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Characterise how credit risk is quantified and what drives it, covering exposure at default, loss given default, probability of default, expected loss and the time horizon.

FRM® Part II · Credit Risk Measurement and Management · Introduction to Credit Risk Modeling and Assessment · FRM2.CR.5.b

The idea

Credit risk is quantified through three inputs and a horizon. Exposure at default (EAD) is the amount owed when default occurs, which for a credit line includes the expected draw on the undrawn part. Loss given default (LGD) is the share of EAD not recovered, equal to one minus the recovery rate. Probability of default (PD) is the chance of default over the horizon, which must match the loss horizon, usually one year, and EAD and LGD must refer to that same horizon. A multi-year cumulative PD (CPD) with a constant annual rate p is not T × p. CPD = 1 − (1 − p)^T. Expected loss is the average loss that a bank prices and provisions for. EL = PD × LGD × EAD. Unexpected loss is the volatility around it and is what capital covers. For one exposure with fixed LGD, UL = EAD × LGD × √(PD × (1 − PD)).

Formulas

Cumulative PD from a constant annual PD

CPD = 1 − (1 − p)ᵗ

p
the constant annual probability of default
t
the number of years

Assumes independence across years. Multiplying p by the number of years overstates it.

Expected 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

Priced and provisioned for. EAD for a credit line includes the expected draw on the undrawn part.

Unexpected loss of one exposure

UL = EAD × LGD × √(PD × (1 − PD))

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

Treats default as a Bernoulli event with LGD known. Capital is held against this volatility, not against EL.

A practice question

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

A bank estimates that, in the event of default, it will recover 42 percent of the exposure. The loss given default is:

  1. 42%Why not: This is the recovery rate itself, not its complement.
  2. 58% - correct
  3. 142%Why not: Adds the recovery rate to 100 percent instead of subtracting it.
  4. 70%Why not: Divides 1 by (1 + recovery rate), a formula that does not exist.

LGD = 1 - recovery rate = 1 - 0.42 = 58%.

A harder one

A borrower has a constant annual PD of 2 percent and defaults are independent across years. The cumulative probability of default over three years is closest to:

  1. 5.88% - correct
  2. 6.00%Why not: Multiplies the annual PD by three, which ignores that a borrower that has defaulted cannot default again.
  3. 3.46%Why not: Scales PD by the square root of time, a rule used for volatility, not for default probabilities.
  4. 94.12%Why not: This is the three-year survival probability, not the cumulative default probability.

Cumulative PD = 1 - (1 - 0.02)³ = 1 - 0.9412 = 5.88%.

The hardest tier

A bank has a single exposure with EAD of USD 20 million, LGD of 50 percent (known with certainty) and a one-year PD of 4 percent. Treating default as a Bernoulli event, the standard deviation of the loss (unexpected loss) is closest to:

  1. USD 3.92 millionWhy not: Omits the 50 percent LGD, so it measures the volatility of the whole exposure rather than the loss.
  2. USD 1.96 million - correct
  3. USD 0.384 millionWhy not: Forgets the square root of PD x (1 - PD), so it reports the variance term scaled by exposure.
  4. USD 0.40 millionWhy not: This is the expected loss, 20 x 0.5 x 0.04, not its standard deviation.

UL = EAD x LGD x sqrt(PD x (1 - PD)) = 10 x sqrt(0.04 x 0.96) = 10 x 0.196 = USD 1.96 million. Expected loss is only 0.40 million; capital is held against the volatility around it.

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