Patient Reader:
There's this thing called Safe Withdrawal Rate which says that, if you withdraw from a portfolio a certain percentage each year (a percentage of the original portfolio), the withdrawal increasing annually with inflation, then there's a SWR which will enable your portfolio to survive umpteen years with a high probability.

For example, if you want your portfolio to survive 40 years with a 95% probability, the SWR might be 4.0%, however, if you only want it to survive 10 years with 80% probability, maybe SWR = 16%.

Of course, the value of SWR will depend upon what you're investing in. However, one shouldn't think that the SWR is a real-life object. It's a mathematical construct and no amount of mathematical sleight of hand can predict the future.

A particular (mathematical) ritual which can generate SWRs is Monte Carlo simulation.
It will give a specific number for the SWR once you define the distribution of returns (example: lognormal with Mean=10% and Volatility=25%), an inflation rate (example: 3.5%), a number of years (example: 40) and a probability of portfolio survival (example: 95%).

My interest is to more clearly understand these Monte Carlo SWRs. The Sam & Sally questions started all this ... and I often don't know how to go about answering certain questions or, indeed, what questions to ask. For this reason, these tutorials are ... well, unrehearsed, unstructured and (perhaps) unreadable. Hopefully, by the end (if there is an end) I will succeed in acquiring a better understanding.

>I doubt it!
One question of interest is:
We follow an investor who's done the Monte Carlo thing and thinks he's got a 95% probability of surviving 40 years. Ten years later, his withdrawals are defined by his assumed inflation rate, but his portfolio may be almost anything. What are the chances that his current withdrawal rate is a Monte Carlo SWR? Because he's only got 30 years left (in his 40-year lifespan) one expects a larger SWR.

What might happen in real-life is, of course, irrelevant to my purposes. I just want to study the progression of SWRs for investors. The figure to the right shows what happens to the Maximum Rate of Withdrawal Rate (MRW) if you started withdrawing from an S&P500 portfolio in the year indicated ... and what the MRW is ten years later. For example, you'll note that had you started in 1960 the maximum withdrawal rate (to last 40 years) is 4.7%. A larger withdrawal rate and your portfolio would not have survived to year 2000. Ten years later (in 1970) the MRW was only 4.2% (to last another 30 years).

>Inflation?
For the S&P example, I used the actual inflation as determined by the Consumer Price Index, but my purpose in noting the S&P stuff is simply to explain that I'm NOT interested in the complications of real-life, but rather in ...

>So, do something.

Here's the problem we want consider:
You have a portfolio of stocks and you need to know the "correct" withdrawal rate so that there's a 95% probability that your portfolio will last for 40 years ... so you do a jillion Monte Carlo simulations (assuming some distribution of returns) and find that the Monte Carlo Rate (MCR) is 4.7%.

Let's call this M₄₀[0.95] so, for this example, M₄₀[0.95] = 0.047 or 4.7%.
>Huh?
We're just introducing our notation, okay?
Here's another example:
To achieve a 97% probability of surviving 25 years, Monte Carlo says you can withdraw 4.3%, so ...
>So M₂₅[0.97] = 0.043, eh?
Yes. Of course, if you withdraw at a lower rate, then the probability will be larger than 97%, so M₂₅[0.97] gives the largest withdrawal rate. Don't withdraw more, else you won't get that 97% probability.

In fact, what we're talking about isn't any ACTUAL, real-life withdrawal rate. It's what you get by assuming some fixed inflation rate and a distribution of stock returns (like lognormal with prescribed Mean and Standard Deviation), doing an infinite number of Monte Carlo simulations and seeing what withdrawal rate ... that's (Initial Withdrawal)/(Initial Portfolio) ... seeing which withdrawal rate results in 85% survival over 30 years. That'd be M30[0.85]. Figure 1 shows what happens to a real-life slice and dice portfolio starting in 1929 and running out of steam in 37 years. It's wild and wooly.

Figure 1

Our withdrawal rates: MN[p] = the rate that'll give a p% survival over N years will be smooth, like maybe Figure 2a where, for a 60% survival over 30 years, we withdraw at 6.0%. >The red dot? The red dot. >What inflation rate did you use? Figure 2a is just an invention ... just to illustrate what we mean by MN[p]. Figure 2b is also an invention, to illustrate how MN[0.95] might change with N, and the yellow dot ... >Wait! Don't tell me! That says that, with 95% probability, a withdrawal rate of ... it looks like 11%, that'll last 10 years. Right? Right. That's what it says, but I just invented ... >I know! I know! You just invented the chart to illustrate.

Figure 2a Figure 2b

In any case, we'll be talking more about M_N[p].

>So what's the problem?
If we follow a particular investor, her withdrawal rate will change as the years go by as her portfolio changes. In the final year (if her portfolio lasts that long!) her withdrawal rate may be 100%. What I mean is, she started with M₄₀[0.95], a rate that would give a 95% survival over 40 years, but in the last year she withdraws everything so her current rate, that last year, is 100%.

>So what's the problem?
We follow a jillion investors who start with a rate of, say, M₄₀[0.95]. That'd give them 95% survival over 40 years. After n years they still have 40-n years to go. How do their current withdrawal rates ... that's (Current Withdrawal)/(Current Portfolio) ... how does that compare to the Monte Carlo rates, as the years go by.

Okay, so we follow K portfolios for 40 years, starting at $1M and investing in assets with a common distribution of returns.
>Huh?
We're going to look at the evolution of these portfolios, to compare them, and we don't want to compare apples to ...
>To oranges?
Yes.

