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Define unexpected loss (UL) and explain what it represents.

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

The idea

Unexpected loss is the potential loss beyond what is expected: the amount by which the loss at a chosen high confidence level, such as the 99.9th percentile over one year, exceeds expected loss. It captures the uncertainty around the average, and it is what capital is held against. Expected loss is covered by pricing and provisions. Unexpected loss is covered by equity. Some texts quote UL as the standard deviation of loss, which is a different measure from the high-percentile figure used for capital. For a single exposure with fixed LGD and EAD, treated as default or no default over one period, that standard deviation is UL = EAD × LGD × √(PD × (1 − PD)). It is not a general portfolio UL or regulatory capital formula. Expected losses add across loans, but unexpected losses do not. With standard deviations σ1 and σ2 and correlation ρ, use Combined = √(σ1² + σ2² + 2ρσ1σ2). This is less than σ1 + σ2 unless ρ is one. Diversification reduces UL, never EL.

Formulas

Unexpected loss on one loan as a standard deviation

UL = EAD × LGD × √(PD × (1 − PD))

EAD
exposure at default, fixed
LGD
loss given default, fixed
PD
the one-period probability of default

Treats the loan as default or no default over one period with known LGD and EAD. It is not the high percentile UL used for capital or a portfolio formula.

Combined unexpected loss of two loan groups

UL = √(s1² + s2² + 2 × ρ × s1 × s2)

s1
the standard deviation of loss of the first group
s2
the standard deviation of loss of the second
ρ
the correlation between their losses

Less than s1 + s2 unless ρ is one. Expected losses add exactly; unexpected losses do not.

A practice question

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

Which statement best defines unexpected loss (UL) on a credit portfolio?

  1. The average loss the bank anticipates over the horizon, which is priced into loan spreads and provisions.Why not: Describes expected loss, not unexpected loss.
  2. The potential loss beyond expected loss at a high confidence level, which is what capital is held to absorb. - correct
  3. The fraction of the exposure that is not recovered after a default has occurred.Why not: Describes loss given default, a parameter of a single loan rather than a portfolio loss measure.
  4. The average of all losses that exceed the percentile loss of the loss distribution at the confidence level.Why not: Describes expected shortfall; UL is the percentile loss minus expected loss, not the average beyond it.

UL = loss at the chosen confidence level - EL. It is the part of the loss that is uncertain and is covered by capital, as opposed to EL, which is covered by pricing and provisions.

A harder one

A bank's loan portfolio is split between a corporate division and a retail division. Which statement about adding up risk from the two divisions is correct?

  1. Unexpected losses add exactly, but expected losses add to less than their sum because of diversification.Why not: Reverses the two: diversification reduces UL, never EL.
  2. Both expected and unexpected losses add exactly, regardless of correlation.Why not: Only expectations are linear; standard-deviation-based UL depends on correlation and sub-adds.
  3. Neither adds, so each division's expected loss must be re-estimated on a combined basis.Why not: Expected loss is a mean and is additive without any re-estimation.
  4. Expected losses add exactly; unexpected losses add to less than their sum unless correlation is perfect. - correct

Mean losses are linear, so ELs add. UL involves variance and covariance, so the portfolio UL is less than the sum of the divisional ULs unless correlation is 1.

The hardest tier

A bank lends USD 100 million to a single borrower for one year. PD is 2%, LGD is a known 50%, and exposure is constant. Taking UL as the standard deviation of the loss, EAD x LGD x sqrt(PD x (1 - PD)), the UL is closest to:

  1. USD 7.0 million - correct
  2. USD 1.0 millionWhy not: Gives the expected loss, PD x LGD x EAD, rather than its standard deviation.
  3. USD 14.0 millionWhy not: Omits LGD, so it measures the standard deviation of the exposure at default rather than of the loss.
  4. USD 3.5 millionWhy not: Squares LGD, treating it as part of a variance, although the standard deviation scales with LGD once.

UL = 100 x 0.5 x sqrt(0.02 x 0.98) = 50 x 0.14 = USD 7.0 million, seven times the expected loss of USD 1.0 million. For one loan UL is large relative to EL.

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