The spot rate of interest is defined by s_t =(.9^t)*.1 for t=1,2,3,4,5. Find the present value of a 5 year annuity due in which the first payment is equal to 1000 and each subsequent payment increases by 5% of the immediately proceeding payment. How would the formula look for this problem? Then how would it look if it was an annuity immediate? How would it look in both cases if there was no price increase? The book does not even go over how to solve questions like this (Theory of Interest by Stephen G kellison) Any help would be greatly appreciated

Also what is the formula for :S_n (s double dot n, future value)?

As I said in the other thread: forget the idea of knowing “the formula”. I doubt anyone here knows “the formula”, though many would know the approach to get the answer. Some questions for you:

a. How much is the first payment?
b. What is the spot rate for that payment?
c. What is the PV of that payment?

Questions 2-5. Exactly the same questions, but for the second, third, fourth and fifth payments.

What is the total present value of the payments?

And the great thing about focusing on the approach, not the “formula”. The questions are exactly the same for your four cases (due, immediate, due with no increase, immediate with no increase).

If you see such a question in any of your study material, ask again then. I don’t expect such a question would be asked. (If it were, I think I know what the intended answer would have to be, but I have doubts as to whether the question really makes sense.)