Safe Withdrawal Rates and Monte Carlo ... continuing from Part I

We're following K investors for N years.
They start together and invest in the same stocks (with prescribed return distribution) for N years.
They all start with the same withdrawal rate, the withdrawal amount increasing with inflation (which we assume is fixed).
They all withdraw at some (initial) rate that (hopefully) will last N years.
Here are our labels:
Mn[p] is the Monte Carlo withdrawal rate which gives a p% probability of surviving n years.
A(n) is the withdrawal amount at year n   (n = 1, 2, 3, ... N)
(It starts at some amount, increases with inflation and is the same for all investors.)
Pj(n) is the size of portfolio at year n for investor #j   (j = 1, 2, 3, ... K)
Wj(n) is the current withdrawal rate, at year n, for investor #j   (j = 1, 2, 3, ... K)
so Wj(n) = A(n)/Pj(n)
FN(n) is the fraction of our K investors that survive to year n.

>So if we follow them for 30 years, then we're talking about N = 30 and F30(n), right?
Right, and here's a sample set of charts, following investor #123, where we assume parameters:
Initial Portfolio = $1M
A normal distribution of annual returns with Mean = 10%, Standard Deviation = 20%
Inflation Rate = 3%
Initial Withdrawal Rate = 4% ... thereafter, the withdrawal amounts increase at 3% per year

Then, at year n:
His Portfolio is P(n)   with a random set of returns
His withdrawal amount is A(n)   $40K increasing at 3%
His current withdrawal rate is W(n) = A(n)/P(n)   ... also random

>But what if somebuddy's portfolio doesn't survive for 40 years?
It'd look like this:

Here, after 31 years, there's not enough portfolio left to accommodate the required withdrawal dollars (about $100K).
The withdrawal rate goes to infinity.
>That investor is dead, eh?
I wouldn't put it that way, but yes. He's out working again.

Okay, here's what we'll do.
Suppose all our investors buy 10,000 shares of stock worth $10.
After one year the distribution of stock prices is like Figure 1a.
Each investor will get one of these stock prices, with many getting prices near $10 ...

>Like Figure 1a.
Yes.
The various portfolios will look like Figure 1b , where, for most portfolios, the 10,000 shares are worth something more than the original $100K.

>But 1b looks like 1a.
Of course. The horizontal axis is just relabelled. Figure 1b shows the various portfolios before the annual withdrawal.

Now they all withdraw, say, 4% of the orginal $100K. That's $4K.
Let's assume that 4% is the Monte Carlo 40-year 95% survival rate.

>4%, increased by a year's inflation?
Sure, if you like ... but I'm just trying to make a point here. The exact amount isn't important. Let's just say it's $4K. Then the umpteen portfolios, after the withdrawal, look like Figure 1c

>That looks like Figure 1b.
Of course. We just shifted the graph to the left by $4K.

>I get it! You just look at the 1-year stock price distribution, relabel the axis then ...
Then shift left by the current withdrawal amount.

>And you do this again and again, right?
But the current withdrawal amount changes. That's your inflation increase. Also, investors have a different portfolio and are withdrawing at a rate defined by their original portfolio. As a percentage of the current portfolio, they're different. Eventually, some of the graph lies in negative territory and ...


Figure 1a

Figure 1b

Figure 1c
>Negative territory?
Yes, with negative portfolios. Well, actually, they're portfolios worth $0.

>Aaah ... they're the dead guys.
Yes, so here's what we do, each year.

  1. Look at the surviving portfolios only.
  2. Apply a random set of annual returns to these survivors (distributed as in Figure 1a).
  3. Subtract the current inflation-adjusted withdrawal amount
    (based upon the initial portfolios and common to all investors, even though their portfolios differ).
  4. Repeat steps 1, 2 and 3.
After 40 years, we see how many of the original portfolios survive and ask:
Have 95% survived?

Yes, but when that occurs, we have fewer investors still buying that stock, fewer surviving portfolios, fewer investors, so ...

>Bury them!
Figure 1d shows what would happen if we continue with this shifting left by the current withdrawal amount, for a few years. Some portfolios haven't survived.


Figure 1d

>But some still have $100K ... or more!
Yes, but now we have to ask:
"Do we continue shifting left, or do we start anew with fewer investors having different portfolios?"

>That's the question, but what's the answer?
If the distribution of portfolios is f(P), after one year but BEFORE a withdrawal of $W (as in Figure 1b) , then AFTER the withdrawal we have the graph of f(P+W). That's Figure 1c. It's Figure 1b shifted left.

>And Figure 1d?
That'd be the graph of f(P+W1+W2+...+Wn), after n withdrawals of amounts W1, W2,... Wn.

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