# Predicting bankruptcy with survival analysis: does it makes sense?

So I would like to implement a bankruptcy prediction model. I have found some work done before which rely on survival analysis. This is however based on the assumption that each subject will die one day (does not have to be during study, those are so called censored data) and that the risk of dying is increasing in time (i.e. survival function is decreasing). IMO this does not make sense in terms of bankruptcy prediction, I do not think that each company has to go bankrupt one day, nor that the chances of going bankrupt are increasing just with time. Moreover, I would assume that the company could even get better in terms of financial health and decrease its chances to go bankrupt, which is not incorporated in survival analysis in any way. So my question is, does it make sense to model bankruptcy by survival analysis, am I missing something? Thanks.

• okay, but in my point of view the company could increase its chances of survival (not going bankrupt) in some time by taking some good steps, which is not possible to show in survival model, as the survival function is strictly decreasing. Or? Commented Jul 7, 2020 at 6:55
• the comment looks weird, because it was an answer to already deleted comment, but I do not want to delete mine anyway.... Commented Jul 14, 2020 at 9:46

This is however based on the assumption that each subject will die one day (does not have to be during study, those are so called censored data) and that the risk of dying is increasing in time (i.e. survival function is decreasing).

Both of these "assumptions" are not necessary to perform survival analysis. In fact, they are not assumptions of survival analysis. I'll explain the fault in the second, and then the first.

risk of dying is increasing in time (i.e. survival function is decreasing).

First note that this is not logically equivalent. Risk in survival analysis is characterized by the hazard function, $$h(t)$$, which is a non-negative function over time. It can increase (which represents higher risk of death at that point in time) or decrease. A hazard of 0 or near 0 means that there is a very small risk of death. So, a company that performs well can have a decreasing hazard, and end up near 0 (i.e. little risk of death). However, this survival function will still decrease, because there did exist a positive probability of death in that interval. Take a look at the relationship between hazard and survival:

$$P(T >t)= S(t) = \exp{\left(-\int_0^t h(s) ds \right)}$$

So any non-zero hazard, whether increasing or decreasing, will cause the survival function to decrease.

that each subject will die one day

I've heard this statement from others, and I'm not sure where it comes from. Considering what we know from above, suppose that after some time, the hazard becomes 0. This could be because a patient is cured of a disease, or a competing death-event occurs first which prohibits our event of interest from occurring. Irregardless of what causes a 0 hazard, this means that the survival function will not converge to 0, and instead some asymptote between 0 and 1. There is class of models called cure models that model this behaviour, and they often look like:

$$S(t) = p + (1-p)S_1(t)$$

where $$p$$ is the probability of never dying.

Also, the assumption that a bank never fails is quite extreme, and I don't think would produce a very reliable model 🙃

• +1 Much better than the answer I had thought earlier about offering. Just want to note for others who might read this that multiple bankruptcies of the same firm are possible, different types of bankruptcies exist, and as a comment on the OP says, "taking some good steps" could influence the hazard. All 3 possibilities can be handled by survival models via recurrent events, competing risks and time-dependent covariates, respectively.
– EdM
Commented Jul 7, 2020 at 20:50
• Just a quick note: there should probably be $P(T>t)$ instead of $P(t>T)$ or? Commented Jul 15, 2020 at 8:12
• Yes, that’s my mistake! I’ll edit that Commented Jul 15, 2020 at 10:44