Tutorials › CAS Exam 9 - Risk Management for Actuaries › Counterparty Credit Stress and Limit Breaches

Home  /  Tutorials  /  CAS Exam 9 / Credit Risk Modeling and Management

CAS Exam 9 · Credit Risk Modeling and Management · Tutorial 742 of 995

Counterparty Credit Stress and Limit Breaches

Apply coherent counterparty stress scenarios and determine whether exposure or expected-loss limits are breached.

Advanced 12 min read

What You'll Learn

  • Convert a default probability stress stated as an odds multiplier into a valid probability
  • Aggregate exposure over time while accounting for survival between periods
  • Recalculate exposure after permitted netting and haircut-adjusted collateral
  • Test counterparty exposure and expected loss against separate approved limits
  • Identify how legal, collateral and probability assumptions can change breach conclusions

Why Stress Counterparty Limits?

A counterparty can appear within its approved limit under current assumptions and exceed it under a plausible stress. Exposure may rise, collateral may lose value, a netting arrangement may not be available, or the counterparty’s probability of default may increase. A useful stress test changes specified assumptions together, recalculates the relevant risk measures, and compares them with the limits that actually govern the relationship.

This tutorial focuses on counterparty default probability, exposure profiles, collateral and netting assumptions, and the identification of limit breaches. The calculations are aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus. They are conditional on the stated scenario; they are not forecasts that a particular counterparty will default.

Definition: For this tutorial, exposure at default (EAD) is the amount exposed to loss if default occurs, after applying only the netting and collateral recognized under the stated assumptions. Loss given default (LGD) is the proportion of EAD lost conditional on default. For a single exposure with default probability \(p\), expected credit loss is \(EAD \times p \times LGD\). A counterparty limit may separately cap EAD, expected loss, or both; each limit must be tested on its own stated basis.

Set Up a Stress Test Before Calculating

A stress result is meaningful only if the assumptions specify what is being stressed and how. At minimum, state the exposure date or period, the default-probability convention, the collateral valuation and haircut, which offsets are eligible for netting, LGD, and the approved limits. If a limit is defined using a measure other than EAD or expected loss, calculate that defined measure rather than substituting one of these.

Default probabilities may be stated as unconditional probabilities over a horizon or as conditional probabilities by period. In a multi-period calculation, a period’s conditional default probability applies only if the counterparty survived to the start of that period. Exposure profiles can vary by period, so simply multiplying the sum of exposures by one horizon probability will generally be incorrect.

An odds multiplier changes the odds of default, not the probability itself. If the unstressed default probability is \(p\), and the stressed odds are \(m\) times the unstressed odds, then:

$$ \frac{p^*}{1-p^*}=m\frac{p}{1-p}, \qquad p^*=\frac{mp}{1-p+mp}. $$

Here \(p^*\) is the stressed probability and \(m\) is a positive odds multiplier. The formula assumes the stress is defined on odds. If a scenario instead specifies a direct probability multiplier, use that instruction and check that the resulting probability is no greater than 1.

Model assumption: For a single period, expected loss is \(EAD \times p \times LGD\), with EAD treated as fixed over the period and LGD as the loss fraction conditional on default. For multiple periods, use conditional default probabilities and survival probabilities so each period’s contribution is counted only when the counterparty has survived to that period. These formulas do not capture additional dependencies between default and exposure unless those dependencies are represented in the stressed inputs.

Worked Example: Stressing an Exposure Profile Over Two Years

Worked Example: Stressing an Exposure Profile Over Two Years

A hypothetical counterparty has year 1 EAD of $600,000 and year 2 EAD of $900,000. Its base conditional default probabilities are 2% in year 1 and 4% in year 2. A stress multiplies the odds of default in each year by 3. Assume LGD is 60% in both years. Calculate the stressed expected loss over the two years.

Method. Convert each base conditional probability to a stressed conditional probability, then multiply each period’s EAD by its conditional default probability, LGD and probability of surviving to the period. This avoids the common mistake of treating year 2’s conditional default probability as an unconditional probability from time 0.

Year 1 stressed probability. With \(p_1=0.02\) and \(m=3\):

$$ p_1^*=\frac{3(0.02)}{1-0.02+3(0.02)} =\frac{0.06}{1.04} =\frac{3}{52} =0.0576923. $$

The year 1 expected-loss contribution is \(600{,}000(3/52)(0.60)=1{,}080{,}000/52=\$20{,}769.23\).

Year 2 stressed probability. With \(p_2=0.04\):

$$ p_2^*=\frac{3(0.04)}{1-0.04+3(0.04)} =\frac{0.12}{1.08} =\frac{1}{9} =0.1111111. $$

The probability of surviving year 1 is \(1-3/52=49/52\). Therefore the year 2 contribution, including survival, is \(900{,}000(49/52)(1/9)(0.60)=26{,}460{,}000/468=\$56{,}538.46\).

