Ok I think I've got a counter example. First of all, I'm only going to consider values where N approaches infinity && r < 1 && r > 0 to make the math easier.

Ok I think I've got a counter example. First of all, I'm only going to consider values where N approaches infinity && r < 1 && r > 0 to make the math easier. lim [n-> inf, r < 1, r > 0 ] of S_0 = E_0 (1-r^n)/(1-r) -> S_0 = E_0 * 1 / (1-r) I propose that the actual relationship is: S_0 = E_0 / (i-f) E_0=1,f=.07,i=.10 - Original: S_0 = 36.666..., My proposal: S_0 = 33.333... E_0=1,f=.80,i=1.00 - Original: S_0 = 10, My proposal: S_0 = 5 So it seems your formula approximates mine at low values of i and f, but diverges strongly at higher values. I believe I can show mine is the correct one with a small amount of intuition when we let f=0: If I make 10% annually in interest, then $1mil will provide $100k/year forever. The same thing more concisely: i * S_0 = E_0 r = F/I = (1+0/1+i) After substitution and division, your formula is E_0 = S_0 / ((i+1)/i) In words: If I make 10% annually in interest, then $1mil will provide $90.9k/year forever, which is intuitively false.