Integrated Credit Risk Metrics Workshop

Why Integrated Credit Metrics Matter

A credit exposure is not described fully by a default probability alone. An insurer also needs to consider the amount exposed if default occurs, the recovery available afterward, and whether a nondefault credit migration creates a loss in value. Portfolio decisions add further questions: how much of the exposure is concentrated in one name, and how large could a counterparty loss become after netting and collateral?

The calculations below use one-period hypothetical scenarios. Probabilities and exposure amounts are supplied rather than estimated. The integrated methods are aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Definition: Exposure at default (EAD) is the amount exposed when default occurs. Probability of default (PD) is the probability of default over the stated horizon. Loss given default (LGD) is the fraction of EAD not recovered after default; if the recovery rate is $r$, then $\mathrm{LGD}=1-r$. In a fixed-EAD, fixed-LGD default model, the loss is $L=\mathrm{EAD}\times\mathrm{LGD}$ if default occurs and zero otherwise. Expected loss is the probability-weighted mean loss; unexpected loss is often summarized by the standard deviation of loss, but it is not a universal capital definition.

A migration from one credit quality to another need not be a default. It may nevertheless change the market value of a bond or another credit-sensitive asset. For a one-period calculation, use the loss amounts and state probabilities that the problem supplies. Do not apply a default recovery assumption to a migration loss that is already expressed as a change in value.

Expected Loss and Loss Variability

For a single exposure with fixed EAD $E$, fixed LGD $g$, and default probability $p$, define the default indicator $I$ to be 1 upon default and 0 otherwise. The loss is $X=EgI$. Since $E[I]=p$, the expected loss is $E[X]=Egp$. The indicator has variance $p(1-p)$, so scaling it by $Eg$ gives $\operatorname{Var}(X)=(Eg)^2p(1-p)$.

E[X]=p\,E\,g,\qquad \operatorname{Var}(X)=p(1-p)(E g)^2,\qquad \operatorname{SD}(X)=E g\sqrt{p(1-p)}.

This result assumes a one-period binary outcome, with fixed exposure and recovery conditional on default. It does not apply unchanged if exposure or recovery varies across default scenarios, or if losses can occur through migration as well as default. In those cases, calculate from the full set of possible loss outcomes.

Discrete loss distribution: If loss $X$ takes values $x_j$ with probabilities $q_j$, where $\sum_j q_j=1$, then $E[X]=\sum_j q_jx_j$ and $\operatorname{Var}(X)=\sum_jq_jx_j^2-(E[X])^2$. This form accommodates default, migration, and conditional exposure scenarios, provided the scenarios are mutually exclusive and collectively exhaustive.

Worked Example: Default and Migration Losses on an Insurer Bond

A hypothetical insurer holds a corporate bond. Over a one-year horizon, the supplied scenario probabilities are: default, 0.03; downgrade without default, 0.12; unchanged credit quality, 0.80; and upgrade, 0.05. The probabilities sum to $0.03+0.12+0.80+0.05=1.00$. EAD at default is $1,200,000 and the recovery rate is 0.40. The estimated value loss from the downgrade is $36,000. For this calculation, an upgrade and an unchanged rating have zero loss. Calculate expected loss and the standard deviation of loss.

Method. First calculate default loss using LGD, then list the loss in every mutually exclusive state. The downgrade amount is already a value loss, so it is not multiplied by LGD.

The default LGD is $1-0.40=0.60$, giving default loss $1{,}200{,}000(0.60)=\$720{,}000$. The loss outcomes are therefore $720,000 with probability 0.03; $36,000 with probability 0.12; and $0 with combined probability $0.80+0.05=0.85$.

Expected loss is $0.03(720{,}000)+0.12(36{,}000)+0.85(0)=21{,}600+4{,}320=\$25{,}920$.

For the second moment, $E[X^2]=0.03(720{,}000^2)+0.12(36{,}000^2)$. The squared amounts are $720{,}000^2=518{,}400{,}000{,}000$ and $36{,}000^2=1{,}296{,}000{,}000$. Thus $E[X^2]=15{,}552{,}000{,}000+155{,}520{,}000=15{,}707{,}520{,}000$.

