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Tell expected loss and unexpected loss apart, illustrating both with examples.

FRM® Part I · Foundations of Risk Management · The Building Blocks of Risk Management · FRM1.FR.1.c

The idea

Expected loss (EL) is the average loss an institution anticipates over a horizon. It is a cost of doing business, so it is priced into spreads and covered by provisions. For credit, the formula is EL = PD × LGD × EAD. Here PD is the probability of default, LGD the loss given default and EAD the exposure at default. Unexpected loss (UL) is the variability of loss around that average. It is often measured as the standard deviation of losses, or as the loss at a high confidence level minus EL. It cannot be priced away, so capital is held against it. EL is roughly additive across positions, while UL is not, because it depends on correlation. A card issuer writing off 4% of balances in a normal year has an expected loss of 4%. The extra losses in a severe recession are unexpected. Do not call the whole bad-year loss unexpected.

Formulas

Expected credit loss

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

The average loss, priced into spreads and covered by provisions. It is not the bad-year loss: capital covers the variability around it, the unexpected loss.

Capital for unexpected loss

unexpected loss = tail loss − EL

tail loss
the loss at the chosen high confidence level
EL
the expected loss

Capital covers only the excess over the mean, because expected loss is already funded by margins and provisions. Holding capital against the whole tail loss double counts EL.

A practice question

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

Expected loss is best described as:

  1. the loss that would be exceeded only in the worst 1% of outcomes over the horizon.Why not: Describes a tail quantile of the loss distribution, as in VaR, not the average loss.
  2. the standard deviation of losses around their average, which economic capital is held to cover in bad years.Why not: That is unexpected loss; expected loss is the mean, covered by pricing and provisions.
  3. the average loss anticipated over a horizon, treated as a cost of doing business and provisioned for. - correct
  4. the largest single loss the institution has ever recorded in its history.Why not: A historical maximum is not a forecast average; expected loss is a mean over the coming horizon.

Expected loss is the mean of the loss distribution. Because it is anticipated, it is built into pricing and provisioning, whereas capital addresses the variability around it.

A harder one

A bank has a USD 5 million exposure to a corporate borrower with a one-year probability of default of 2% and a loss given default of 45%. The expected loss is closest to:

  1. USD 45,000. - correct
  2. USD 100,000.Why not: Multiplies PD by exposure and omits the loss given default.
  3. USD 55,000.Why not: Uses the recovery rate (1 - LGD = 55%) where LGD of 45% is required.
  4. USD 2,250,000.Why not: Multiplies LGD by exposure and omits the probability of default.

EL = PD x LGD x EAD = 0.02 x 0.45 x 5,000,000 = USD 45,000. It is the average loss anticipated for this exposure over the year.

The hardest tier

A bank lends USD 10 million to a single borrower with a one-year default probability of 4% and a loss given default of 60%. Treating the loss as 6 million if default occurs and zero otherwise, the standard deviation of the loss (its unexpected loss) is closest to:

  1. USD 1.18 million. - correct
  2. USD 1.38 million.Why not: Reports the variance (in millions squared) instead of taking its square root.
  3. USD 1.96 million.Why not: Uses the full exposure of 10 million and ignores the loss given default.
  4. USD 0.24 million.Why not: Reports the expected loss (0.04 x 6) rather than the standard deviation of the loss.

The loss is 6 million with probability 0.04, so the standard deviation is 6 x sqrt(0.04 x 0.96) = 6 x 0.196 = about 1.18 million. Expected loss is only 0.24 million, so unexpected loss is far larger for a single name.

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