Stress Testing Credit Portfolios

What Credit Stress Testing Measures

A credit stress test asks how a portfolio’s loss estimate changes under a specified adverse scenario. A recession, a shock to a particular industry, or a deterioration in borrower ratings may affect both the likelihood of default and the amount recovered after default. The calculation is useful only if those effects are translated consistently into stressed inputs and applied to the appropriate exposures.

The central calculation is expected loss over a stated horizon. A stress-test result is generally a conditional estimate: it answers what the expected loss would be if the specified scenario assumptions applied. It is not automatically a forecast, a tail-loss measure, or a probability-weighted estimate across all possible economic conditions. This treatment is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Definition: For an exposure with exposure at default $EAD$, default probability $PD$, and recovery rate $R$, the loss given default is $LGD=1-R$. Under a one-period model with fixed $EAD$, the expected loss is $EAD\times PD\times LGD$. A stressed expected loss applies default and recovery assumptions conditional on the specified stress scenario.

Use one consistent horizon for every input. For example, a one-year default probability must be paired with exposure and recovery assumptions relevant to the same one-year loss calculation. Unless the scenario explicitly specifies otherwise, assume the stated recovery rate applies to the exposure at default and that there are no additional costs, discounting effects, or recoveries after the modeled horizon.

Translating Scenario Shocks Into Inputs

A scenario description does not calculate a loss by itself. First convert each stated shock into a stressed default probability and recovery rate. Then apply the resulting loss-given-default assumption to the exposure in the scenario.

Fix the horizon and exposure basis.
Identify the period being stressed and the exposure at default for each portfolio segment. Confirm whether the exposure changes under the scenario.

Translate default shocks.
Distinguish an additive percentage-point change from a relative percentage change or a multiplier. Apply the specified operation to the base probability.

Translate recovery shocks.
Determine whether recovery falls by percentage points or proportionally. Convert the stressed recovery to $LGD=1-R$.

Calculate and compare losses.
For each segment, multiply exposure by stressed default probability and stressed loss given default. Sum segment losses and compare with the base case.

A percentage-point change is additive: an increase of 2 percentage points from 1% gives 3%. A relative increase of 2% instead gives $1\%\times1.02=1.02\%$. Similarly, a recovery reduction of 10 percentage points from 40% gives 30%; a relative reduction of 10% gives 36%. Scenario wording determines which calculation is appropriate.

Why the Expected-Loss Formula Holds

Consider one exposure with fixed $EAD$, recovery rate $R$, and default indicator $I$, where $I=1$ if default occurs during the horizon and $I=0$ otherwise. Its modeled loss is $L=EAD(1-R)I$. If the probability of default is $PD$, then $E[I]=PD$, so:

E[L]=E[EAD(1-R)I]=EAD(1-R)E[I]=EAD\times PD\times LGD.

For several exposures, the expected total loss is the sum of their expected losses because expectation is additive. This does not require defaults to be independent. Dependence matters for the distribution of total loss, including its variability and tail, but not for this sum of individual expected losses when the marginal default probabilities and loss amounts are specified.

Important distinction: A scenario-conditioned expected loss uses the default probabilities and recoveries assumed to prevail in that scenario. If a scenario has probability $w$, its loss estimate is not multiplied by $w$ unless the question asks for a probability-weighted estimate across scenarios. A weighted estimate requires scenario probabilities that sum to 1 and a loss estimate for each scenario.

Worked Calculations

Worked Example: Percentage-Point Shocks to Defaults and Recoveries

A hypothetical portfolio has $10 million of BBB exposure and $5 million of BB exposure. The one-year base assumptions are: BBB $PD=1\%$ and recovery $R=40\%$; BB $PD=3\%$ and recovery $R=30\%$. In a recession scenario, BBB default probability rises by 2 percentage points and BB default probability rises by 4 percentage points. Recovery for both ratings falls by 10 percentage points. Calculate base and stressed expected loss, and the increase.

Method. Calculate each segment’s loss as $EAD\times PD\times(1-R)$, first using the base assumptions and then using the scenario assumptions. The exposure and one-year horizon remain unchanged.

Base case. BBB has $LGD=1-0.40=0.60$, so its expected loss is $10{,}000{,}000\times0.01\times0.60=\$60{,}000$. BB has $LGD=1-0.30=0.70$, so its expected loss is $5{,}000{,}000\times0.03\times0.70=\$105{,}000$. Total base expected loss is $\$60{,}000+\$105{,}000=\$165{,}000$.

