Interpretation of Logistic Regression output in Credit Scoring

Question

Currently I'm working on a project involving loan data where I am trying to build a model that will predict a probability of an individual defaulting on a loan. I have binary dependent variable where defaulted loans are denoted with 1. I decided to use logistic regression for this task, but However, I have difficulty interpreting the results of the regression and would greatly appreciate some clarification.

Namely, I would like to know, whether the probabilities calculated by logistic regression should be interpreted as a probability of default? For example, if for some observation, the predicted value is 0.8, should I assume that according to the model, this individual has 80% chance of default?

Answer 1

Yes, as long as the input for the outcome is failure (0,1) where 0-No 1-Yes (default), then the predicted probability from your logistic regression model will be the probability of defaulting. For your use case of 0.8 or 80% that person does have a default prediction of 80%, so probably not approved. I would think a cutoff of 50% (0.5) is used, but this would depend on the model, the credit agency, and the input predictors.

Please do consider that a lot of times in finance several models could be used. It may work out that a model for predicting default with outcome coded 0-no default, 1-default will sometimes employ features which are different from optimal features used for a model for predicting no default with outcome coded 0-default, 1-no default.

Reason I am raising this is because a lot of people try to front-load everything into one model, when in fact if you "divide and conquer" to break up a complex problem into multiple smaller problems your prediction may be better.

In fact, I would never approach a bank or credit agency with results of one model!

Answer 2

For the sake of notation, I am using the model with only one independent variable $X$. All of the results hold true with more independent variables as well.

In logistic regression, we use the logistic function

$$ p(X) = \frac{e^{\beta_0 + \beta_1 X}}{1 + e^{\beta_0 + \beta_1 X}}$$

to model the relationship between $p(X) = \mathbb{P}(Y = 1|X)$ and $X$. In your case, $p(X)$ denotes the probability of the default conditional on $X$.

For the predictions, for an individual with the value of the independent variable $x_0$, we predict the probability of default by just plugging it in the estimated function:

$$ \hat{p}(x_0) = \frac{e^{\hat{\beta}_0 + \hat{\beta}_1 x_0}}{1 + e^{\hat{\beta}_0 + \hat{\beta}_1 x_0}}.$$

If you get $\hat{p}(x_0) = 0.8$, then you predict the probability of default for this individual to be $0.8$.