# Probability of default

I have to calculate probability of default (PD) rates for our clients (I am working in a Bank) based on clients' financials. Could you, please, advise me how to do that?

I think we have two Options: 1. Calculate PDs for each client based on their financial Statements, 2. We have internal ratings (again based on financials) that we are using for assessment of the clients, and we can calculate PD for every internal rating.

Could you please advise, is it possible to do that with one of the above-mentioned Options and how?

Thanks in advance,

• The Altman Z-score is the best known measure and it has many variants. If you are looking for default probability of bank, for example, the original Altman Z must be modified to fit banks' financial reporting schema. If you could provide a little more detail on what types of companies and/or individuals you have to evaluate, that would help us give a more specific answer. Mar 14, 2017 at 0:35

## 3 Answers

Merton model will be a bit more quantitiative.

Z-Score is an option, as is Ohlson.

In the end you are going to want some non-defaulted->defaulted transition mapping based on factors you identify as meaningful.

Yes, you can. Also, do not use Altman's Z. The extreme scores are predictive, but a load of empirical research shows the intermediate values are not predictive.

The best solution is a Bayesian solution because you are gambling money. Bayesian methods are coherent.

Coherence is the statistical property by which fair gambles can be placed. Frequentist methods are not coherent. They do not generate probability distributions that can safely be gambled with.

In addition, because of the high level of correlation by design in financial statements, the assumptions of logistic regression are strongly violated.

I have solved this problem before. It's actually hard.

The best solution that I have found is to create Pearson-Tukey groups and to calculate the binomial over each group. For example, one group could be the probability of default given that the acid ratio is between the 40th and 60th percentile of firms, the turnover is between the 80th and 100th percentile and the prior quarter's change in GDP was between the 40th and 60th percentile. You would then construct the predictive distribution and take its expectation.

Because you do not know which combination of variables is optimal, you will need to do a somewhat combinatoric solution using the Bayesian posterior to determine which models to keep.

I ended up keeping 2 of 78 models. One had a posterior probability of roughly 54% and the other had a posterior of approximately 46%. The other 76 models had a combined posterior probability of 1/10,000 of one percent chance of being true.

I then combined the two models using Bayesian model averaging.

If I were you I would grab a commercial data set such as Compustat to expand your set so that you can condition on a larger set than your internal set.

Now as to your internal ratings you are likely getting at least part of the rating from financials. It is not clear that your rating system is internally consistent or consistent across time.

You can use your rating system as an alternative Bayesian model. Add it into your other models. It will also have a posterior probability.

You will want to carefully think about your priors. The prior should not be uninformative. You know that 93% of all firms are not going to go bankrupt in the next 12 months. The prior does matter.

Finally, collateral requirements should be considered as well.

Take a look at the Altman Z Score, sounds like it is what you are looking for - https://en.wikipedia.org/wiki/Altman_Z-score