Calibrating Altman Z-Scores to Default Probabilities and Ratings
Summary
The document asks whether the private-company Altman Z′ score can be translated directly into a bond rating or a one-year probability of default. It explains that the score’s original coefficients and zone thresholds were calibrated on an older sample, so those fixed cutoffs should not be assumed to represent current default probabilities or ratings reliably.
One proposed method is to collect a sample of private firms with known default outcomes and estimate a probit model using Z′ as the predictor. The fitted model maps scores to estimated default probabilities; those probabilities can then be compared with historical default rates associated with rating categories. A second option is to use coefficients from a relevant published probit or logit study, ideally one based on companies and conditions similar to the target population. That shortcut is described as arbitrary, especially without a local historical dataset. The mapping therefore requires empirical calibration and should be treated as sample- and context-dependent.
Key ideas
- Altman Z′ scores do not provide a universal direct mapping to ratings or default probabilities.
- Historical default outcomes can be used to fit a probit model that maps Z′ to estimated default probability.
- Estimated default probabilities can be related to rating categories using historical rating default rates.
- Published model coefficients may be a fallback, but their relevance depends on the firms and conditions studied.
Tags
Full text
# is there a mapping from Altman Z-score for private companies to bond ratings or probability of default?
# is there a mapping from Altman Z-score for private companies to bond ratings or probability of default?
On wikipedia, there is a formula to calculate the Altman Z-score for private companies:
Z-score estimated for private firms:
T1 = (Current Assets − Current Liabilities) / Total Assets T2 = Retained Earnings / Total Assets T3 = Earnings Before Interest and Taxes / Total Assets T4 = Book Value of Equity / Total Liabilities T5 = Sales/ Total Assets
Z' Score Bankruptcy Model:
```
Z' = 0.717T1 + 0.847T2 + 3.107T3 + 0.420T4 + 0.998T5
```
Zones of Discrimination:
Z' > 2.9 -“Safe” Zone 1.23 < Z' < 2. 9 -“Grey” Zone Z' < 1.23 -“Distress” Zone
Is there a mapping from this Z', a dimensionless number, to something like, "BBB" or "12% of default in one year"?
## Answer by alexbougias (score 1)
https://quant.stackexchange.com/a/48616
Note that Altman Z-Scoring model is calibrated on a sample many years ago. Therefore, a discrimination with these specific values for the coefficients is quite arbitrary. In that situation I think there are 2 options
Option 1: Use the Altman's calibrated Z-Score as an indicator
Suppose that you have a sample of $N$ private companies, where $D$ of them have defaulted. Define the variable $Y$ which is your dependent variable, taking $1$ if firm defaulted and $0$ otherwise. Your independent variable $Z$ is the Z-score of the corresponding private company. By estimating the following $Probit$ model you link the Z-Score with the Probability of Default (PD)
$$ \mathbb{P}(Y=1|Z)=\Phi(\beta Z)$$
Probit provides a map from Z-Score to the PD. The following step is to map PD with a credit rating using historical default rates (e.g Moodys)
Option 2: Use a calibrated Probit/Logit model
A choice of last resort is to use the results of a research paper that examines defaults of private companies, preferably in your country. If authors estimate a Probit/Logit model, use their output (the coefficients), creating a new score that maps firm's fundamentals with the PD. Be careful that this option is highly arbitrary but if you lack a dataset of historical defaults, this might be the only feasible path.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.