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FRM Exam Part I · Measuring Credit Risk

Expected and Unexpected Credit Losses: PD, LGD and EAD

Updated 11 October 2026 · Fact-checked

Expected loss is the average credit loss you predict: EL = PD × LGD × EAD. Unexpected loss is the volatility (standard deviation) of credit loss around that average. You cover expected loss with pricing and provisions, and unexpected loss with capital. Solve questions by finding each input, then applying the formula.

Understand Expected and Unexpected Credit Losses

Credit risk is the chance that a borrower fails to pay what it owes. To measure it you break it into three pieces.

Probability of default (PD) is the chance the borrower defaults over a stated horizon, usually one year. Exposure at default (EAD) is the amount you expect to be owed when default happens. For a term loan it is the outstanding balance. For a credit line it includes the drawn amount plus the share of the undrawn amount you expect the borrower to draw before default. Loss given default (LGD) is the fraction of EAD you lose after recoveries. LGD = 1 − recovery rate.

Multiply them and you get expected loss (EL): EL = PD × LGD × EAD. This is a cost of doing business. Banks price it into loan spreads and hold loss provisions against it. Expected loss is not a risk in itself, because you know it is coming on average.

Unexpected loss (UL) is the risk that actual losses differ from the expected figure. It is measured as the standard deviation of the loss. Banks hold capital against it. For a single loan, with EAD and LGD fixed, the loss is EAD × LGD with probability PD and zero otherwise. This is a Bernoulli-type outcome, so UL = EAD × LGD × √(PD × (1 − PD)).

If LGD is also random, UL grows. For a portfolio, UL depends on correlation. Defaults that move together make portfolio UL larger. Expected losses simply add across loans, but unexpected losses do not add unless correlation is perfect.

Key formulas to remember

Loss given default
LGD = 1 − Recovery rate
Recovery rate is the fraction of exposure recovered after default.
Expected loss
EL = PD × LGD × EAD
Use the same horizon for PD as for the loss you want. Expected losses add across loans.
Unexpected loss, single loan (fixed LGD)
UL = EAD × LGD × √(PD × (1 − PD))
Standard deviation of loss when LGD is known with certainty.
Unexpected loss, single loan (random LGD)
UL = EAD × √(PD × σLGD² + LGD² × PD × (1 − PD))
σLGD is the standard deviation of LGD, assumed independent of default and EAD fixed.
Portfolio unexpected loss, two loans
ULp = √(UL1² + UL2² + 2ρ × UL1 × UL2)
ρ is the correlation between the two loss outcomes. Perfect correlation gives UL1 + UL2.
Exposure on a credit line
EAD = Drawn + CCF × Undrawn
CCF is the credit conversion factor, the expected fraction of the undrawn amount drawn by default.

How to solve Expected and Unexpected Credit Losses questions

Use this order for any question on expected or unexpected credit loss.

  1. 1Identify the horizon and make sure PD is for that horizon.
  2. 2Find EAD. For credit lines, add the drawn amount to CCF times the undrawn amount.
  3. 3Find LGD. If you are given a recovery rate, compute LGD = 1 − recovery.
  4. 4Compute EL = PD × LGD × EAD.
  5. 5If asked for risk or capital, compute UL as a standard deviation. Check whether LGD is fixed or random.
  6. 6For several loans, add the ELs. Combine the ULs with the correlation formula, not by simple addition unless ρ = 1.
  7. 7Sanity check: EL should be smaller than EAD × LGD, and UL for a low-PD loan is usually larger than its EL.

Quickest way: Multiply, then take the square root

When to use it: Use this for single-loan questions with fixed LGD when time is short.

  1. Compute the loss if default occurs: L = EAD × LGD.
  2. EL = PD × L.
  3. UL = L × √(PD × (1 − PD)).
  4. For small PD, √(PD × (1 − PD)) is slightly below √PD, which gives a quick estimate to eliminate options.
  5. With a calculator, key PD × (1 − PD) first, press the square root, then multiply by L.