Adding the unrounded contributions gives \(20{,}769.2308+56{,}538.4615=77{,}307.6923\). The stressed expected loss is $77,307.69, rounded to the nearest cent. As a check, the year 2 contribution is less than \(900{,}000(1/9)(0.60)=\$60{,}000\), because it includes the probability of surviving year 1.

Why the Multi-Period Formula Works

Let \(q_t\) be the stressed conditional probability of default in period \(t\), given survival to that period, and let \(EAD_t\) and \(LGD_t\) be the exposure and loss fraction for that period. The probability of reaching period \(t\) without default is the product of the preceding survival probabilities. Thus, under fixed period inputs, the expected-loss contribution for period \(t\) is:

$$ \left(\prod_{j=1}^{t-1}(1-q_j)\right)q_t EAD_t LGD_t. $$

To see why, the event that creates a period \(t\) loss requires two things: survival through each earlier period and default in period \(t\). Under the conditional-probability definition, the probability of that event is the probability of surviving the earlier periods multiplied by the probability \(q_t\) of default in period \(t\), conditional on reaching it. Multiplying this event probability by the loss \(EAD_t LGD_t\) gives its expected loss. Summing across periods gives total expected loss when the periods represent mutually exclusive default times and no other loss is being counted.

This calculation does not mean that exposures are necessarily independent of default conditions. If exposure changes with the counterparty’s credit condition, a stress should use exposure assumptions appropriate to the scenario; the fixed-input calculation alone does not model that relationship.

Netting, Collateral and Limit Tests

Netting and collateral reduce exposure only to the extent they are recognized in the applicable limit calculation. Netting should reflect eligible transactions under the assumed enforceable arrangement; do not offset amounts merely because they are with the same counterparty. For collateral, apply any stated market-value stress and then the applicable haircut to determine recognized collateral. A simple calculation is:

$$ \text{Recognized collateral} =(\text{stressed collateral market value})(1-\text{haircut}), $$
$$ EAD=\max\{0,\ \text{exposure after permitted netting} -\text{recognized collateral}\}. $$

This is a stated exposure convention, not a universal rule for every limit. Follow the approved limit’s definition if it specifies different treatment. Also avoid counting collateral twice: if collateral has already reduced EAD, do not reduce LGD again for the same collateral effect unless the model explicitly defines LGD that way.

Worked Example: Collateral and Netting Under a Probability Stress

A hypothetical counterparty has positive exposure of $1,400,000 and $550,000 of payable amounts eligible for netting. It has collateral with a market value of $600,000, subject to a 20% haircut. The base default probability is 2%, the odds stress multiplier is 4, and LGD is 75%. The approved EAD limit is $400,000 and the expected-loss limit is $20,000. Test both limits.

Method. Apply only the eligible netting, haircut the collateral, calculate EAD, then apply the stressed probability and LGD. The trap is to subtract the full collateral market value rather than its recognized value.

Exposure after permitted netting is \(1{,}400{,}000-550{,}000=\$850{,}000\). Recognized collateral is \(600{,}000(1-0.20)=\$480{,}000\). Therefore \(EAD=850{,}000-480{,}000=\$370{,}000\).

The stressed probability is:

$$ p^*=\frac{4(0.02)}{1-0.02+4(0.02)} =\frac{0.08}{1.06} =\frac{4}{53} =0.0754717. $$

Expected loss is \(370{,}000(4/53)(0.75)=1{,}110{,}000/53=\$20{,}943.40\), rounded to the nearest cent.

EAD of $370,000 is below its $400,000 limit, so the EAD limit is not breached. Expected loss of $20,943.40 exceeds its $20,000 limit, so the expected-loss limit is breached. The two conclusions differ because the limits measure different things. As a check, expected loss is less than EAD, as it must be here because both the stressed probability and LGD are less than 1.

Worked Example: Separating Collateral Stress From Netting Recognition

A hypothetical counterparty has positive exposure of $1,000,000, with $400,000 of payable amounts eligible for netting under the normal assumption. Collateral has a market value of $300,000. In a combined stress, its market value falls by 30%, the collateral haircut rises to 20%, and the netting arrangement is not recognized. The base default probability is 1.5%, the odds multiplier is 5, and LGD is 60%. The approved limits are $700,000 for EAD and $30,000 for expected loss. Compare the normal-netting exposure with the combined stress and test the stressed limits.

Method. Hold the probability stress and LGD fixed, then show how the exposure assumptions change EAD. This makes clear that the result depends on the netting recognition assumption, not merely on the probability stress.

Under the normal-netting assumption, exposure after netting is \(1{,}000{,}000-400{,}000=\$600{,}000\). With the original 10% haircut, recognized collateral is \(300{,}000(0.90)=\$270{,}000\), giving EAD of \(600{,}000-270{,}000=\$330{,}000\).

