Withdrawals ... and Life Expectancy

The other evening I was watching a bad TV movie and started thinking ...
>Now that was a mistake!
... about Safe Withdrawal Rates and the
notorious 4% rule and ...
>Not again! You've done that,
here and
here and ...
Yeah, but here's what I was thinking. You can withdraw 100% per year, from your retirement portfolio, and have
it last until you die. Did you know that?
>Sure, if you drop dead very soon.
Exactly! Remember that stuff we did about Life Expectancy?
Well, suppose we mix that stuff with the other stuff and ...
>And just add water and a pinch of salt.
Pay attention. Here we go:
 We begin our retirement with a portfolio worth $P and we withdraw $A every year.
 The withdrawal amount increases at the Inflation Rate which we assume is I (for 3% inflation, I = 0.03).
 Our annualized Portolio Return is R (for 9% return, R = 0.09).
>I hope you're not talking constant Inflation and Return?
Yes, for the time being. Let's continue:
 If we withdraw for just N years, the Maximum Rate of Withdrawal,
that's MRW = A/ P,
is given by the magic formula we obtained
here:
MRW = 1 / (x + x^{2} + x^{3} + ... + x^{N})
where x = (1+I)/(1+R)
 There's a formula for that sum; it can be rewritten:
MRW = (1/x) (1  x) / (1  x^{N})
Now the problem is to connect this Maximum Rate of Withdrawal (MRW)
to N, how long you'll live,
and for that we turn to another magic formula given here
from which we can generate a chart like Figure 1 (for males or females of any age) and ...
>Wait! What's that chart saying? I can't ...
It says that, starting with a thousand 65yearolds, some of them will die after 1 year or 2 years or 3 years or ...
>Yeah, yeah, I can see that, but it says that only about 44 will die by age 81. Are you kidding?
No! It says that 44 will die in their 81st year, and another 25 will die in their 90th year ... and so on.
>And if I add all those numbers I'd get 1000.
Well ... uh, 999 actually. There's one guy still hangin' around at age 105.
 Figure 1

Okay, so we superimpose the Number Dying and the MRWrates versus
N ... or, for a 65yearold, your age, as in Figure 2.
>You're assuming some return on your portfolio, eh?
Yes, and ...
>Looks like that MRW is approaching ... looks like 5%.
It's actually approaching 4.3% for this set of parameters.
>That 4.3% looks like the notorious safe withdrawal rate!
Yes, that's worstcase. Now listen carefully.
>That's for a male, eh? What about females or bigger inflation or ...?
 Figure 2

Patience. Here's what we want to do ...
When one uses Monte Carlo simulation one likes to say
things like:
"With a 99% probability such and such will happen."
We'll do the same here. We'll say:
"With a 99% probability, your portfolio will survive you, if you withdraw at such and such a rate."
Here's a picture of a spreadsheet which does this:
To play with this online spreadsheet, click on the picture above.
>But it's still got those constant Mean and Standard Deviation.
Yes ... so far. But I should point out that the portfolio gains are the Annualized return, not the Mean.
To get this we use the approximation noted
here, namely:
Annual Return = SQRT[ (1+Mean)^{2}  SD^{2} ]  1
How good is this approximation? Check this online spreadsheet: click!
>And what about random returns and ...
Check this online spreadsheet: click!
