is there a mapping from Altman Z-score for private companies to bond ratings or probability of default?

Question

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 1

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.