Questions: Default Probability and Recovery Rate Estimation

Question 1

A corporate bond yields 5% while a risk-free Treasury of the same maturity yields 2%, implying a credit spread of 3%. An analyst concludes that the market-implied default probability is 3%. Why is this likely an overestimate?

• A. Credit spreads undercount default probability because they don't include LGD
• B. Credit spreads embed a liquidity premium on top of expected loss, inflating the implied PD beyond the actual default probability
• C. Treasury yields already include a credit component, so the spread is not a pure default signal
• D. Default probability can only be estimated from equity prices, not bond prices

Answer: (B)

Under risk-neutral pricing, credit spread ≈ PD × LGD for short maturities. But observed spreads also include a liquidity premium — investors demand extra yield for holding less liquid instruments, regardless of default risk. This means the spread is higher than expected loss alone, so backing out PD from the spread produces an overestimate of actual default probability. In practice, market-implied PDs from bond spreads are systematically higher than historically realized default rates, partly for this reason.

Question 2

A bank has a $2 million loan outstanding. The borrower has a 5% annual probability of default, and the bank expects to recover 40% of the loan if default occurs. What is the expected annual credit loss?

• A. $60,000
• B. $100,000
• C. $40,000
• D. $4,000

Answer: (A)

Expected Loss = PD × LGD × EAD. LGD = 1 − recovery rate = 1 − 0.40 = 0.60. EL = 0.05 × 0.60 × $2,000,000 = $60,000. Note that option B ($100,000) is the result of using 5% × $2M, ignoring recovery — a common error that conflates PD with loss rate. Banks price this $60,000 expected loss into their lending spread and hold regulatory capital for it.

Question 3

During a severe economic downturn, default rates and recovery rates tend to move in opposite directions — as defaults rise, recoveries fall — amplifying credit losses beyond what simple expected-loss calculations predict.

• T. True
• F. False

Answer: True

This negative correlation between PD and recovery is one of the most important (and most often underestimated) features of credit risk. In a crisis, more borrowers default simultaneously AND assets are sold at fire-sale prices, reducing recovery values. A bank that calculated EL = PD × LGD using average historical values for each separately would understate crisis losses because those averages apply to normal times. The positive correlation of losses across borrowers (systematic risk) combined with this PD/recovery correlation is why bank capital requirements must account for concentration risk and systemic scenarios.

Question 4

Market-implied default probabilities derived from credit spreads tend to underestimate actual default probabilities because bond investors are overly optimistic.

• T. True
• F. False

Answer: False

The direction is reversed. Market-implied PDs derived from credit spreads tend to *overestimate* actual default probabilities. This is because credit spreads include a liquidity premium — extra yield demanded for illiquidity — in addition to compensation for expected loss. When you back out PD from a spread assuming it reflects only default risk, the PD is inflated. Empirically, realized default rates are typically well below market-implied PDs from spreads, especially for investment-grade bonds where liquidity premiums are proportionally large.

Question 5

Explain why using the formula EL = PD × LGD × EAD to estimate a credit portfolio's total expected loss may significantly understate actual losses during a financial crisis.

Model answer: Two reasons compound: (1) PD and recovery rates are negatively correlated in crises — both worsen simultaneously. Using average PD and average LGD from normal times misses this. (2) Defaults cluster — many borrowers default together in a systemic shock, meaning portfolio losses are correlated, not independent. The formula gives each loan's expected loss correctly but treating them as independent ignores correlation, which drives unexpected portfolio-level losses far above the sum of individual ELs.

The formula EL = PD × LGD × EAD is accurate for a single loan in normal times. The problem at the portfolio level is correlation: if a recession causes 10 borrowers to all default at once, the portfolio loss isn't 10 × EL (individual) — it's much higher than what the formula implies when applied independently to each loan. This is why regulatory frameworks (Basel) require banks to hold capital for unexpected losses (the variance around EL) not just expected losses, and why stress testing under correlated adverse scenarios is essential for sound credit risk management.