I am looking at the clinical risk of a particular drug being taken off the market by the FDA. The purpose of this exercise is to apply it to a financial model and get a risk adjusted NPV. Specifically, I am looking at the chances of the drug being taken off market each year, applying that chance to the net present value of that year, and summing those up to get the value of the drug.

There are two actions the FDA can take to remove the drug from the market. The first is a "summary judgment" which would remove the drug immediately. For simplicity, this action can only happen at the end of the year. The second action they can take is request a hearing. Again, for simplicity, the hearing would be requested at the end of the year. When a hearing is requested, there are multiple possibilities of when the hearing would conclude, each with their own probability. Finally, when the hearing does conclude, there is a chance that the conclusion would be to remove the drug. If a hearing is requested, no summary judgment can be made. If a hearing is concluded positively, the drug will stay on until the end. By the last period (19) the drug will definitely be taken off the market regardless.

So these are the actual probabilities I am currently using, for each period (1-19), respectively:

Summary judgment: 5% (every year)

Hearing: 10%, 20%, 30%, 20%, 10% (from period 5 to 19)

Conclusion being reached after x years: 0% (after 1 year), 25%, 50%, 25%, 0% (I left the zeroes in so I could move these percentages around)

Conclusion removing drug from market: 90%

So for a hearing, the first year the drug can possibly be taken off market due to the conclusion of a hearing is period 3 (hearing requested in year 1, 0% 1 year after that, and 25%*90% chance 2 years after that - in period 3).

Note that these probabilities are independent, but should be made dependent (at least in my mind). Meaning, since you cannot remove the drug more than once, then the chance of the drug being removed in the second year, is the chance it is not removed in the first times the chance it is removed in the second. So again, these are only the chances of it being removed each year.

I think the complication for me is that the chance of a summary judgment being made is the chance a hearing is not requested cumulatively prior (as opposed to concluded) time the chance a summary judgment hasn't been made cumulatively prior. I may be wrong on this, but I feel pretty confident.

So one last example for illustrative purposes:

Period 1: total chance of being taken off market is 5% = Summary judgment

Period 2: total chance of being taken off market is (1-10%)*(1-5%)*5% = summary judgment times the chance there was not a hearing requested and no summary judgment the period prior

Period 3: total chance of being taken off market is (1-10%)*(1-20%)*(1-5%)^2*5%+10%*25%*90% = summary judgment in period 3 times the chance there was no hearing requested in first 2 years and no summary judgment in first 2 years plus the chance the drug is taken off market due to a hearing being requested in year one times the chance the hearing was concluded in 2 periods times the chance the conclusion was negative

Period 4: verbally ... the chance nothing has happened (hearings nor summary judgments in 3 periods prior) times the chance of summary judgment in period 4 plus the chance there was a hearing in period one but concluded in 3 periods plus the chance there was a hearing requested in period 2 and concluded in 2 periods both times the chance the hearing is negative and an adjusted chance (basically 1- (5%*25%)) that a summary judgment wasn't reached in the first year (the adjustment is to compensate for the fact that it only effects the second hearing period but not the first, but that the hearings can still be ongoing ...)

so these first 4 percentages come out to ... chance of being removed in that year:

1: 5%

2: 4.28%

3: 5.50%

4: 10.85%

I have attached a censored excel for the work I've done on this so far for anyone who wants to give it a shot!