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?

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.

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:
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 ...

>Can you just jump to the end ... please!
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?
Try this spreadsheet where, by using the sliders, you can vary the parameters. (Click on the picture to download the spreadsheet.)

>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.

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?