Common mistakes in Expected and Unexpected Credit Losses

  • Using LGD as the recovery rate.

    Questions often give recovery first, and students plug it straight in.

    Fix: Always convert: LGD = 1 − recovery. Underline which one the question gives.

  • Calling unexpected loss the loss above the expected loss in a bad scenario.

    Confusion with VaR minus EL.

    Fix: In this topic UL is the standard deviation of loss. Credit VaR minus EL is a separate quantity that is sometimes used as economic capital.

  • Using only the drawn balance as EAD on a credit line.

    Students forget borrowers draw more as they approach default.

    Fix: Add CCF × undrawn amount to the drawn amount.

  • Adding unexpected losses across loans.

    Expected losses add, so students assume UL does too.

    Fix: Only EL adds. UL needs the correlation formula. Diversification lowers portfolio UL when ρ < 1.

  • Forgetting the (1 − PD) term in UL.

    Students use √PD or PD alone.

    Fix: Treat default as a Bernoulli event with variance PD × (1 − PD).

  • Mixing horizons, such as a monthly PD with an annual loss.

    Data are quoted in different periods.

    Fix: Align the horizon before multiplying.

Worked examples

Example 1

A bank has a term loan with outstanding balance $5,000,000. The one-year PD is 2%, and the expected recovery rate is 40%. Calculate the expected loss and the unexpected loss, assuming LGD is fixed.

Show the solution
  1. EAD = $5,000,000.
  2. LGD = 1 − 0.40 = 0.60.
  3. Loss if default: 5,000,000 × 0.60 = $3,000,000.
  4. EL = 0.02 × 3,000,000 = $60,000.
  5. PD × (1 − PD) = 0.02 × 0.98 = 0.0196.
  6. √0.0196 = 0.14.
  7. UL = 3,000,000 × 0.14 = $420,000.

Answer: EL = $60,000 and UL = $420,000.

Example 2

A bank has a credit line with limit $10,000,000, of which $4,000,000 is drawn. The CCF on the undrawn part is 50%. PD is 1% and LGD is 45%. Find the EAD and expected loss.

Show the solution
  1. Undrawn = 10,000,000 − 4,000,000 = $6,000,000.
  2. EAD = 4,000,000 + 0.50 × 6,000,000 = $7,000,000.
  3. EL = PD × LGD × EAD = 0.01 × 0.45 × 7,000,000.
  4. 0.01 × 0.45 = 0.0045.
  5. 0.0045 × 7,000,000 = $31,500.

Answer: EAD = $7,000,000 and expected loss = $31,500.

Exam tips

  • Read whether the question gives recovery rate or LGD before doing anything else.
  • Expect at least one question that tests whether you know EL is covered by provisions and pricing, while UL is covered by capital.
  • For two-loan portfolios, check whether the correlation is given. If ρ = 1, simply add the ULs; if ρ = 0, use the square root of the sum of squares.
  • Eliminate options by size: EL is PD × loss given default, so it must be far smaller than EAD for low PD.

Practice questions from Measuring Credit Risk

Expected and Unexpected Credit Losses in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Expected and Unexpected Credit Losses: frequently asked questions

What is the difference between expected loss and unexpected loss?

Expected loss is the average loss you forecast, equal to PD × LGD × EAD. Unexpected loss is the standard deviation of losses around that average. Provisions and pricing cover EL, while capital covers UL.

How do I calculate the unexpected loss of a loan?

With fixed LGD, UL = EAD × LGD × √(PD × (1 − PD)). If LGD is random, add its variance using the extended formula. Always compute the loss given default first.

Is exposure at default the same as the loan balance?

Not always. For a term loan it is usually the outstanding balance. For credit lines it includes the drawn amount plus the expected draw on the undrawn amount, using a credit conversion factor.

Do expected losses add across a portfolio?

Yes, portfolio expected loss is the sum of the individual expected losses. Unexpected losses add only if the loans are perfectly correlated. Otherwise, diversification makes portfolio UL smaller than the sum.