The squared mean is $25{,}920^2=671{,}846{,}400$. Hence the variance is $15{,}707{,}520{,}000-671{,}846{,}400=15{,}035{,}673{,}600$, and the standard deviation is $\sqrt{15{,}035{,}673{,}600}\approx\$122{,}620$, rounded to the nearest dollar.

Check and trap avoided. The probabilities total 1, and the loss distribution includes both default and downgrade. Applying the 0.60 LGD again to the $36,000 downgrade loss would incorrectly reduce a value loss that was already specified in dollars.

Aggregating Exposures and Reading Concentration

For portfolio loss $X=\sum_iX_i$, expected losses add whether or not obligors are independent: $E[X]=\sum_iE[X_i]$. Variances add only when the losses are independent. More generally, $\operatorname{Var}(X)=\sum_i\operatorname{Var}(X_i)+2\sum_{i<j}\operatorname{Cov}(X_i,X_j)$. Positive dependence can therefore make portfolio variability larger than an independence calculation suggests.

Concentration measures answer a different question from expected loss. For example, the Herfindahl index based on exposure shares $w_i$ is $\sum_iw_i^2$. It summarizes how exposure is distributed across names; it is not a loss probability or a capital amount. A portfolio may have modest expected loss and still be concentrated in a small number of obligors.

Worked Example: Independent Investment Exposures and Name Concentration

A hypothetical insurer has two issuer exposures. Issuer A has EAD $1,500,000, default probability 0.02, and LGD 0.60. Issuer B has EAD $500,000, default probability 0.08, and LGD 0.40. Assume fixed EAD and LGD, and independent defaults over the same one-year horizon. Calculate total expected loss, the standard deviation of aggregate loss, and the Herfindahl index based on EAD shares.

Expected losses. Issuer A's default loss is $1{,}500{,}000(0.60)=\$900{,}000$, so its expected loss is $0.02(900{,}000)=\$18{,}000$. Issuer B's default loss is $500{,}000(0.40)=\$200{,}000$, so its expected loss is $0.08(200{,}000)=\$16{,}000$. Total expected loss is $18{,}000+16{,}000=\$34{,}000$.

Aggregate standard deviation. Issuer A's variance is $0.02(0.98)(900{,}000^2)=0.0196(810{,}000{,}000{,}000)=15{,}876{,}000{,}000$. Issuer B's variance is $0.08(0.92)(200{,}000^2)=0.0736(40{,}000{,}000{,}000)=2{,}944{,}000{,}000$. Independence permits adding these variances, giving $18{,}820{,}000{,}000$. The standard deviation is $\sqrt{18{,}820{,}000{,}000}\approx\$137{,}186$, rounded to the nearest dollar.

Concentration. Total EAD is $2,000,000. The shares are $w_A=1{,}500{,}000/2{,}000{,}000=0.75$ and $w_B=0.25$. Thus the Herfindahl index is $0.75^2+0.25^2=0.625$, or 6,250 on a 0-to-10,000 scale.

Check and trap avoided. Adding the two variances and then taking a square root gives the portfolio standard deviation under the stated independence assumption. Adding the individual standard deviations instead would be incorrect. If defaults were dependent, a covariance term would be needed; independence cannot be inferred merely because the issuers are different.

Counterparty Exposure with Variable EAD

Counterparty calculations require clarity about what an exposure amount represents. Current exposure, potential future exposure, collateral, and netting are not interchangeable. If a problem supplies EAD already adjusted for netting and collateral, use that amount as given and do not subtract the same collateral a second time. If EAD varies conditional on default, expected loss can be calculated by weighting each conditional default scenario by its probability of default.

E[X]=\sum_j P(\text{default and scenario }j)\,EAD_j\,LGD_j.

This expression follows by splitting the loss expectation across mutually exclusive scenarios: each scenario contributes its probability multiplied by its loss. If every scenario has the same PD and LGD but different EAD, it is equivalent to multiplying PD and LGD by the expected EAD conditional on default. For variance, however, the separate outcomes must be retained; replacing variable EAD by its conditional average can discard real variability.

Worked Example: Conditional Exposure to a Derivative Counterparty

A hypothetical insurer has a derivative counterparty exposure after netting and collateral. Conditional on counterparty default during the one-year horizon, EAD is $300,000 with probability 0.60 or $500,000 with probability 0.40. The counterparty's default probability is 0.025 and its recovery rate is 0.45. Assume the conditional EAD scenario and recovery rate are as stated, and there are no additional migration losses. Calculate expected loss and the standard deviation of loss.

