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.
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?

That's one of them.
>How many questions do you have?
How high can you count?

Before we continue, let's do something else ...
>Do we have to?
Pay attention:

Let's follow just investor #j, from year n = 0 to year n = N:
To make the notation more sanitary, we'll use the following labels:

Pn is the sequence of portfolios, starting with some P0, like P0=\$1M   (for n = 1, 2, 3, ... N)
gn is the sequence of annual Gain Factors for this investor   (which varies, investor to investor)
in is the sequence of annual Inflation Factors   (which is common to ALL investors

For example:
Suppose W is the initial withdrawal rate (like W = 0.04 for 4%) so WP0 would be the initial withdrawal amount.
At the end of year 1, P0 has increased by the factor g1 (the first year gain factor) ... to g1P0.
Then our investor makes a withdrawal of  Wi1P0 (having been increased by the first year inflation factor i1).
Hence, at the end of year 1, P1 = g1P0 - Wi1P0
Simlarly, we can get the investor's portfolio at the end of year 2 (after a withdrawal of Wi2i1P0), namely:
P2 = g2g1P0 - Wg2i1P0 -Wi2i1P0 = g2g1P0 - W { g2i1 + i2i1 }P0

>I don't get it.

Since we're considering a constant inflation factor, all the in are equal: say in = I, so we continue and we get, at year n:

Pn = gngn-1...g1P0 - W { gngn-1...g2I + gngn-1...g3I2 + ... + In }P0

It's convenient to write this in millions of dollars, so P0 = 1.
Okay, we'll assume that every investor starts with 1 gummyBuck so we'll measure our portfolio value in gummyBucks.
As it happens, 1 gummyBuck = \$100K.
Remember, it doesn't matter whether we measure our portfolios in dollars or yen of lira or ...
>Okay, we put P0 = 1. Then?
Then:

[1]     Pn = gngn-1...g1 [1 - W { g1-1 I + g1-1g2-1 I2 + g1-1g2-1g3-1 I2 + ... + g1-1g2-1...gn-1 In } ]

Let's define Gn = gngn-1...g1 = the n-year Gain Factor.
Then Gn = (1+r1)(1+r2)...(1+rn) is the product of annual gain factors, r1, r2, ... rn being the annual returns.
It has nothing to do with the investor. It depends upon what sequence of (random) returns she got, over n years.

We can rewrite the above equation like so:

[2]     Pn = Gn [1 - W { I / G1 + I2 / G2 + I3 / G3 + ... + In / Gn } ]

Here we defined a Magic Sum:

[3]     gMS(n) = I / G1 + I2 / G2 + I3 / G3 + ... + In / Gn

We then rewrite like so:

[4]     Pn = Gn {1 - W gMS(n)}

For this investor, her withdrawal at year n is W In
Her current withdrawal rate is then:

[5]     W(n) = W In / Pn = W In / [Gn {1 - W gMS(n)}] = W { In / Gn} / {1 - W gMS(n)}

To simplify the notation ...
>zzzZZZ
Patience. We're getting close.

We define xk = W Ik / Gk so that
W gMS(n) = x1 + x2 + ... + xn = Σxk

Finally then:
 If r1, r2, ... rn are n annual returns (example: 0.12 for a 12% return) and I is the annual inflation factor (example: 1.035 for a 3.5% annual inflation) and W is the initial withdrawal rate (example: 0.04 for a 4% initial withdrawal rate) then: the current withdrawal rate, at year n, is:       W(n) = xn / {1 - Σxk} where xk = W Ik / Gk and Gk = (1+r1)(1+r2)...(1+rk) is the product of k successive annual gain factors, hence is the total k-year Gain Factor.

Uh ... one other thingy:
It may happen (for heavy losses) that, at some point, Σxk is greater than (or equal to) "1". This is a portfolio that hasn't survived, so we should ignore that portfolio ... and stop the calculations.

>Huh? Is that formula useful?
I have no idea.