Patient Reader:

If these SafeWithdrawalRates/MonteCarlo tutorials seem to wander about, it's because of ...

>Creeping senility?

Well ... yes. But, as KenM, says:
"The zen of SWRs is the contemplation of one's essential nature to the exclusion of all else as the only way of achieving pure enlightenment".

We're talking (among other things ) about Monte Carlo consistency.
>Consistency?
Okay. Here's the problem:

• Monte Carlo assumes some distribution of annual returns and annual withdrawals (increasing with inflation).
• After a jillion simulations, Monte determines the percentage of portfolios that survive for 40 years.
  (For example, suppose that a 4% initial withdrawal rate will result in a 95% survival, over 40 years.)
• To do this, Monte starts a jillion portfolios and applies a random return to each, deducts the withdrawal, then repeats this 40 times.
• Instead, we follow N investors, all with $1M portfolios, investing in the same stocks (with the prescribed return distribution).
• All N investors withdraw, initially, say, 4% which (according to Monte) gives a 95% survival for 40 years.
• Withdrawals increase with inflation, but are the same annual withdrawals for all investors.
• After 40 years, there should be 0.95N surviving portfolios (according to Monte) .
• Each year we ignore those investors whose portfolios have been reduced to $0. (They've "died".)
• Each year, for the portfolios that are still "alive", we apply a random return (from the prescribed return distribution).
• We do this for the remaining years and determine the number who are still alive (after 40 years).
• It should be 0.95N, right?
• If not, then we increase N to a jillion and do it again.
• It should be 0.95N, right?
• If not, then maybe Monte is inconsistent, right?

Okay, let's assume that Monte IS consistent.
We've already introduced some notation:

Mn[p] = the MC withdrawal rate that'll give a p% survival over n years F40(n) is the fraction of our original 40-year investors that survive to year n.

We conclude that:
With a M_n[p] withdrawal rate, p% of the original portfolios will survive n years... and (1-p)% will not.
In particular, if 95% survive 40 years ... then 5% will not.

At year n, there are still F₄₀(n)N of the original portfolios "alive".
We know the number which survive the remaining 40-n years. It's 0.95N.
Hence, of those portfolios that are still alive at year n,
the fraction that will survive the remaining 40-n years is: 0.95N / F₄₀(n)N = 0.95/F₄₀(n).
(We got this earlier, in Part I, but repeat it so we don't have to go back and peek.)

Now M_n[0.95] gives a withdrawal rate if we specify n, the number of years.
We should also be interest in determining the number of years when we specify the withdrawal rate, like so:

>So where's the 40 years stuff?
Well ... uh, the charts have nothing to do with our 40 years. They're inventions(!), but are meant to illustrate the relationship between "safe" withdrawal rates and years, n: "safe" meaning 95% probability of surviving n years.

If Sam has just retired and expects to drop dead in n years, he looks at the left chart.
It says that (for example), to last n = 15 years, he can withdraw 9.6% (initially).

On the other hand, if Sally wants to withdraw x% (initially), how many years can she count on (with 95% probability of surviving)? Sally would look at the right chart.
It says that (for example), if she withdraws 9.6% (initially), it'll last n = 15 years (with 95% probability of surviving).
Both charts are useful.
>They are?
Well, if not useful, then interesting.
>To whom?

There's a "DRAFT" spreadsheet which may be interesting. It looks like this with an explanation which looks like this. To download a ZIPd version, RIGHT-click and Save Target here. One interesting thing (for me) is that, if we choose a sufficiently large collection of investors (like 10,000), then the current withdrawal rates (for those portfolios that survive to year 10 or 20 or 30 or, eventually, to year 40) ... these withdrawal rates first increase (from, say, 4%) then decrease, like Figure 1. The chart assumes a constant inflation of 2% and random returns (for 10,000 investors) selected from a lognormal distribution with Mean = 9% and Volatility=15%. In Figure 2, the distribution of final, 40-year portfolios (for a $1.00 initial portfolio). In this Monte Carlo run, there was one lucky investor who ended up with $500, starting with $1.00. Alas, there were about a thousand investors who ended up with $0.00

Figure 1 Figure 2

It isn't surprising that, in the early years, the withdrawal rates are similar to the initial rate (since neither the withdrawals not the portfolios will have changed a great deal), but to see the withdrawal rates at the 40-year mark (for the 40-year survivors) looking similar to the initial rate ... now that's interesting, eh?
>Does the spreadsheet do the distribution, too?
No. If you unZIP the file, there'll be two spreadsheets.
The first (called SWR-MC.xls) generates a bunch of data, including withdrawal rates and portfolios for umpteen investors. The second, called SWR-MC-distribution.xls, will plot the withdrawal rates and portfolios.
>Huh?
You do this:

1. Run SWR-MC and stare in awe and amazement at the pretty pictures.
2. Copy a column (of eight available columns ... for portfolios or withdrawal rates)
3. Open SWR-MC-distribution and Paste the copied data into column B (starting at B2).
4. Click a GO! button and see the distribution develop (I hope).

The "distribution" spreadsheet looks like this (with the current withdrawal rates at the 30-year mark).
It may be more interesting to see the (current) withdrawal rate at the 40-year mark, for our 10,000 investors.

It looks like this:

They look like this:

That means, if x% of investors have a portfolio of P (from the lower chart), then x% will have a withdrawal rate of W/P (that's the upper chart).
>So?
So if I take the Portfolios from the lower chart and calculate 1/P₁, 1/P₂, etc. then the distribution of these numbers will look like the upper chart.
>So?
I'm thinking.
>About what?
Reciprocals ...

to continue