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Define the recovery rate and, with it, compute the loan's expected loss.

FRM® Part I · Valuation and Risk Models · External and Internal Credit Ratings · FRM1.VR.4.d

The idea

The recovery rate is the fraction of the claim that the lender recovers after default, measured as a percentage of the exposure or face value. It depends on seniority, collateral and the state of the economy, and it is often estimated from the bond price a month or so after default. Loss given default is 1 − recovery rate. Expected loss = PD × (1 − R) × EAD. Here PD is the probability of default, R is the recovery rate and EAD is the exposure at default. Exposure at default includes expected drawdowns on undrawn commitments, not just the amount currently drawn. Recovery rates are lower in recessions, when default rates are higher, so an average recovery rate understates stress losses. The classic mistakes are multiplying by the recovery rate instead of 1 − R and forgetting exposure at default.

Formulas

Expected credit loss

EL = PD × (1 − R) × EAD

PD
the probability of default
R
the recovery rate
EAD
the exposure at default, including expected drawdowns on undrawn commitments

Multiply by 1 − R, the loss given default, not by R. Recoveries fall when defaults rise, so stress losses exceed what average inputs suggest.

A practice question

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

The recovery rate on a defaulted loan is best defined as:

  1. the proportion of the claim that the lender loses, expressed as a percentage of the exposure.Why not: This is loss given default, which equals 1 minus the recovery rate.
  2. the share of the claim the lender eventually recovers, as a percentage of exposure. - correct
  3. the probability that the borrower repays the loan in full before maturity.Why not: This is a survival probability, which concerns whether default happens, not what is recovered after it.
  4. the share of the portfolio that returns to performing status after a missed payment.Why not: Describes a cure rate; recovery rate is measured on loans that have actually defaulted.

The recovery rate is the fraction of the exposure recovered after default. Loss given default is its complement, 1 - R.

A harder one

A bank has USD 5 million lent to a borrower with a one-year probability of default of 2% and an expected recovery rate of 40%. The one-year expected loss is closest to:

  1. USD 60,000 - correct
  2. USD 100,000Why not: Ignores recovery: 5 million x 2% treats the entire exposure as lost on default.
  3. USD 40,000Why not: Multiplies by the recovery rate (40%) instead of loss given default (60%).
  4. USD 3,000,000Why not: Multiplies exposure by loss given default only (5 million x 60%), omitting the probability of default.

EL = PD x (1 - R) x EAD = 0.02 x 0.60 x 5,000,000 = USD 60,000.

The hardest tier

In a recession a loan's default probability rises from 2% to 3% and the recovery rate falls from 40% to 25%, with exposure unchanged. By what percentage does the expected loss rise?

  1. 50.0%Why not: Captures only the rise in default probability (2% to 3%) and ignores the lower recovery rate.
  2. 25.0%Why not: Captures only the rise in loss given default (60% to 75%) and ignores the higher default probability.
  3. 87.5% - correct
  4. 75.0%Why not: Adds the separate 50% and 25% effects instead of compounding them.

EL before = 2% x 60% = 1.2%; after = 3% x 75% = 2.25%. Ratio = 1.875, a rise of 87.5%. The effects multiply, and recoveries tend to fall exactly when default rates rise.

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