Why Credit Loss Calculations Go Wrong
A credit loss estimate can be arithmetically correct and still be wrong because its inputs do not describe the same event. A default probability for one year cannot be used without adjustment for a multi-year loss horizon; a recovery rate is not the fraction lost; and an exposure already calculated at default must not be added to its components again. A further interpretive error is to treat expected loss as the amount that will necessarily be lost.
The central calculation is simple when its terms are aligned. Let \(EAD\) be exposure at default, \(PD\) the probability of default over the stated horizon, and \(LGD\) the fraction of exposure lost if default occurs. Under assumptions that \(EAD\) and \(LGD\) are fixed for this calculation, the expected credit loss is their product with \(PD\). Before using that expression, specify the horizon, what exposure includes, and how recoveries are represented.
This expression describes an expected loss over the specified horizon, not a present value unless the model explicitly discounts future losses. It assumes the stated EAD and LGD are appropriate for default during that horizon. If exposure or severity varies by default scenario, expected loss must account for that variation rather than multiply by arbitrary averages. The examples below use fixed values unless stated otherwise.
Deriving the Basic Expected Loss Expression
The calculation follows directly from writing the loss as a random variable. Suppose default occurs with probability \(PD\), and a default produces loss \(EAD\times LGD\); otherwise the credit loss is zero. This is a two-outcome model with a Bernoulli default indicator.
The mean of a discrete random variable is the sum of each possible outcome multiplied by its probability. Therefore:
The derivation also explains a key limitation: the product is a mean across possible outcomes. It does not say which outcome will occur for a particular exposure. If default severity varies, the mean loss conditional on default must be used; if severity and default risk vary together, the joint outcomes must be represented consistently.
Four Checks Before Multiplying
Write down the period over which default probability and loss are being measured. Confirm that the PD is for that same period.
If the input is a recovery rate, subtract it from one to obtain LGD. Do not use the recovered fraction as the loss fraction.
Determine whether it already includes drawn balances, converted undrawn commitments, accrued amounts, or other components. Do not count included items a second time.
Expected loss is an average across possible outcomes. Check whether the question asks for an expectation, a possible realized loss, or a measure of variability.
Horizon Mismatch
A PD is meaningful only with its time period. If a problem gives a one-year conditional default probability and asks for the probability of default within several years, the one-year probability generally cannot simply be copied across as the cumulative probability. Under the specific assumption of a constant conditional PD \(p\) in each year, with survival in one year followed by the same conditional probability in the next, the probability of surviving \(n\) years is \((1-p)^n\). Thus cumulative default probability is \(1-(1-p)^n\). This conversion depends on the stated constant conditional probability assumption; it is not a universal rule for every PD input.
Worked Example: Aligning a Three-Year PD
A hypothetical insurer holds a corporate note with EAD of $2,000,000. The conditional probability of default in each year is 1.5%, conditional on surviving to that year. Assume this probability is constant for three years, LGD is fixed at 60%, and a default can occur only once. Estimate expected loss over three years. Ignore discounting, and assume loss is measured without discounting at the time of default.
Method and assumptions. Model annual default as a constant conditional probability \(p=0.015\), so the probability of surviving three years is \((1-p)^3\). EAD and LGD are fixed at $2,000,000 and 0.60, respectively. The three-year expected loss uses the three-year cumulative PD.
Step 1: Calculate survival probability. \(1-0.015=0.985\) is the probability of surviving one year. Then \(0.985^2=0.970225\) and \(0.985^3=0.955671625\).
Step 2: Calculate three-year PD. The cumulative probability of default is \(1-0.955671625=0.044328375\), or approximately 4.4328%.
Step 3: Calculate expected loss. The loss if default occurs is \(2{,}000{,}000\times0.60=\$1{,}200{,}000\). Expected loss is \(0.044328375\times\$1{,}200{,}000=\$53{,}194.05\). Reported to the cent, the three-year expected loss is $53,194.05.
