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Define expected loss (EL) and calculate it.

FRM® Part I · Valuation and Risk Models · Measuring Credit Risk · FRM1.VR.6.c

The idea

Expected loss is the average credit loss a bank anticipates over a horizon, usually one year. For a single exposure it equals the probability of default times the loss given default times the exposure at default. Expected loss = PD × LGD × EAD. The loss given default is one minus the recovery rate. For a partly drawn facility, exposure at default is the drawn amount plus a conversion factor times the undrawn commitment. Expected losses of different loans simply add, whatever the correlation between them, because expectations are linear. Expected loss is a cost of doing business, recovered through loan spreads and covered by provisions rather than capital. The credit spread is roughly PD × LGD, so PD is about the spread ÷ (1 − recovery). Over several years use the cumulative default probability, not summed annual ones. A common mistake is using the recovery rate in place of LGD.

Formulas

Expected credit loss on one exposure

EL = PD × LGD × EAD

PD
the probability of default over the horizon
LGD
loss given default, as a share of the exposure
EAD
the exposure at default

Over several years use the cumulative PD, not the sum of annual ones. The commonest slip is putting the recovery rate where LGD belongs.

Exposure at default on a partly drawn facility

EAD = drawn + CCF × undrawn

drawn
the amount already drawn
CCF
the conversion factor, the share of the undrawn amount expected to be drawn by default
undrawn
the undrawn commitment

Borrowers draw down as they weaken, so ignoring the undrawn line understates exposure.

A practice question

From the question bank, with the reason each wrong option is wrong.

A bank lends USD 8 million to a firm with a one-year default probability of 2%. If default occurs the bank expects to recover 35% of the exposure. The expected loss is closest to:

  1. USD 104,000 - correct
  2. USD 160,000Why not: Ignores recovery and treats the whole exposure as lost on default.
  3. USD 56,000Why not: Uses the recovery rate of 35% as if it were the loss given default.
  4. USD 5,200,000Why not: Multiplies exposure by LGD only, which is the loss if default occurs, and leaves out the 2% probability.

EL = 8,000,000 x 0.02 x (1 - 0.35) = USD 104,000. LGD is 65% because only 35% is recovered.

A harder one

A bank has a USD 20 million committed credit line of which USD 12 million is drawn. It estimates that 50% of the undrawn amount will be drawn at the time of default. The borrower's one-year PD is 1.5% and the LGD is 45%. The expected loss is closest to:

  1. USD 0.081 millionWhy not: Uses the drawn balance only and ignores the portion of the undrawn commitment likely to be drawn by default.
  2. USD 0.135 millionWhy not: Treats the full commitment as exposed, although only part of the undrawn amount is expected to be drawn.
  3. USD 0.0675 millionWhy not: Applies the conversion factor to the whole commitment and ignores the drawn balance.
  4. USD 0.108 million - correct

EAD = 12 + 0.5 x 8 = USD 16 million. EL = 16 x 0.015 x 0.45 = USD 0.108 million.

The hardest tier

A one-year loan is priced 240 basis points over the risk-free rate. Assume the whole spread compensates for expected credit loss, and that recovery on default is 40%. The implied one-year default probability is closest to:

  1. 4.0% - correct
  2. 2.4%Why not: Equates the spread with the PD and ignores recovery; the spread is PD x LGD.
  3. 6.0%Why not: Divides by the recovery rate instead of by LGD (1 - recovery).
  4. 1.44%Why not: Multiplies by LGD instead of dividing: the spread equals PD x LGD, so PD = spread / LGD.

Spread = PD x LGD with LGD = 1 - 0.40 = 0.60, so PD = 2.4% / 0.60 = 4.0%.

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