What is the expected loss formula?
Expected loss is probability of default multiplied by loss given default multiplied by exposure at default: EL = PD x LGD x EAD. PD is the chance the borrower defaults within a set period, LGD is the share of the exposure lost if it does, and EAD is the amount owed at the time of default.
The result is a dollar amount: the average loss a lender should expect on that exposure over the period, usually one year. It is not a prediction of what will happen to any single loan. A loan either defaults or it does not. Expected loss is the average across many loans like it, which is why it is useful for pricing, reserving and comparing risk across a portfolio.
What do PD, LGD and EAD each mean?
Each input answers a different question, and each is estimated from different data. Keeping them separate is the point of the formula: a loan to a strong borrower with weak collateral and a loan to a weaker borrower with strong collateral can carry similar expected loss for very different reasons, and the response to each is different.
- Probability of default (PD): how likely is the borrower to default within the period? Usually expressed as a percentage over one year, and driven by the borrower's risk grade or score.
- Loss given default (LGD): if the borrower defaults, what share of the exposure will not be recovered? Equal to 1 minus the recovery rate, after collection costs and time.
- Exposure at default (EAD): how much will be owed when default happens? For a term loan this is close to the balance. For a line of credit it includes some of the unused commitment, because borrowers tend to draw down as they weaken.
- Expected loss (EL): the product of the three, in dollars, or as a percentage of exposure when EAD is left out (PD x LGD).
How do you calculate expected loss on a term loan?
Example only, with made-up round numbers. A borrower has a $2,000,000 term loan. Its risk grade maps to a one-year PD of 2.0%. The loan is secured by equipment and real estate, and the lender's LGD estimate for this collateral type is 40%. Because the loan is fully drawn and amortizing, EAD for the next year is taken as the current balance of $2,000,000.
Expected loss is 0.02 x 0.40 x $2,000,000 = $16,000. Put another way, the loan is expected to lose 0.8% of its balance per year on average (2.0% x 40%). If the loan earns a credit spread of 2.5%, the expected loss uses up about a third of that spread before any operating cost or capital charge.
- 1Find the one-year PD for the borrower's risk grade or score.
- 2Estimate LGD from the collateral, seniority and the lender's recovery history.
- 3Estimate EAD: the balance for a fully drawn term loan, or balance plus expected further draws for a line.
- 4Multiply the three: EL = PD x LGD x EAD.
- 5Express it as a percentage of exposure (PD x LGD) to compare with pricing and with other loans.
How is EAD calculated for a line of credit?
For a revolving line, the borrower can draw more before defaulting, and borrowers under stress usually do. EAD is therefore estimated as the drawn balance plus a share of the undrawn commitment. That share is called the credit conversion factor (CCF) or loan equivalent factor, and lenders estimate it from how much of the unused line past defaulters drew in the period before default.
Example only. A borrower has a $1,000,000 line with $600,000 drawn, leaving $400,000 unused. If the lender's CCF for this product is 50%, EAD is $600,000 + (0.50 x $400,000) = $800,000. With a PD of 2.0% and an LGD of 40%, expected loss is 0.02 x 0.40 x $800,000 = $6,400. Using only the drawn balance would have understated it at $4,800.
| Input | Term loan | Line of credit |
|---|---|---|
| Commitment | $2,000,000 | $1,000,000 |
| Drawn balance | $2,000,000 | $600,000 |
| Credit conversion factor | Not applicable | 50% of unused |
| EAD | $2,000,000 | $800,000 |
| PD (one year) | 2.0% | 2.0% |
| LGD | 40% | 40% |
| Expected loss | $16,000 | $6,400 |
How is probability of default calculated?
Most lenders do not calculate PD loan by loan from scratch. They assign each borrower a risk grade or score, then attach a PD to each grade based on how loans in that grade have actually performed. The simplest method is the historical default rate by grade: for each year, count the loans in a grade at the start of the year and how many of them defaulted within twelve months.
