FINC6023 Chap.10 Credit Risk: Estimating Default Probabilities
Credit Risk: Estimating Default Probabilities
Credit Risk: Estimating Default Probabilities turns ratings, market prices, and firm value into measures of default risk. The building blocks are probability of default, exposure at default, and loss given default. Recovery reduces the loss conditional on default but does not change the probability that default occurs.
Keeping those quantities separate is essential before expected loss or Credit VaR can be calculated.
Rating transition matrices describe movement among credit states and default. Cumulative default probability measures the chance of default by a horizon.
A year-specific unconditional increment is the increase in cumulative probability, while the hazard or conditional rate divides that increment by the probability of surviving to the start of the year. When annual conditional default rates are supplied, multiply survival probabilities and subtract the product from one.
Market prices imply risk-neutral rather than real-world probabilities.
A credit spread can provide an approximate default intensity after adjusting for recovery. A risky bond can be compared with an otherwise similar risk-free bond: the price gap equals the present value of probability-weighted losses under the model assumptions, allowing the common risk-neutral default probability to be solved.
These pricing probabilities include risk premia and are used for valuation; historical or physical probabilities are used for scenario analysis and Credit VaR.
The Merton model takes a structural view. Equity is a call option on firm assets, default occurs when asset value is insufficient to meet debt, and distance to default is converted into a probability.
The available final paper tests cumulative probability arithmetic in Part A and assigns a full Part B calculation to risk-neutral default probability from a coupon bond. Clear timing of possible default and recovery cash flows is as important as the final probability.
What this chapter covers
- 01
The three drivers: PD, recovery/LGD, exposure (EAD)
- 02
Ratings, transition matrices and two-period default via matrix powers
- 03
Cumulative vs marginal PD; hazard (conditional) = [cum(t) − cum(t−1)]/[1 − cum(t−1)]
- 04
N-year cumulative from marginals: PD = 1 − Π(1 − dᵢ) [product is OFF-SHEET]
- 05
Continuous survival and spread-implied intensity: V(t) = e^(−λ̄t), Q(t) = 1 − e^(−λ̄t), λ̄ ≈ s/(1 − R)
- 06
PD from bond prices and the separate jobs of physical and pricing probabilities
- 07
Merton structural model: equity as a call, PD = N(−d₂); Altman Z
Worked example · free
Cumulative PD from marginals, plus the year-2 hazard rate
- 2Survival probability over 3 years = (1 − 0.02)(1 − 0.03)(1 − 0.04) = 0.98 × 0.97 × 0.96 = 0.912576.
- 13-year cumulative PD = 1 − 0.912576 = 0.087424 ≈ 8.74%.
- 2Cumulative PD to year 1 = 0.02; cumulative PD to year 2 = 1 − (0.98 × 0.97) = 1 − 0.9506 = 0.0494.
- 1Year-2 hazard = [cum(2) − cum(1)]/[1 − cum(1)] = (0.0494 − 0.02)/(1 − 0.02) = 0.0294/0.98 = 0.03.
- 1This 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.
- Recovery rate
- The recovery rate is the fraction of exposure recovered after default; one minus recovery is loss given default and determines the size rather than the probability of loss.
- Distance to default
- Distance to default measures how many asset-volatility units separate expected firm asset value from the debt threshold in a structural credit model.
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.
How does the available final test default probabilities?
Part A asks for cumulative default probability from repeated annual rates and for updating an existing cumulative probability with a new conditional rate. Part B asks for a common risk-neutral default probability from a coupon bond. These items reward correct survival arithmetic, timing of default, recovery treatment, and a clear distinction between pricing and real-world probabilities.
Why is recovery separate from probability of default?
Default probability describes whether the credit event occurs. Recovery describes how much value remains after the event. A higher recovery rate reduces loss given default and expected credit loss but does not, by itself, lower the probability of default. Keeping the inputs separate prevents spread and bond-price calculations from double-counting the same effect.
Exam move
Maintain a three-column probability table. Record cumulative default by each horizon, the unconditional increment during the year, and the conditional hazard among survivors. Recompute one column from the other two until the denominators become automatic. When conditional annual rates are given directly, write the survival product before subtracting from one.
For spread and bond-price problems, label the resulting probability as risk-neutral and state that recovery changes loss given default. Build the risky and risk-free cash-flow timelines before solving a bond-implied probability, and mark exactly when default can occur. For Merton questions, draw the asset-value distribution, the debt threshold, and the distance-to-default region so the option interpretation remains clear.
Use the available paper's emphasis to allocate practice: short cumulative-probability calculations for Part A and a complete coupon-bond pricing calculation for Part B. End each problem by stating whether the probability is suitable for valuation or for real-world risk measurement and why.
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