Check and mistake avoided. The result is less than the $1,200,000 loss in the default outcome, as an expectation should be here. Using the one-year PD directly would give \(0.015\times\$1{,}200{,}000=\$18{,}000\), which understates the three-year estimate because it ignores defaults in years two and three.
The reverse mistake is also possible: a cumulative PD for the entire horizon might be treated as an annual probability and compounded again. Always identify whether the supplied probability is annual, conditional on survival, or already cumulative for the measurement period before applying any conversion.
Recovery Rate Is Not Loss Given Default
Recovery and loss describe opposite portions of EAD. If 35% of EAD is recovered, the loss fraction is 65%, not 35%. This assumes recovery is defined as a fraction of the same EAD used in the loss calculation. If a source instead reports a recovery amount in dollars or defines recovery against a different base, convert it to the appropriate fraction before using the formula.
Worked Example: Converting a Recovery Rate to LGD
A hypothetical insurer has a $800,000 credit exposure to a service provider. The one-year PD is 2.5%, and the estimated recovery rate following default is 35% of EAD. Assume EAD and recovery rate are fixed, and measure undiscounted loss over one year. Calculate expected loss.
Method and assumptions. Use the one-year PD with one-year EAD. Convert the recovery rate to LGD before applying the expected loss formula. The default loss is modeled as a fixed fraction of EAD.
Step 1: Calculate LGD. \(LGD=1-0.35=0.65\), or 65%.
Step 2: Calculate loss if default occurs. \(\$800{,}000\times0.65=\$520{,}000\).
Step 3: Weight by PD. \(0.025\times\$520{,}000=\$13{,}000\). Expected one-year loss is $13,000.00.
Check and mistake avoided. The recovery amount in the default outcome is \(\$800{,}000\times0.35=\$280{,}000\); recovery plus loss is \(\$280{,}000+\$520{,}000=\$800{,}000\). The incorrect calculation \(0.025\times\$800{,}000\times0.35=\$7{,}000\) weights the recovered portion, not the lost portion, and understates expected loss.
Count Exposure Once
EAD can be assembled from multiple items, but an assembled total is still one exposure amount. For a hypothetical revolving facility, a simple exposure estimate might add the drawn balance to a credit-conversion factor times the undrawn commitment. If that EAD has already been calculated, adding the drawn balance or undrawn commitment again duplicates exposure. The conversion factor and the items it applies to must be specified; a model should not silently replace a converted amount with the full commitment.
Worked Example: Avoiding Double-Counting a Facility
A hypothetical insurer has a $600,000 drawn balance and a $400,000 undrawn commitment to one borrower. For this calculation, use a 50% credit-conversion factor on the undrawn amount. The one-year PD is 3%, and LGD is 40%. Assume these amounts and rates are fixed, and calculate one-year undiscounted expected loss.
Method and assumptions. First construct EAD as the drawn balance plus the converted undrawn commitment. Then use that total exactly once in the expected loss formula.
Step 1: Convert the undrawn amount. \(0.50\times\$400{,}000=\$200{,}000\).
Step 2: Calculate EAD. \(\$600{,}000+\$200{,}000=\$800{,}000\).
Step 3: Calculate the loss if default occurs. \(\$800{,}000\times0.40=\$320{,}000\).
Step 4: Calculate expected loss. \(0.03\times\$320{,}000=\$9{,}600\). Expected one-year loss is $9,600.00.
Check and mistake avoided. EAD of $800,000 is $600,000 drawn plus $200,000 of converted undrawn exposure. Adding the original $600,000 drawn amount to the $800,000 EAD would incorrectly produce $1,400,000 of exposure and expected loss of \(0.03\times0.40\times\$1{,}400{,}000=\$16{,}800\). The drawn amount was already included.
