Safe Withdrawal rates

I was thinking ...

>That was a mistake!
... that there should be some quick-and-dirty way to calculate a Safe Withdrawal Rate, a Rule of Thumb if you will.
Something that depended upon a Mean Portfolio Return, its Standard Deviation, some Inflation Rate and the Number of Years that you wanted your portfolio to last.

>Haven't you done this a jillion times?
Uh ... yes, but I'd like to get a feel for how the SWR varies with those parameters, like Standard Deviation and Inflation.
Remember the Magic Sum?

>No!
Then pay attention:
 gMS = I1/G1 + I2/G2 + I3/G3 + ... + IN/GN where In is the cumulative Inflation Factor at year n (e.g. for a constant 2.5% inflation, In = 1.025n) and Gn is the cumulative Gain Factor at year n (e.g. for a constant 6.0% annual gain, Gn = 1.06n)

So let's assume that inflation is constant at the rate i. (e.g. for 2.5% annual inflation, i = 0.025)
Further, [Gn]1/n = 1 + Annualized Return   ... over N years.

>Huh?
If the annual returns for n years are r1, r2, r3, ... rn, then
Gn = (1+r1) (1+r2) ... (1+rn) = (1 + Annualized Return)n.

Okay, let's assume this Annualized Return, R, is also constant over N years.

>What! You're kidding, right?
Did I mention quick-and-dirty?

Okay, then In = (1+i)n and Gn = (1+R)n

Anyway, our Magic Sum would be:
 quick&dirty gMS = I1/G1 + I2/G2 + I3/G3 + ... + IN/GN = Σ[(1+i)/(1+R)]n = X (1 - XN) / (1 - X) where X = (1+i)/(1+R).
Now we know that, in order to last N years, the Withdrawal Rate f must be less than 1/gMS.
Or, to put it differently:   f gMS < 1
or, to put it quick&dirty :   f X (1 - XN) / (1 - X) < 1
or, to put it differently:   XNX) / fX
or, to put it differently:   N < log[1 - (1 - X) / fX ] / log[X]
or, to put it differently ...

 quick&dirty A portfolio will survive N years where: N = log[1 - (1 - X) / fX ] / log[X] and X = (1+i)/(1+R) and f is the Withdrawal Rate.
We can do another quick&dirty Annualized Return calculation using the stuff here, namely:

R = Annualized Return = Mean Return - (1/2) (Standard Deviation)2

>So is that quick&dirty thing any good?
Here's an example:

>And I'm supposed to get a feel for how SWR changes with the parameters just by looking at that chart? You kidding?

>That's it?
That's it. I'm happy.
 >But shouldn't there be some distribution of possible portfolios. Some which die and some which survive and some which thrive? Yes, of course. We could, for example, do a jillion Monte Carlo simulations and display the average portfolio value after 1 year, 2 years, 3 years etc. etc. Then we could look at the portfolio that's generated by a constant Annualized return and compare. Here's an example: We did 5000 simulations and calculated the average portfolio value after each of 40 years. (Portfolios that failed to survive counting as 0). That's the blue curve. Then we plot a portfolio that has a constant annual return equal to the Annualized return. That's the red curve. >And all that other stuff? Just to give you an idea of the variability, them's 30 portfolios selected at random from the 5000 simulations.
P.S.
Did I mention that we used a set of annual returns for the S&P500, from 1928 to 2000, selected at random to generate the 5000 portfolios.
We also assumed an annual inflation of 2.5%.
See? The constant annual return portfolio is in there among the others ... somewhere.

>And that makes you happy?
Yes.