Stressed case. The BBB default probability is $1\%+2\%=3\%$, and its stressed recovery is $40\%-10\%=30\%$, giving $LGD=70\%$. Its loss is $10{,}000{,}000\times0.03\times0.70=\$210{,}000$. The BB default probability is $3\%+4\%=7\%$, and its stressed recovery is $30\%-10\%=20\%$, giving $LGD=80\%$. Its loss is $5{,}000{,}000\times0.07\times0.80=\$280{,}000$. Total stressed expected loss is $\$210{,}000+\$280{,}000=\$490{,}000$.

The increase is $\$490{,}000-\$165{,}000=\$325{,}000$. The stressed-to-base ratio is $\$490{,}000/\$165{,}000\approx2.970$, to three decimal places. The result is consistent: both default probabilities rise and both recoveries fall, so each segment’s loss increases.

Common mistake avoided. Treating the BBB shock as a relative 2% increase would give $1\%\times1.02=1.02\%$, not the specified 3%. If that error were made for both ratings, the stressed loss would be calculated using the wrong default assumptions. The scenario explicitly says percentage points, so the correct changes are additive.

Worked Example: Rating Migration Under a Stress Scenario

A hypothetical portfolio has $10 million initially rated A and $6 million initially rated BBB. One-year rating assumptions are: A, $PD=0.5\%$ and recovery 50%; BBB, $PD=2\%$ and recovery 40%; BB, $PD=5\%$ and recovery 25%. In the scenario, 20% of the original A exposure migrates to BBB, and 25% of the original BBB exposure migrates to BB. Use the destination rating’s default and recovery assumptions for migrated exposure. Find the base and stressed expected losses.

Method. First calculate the initial loss by rating. For the stress case, reassign exposure according to the stated migration proportions, then use destination-rating assumptions. The migration percentages apply to the original exposure in each source rating.

Base case. A has $LGD=1-0.50=0.50$, giving $10{,}000{,}000\times0.005\times0.50=\$25{,}000$. BBB has $LGD=1-0.40=0.60$, giving $6{,}000{,}000\times0.02\times0.60=\$72{,}000$. Total base expected loss is $\$25{,}000+\$72{,}000=\$97{,}000$.

Stressed exposures. Of the original $10 million A exposure, $0.20\times\$10{,}000{,}000=\$2{,}000{,}000$ migrates to BBB, leaving $8 million rated A. Of the original $6 million BBB exposure, $0.25\times\$6{,}000{,}000=\$1{,}500{,}000$ migrates to BB, leaving $4.5 million rated BBB. Thus stressed exposures are $8 million A, $\$2{,}000{,}000+\$4{,}500{,}000=\$6{,}500{,}000$ BBB, and $1.5 million BB.

Stressed loss. A loss is $8{,}000{,}000\times0.005\times0.50=\$20{,}000$. BBB loss is $6{,}500{,}000\times0.02\times0.60=\$78{,}000$. BB loss is $1{,}500{,}000\times0.05\times0.75=\$56{,}250$. Total stressed expected loss is $\$20{,}000+\$78{,}000+\$56{,}250=\$154{,}250$. The increase is $\$154{,}250-\$97{,}000=\$57{,}250$.

Check and common mistake avoided. The stressed exposures total $\$8{,}000{,}000+\$6{,}500{,}000+\$1{,}500{,}000=\$16{,}000{,}000$, equal to the original portfolio. A common error is to count migrated exposure in both its source and destination rating, or to use the source assumptions after migration. The destination assumptions apply here, and the total exposure is conserved.

Worked Example: Sector-Specific Recession Shocks

A hypothetical lender has $12 million of exposure to sector X and $8 million to sector Y. Base one-year assumptions are: sector X $PD=2\%$, recovery 45%; sector Y $PD=4\%$, recovery 35%. In a recession scenario, sector X default probability doubles and recovery falls by 10 percentage points. Sector Y default probability rises by 50% relative to its base rate and recovery falls by 5 percentage points. Calculate and compare base and stressed expected losses.

Method. Apply each sector’s stated default transformation and recovery-point reduction separately. A doubling and a 50% relative increase are multipliers; the recovery shocks are additive percentage-point reductions.

Base case. Sector X has $LGD=1-0.45=0.55$, so loss is $12{,}000{,}000\times0.02\times0.55=\$132{,}000$. Sector Y has $LGD=1-0.35=0.65$, so loss is $8{,}000{,}000\times0.04\times0.65=\$208{,}000$. Total base expected loss is $\$340{,}000$.