Expected Loss Is Not the Total Loss
In the fixed-severity, one-period model, the realized credit loss has two possible outcomes: zero if there is no default, or \(EAD\times LGD\) if default occurs. Expected loss is the probability-weighted average of those outcomes. It can be much smaller than the loss conditional on default and is not a promise that the actual loss will equal the calculated amount. Nor is it a maximum: the loss in the default outcome can exceed expected loss.
Worked Example: Separating the Mean From Possible Outcomes
A hypothetical insurer has a $1,500,000 exposure with a one-year PD of 4% and fixed LGD of 50%. Assume a one-period Bernoulli default model: no default produces zero loss, and default produces the stated fixed severity. Ignore discounting. Calculate expected loss, identify the possible losses, and calculate the standard deviation.
Method and assumptions. The loss \(X\) is zero with probability 0.96 and equals \(EAD\times LGD\) with probability 0.04. Thus it is a two-outcome loss distribution, not a fixed $30,000 loss.
Step 1: Find loss conditional on default. \(\$1{,}500{,}000\times0.50=\$750{,}000\).
Step 2: Find expected loss. \(0.04\times\$750{,}000+0.96\times\$0=\$30{,}000\).
Step 3: State possible realized losses. The realized loss is $0 with probability 96% or $750,000 with probability 4%. It is not necessarily $30,000.
Step 4: Find standard deviation. For a Bernoulli loss that equals severity \(L\) with probability \(p\) and zero otherwise, variance is \(p(1-p)L^2\). Here, \(0.04\times0.96\times(\$750{,}000)^2=21{,}600{,}000{,}000\) dollars squared. The standard deviation is \(\sqrt{21{,}600{,}000{,}000}=\$146{,}969.38\), rounded to the nearest cent.
Check and mistake avoided. The mean is between the two possible outcomes, as it must be, and the variance is in squared dollars before taking its square root. Calling $30,000 the total loss confuses an average over outcomes with a realized outcome.
Common Error Checklist
| Error | Why it fails | Correction |
|---|---|---|
| Using a one-year PD for a multi-year loss | The probability and loss horizons differ. | Use a cumulative PD for the horizon or derive one under an explicit model. |
| Multiplying by recovery rate | Recovery is the fraction regained, not the fraction lost. | Calculate \(LGD=1-R\), using consistent definitions. |
| Adding components to an EAD total | Some or all components may already be included. | Write the EAD construction and use the resulting total once. |
| Treating expected loss as the outcome | An expectation averages over possible outcomes. | Separate the mean from default severity and loss variability. |
Check Your Understanding
For each problem, state the relevant horizon and assumptions before calculating.
- A hypothetical $450,000 exposure has a one-year PD of 1.2% and a recovery rate of 25% of EAD. Assume fixed EAD and recovery, and no discounting. What is one-year expected loss? (Answer: $4,050.00)
- A hypothetical $900,000 loan has a 3% conditional PD in each year, conditional on survival, for two years. Assume the conditional PD is constant, LGD is fixed at 40%, and loss is undiscounted. What is two-year expected loss? (Answer: $21,276.00)
- A hypothetical facility has $500,000 drawn and $300,000 undrawn. Use a 40% conversion factor on the undrawn amount, a one-year PD of 2.5%, and LGD of 40%. What is EAD and expected loss? (Answer: EAD is $620,000; expected loss is $6,200.00)
- A hypothetical $2,000,000 exposure has a one-year PD of 1.5% and a fixed 50% recovery rate. In a one-period Bernoulli model, what are the possible losses and the expected loss? (Answer: Loss is $0 with probability 98.5% or $1,000,000 with probability 1.5%; expected loss is $15,000.00.)
- A hypothetical portfolio exposure of $3,000,000 has a 5-year cumulative PD of 8% and fixed LGD of 55%. Assume the supplied PD is already cumulative over the full five years and use undiscounted loss. What is five-year expected loss, and should the PD be compounded again? (Answer: $132,000.00; no, the PD is already cumulative.)