Deriving Expected Credit Loss

Why Expected Credit Loss Is a Product

A credit exposure produces a credit loss only if a defined default event occurs. If default does occur, the loss depends on the amount outstanding at that time and on the portion that is not recovered. Expected credit loss combines these ideas into a single expected dollar amount for a stated exposure and time horizon.

The familiar formula is useful, but it is not automatically valid with any three numbers labelled PD, EAD, and LGD. The probability must refer to the same default definition and horizon as the exposure and loss severity. Further, if exposure and loss severity vary by default scenario, their dependence can affect the answer.

Definition: For a specified horizon, probability of default (PD) is the probability that the borrower defaults during that horizon. Exposure at default (EAD) is the amount exposed when default occurs. Loss given default (LGD) is the fraction of EAD lost after recoveries, on the stated loss-measurement basis. Expected credit loss is the expected dollar loss from the exposure over that horizon.

When stated as a fraction, LGD ranges from 0 to 1: an LGD of 0.40 means that 40% of EAD is lost, not that 40% is recovered. If recovery is expressed as a fraction of EAD and there are no other loss adjustments, LGD equals one minus the recovery fraction.

Deriving the Expected Loss Formula

Let $D_h$ denote the event that default occurs during horizon $h$, and let $I_{D_h}$ equal 1 when $D_h$ occurs and 0 otherwise. Let $EAD$ and $LGD$ describe the exposure and severity associated with default during that horizon. For an undiscounted loss measure with no loss in the absence of default, the loss is $L=I_{D_h}EAD\times LGD$.

Write loss by outcome.
If default does not occur, the indicator is zero and this exposure produces no default loss. If default occurs, the loss is EAD multiplied by LGD.

Take the expectation.
By the definition of conditional expectation, the expected loss is the probability of default multiplied by the average loss conditional on default.

Apply the product assumptions.
If EAD and LGD are fixed given default, or are conditionally independent given default, the conditional average of their product is the product of their conditional averages.

\begin{aligned} E[L] &= E[I_{D_h}EAD\times LGD] \\ &= P(D_h)\,E[EAD\times LGD\mid D_h]. \end{aligned}

The first equality follows because $L=I_{D_h}EAD\times LGD$. For the second, the loss is zero outside $D_h$; on $D_h$, it is $EAD\times LGD$. Thus the expected loss is the probability of that event times the conditional expected loss given that it occurred.

E[L]=PD\times EAD\times LGD

The last formula is exact when EAD and LGD are fixed for default, or when the EAD and LGD values used are their conditional means and the two quantities are independent given default. More generally, it is exact if the EAD and LGD factors are defined so their product equals $E[EAD\times LGD\mid D_h]$. If both are variable and dependent, multiplying their separate conditional means is not generally correct.

A useful way to report the exact result when EAD varies is to use $E[EAD\mid D_h]$ as the average exposure and define an exposure-weighted LGD as $E[EAD\times LGD\mid D_h]/E[EAD\mid D_h]$, provided the average exposure is positive. Their product is the conditional expected loss. This definition makes clear why an unweighted average LGD may not be suitable when larger exposures tend to have different loss severities.

Worked Examples

Worked Example: Fixed Exposure and Recovery

A hypothetical lender has a one-year loan exposure of $250,000. The one-year probability of default is 2.4%. If default occurs, the lender expects to recover 55% of EAD, with no additional costs or discounting. Calculate the one-year expected credit loss.

Method. Convert the recovery fraction to LGD, then multiply the one-year PD, EAD, and LGD. The exposure and recovery fraction are fixed for this calculation.

Step 1: Find LGD. The recovery fraction is 0.55, so $LGD=1-0.55=0.45$. The conditional loss if default occurs is $250{,}000\times0.45=\$112{,}500$.

Step 2: Convert PD to a probability. $PD=2.4\%=0.024$.

Step 3: Weight the conditional loss by PD.

E[L]=0.024\times \$112{,}500=\$2{,}700.00

Check. The expected recovery conditional on default is $0.55\times\$250{,}000=\$137{,}500$. The conditional loss is $\$250{,}000-\$137{,}500=\$112{,}500$, consistent with the LGD calculation. Multiplying that loss by the 0.024 default probability gives $2,700.00. The common mistake is to multiply by the 55% recovery rate instead of the 45% LGD, which would give $0.024\times\$250{,}000\times0.55=\$3{,}300.00$, an expected recovery-weighted amount rather than expected loss.

Worked Example: EAD Varies at Default

A hypothetical lender estimates a one-year PD of 1.5% for a borrower with a revolving credit facility. Conditional on default, the balance is expected to be $80,000 with probability 0.60 and $140,000 with probability 0.40. LGD is fixed at 35% in either case. Find expected credit loss for the year. Ignore discounting and costs.

Method. First calculate average EAD conditional on default. Since LGD is fixed, multiply that average exposure by LGD and then weight by PD.

Step 1: Calculate conditional average EAD.

E[EAD\mid D_h] =(0.60\times\$80{,}000)+(0.40\times\$140{,}000) =\$48{,}000+\$56{,}000 =\$104{,}000.

