FINC6023 · Financial Risk Management
Credit Risk: Estimating Default Probabilities
Credit Risk: Estimating Default Probabilities is about turning ratings, spreads and bond prices into a probability of default. You decompose credit risk into PD, recovery (LGD = 1 − recovery) and exposure, build cumulative default probabilities from yearly marginals, and distinguish the unconditional PD from the conditional hazard rate. The two exam workhorses are extracting PD from a credit spread (λ ≈ s/(1−R), off-sheet) and from a bond-price gap (risk-neutral), plus the structural Merton model where equity is a call option on firm assets and PD = N(−d₂).
What this chapter covers
- 01The three drivers: PD, recovery/LGD, exposure (EAD)
- 02Ratings, transition matrices and two-period default via matrix powers
- 03Cumulative vs marginal PD; hazard (conditional) = [cum(t) − cum(t−1)]/[1 − cum(t−1)]
- 04n-year cumulative from marginals: PD = 1 − Π(1 − dᵢ) [product is OFF-SHEET]
- 05Continuous survival V(t) = e^(−λ̄t), Q(t) = 1 − e^(−λ̄t)
- 06PD from spread: λ̄ ≈ s/(1 − R) [OFF-SHEET]
- 07PD from bond prices (risk-neutral) and real-world vs risk-neutral PD
- 08Merton structural model: equity as a call, PD = N(−d₂); Altman Z
Cumulative PD from marginals, plus the year-2 hazard rate
- 2 marksSurvival probability over 3 years = (1 − 0.02)(1 − 0.03)(1 − 0.04) = 0.98 × 0.97 × 0.96 = 0.912576.
- 1 mark3-year cumulative PD = 1 − 0.912576 = 0.087424 ≈ 8.74%.
- 2 marksCumulative PD to year 1 = 0.02; cumulative PD to year 2 = 1 − (0.98 × 0.97) = 1 − 0.9506 = 0.0494.
- 1 markYear-2 hazard = [cum(2) − cum(1)]/[1 − cum(1)] = (0.0494 − 0.02)/(1 − 0.02) = 0.0294/0.98 = 0.03.
- 1 markThis equals the year-2 marginal of 3%, confirming the relationship.
Key terms
- PD, LGD, EAD
- The three credit-risk drivers: probability of default, loss given default (= 1 − recovery), and exposure at default. Expected credit loss multiplies the three.
- Hazard rate (default intensity)
- The conditional probability of defaulting in a period given survival to its start: [cum(t) − cum(t−1)]/[1 − cum(t−1)]. In continuous time it gives survival V(t) = e^(−λ̄t).
- Cumulative vs marginal PD
- The marginal (unconditional) PD in year t is cum(t) − cum(t−1); the cumulative PD over n years built from yearly marginals is 1 − Π(1 − dᵢ). The product form is OFF the provided sheet.
- Risk-neutral vs real-world PD
- Risk-neutral PDs (from spreads/bond prices) are used for valuation and pricing; real-world (physical) PDs (from historical data, KMV) are used for Credit VaR and scenario analysis. They differ because the risk-neutral figure embeds a risk premium.
- Merton model
- A structural model treating equity as a call option on firm assets, E₀ = V₀N(d₁) − De^(−rT)N(d₂), giving PD = N(−d₂). It backs out unobservable asset value and volatility from observed equity.
Credit Risk: Estimating Default Probabilities FAQ
What is the difference between the marginal, hazard and cumulative PD?
The marginal (unconditional) PD is the chance of defaulting IN year t viewed from today, cum(t) − cum(t−1). The hazard (conditional) PD assumes survival to the start of year t and divides by the survivors. The cumulative PD is the chance of defaulting by year t, built as 1 − the product of yearly survival probabilities.
How do I get a default probability from a credit spread?
Use the off-sheet relation λ̄ ≈ s/(1 − R), where s is the continuously-compounded spread and R the recovery rate. For example a 180bps spread with 40% recovery implies an annual default intensity of about 0.018/0.6 = 3%. This is a risk-neutral PD, suitable for valuation.
Why does the course distinguish risk-neutral and real-world PD?
Because they are used for different jobs. PDs implied by market spreads or bond prices are risk-neutral and embed a risk premium, so you use them to value and price credit. Historical/physical PDs are smaller and are the right input for Credit VaR and stress scenarios.
Exam move
Build a small table that separates marginal, hazard and cumulative PD and practise converting between them, because that conversion is examined repeatedly. Memorise the off-sheet PD-from-spread relation and the 1 − Π(1−dᵢ) product, and know which PD (risk-neutral or real-world) each use case needs.