Stressed case. Sector X has stressed $PD=2\%\times2=4\%$, recovery $45\%-10\%=35\%$, and $LGD=65\%$. Its loss is $12{,}000{,}000\times0.04\times0.65=\$312{,}000$. Sector Y has stressed $PD=4\%\times1.50=6\%$, recovery $35\%-5\%=30\%$, and $LGD=70\%$. Its loss is $8{,}000{,}000\times0.06\times0.70=\$336{,}000$. Total stressed expected loss is $\$312{,}000+\$336{,}000=\$648{,}000$.

The increase is $\$648{,}000-\$340{,}000=\$308{,}000$, and the stressed-to-base ratio is $\$648{,}000/\$340{,}000\approx1.906$, to three decimal places. The calculation shows why a sector scenario should preserve sector-specific assumptions rather than apply one portfolio-wide shock without justification.

Common mistake avoided. The 50% relative increase in sector Y’s default probability means $4\%\times1.50=6\%$, not an increase of 50 percentage points. Conversely, the stated recovery shocks are percentage-point reductions, not relative reductions.

Comparing Results and Interpreting Them

A concise comparison should show the base estimate, the stressed estimate, the absolute increase, and—when useful—the stressed-to-base ratio. The absolute increase is often easier to connect to a financial impact; the ratio summarizes the proportional change but can be misleading when the base loss is very small.

Portfolio segment	Base expected loss	Stressed expected loss	Increase
BBB and BB rating portfolio	$165,000	$490,000	$325,000
Rating-migration portfolio	$97,000	$154,250	$57,250
Sector portfolio	$340,000	$648,000	$308,000

A stressed expected loss is not the same as unexpected loss or a high-quantile portfolio loss. The expected-loss calculation uses marginal default probabilities and loss severities. To estimate a portfolio tail loss, the model also needs a joint distribution of defaults or losses; stress testing expected loss alone does not supply that distribution. Likewise, a scenario result should not be called probability-weighted unless scenario likelihoods are explicitly provided and used.

Common Exam Traps

Confusing percentage points with relative changes. From a 2% base rate, a 3-percentage-point increase gives 5%; a 3% relative increase gives 2.06%.
Shocking recovery but not LGD. Since $LGD=1-R$, a fall in recovery increases the loss fraction. A 10-percentage-point recovery reduction increases LGD by 10 percentage points.
Applying a migration twice. Migrated exposure leaves its original rating group and enters the destination group. It must not remain in both.
Using the wrong destination assumptions. If the scenario specifies destination-rating assumptions, migrated balances use those assumptions—not their original rating’s assumptions.
Interpreting conditional loss as a probability-weighted forecast. A scenario weight is applied only when combining scenarios into an overall weighted estimate.
Assuming independence is needed for expected loss. Additivity of expectation does not require independent defaults. Dependence is essential for portfolio loss variability and tail outcomes, not for summing marginal expected losses.

Key takeaway: State the horizon, exposure basis, and meaning of every shock before calculating. Convert default probabilities and recoveries exactly as specified, calculate loss segment by segment, check that any migrated exposures reconcile to the original total, and label the result as conditional or probability-weighted as appropriate.

Check Your Understanding

For each problem, use the stated one-year assumptions and calculate expected loss without discounting. Treat all scenarios as hypothetical.

A $4 million exposure has base $PD=1.5\%$ and recovery 40%. Under stress, default probability rises to 3% and recovery falls to 30%. Find base and stressed expected loss. (Answer: Base $36,000; stressed $84,000.)
A portfolio has $7 million of exposure with $PD=2\%$, recovery 50%, and $4 million with $PD=5\%$, recovery 25%. Stress increases the first default probability by 1 percentage point and the second by 2 percentage points; both recoveries fall by 5 percentage points. Find the total base and stressed expected loss. (Answer: Base $220,000; stressed $339,500.)
A portfolio has $5 million of BBB exposure with $PD=2\%$, recovery 40%, and $3 million of BB exposure with $PD=6\%$, recovery 25%. Ten percent of the BBB exposure migrates to BB, using destination assumptions. Find base and stressed expected loss. (Answer: Base $195,000; stressed $211,500.)
A $9 million sector A exposure has $PD=3\%$, recovery 40%; a $6 million sector B exposure has $PD=2\%$, recovery 50%. A recession multiplies sector A’s default probability by 1.5 and reduces its recovery by 10 percentage points; it doubles sector B’s default probability and reduces its recovery by 5 percentage points. Find base and stressed expected loss. (Answer: Base $222,000; stressed $415,500.)
A hypothetical portfolio has a $200,000 expected loss in the base scenario and a $650,000 expected loss in a recession scenario. The recession scenario has probability 10%, and the base scenario has probability 90%. Find the probability-weighted expected loss across these two scenarios. (Answer: $245,000.)

What You'll Learn