Step 2: Calculate average loss conditional on default. With fixed LGD of 0.35, this is $\$104{,}000\times0.35=\$36{,}400$.

Step 3: Apply the one-year PD. $0.015\times\$36{,}400=\$546.00$.

Check. In the $80,000 balance scenario, loss on default is $\$80{,}000\times0.35=\$28{,}000$, contributing $0.60\times\$28{,}000=\$16{,}800$ to the average conditional loss. In the $140,000 scenario, loss is $\$140{,}000\times0.35=\$49{,}000$, contributing $0.40\times\$49{,}000=\$19{,}600$. Their sum is $36,400, and $0.015\times\$36{,}400=\$546.00$, as above. A common error is to use the facility limit instead of the stated balances at default; EAD is the amount exposed at default, not automatically the full commitment.

Worked Example: Dependent EAD and LGD

For a hypothetical one-year exposure, PD is 3%. Conditional on default, there are two possible outcomes: with probability 0.70, EAD is $100,000 and LGD is 20%; with probability 0.30, EAD is $200,000 and LGD is 60%. Find expected credit loss. Assume there is no loss outside default and no discounting.

Method. Calculate the loss in each default scenario, average those losses using their conditional probabilities, and then multiply by PD. Do not multiply separate average EAD and average LGD unless the conditions for doing so hold.

Step 1: Calculate each loss given default. In the first scenario, $\$100{,}000\times0.20=\$20{,}000$. In the second, $\$200{,}000\times0.60=\$120{,}000$.

Step 2: Average the scenario losses conditional on default.

E[EAD\times LGD\mid D_h] =(0.70\times\$20{,}000)+(0.30\times\$120{,}000) =\$14{,}000+\$36{,}000 =\$50{,}000.

Step 3: Multiply by PD. $0.03\times\$50{,}000=\$1{,}500.00$.

Check the dependence issue. Average EAD conditional on default is $0.70\times\$100{,}000+0.30\times\$200{,}000=\$130{,}000$. Average LGD is $0.70\times0.20+0.30\times0.60=0.32$. Their product is $\$130{,}000\times0.32=\$41{,}600$, not the correct conditional loss of $50,000. Here larger EAD occurs in the scenario with larger LGD, so multiplying the separate averages understates expected loss. The correct expected credit loss is $1,500.00.

Conditions, Horizon, and Interpretation

The product formula is a compact calculation of an expectation, not a rule that removes the need to define the inputs. Before using it, check that:

The default event is specified. PD must correspond to the relevant definition of default and to the exposure being measured.
The horizon matches. A one-year PD should not be combined without adjustment with an exposure or LGD estimate intended for a different horizon. If the horizon changes, default probability and the conditions of exposure and recovery may also change.
EAD is measured at default. For facilities with changing balances, EAD may depend on when default occurs. A representative EAD must reflect the default scenario, rather than simply the current balance or facility limit.
LGD reflects loss after recovery. If LGD is already net of recoveries, do not subtract recovery a second time. If LGD is stated as a percentage, use it as a fraction in the calculation.
The loss basis is clear. The formulas above calculate undiscounted loss and assume no loss from this exposure in the absence of default. A present-value measure requires the timing and discounting of cash flows to be included consistently.

Expected credit loss is an average across possible outcomes. It is not a forecast that the specified dollar amount will be lost in the period, nor does it describe the range of possible losses. Its value is that it provides a consistent expected-loss measure when the event, horizon, and conditional loss assumptions are aligned. The same reasoning is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Key takeaway: The general identity is $E[L]=PD\times E[EAD\times LGD\mid D_h]$. The familiar $PD\times EAD\times LGD$ form follows when the conditional product can be represented by the stated EAD and LGD factors. When EAD and LGD vary together, average scenario losses conditional on default before applying PD.

Check Your Understanding

For each problem, use the stated horizon and assumptions. Treat all losses as undiscounted unless the problem says otherwise.

A hypothetical one-year exposure has PD 0.8%, fixed EAD of $500,000, and LGD of 20%. Calculate one-year expected credit loss. (Answer: $800.00)
A one-year exposure has PD 2.5%. Conditional on default, EAD is $60,000 with probability 0.75 and $180,000 with probability 0.25. LGD is fixed at 40%. Calculate expected credit loss. (Answer: $900.00)
A hypothetical one-year exposure has PD 4%. Conditional on default, there is a 50% chance of EAD $120,000 and LGD 25%, and a 50% chance of EAD $80,000 and LGD 50%. Calculate expected credit loss using the scenario losses. (Answer: $1,400.00)
A one-year exposure has fixed EAD of $140,000 and PD 1.8%. What LGD produces expected credit loss of $1,260? State the implied recovery fraction if recoveries are the only adjustment. (Answer: LGD = 50%; recovery fraction = 50%)
A hypothetical exposure has an annual default probability of 2% in each year conditional on surviving to that year. Assume the annual conditional probability is constant, EAD remains $300,000, and LGD remains 40%. Ignoring discounting and assuming loss occurs at most once, calculate expected loss over three years using cumulative default probability. (Answer: cumulative PD = 0.058808; expected loss = $7,056.96)

What You'll Learn