How does probability of default evolve over time? Say I have a probability of default of 0.02 (which is annual so over next year) for a certain client. Then say this client takes out a 180 day loan, how can I adjust my probability of default for this time horizon? At first I thought 180/365 * 0.02 but then i am assuming that how probability of default evolves over time is linear ...
I have read there are many risk models online e.g. cox etc but these seem complex, how can I go about finding a PD (Probability of default) over a certain time horizon which makes sense?
 A: You don't.  You are assuming that default rates are scale-invariant.  In other words, there is a function that maps the one minute rate to the one day rate to the one month rate to the annual rate to the decadal rate.  Logically and experientially, there isn't.
Defaults are a function of cash flows.  Enron remained profitable, for GAAP purposes, eighteen months into its bankruptcy.
I did research on 78 models of bankruptcy and default.  There are shockingly many actually.
To understand why you cannot just find a function, imagine that the borrower is the largest maker of post office boxes in the world.  They just received a large order for new boxes.  They have taken out a loan to cover the materials and equipment and the loan will be paid by revenues with a large operating margin.
Your one-year probability of default likely fell.  OTC, the margin would have increased equity, assuming there is no greater incentive for embezzlement or programmatic failure.  Nonetheless, your short-term default rate may have increased due to the timing of investment cash demands to meet the contract terms.  Any change of that nature creates new, intermediate-term risks.
Now imagine the loan is to pay rents due to collapsed sales due to COVID.  Also, imagine that it has onerous covenants.  For example, let us imagine that the existing credits were debentures and the new ones are secured with first liens.  Also, imagine that the cash flows due at the end of the period will not be renewed and the firm must pony up the cash or collapse.
That is the reason that U.S. securities law requires the disclosure of material cash flows from debt and leases by relevant period.
In this example, the short term (less than six months) probability of default may be nearly zero due to the cash infusion, but the likelihood of survival of cash recovery over the next twelve months may have materially worsened.
For this type of problem, you need to have an existing credit model with validated probabilities.  You need to read the terms of the loan and the covenants.  There won't be a function that maps six month default rates to annual ones and vice versa.