Method. The default LGD is $1-0.45=0.55$. The default losses are $300{,}000(0.55)=\$165{,}000$ and $500{,}000(0.55)=\$275{,}000$. Their unconditional probabilities are $0.025(0.60)=0.015$ and $0.025(0.40)=0.010$. No default has probability $1-0.025=0.975$ and loss zero.

Expected loss is $0.015(165{,}000)+0.010(275{,}000)=2{,}475+2{,}750=\$5{,}225$. Equivalently, expected EAD conditional on default is $0.60(300{,}000)+0.40(500{,}000)=\$380{,}000$, and $0.025(0.55)(380{,}000)=\$5{,}225$.

The second moment is $0.015(165{,}000^2)+0.010(275{,}000^2)$. Since $165{,}000^2=27{,}225{,}000{,}000$ and $275{,}000^2=75{,}625{,}000{,}000$, the second moment is $408{,}375{,}000+756{,}250{,}000=1{,}164{,}625{,}000$. The squared mean is $5{,}225^2=27{,}300{,}625$. Therefore variance is $1{,}164{,}625{,}000-27{,}300{,}625=1{,}137{,}324{,}375$, and standard deviation is approximately $\sqrt{1{,}137{,}324{,}375}=\$33{,}724$, rounded to the nearest dollar.

Check and trap avoided. The two default-scenario probabilities sum to 0.025, the supplied PD. Using average conditional EAD gives the correct expected loss here, but treating $209,000 ($380{,}000\times0.55$) as a fixed default loss would give a standard deviation of only about $32,630. That shortcut omits variability between the two EAD outcomes; the full discrete distribution gives about $33,724.

Common Exam Traps and Practical Interpretation

Confusing expected loss with unexpected loss. Expected loss is the probability-weighted mean. A standard deviation describes dispersion around that mean; it is not automatically a regulatory capital requirement or a tail quantile.
Applying LGD twice. Multiply EAD by LGD to obtain a default loss. Do not multiply by LGD again if a migration or valuation loss is already stated as a dollar amount.
Using an average where variability matters. Conditional average EAD can be sufficient for expected loss when the other assumptions are fixed. It need not preserve variance or tail outcomes.
Adding standard deviations. Under independence, add variances, then take the square root. Under dependence, include covariance terms or use the supplied joint scenarios.
Double-counting collateral or exposure components. Confirm whether EAD is already net of collateral and netting before making adjustments.
Reading concentration as a loss measure. A concentration index describes exposure distribution, not the chance or dollar amount of loss by itself.

Key takeaway: Identify the horizon, loss outcomes, and scenario probabilities first. Calculate each state's loss using the assumptions that apply to that state; then compute the mean and, when requested, the variance from the complete loss distribution. Aggregate exposures only with an explicit dependence assumption.

Check Your Understanding

For each hypothetical problem, state any required assumption before calculating. Amounts are in dollars unless stated otherwise.

A fixed exposure has EAD $900,000, one-year PD 0.04, and recovery rate 0.70. Find expected loss and the standard deviation of loss, rounded to the nearest dollar. (Answer: Expected loss = $10,800; standard deviation = approximately $52,909.)
A hypothetical insurer has fixed EAD $2,000,000, PD 0.015, and LGD 0.50. Calculate the default loss, expected loss, and standard deviation, rounded to the nearest dollar. (Answer: Default loss = $1,000,000; expected loss = $15,000; standard deviation = approximately $121,552.)
Two independent fixed-loss exposures have (default loss, PD) of ($300,000, 0.04) and ($500,000, 0.02). Find aggregate expected loss and standard deviation, rounded to the nearest dollar. (Answer: Expected loss = $22,000; standard deviation = approximately $91,411.)
A counterparty has PD 0.01 and LGD 0.40. Conditional on default, EAD is $200,000 with probability 0.70 or $600,000 with probability 0.30. Find expected loss. (Answer: $1,280.)
A hypothetical insurer's exposures to three names are $500,000, $300,000, and $200,000. Find the Herfindahl index on both the 0-to-1 and 0-to-10,000 scales. (Answer: 0.38; 3,800.)

What You'll Learn