Predicting bankruptcy with survival analysis: does it makes sense?

Question

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.

Answer 1

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.

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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.

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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.

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