Work out expected loss using default probability and loss given default

CFA® Level I · Fixed Income · Credit Risk · FI.5.e

The idea

Expected loss is the probability of default multiplied by the loss given default, where loss given default is one minus the recovery rate applied to the exposure. The decomposition matters because the two components respond to different things and need not move together: a secured lender may face a rising default probability with a stable recovery, while a subordinated lender to the same issuer sees both deteriorate together. Recovery rates are also correlated with the economic cycle, so defaults and low recoveries arrive at the same time, which makes expected loss understate the risk of the tail.

Formulas

Expected loss

expected loss = POD × LGD

POD: the probability of default over the period
LGD: loss given default, the amount lost if it does default

Both inputs are expectations, so this is an average over outcomes rather than a forecast for one bond. A bond either defaults or it does not; no bond loses exactly its expected loss.

Loss given default

LGD = exposure × (1 − recovery rate)

exposure: the amount at risk when the default happens
recovery rate: the share of that exposure expected back through the workout

Recovery depends on seniority and on what the issuer owns, so the same default produces a different loss for a secured lender and for a subordinated bondholder.

A practice question

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

How is expected loss calculated?

Probability of default multiplied by loss given default — correct
Probability of default divided by the recovery rateWhy not: Dividing produces no meaningful quantity, and a recovery approaching zero would send the result to infinity rather than to the full exposure.
Exposure multiplied by the probability of defaultWhy not: That gives the loss if nothing at all were recovered. It is the ceiling on expected loss rather than the figure.

Both the chance of the event and the size of the loss if it happens are needed. Either alone describes only half the risk.

A harder one

A bond has a 4% probability of default and an expected recovery of 40%. What is the expected loss as a percentage of exposure?

1.6%Why not: 0.04 x 0.40 — the recovery rate used as the loss rate. What is lost is the 60% not recovered.
2.4% — correct
4.0%Why not: The default probability alone, which assumes nothing is recovered.

Loss given default is 1 - 0.40 = 0.60, and 0.04 x 0.60 = 0.024. Using the recovery rate instead of the loss rate gives the 1.6% distractor.

The hardest tier

Why is an average recovery rate a poor input for a stress scenario?

Because recovery rates are fixed by contractWhy not: Nothing fixes them. Seniority and collateral shape them, but the realised figure depends on what the assets fetch.
Because recoveries are unobservable until a default occursWhy not: True of any single case, and the reason averages are used at all. The objection is that the average is drawn from conditions the stress case is not in.
Because recoveries fall when defaults rise — correct

The average is measured across a whole cycle, and the stress case is not in that cycle. Multiplying an average probability by an average recovery assumes an independence the cycle does not deliver.

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