I was lying awake last night thinking about all the financial math problems I couldn't solve and ...
>Couldn't solve? That's not unusual. Besides, I thought you were giving up on financial stuff.
Pay attention.
In an earlier tutorial (Equation [4A]), we ran across the expression:
[4A]    SD = M SQRT[(1+S²/M²) - 1] +M² SQRT[(1+S²/M²)² - 1] +...+Mⁿ SQRT[(1+S²/M²)ⁿ - 1].

I never could sum this series, so tried to simplify and approximate and ...
>Why?

That's the question I asked myself. My approximations resulted in another series which was no better than the one above.

>And that kept you awake? Besides, weren't you through with financial stuff?
Don't keep reminding me!
Anyway, it occurred to me that I might just as well use the series [4A].

>What was the approximation?
It was:
[4B]    SD(gMS(n)) = S { 1+M SQRT[2]+M² SQRT[3]+...+M^n-1SQRT[n] }

Anyway, since the formula was my attempt to calculate a Safe Withdrawal Rate (avoiding a jillion Monte Carlo simulations), I decided to modify the spreadsheet by allowing the user to pick either [4B] or [4A]
... which, in the modified spreadsheet, I call Formula #1 and Formula #2, respectively.

>And does Formula #2 give a better value for SWR?
Who knows. Try it yourself ... and decide. The (modified) spreadsheet looks like this:

To download a .ZIPd file, RIGHT-click on the picture above and Save target.

>What's the point of all this?
Well ... uh, I just figured that if you wanted to see how the SWR changed when you changed from 30 years to 35 or maybe 8% return to 9% or 3% inflation to ...
>Yeah, so?
So you wouldn't have to do a jillion Monte Carlo simulations for each new set of parameters.
>And that's good?
Sure. Don't you think so?
>No.

Here's some examples, with various inflation rates (that's i), Years (that's N) and Standard Deviations (that's S) :