0.01(2.00)=0.084+0.160+0.020=0.264\) million. The action lowers modeled loss by reducing the largest exposure; it does not establish that the action is optimal, since costs, replacement investments, and other constraints are not modeled here.
Interpreting an Integrated Credit Result
An integrated portfolio assessment is strongest when each output is labeled by what it measures. Rating migration probabilities describe transitions over a specified horizon. Structural estimates depend on asset value, debt, drift, volatility, and the assumed asset-value model. Expected loss is an average over modeled default outcomes. Dependence assumptions shape the joint outcomes and therefore the loss distribution. Concentration measures summarize how exposure is distributed. Stress results show the effect of a chosen adverse scenario, not a forecast.
Before using a result to support a management decision, check the time horizon, the meaning of exposure and recovery inputs, collateral eligibility and haircuts, and whether default probabilities are marginal or conditional. Recalculate loss severity when collateral assumptions change. Also distinguish a change in expected loss from a change in tail risk: a management action can lower the mean, reduce concentration, or alter dependence, and these effects are not interchangeable.
Check Your Understanding
Work through each hypothetical problem, keeping exposure, severity, probability, and scenario assumptions distinct.
- A hypothetical A-rated borrower has a one-year migration default probability of 1.5%, an exposure of 12 million, and an LGD of 55%. What is its expected loss? (Answer: 0.099 million.)
- Two obligors default conditionally independently given a shared regime. The normal regime has probability 0.75 and conditional default probabilities 0.01 and 0.02; the stress regime has probability 0.25 and conditional default probabilities 0.05 and 0.10. Find their marginal default probabilities and joint default probability. (Answer: marginal probabilities 0.0200 and 0.0400; joint probability 0.0014.)
- A 7.5 million exposure has collateral worth 0.80 million, subject to a 25% haircut. Recovery on the remaining unsecured balance is 30%, and the default probability is 3%. Find the default loss and expected loss. (Answer: default loss 4.830 million; expected loss 0.1449 million.)
- A three-obligor portfolio has exposures of 6, 5, and 4 million, with LGDs of 50%, 40%, and 75%. Base default probabilities are 0.02, 0.03, and 0.01; stressed probabilities are 0.06, 0.08, and 0.04. Calculate base and stressed expected loss and the increase under stress. (Answer: base 0.150 million; stressed 0.460 million; increase 0.310 million.)
- A portfolio has four exposures of 4, 3, 3, and 2 million. Calculate the exposure Herfindahl index. If the 4 million exposure is reduced to 2 million and the other exposures are unchanged, calculate the new index. (Answer: original \(38/144\approx0.2639\); after reduction \(26/100=0.2600\).)