• Each investor withdraws at the notorious Monte Carlo Safe rate of 4%, or $40K ... increasing with inflation.
  (At this rate, the probability of surviving 40 years is, say^*, 95%, so M₄₀[0.95] = 0.04 )
  * or 99% or 94.567% or whatever.
• Let's assume that Monte is right ... and exactly 0.95K do survive 40 years.
• We assume K possible sequences of stock returns with each investor assuming one of these sequences.
  (This gives us a look into possible future evolutions of a $1M portfolio.)
• At year n, all investors will be withdrawing the same amount
  (the $40K, increased by some common inflation factor).
  We'll call this amount $A(n).
  But their portfolios will be different (depending upon which of the K return sequences that portfolio was following).
  For the jth investor, we'll call his portfolio P_j(n)   for j = 1, 2, 3, ... K.
• At year n we look at what the current withdrawal rate is, for each of our K investors.
• The K withdrawal rates are then: W_j(n) = $A(n)/P_j(n)   j = 1, 2, 3, ...K.
        For example, at year 10, investor j = 123 may have a portfolio P₁₂₃(10) = $456,000
        and the withdrawal amount at year 10 (common to all investors) might be
        $A(10) = $44,185 (which is $40K with 10 years of 2% inflation)
        giving a current withdrawal rate for investor 123 at year 10:
        W₁₂₃(10) = 44185/456000 = 0.0969 or 9.69%

Now come the big questions:

1. At year n, 0.95K portfolios are still there (since they last for the full 40 years, eh?)
   and some of the original K portfolios have already died
   and some will die before the remaining 40-n years have elapsed.
2. Suppose that D₄₀(n) have already died, so there are still K-D₄₀(n) portfolios alive (at year n).
   (Remember: Only 0.95K of these K-D₄₀(n) will survive to year 40.)
3. Note: We call it D₄₀(n) because it's the number who died after n years in a 40-year sequence of stock returns.
4. The fraction of these K-D₄₀(n) portfolios that survive to year 40 is then 0.95K/(K-D₄₀(n)).

  0.95K/(K-D₄₀(n)) = 0.95/(1 - D₄₀(n)/K)
  and since D₄₀(n)/K is the fraction of original portfolios that died by year n (from the original K portfolios)
  then F₄₀(n) = 1 - D₄₀(n)/K is the fraction of original portfolios that survived to year n
  hence, at the n-year mark, the fraction of these portfolios that survive an additional 40-n years is 0.95/F₄₀(n).

We make a big thing of this:

If F40(n) is the fraction of 40-year portfolios that have survived to year n, then the fraction that will survive the additional 40-n years is 0.95/F40(n)

>Huh?
If 97% of the original portfolios survived to year 10, then 0.95/0.97 = 0.979 so we can conclude that 97.9% of these portfolios will survive to the 40th year. That's an additional 30 years.

Now it's fair to ask how the current withdrawal rates for these portfolios (that have survived for 10 years) ... how these withdrawal rates compare to M₃₀ [0.979].

In general:

• We now look at those remaining K - D₄₀(n) portfolios and ask:
  "What is the Monte Carlo withdrawal rate so that the probability of surviving 40-n years is 0.95/F₄₀(n)?"
• We already have a label for this MC rate. It's M_40-n[0.95/F₄₀(n)].

Which brings us to our questions:

1. How does M_40-n[0.95/F₄₀(n)] compare to the investors' withdrawal rates: W_j(n) ?
2. What FRACTION of investors that have survived to year n have a rate less than M_40-n[0.95/F₄₀(n)] ?
3. How does this FRACTION compare to 0.95/F₄₀(n) ?

• F₄₀(n), the fraction surviving, starts at n = 0 with F₄₀(0) = 1.00 (since all will survive 0 years!)
  and ends at n = 40 with F₄₀(40) = 0.95 (since we've assumed that 95% survive all 40 years).
• Your definition of a "Safe Withdrawal Rate" may change the 0.95
• And you may be interested in 30 instead of 40 years
  ... so feel free to change these numbers, eh?  

Now that we've introduced all the symbols, like $A(n), P_j(n), W_j(n), M_n(p) etc. we'll investigate these questions.
>zzzZZZ
Aren't you going to ask: "What questions?"
>zzzZZZ

Okay, here's some pictures which illustrate what these guys might look like for a 99% survival rate for 40 years
(tho' I haven't yet done the analysis so they're inventions ... to illustrate):

F40(n) the fraction surviving to year n (example: 99.6% survive to year n = 29)

0.99/F40(n) fraction of year n survivors that'll survive to year 40 (example: 99.4% of year 29 survivors will last to year 40)

M40-n[0.99/F40(n)] MC withdrawal rates to last 40-n years with probability 0.99/F40(n)

>zzzZZZ
Although I've just invented these examples, notice that 99.6% of the original portfolios will last to year 29.
Of those who have lasted this far, 99.4% will last the additional 11 years to year 40.
That makes 0.996*0.994 = 0.990 of the original portfolios that last to year 40.
See?
>zzzZZZ
Look at the last chart, at year 29.
The fraction of original portfolios that last this long is 99.6% - that's the first chart.
99.4% of these will last the additional 11 years - that's the second chart.
The Monte Carlo withdrawal rate which gives a 99.4% probability of lasting that 11 years is 19% - the last chart.
You can withdraw 19% per year and there's a 99.4% probability that your portfolio will last 11 years.
Don't take these numbers seriously. They're inventions, remember? Just to illustrate what we're talking about.

BUT, if the Monte Carlo withdrawal rate really were 19% (to last 11 years with 99.4% probability), what are our survivors withdrawing, at year 29? Are the survivors withdrawing more or less ... or what?
>Huh? Survivors? Is that some TV reality show?
Go back to sleep.

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