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Break expected loss into its components, then show how it differs from unexpected loss.

FRM® Part II · Credit Risk Measurement and Management · Credit Risk Management · FRM2.CR.3.f

The idea

Expected loss is the average credit loss over a horizon. PD is the probability of default. LGD is the fraction of exposure lost after recoveries, which is one minus the recovery rate. EAD is exposure at default. Multiplying the three on one horizon gives the expected loss. EL = PD × LGD × EAD. EL is a cost of doing business. It is priced into spreads and covered by provisions and reserves. Unexpected loss is the variability of losses around that average, which a simplified model measures as a standard deviation, whereas a quantile-based capital measure is credit VaR at a stated confidence level minus EL, a different quantity. UL is covered by capital, and for a portfolio it depends on default correlation. Expected losses add across loans. Unexpected losses do not, unless defaults are perfectly correlated. For one loan with fixed LGD and EAD, in standard deviation terms, UL = LGD × EAD × √(PD × (1 − PD)).

Formulas

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

All three inputs on one horizon. EL adds across loans; unexpected loss does not.

Unexpected loss of a single loan

UL = EAD × LGD × √(PD × (1 − PD))

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

Fixed LGD and EAD, default the only random element. Portfolio UL is not the sum of these unless defaults are perfectly correlated.

A practice question

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

Expected loss on a single credit exposure over a given horizon is:

  1. Probability of default x (1 - LGD) x exposure at default.Why not: Uses the recovery rate (1 - LGD) where the loss rate is needed.
  2. Probability of default x LGD / exposure at default.Why not: Divides by EAD, which turns a currency loss into a meaningless ratio.
  3. Probability of default x loss given default x exposure at default. - correct
  4. LGD x exposure at default, with no default probability.Why not: Omits PD, giving the loss if default occurs rather than the expected loss.

EL multiplies the three drivers: how likely default is, what fraction of exposure is lost if it happens, and how much is exposed. LGD equals 1 minus the recovery rate.

A harder one

A bank has a USD 5 million exposure to a borrower with a one-year PD of 3 percent and an expected recovery rate of 40 percent. Expected loss over the year is:

  1. USD 60,000Why not: Uses the recovery rate in place of LGD.
  2. USD 150,000Why not: Ignores recoveries (treats LGD as 100 percent).
  3. USD 3,000,000Why not: Applies LGD without PD, giving loss given default rather than EL.
  4. USD 90,000 - correct

LGD = 1 - 40% = 60%. EL = 3% x 60% x 5,000,000 = 90,000.

The hardest tier

A portfolio has two loans. Loan A: exposure USD 10 million, PD 2 percent, LGD 50 percent. Loan B: exposure USD 6 million, PD 4 percent, LGD 40 percent. The default correlation is 0.20, and LGD and exposure are fixed. The portfolio's unexpected loss (standard deviation of loss) is closest to:

  1. USD 1.17 millionWhy not: Adds the two ULs, which assumes a correlation of 1.
  2. USD 0.84 millionWhy not: Ignores the correlation term, assuming independent defaults.
  3. USD 0.92 million - correct
  4. USD 1.11 millionWhy not: Adds the portfolio expected loss to the unexpected loss; the two are different measures.

UL_A = 0.700, UL_B = 0.470. Portfolio UL = sqrt(0.700² + 0.470² + 2 x 0.20 x 0.700 x 0.470) = 0.92. Unlike expected loss, UL is not additive: it depends on correlation.

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