Expected loss from PD, LGD and EAD: the credit risk building blocks
Credit risk is the chance that a borrower does not pay what it owes. Most credit models split it into three questions: how likely is a default, how much is owed when it happens, and how much of that is lost. This guide covers the three inputs, how they combine, and why the average loss is not what capital is held against. It is one of the topics on the FRM guides page; for market risk, see value at risk.
The three building blocks
* Probability of default (PD) is the chance that the borrower defaults over a stated horizon, usually one year. * Exposure at default (EAD) is the amount owed at the moment of default. For a credit line it includes part of the undrawn commitment, because borrowers in trouble tend to draw down. * Loss given default (LGD) is the share of EAD that is lost after recoveries and costs. It is the mirror image of the recovery rate: LGD = 1 - recovery rate.
Expected loss (EL) multiplies them:
EL = PD x LGD x EAD
A 2% PD, 50% LGD and 10 million EAD give 0.02 x 0.5 x 10 = 0.1 million. The formula treats the inputs as independent. In practice recoveries tend to be lower in recessions, when defaults are higher.
Expected versus unexpected loss
Expected loss is the average annual loss. It is a cost of doing business, so lenders normally recover it through pricing (the credit spread) and provisions, not capital.
Actual losses vary from year to year. Unexpected loss (UL) measures that variation, either as a standard deviation or as the gap between a high percentile of the loss distribution and EL. Capital is the buffer that absorbs a bad year without insolvency. For one loan with a fixed LGD, the standard deviation of loss is EAD x LGD x sqrt(PD x (1 - PD)).
Correlation and concentration in plain words
If defaults were independent, a large portfolio would behave predictably, with losses near EL. Real defaults cluster because borrowers share an economy, sector or region. Correlation is that tendency to default together. Higher correlation fattens the tail of the loss distribution, so UL rises while EL does not change.
Concentration means a few large exposures, or many exposures to one industry, dominate the book. It also lifts the chance of a very bad year. Portfolio UL is below the sum of single-loan ULs unless default correlation is perfect, which is why diversification matters in credit as in market risk.
Worked example: a four-loan book
An invented lender, Fenwick Bank, holds four facilities (millions). Undrawn commitments are assumed 50% drawn at default (a credit conversion factor, CCF, of 50%).
```python loans = [ # name, drawn, undrawn, ccf, pd (1y), recovery ("Harbour Foods", 4.0, 2.0, 0.5, 0.010, 0.60), ("Brightwell Textiles", 3.0, 0.0, 0.0, 0.040, 0.40), ("Northgate Logistics", 5.0, 1.0, 0.5, 0.020, 0.50), ("Pennine Retail", 2.0, 3.0, 0.5, 0.080, 0.30), ] el = ead_total = 0 for name, drawn, undrawn, ccf, pd, rec in loans: ead = drawn + ccf * undrawn el += pd * (1 - rec) * ead ead_total += ead assert abs(el - 0.343) < 1e-9 and abs(ead_total - 17.0) < 1e-9 ```
| Loan | EAD | PD | LGD | EL |
|---|---|---|---|---|
| Harbour Foods | 5.0 | 1.0% | 40% | 0.020 |
| Brightwell Textiles | 3.0 | 4.0% | 60% | 0.072 |
| Northgate Logistics | 5.5 | 2.0% | 50% | 0.055 |
| Pennine Retail | 3.5 | 8.0% | 70% | 0.196 |
Expected loss is 0.343 million on 17.0 million of EAD, about 2.02%. Pennine Retail is about a fifth of EAD but over half of EL. Using drawn balances only would give 0.250 million, understating EL by about 27%.
Adding single-loan standard deviations of loss gives 1.60 million if defaults were perfectly correlated; the square root of the sum of squares gives 0.87 million if independent. Real books sit between.
Usual mistakes
* Mixing up horizons. A 10% five-year cumulative PD is not a 10% one-year PD; with a constant hazard it is 1 - 0.91/5, about 2.1% a year. * Confusing recovery and LGD. A 40% recovery rate means a 60% LGD. * Ignoring undrawn commitments in EAD. The example shows the understatement. * Treating EL as the capital requirement. Capital addresses unexpected loss.
How the exam typically tests it
Questions typically give PD, LGD or recovery, and exposure details, and ask for an expected loss, often with a trap in the horizon, the recovery rate or the undrawn amount. Conceptual questions typically ask whether pricing or capital covers EL and UL, or what rising correlation does to portfolio risk. For planning, see how to plan your FRM study. Charta's practice questions for this topic let you rehearse these calculations.
Common questions
What is the formula for expected loss? PD x LGD x EAD, with PD and LGD as decimals. The PD horizon sets the loss horizon.
How is loss given default related to the recovery rate? LGD = 1 - recovery rate. If a lender typically recovers 35% of the exposure after a default, the LGD is 65%.
Why is capital held against unexpected loss and not expected loss? Lenders price and provision for expected loss. Capital protects against the uncertain losses above the average.
Does higher correlation change expected loss? No. EL is a sum of individual averages. Higher correlation makes bad years worse, so UL and capital rise.
Sources
- GARP, the FRM programme: what the exam is and its two parts.
- Basel Committee on Banking Supervision, bis.org/bcbs: the public home of the international banking standards that use PD, LGD and EAD.
The exam and the syllabus can change. Check GARP's current details before you book.