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State what unexpected loss (UL) is and compute it.

FRM® Part II · Credit Risk Measurement and Management · Capital Structure in Banks · FRM2.CR.4.d

The idea

Unexpected loss (UL) is the standard deviation of credit losses around the expected loss, the measure of how far actual losses may stray from the average. Take the case where EAD and LGD are fixed and only default is random. Then UL = EAD × LGD × √(PD × (1 − PD)). Now let the loss rate be random too, with standard deviation σLGD. The result is UL = EAD × √(PD × σLGD² + LGD² × PD × (1 − PD)). Both terms matter. One covers uncertainty about whether default occurs, the other uncertainty about how much is lost. Typical errors are leaving out the square root, leaving out the (1 − PD) factor, using the recovery rate for LGD, and forgetting that the LGD variance term adds inside the root, not outside it.

Formulas

Unexpected loss with a fixed loss rate

UL = EAD × LGD × √(PD × (1 − PD))

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

Binomial default with LGD and EAD fixed. Forgetting the square root or the (1 − PD) factor are the usual errors.

Unexpected loss with a random loss rate

UL = EAD × √(PD × σLGD² + LGD² × PD × (1 − PD))

EAD
the exposure at default
PD
the probability of default
σLGD
the standard deviation of the loss rate
LGD
the mean loss rate

Default and loss rate independent. The loss-rate variance term goes inside the root; with σLGD zero it reduces to the fixed-LGD formula.

A practice question

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

Which description of unexpected loss (UL) for a credit portfolio is correct?

  1. The standard deviation of credit losses around the expected loss, a measure of how much actual loss may vary from average. - correct
  2. The average credit loss the bank expects to incur over the horizon.Why not: Describes expected loss, the mean, not the dispersion.
  3. The loss on the single worst loan in the portfolio.Why not: Confuses UL with a worst-case single-name loss.
  4. The difference between the loss rate and the recovery rate.Why not: Confuses UL with a property of LGD; the loss rate is already one minus the recovery rate.

UL is the volatility of losses around EL; EL is the mean. Capital is held against variability, not the average.

A harder one

A USD 10 million loan has a one-year PD of 2 percent and a loss rate with mean 50 percent and standard deviation 20 percent. Using the binomial default assumption and treating default and loss rate as independent, the unexpected loss is closest to:

  1. USD 700,000.Why not: Treats the loss rate as fixed, ignoring its volatility.
  2. USD 282,843.Why not: Keeps only the loss-rate variance term and drops the default-variance term.
  3. USD 754,983. - correct
  4. USD 982,843.Why not: Adds the two standard-deviation terms instead of adding the variances and then taking the root.

UL = EAD x sqrt(PD x sigma(LGD)² + LGD² x PD(1-PD)) = 10m x sqrt(0.0008 + 0.0049) = 754,983.

The hardest tier

A USD 20 million loan has an expected loss of USD 0.6 million and a fixed loss rate of 50 percent. Assuming binomial default, the unexpected loss is closest to:

  1. USD 2.37 million. - correct
  2. USD 1.71 million.Why not: Treats EL over exposure, 3 percent, as the PD instead of dividing EL by exposure x loss rate.
  3. USD 4.75 million.Why not: Forgets to multiply by the loss rate.
  4. USD 0.564 million.Why not: Omits the square root, using the variance term.

PD = EL/(EAD x LGD) = 0.6/(20 x 0.5) = 0.06. UL = 20m x 0.5 x sqrt(0.06 x 0.94) = 2.37 million.

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