Under the combined stress, collateral market value after its 30% decline is \(300{,}000(0.70)=\$210{,}000\). Applying the stressed 20% haircut gives recognized collateral of \(210{,}000(0.80)=\$168{,}000\). Because netting is not recognized in this stress, EAD is \(1{,}000{,}000-168{,}000=\$832{,}000\).

The stressed probability is:

$$ p^*=\frac{5(0.015)}{1-0.015+5(0.015)} =\frac{0.075}{1.06} =\frac{15}{212} =0.0707547. $$

Stressed expected loss is \(832{,}000(15/212)(0.60)=7{,}488{,}000/212=\$35{,}320.75\), rounded to the nearest cent. EAD exceeds $700,000 and expected loss exceeds $30,000, so both limits are breached under the combined stress. The normal-netting EAD of $330,000 is not a substitute for the combined-stress result: the two figures rely on different collateral and netting assumptions.

Common Mistakes and Practical Interpretation

A limit breach is a comparison between a scenario result and a specified threshold. It is not, by itself, a conclusion about what management action must follow. The appropriate response depends on the approved limit framework and the scenario assumptions. Record which assumptions caused the breach so that a reviewer can distinguish a credit-probability effect from an exposure, collateral, or netting effect.

  • Multiplying probability instead of odds. If the stress says “multiply odds by \(m\),” use \(mp/(1-p+mp)\), not \(mp\). For example, a 2% probability stressed by an odds multiplier of 4 becomes \(4(0.02)/(0.98+4(0.02))=7.5472\%\), not 8%.
  • Omitting survival in later periods. A period’s conditional default probability applies only after survival to that period. Multiply by the preceding survival probability when calculating an unconditional period contribution.
  • Using gross collateral value. Apply market-value stresses and haircuts before subtracting collateral. A haircut is not an extra deduction from EAD after the recognized amount has already been calculated.
  • Assuming every offset is nettable. Use only the amounts permitted by the stated netting assumption. When enforceability or eligibility is stressed, calculate an alternative scenario rather than silently retaining the normal offset.
  • Testing only one limit. EAD and expected loss answer different questions. Compare each calculated measure with its corresponding approved limit, and state separately whether each is breached.

A transparent stress result therefore shows the chain from assumptions to decision: stress the relevant inputs, recalculate EAD, calculate expected loss using the appropriate default probability and LGD, and compare each measure with its own threshold. A useful sensitivity is to recompute the result under alternative netting or collateral recognition assumptions, without presenting those alternatives as if they were all simultaneously valid.

Check Your Understanding

For each problem, use only the assumptions stated and round money to the nearest cent and probabilities to four decimals.

  1. A hypothetical counterparty has a 2% base default probability, stressed by an odds multiplier of 3. Its EAD is $450,000, LGD is 40%, and its expected-loss limit is $15,000. Find the stressed probability and determine whether the limit is breached. (Answer: Stressed probability \(3/52=0.0577\); expected loss is $10,384.62, so the limit is not breached.)
  2. Conditional base default probabilities are 3% in year 1 and 5% in year 2. An odds multiplier of 2 applies in both years. EAD is $200,000 in year 1 and $500,000 in year 2; LGD is 50% in both years. Calculate two-year expected loss, accounting for survival. (Answer: Stressed conditional probabilities are \(6/103=0.0583\) and \(2/21=0.0952\); total expected loss is $28,247.80.)
  3. Exposure after permitted netting but before collateral is $620,000. Collateral market value is $500,000 with a 25% haircut. Base default probability is 1%, odds multiplier is 5, and LGD is 80%. The EAD limit is $500,000 and the expected-loss limit is $19,000. Test both limits. (Answer: Recognized collateral is $375,000 and EAD is $245,000, so the EAD limit is not breached. Stressed probability is \(5/104=0.0481\); expected loss is $9,423.08, so the expected-loss limit is not breached.)
  4. Positive exposure is $900,000 and $250,000 is eligible for netting. Collateral market value of $400,000 falls by 25% in stress, and a 10% haircut applies after that decline. Base default probability is 2%, odds multiplier is 4, and LGD is 70%. The EAD limit is $400,000 and the expected-loss limit is $20,000. Test both. (Answer: Stressed collateral is $300,000; recognized collateral is $270,000; EAD is $380,000. Stressed probability is \(4/53=0.0755\), expected loss is $20,075.47; only the expected-loss limit is breached.)
  5. Positive exposure is $1,200,000, with $500,000 eligible for netting only if the arrangement is recognized. Collateral market value is $200,000 with a 15% haircut. Base default probability is 1%, odds multiplier is 3, and LGD is 60%. The EAD limit is $900,000 and the expected-loss limit is $20,000. Compare results with and without netting recognition. (Answer: Recognized collateral is $170,000 and stressed probability is \(1/34=0.0294\). With netting, EAD is $530,000 and expected loss is $9,352.94; neither limit is breached. Without netting, EAD is $1,030,000 and expected loss is $18,176.47; the EAD limit is breached but the expected-loss limit is not.)

Previous

Next

Back to Tutorials
Clinical Biostats