Assess the quality of default probabilities that come out of the Kamakura model, the Merton model and the Moody's KMV model.
The idea
Three well-known default probability models differ in structure. The Merton model is structural. It treats equity as a call on firm assets and backs asset value and volatility out of equity prices. Let DD be the distance to default. Under normality, the Merton default probability is then given by the standard normal cumulative distribution function: PD = N(−DD). The model is simple, but it assumes a single zero-coupon debt, default only at maturity and constant volatility, and its short-horizon probabilities for healthy firms are usually too low. Moody's KMV keeps the structural distance to default, but uses a default point of short-term debt plus half of long-term debt. It maps the distance to an expected default frequency using a large empirical default history instead of the normal distribution. Kamakura is reduced-form: a statistical hazard or logistic model on firm, market and macro variables, including a distance to default. Quality is judged by discrimination, meaning the ranking of defaulters, and by calibration, meaning whether predicted probabilities match realised default rates.
A practice question
How does Moody's KMV convert a distance to default into a default probability?
- It applies the normal cumulative distribution function, EDF = N(-DD).Why not: That is the pure Merton conversion; KMV replaces it with an empirical mapping.
- It maps distance to default to an expected default frequency using a large database of historical defaults, instead of the normal distribution. - correct
- It uses a logistic regression on accounting ratios and macro variables.Why not: Describes the Kamakura reduced-form approach.
- It sets the EDF equal to the CDS-implied hazard rate for the same maturity.Why not: KMV EDFs are not derived from CDS spreads.
KMV keeps the structural distance to default but calibrates it to realised default frequencies, which handles fat tails the normal distribution misses.
A harder one
To apply the Merton model to a listed firm, asset value and asset volatility cannot be observed. How are they estimated?
- By solving two equations together: the equity pricing equation and the link between equity volatility and asset volatility, using the observed equity value and volatility. - correct
- By using the book value of assets and the standard deviation of historical changes in book value.Why not: Book values are infrequent and smoothed; the method relies on market-based information.
- By taking the credit spread on the firm's bonds and dividing by the recovery rate.Why not: Bond spreads are not used as inputs to estimate asset value and volatility in the model.
- By assuming asset volatility equals equity volatility and asset value equals equity value plus debt.Why not: Asset volatility is lower than equity volatility because of leverage.
Two unknowns (V0 and sigma_V) are solved from E0 = call(V0, sigma_V) and sigma_E E0 = N(d1) sigma_V V0, usually iteratively.
The hardest tier
A default model is tested on 2,000 firms, of which 50 defaulted. At a chosen cut-off the model flags 150 firms, and 30 of those flagged firms did default. What are the hit rate (share of defaulters flagged) and the false alarm rate (share of non-defaulters flagged)?
- Hit rate 20%, false alarm rate 80%Why not: Divides by the number of flagged firms and so reports the share of flags that were right and wrong, not hit and false alarm rates.
- Hit rate 60%, false alarm rate 6.2% - correct
- Hit rate 60%, false alarm rate 80%Why not: Computes the hit rate correctly but divides false alarms by the number of flagged firms.
- Hit rate 20%, false alarm rate 6.2%Why not: Divides the correctly flagged firms by flagged firms instead of by actual defaulters.
Hit rate = 30/50 = 60%. False alarms = 150 - 30 = 120 non-defaulters out of 1,950 = 6.2%. These are the two points that make up one point on the ROC curve.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
Start free →More in Estimating Default Probabilities
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