Example only. Over five years, a lender had an average of 400 loans per year in grade 5, and 30 of them defaulted within a year of being in that grade, or 6 per year on average. The one-year default rate for grade 5 is 6 / 400 = 1.5%. Pooling several years smooths out one good or bad year, and the lender should check that PD rises steadily from the best grade to the worst. If it does not, the grades are not separating risk well.
| Risk grade | Average loans per year | Average defaults per year | One-year default rate |
|---|---|---|---|
| 3 (satisfactory) | 900 | 2.7 | 0.3% |
| 4 (acceptable) | 700 | 4.9 | 0.7% |
| 5 (watch) | 400 | 6.0 | 1.5% |
| 6 (special mention) | 120 | 6.0 | 5.0% |
| 7 (substandard) | 50 | 9.0 | 18.0% |
What other methods do lenders use to estimate PD?
Historical rates by grade work when the lender has enough loans and enough defaults. Where data is thin, such as among the best grades, which rarely default, lenders combine several approaches and use judgment to keep the results sensible.
- Scorecards: a statistical model, often a logistic regression, that turns borrower data such as leverage, coverage, liquidity, size and payment behavior into a score, then maps the score to a PD. Common for small business and retail lending, where volumes are large.
- Migration matrices: a table showing the share of loans that moved from each grade to every other grade, including default, over a year. Multiplying the one-year matrix by itself gives multi-year default probabilities, which helps with longer-dated loans.
- External benchmarks: published default studies by rating agencies or pooled industry data, mapped to the lender's own grades. Useful as a sanity check when internal defaults are few.
- Expert adjustment: documented overlays for factors the data does not capture, such as a new product or an industry under stress.
What is the difference between through-the-cycle and point-in-time PD?
A through-the-cycle (TTC) PD is a long-run average that stays fairly stable as the economy moves. It answers: across good years and bad, how often do borrowers in this grade default? Lenders often use TTC PDs for risk grading, limits and capital planning, because they do not swing with every quarter.
A point-in-time (PIT) PD reflects current conditions. It rises in a downturn and falls in a recovery, even if a borrower's grade does not change. PIT estimates are better for near-term loss forecasting and reserving. Many lenders keep a TTC PD by grade and adjust it to a PIT view using economic indicators such as unemployment, GDP growth or sector-specific data. Whichever view is used, label it, so nobody mixes a long-run average with a current forecast.
How do you estimate loss given default?
LGD is 1 minus the recovery rate, where recovery is everything collected after default (collateral sales, guarantor payments, ongoing payments) minus the cost of collecting it, ideally discounted for the time it took. Lenders estimate it from their own workout history, grouped by collateral type and lien position.
Example only. A defaulted loan had an EAD of $800,000. Over two years the lender collected $560,000 from selling equipment and from the guarantor, and spent $80,000 on legal and liquidation costs. Net recovery is $480,000, so the recovery rate is 60% and LGD is 40%. Discounting the recoveries back to the default date would raise LGD somewhat, which is why slow workouts cost more than the headline recovery suggests.
How does expected loss relate to CECL?
Under CECL, the current expected credit loss standard in US GAAP (ASC 326), lenders reserve for expected losses over the full contractual life of a loan, not just the next year. PD x LGD x EAD is one accepted way to do that, though CECL does not require any particular method, and many lenders use loss rate, vintage or discounted cash flow approaches instead.
Example only. Using the PD, LGD and EAD approach for the three-year term loan above, apply the annual PD to the balance expected at the start of each year, adjusting for the chance the loan has already defaulted. With a 2.0% annual PD and a 40% LGD, the lifetime expected loss is about $31,960, compared with $16,000 for one year. A full CECL estimate also needs a reasonable and supportable forecast period, reversion to history and qualitative adjustments, which are beyond this guide.
| Year | Balance at start (EAD) | Chance of default in that year | LGD | Expected loss |
|---|---|---|---|---|
| 1 | $2,000,000 | 2.00% | 40% | $16,000 |
| 2 | $1,350,000 | 1.96% | 40% | $10,584 |
| 3 | $700,000 | 1.92% | 40% | $5,378 |
| Lifetime | $31,962 |
How Aarvion helps
Aarvion Risk OS Commercial Credit keeps the inputs to expected loss tied to the credit file. Risk grades are proposed from the confirmed spread and the bank's own rating definitions, a named reviewer assigns the final grade, and the grade is marked stale when the facts behind it change, so the PD it maps to stays current.
Exposure, collateral and payment data come from the bank's core system, and every grade change, override and approval is recorded with who made it and which rule applied. That gives the team preparing loss estimates a clean, traceable set of grades and exposures to work from.
