Monte Carlo consistency
thanks to bpp for an interesting discussion

Okay, here's the problem:

1. We assume that withdrawals from a portfolio is initially a percentage W% of the initial portfolio, increasing with inflation at i%.
2. Further, we assume that annual returns are selected from some distribution with prescribed Mean, Standard Deviation, etc..
We'll refer to the chosen Distribution simply as D.
3. Further, we assume that the Monte Carlo probability of survival for k years is given by some magic formula: MC(D, W%, i%, k).
Since the survival rate would normally depend upon the number of Monte Carlo simulations,
we'll assume that the MC Probability MC is the result of an infinite number of simulations
This is what we do:
• We start a 100 jillion portfolios with the same starting value and withdraw from each the same amounts each year
starting with a withdrawal rate of, say 4%.
• Annual returns for each portfolio are selected from the same distribution of returns: D.
• The annual withdrawals increase with inflation - the same for each portfolio, say 3%.
• We do Monte Carlo simulations and find that, of the 100 jillion that start, only 95 jillion survive for 40 years.
• This Monte Carlo probability is MC(D, 4%, 3%, 40)
where D refers to the chosen return distribution and the 4%, 3% and 40 are the other pertinent parameters
(namely initial withdrawal rate, inflation and number of years of withdrawals).
• We then have:   MC(D, 4%, 3%, 40) = 0.95 or 95%.
• Suppose that the MC probability of surviving just the first10 years is 99%. That means:   MC(D, 4%, 3%, 10) = 0.99 or 99%.
• Of the 100 jillion portfolios that survive 40 years, only 99 jillion have survived 10 years.
• That means that 95 jillion or these 99 jillion surviving portfolios will last another 30 years.
• That means that, if we select one of these 99 jillion portfolios at random, the probability that it will be one of the surviving 95 jillion is 95/99 or 96%.
• Hence, the probability of picking a portfolio that survives another 30 years (from the 99 jillion still alive at the 10 year mark) is 96%.
 >I haven't the faintest idea what you're talking about! Okay, look at the 100 jillion portfolios Pick any one of the starting portfolios at random. The chances that it'll be one of the 40-year survivors is 95%. Now pick any one of the 100 jillion at the 10 year mark. The chances that it'll be one of the survivors (for the remaining 30 years) is still 95%. Now pick one of the 99 jillion portfolios that are still alive at the 10-year mark. The chances that it'll be a 40-year survivor is 96%. >So? My question is this:
We look at only those 99 jillion portfolios that survived the first 10 years.

We do Monte Carlo simulations for each of these 99 jillion portfolios (that'd be 99 jillion simulations) where we start with the 99 jillion portfolio values (at the 10-year mark) and the 99 jillion withdrawal rates (as a percentage of the 10-year portfolio value!) and keep the 3% annual inflation. Then we'd get 99 jillion survival rates for 30-years of withdrawals, one for each of the 99 jillion portfolios.

We'll call these   MC(D, W1, 3%, 30), MC(D, W2, 3%, 30), MC(D, W3, 3%, 30), etc.   where W1, W2, etc. are the withdrawal rates.

>Huh?
If one of these 99 jillion portfolios is \$123,400 (at the 10-year mark) and the withdrawal amount (at the 10-year mark - this is the same for ALL portfolios) is \$5,678, then the withdrawal rate, as a percentage of the portfolio value is 5,678/123,400 or 4.6% and we'd follow this particular portfolio for another 30 years (a la Monte Carlo) and get a Monte Carlo survival rate of MC(D, 4.6%, 3%, 30).

>For just that one portfolio?
Yes, but how many of these 99 jillion will survive another 30 years?

>I'd say 95 jillion. Am I right?
Actually I'm interested in what ol' Monte would say. When we asked him at t = 0, he said 95 jillion would survive until t = 40 years.
Now we ask him again, at t = 10 years.

>And what does he say?
There's a probability MC(D, W1, 3%, 30) that the first of the 99 jillion portfolios will survive the remaining 30 years,
and a probability MC(D, W2, 3%, 30) that the second portfolio will survive the remaining 30 years,
and a probability MC(D, W3, 3%, 30) that the third portfolio will survive the remaining 30 years
... and so on.

The fraction surviving the additional 30 years should be:
[MC(D, W1, 3%, 30)+MC(D, W2, 3%, 30)+MC(D, W3, 3%, 30)+...] / [99 jillion]

Of course, that 99 jillion in the denominator is just the 10-year survivors, namely MC(D, 4%, 3%, 10) x 100 jillion (which, in our example, is 99 jillion).

So we have: The fraction surviving the additional 30 years should be:
[MC(D, W1, 3%, 30)+MC(D, W2, 3%, 30)+MC(D, W3, 3%, 30)+...] / [MC(D, 4%, 3%, 10) x 100 jillion]

>I'm completely confused!
Maybe it's simpler if we start with P portfolios (at t = 0 years).
The number surviving n years is, according to Monte:   N = MC(D, 4%, 3%, n)P
Example n = 40 : MC(D, 4%, 3%, 40)P = 0.95P or 95% of them survive all 40 years.
After only m years the number surviving is (according to Monte): M = MC(D, 4%, 3%, m)P
Example m = 10 : MC(D, 4%, 3%, 10)P = 0.99P or 99% of them survive the first 10 years.
Of these M survivors (at year m), N should survive the remaining n-m years ... to year n.
That means that, if we pick one of these M survivors (at year m) at random, the probability that it's one of the N survivors (at year n) is N / M.

If we consider each of these m-year survivors (and do the Monte Carlo thing), we'd get a survival rate for each (for the remaining n-m years), namely:
MC(D, W1, 3%, n-m), MC(D, W2, 3%, n-m), MC(D, W3, 3%, n-m), ... MC(D, WM, 3%, n-m)
where M is the number of m-year survivors, namely M = MC(D, 4%, 3%, m)P ... as we indicated above.

The fraction of m-year survivors (that's the number M) that last another n-m years should then be:
[MC(D, W1, 3%, n-m)+MC(D, W2, 3%, n-m)+MC(D, W3, 3%, n-m)+ ... +MC(D, WM, 3%, n-m)] / M.

According to Monte, we KNOW the fraction of these M that survive the additional n-m years, namely   N / M.

In other words, we should have:
[MC(D, W1, 3%, n-m)+MC(D, W2, 3%, n-m)+MC(D, W3, 3%, n-m)+ ... +MC(D, WM, 3%, n-m)] / M = N / M.

In other words, we should have:
 [A]     MC(D, W1, 3%, n-m)+MC(D, W2, 3%, n-m)+MC(D, W3, 3%, n-m)+ ... +MC(D, WM, 3%, n-m) = N = MC(D, 4%, 3%, n)P     where M = MC(D, 4%, 3%, m)P

That gives us a functional equation that should be satisfied by the Monte Carlo function: MC(D, W, i, k).

>A what?
A functional equation.
If f(xy) = f(x) + f(y), what's the function f?
>Log(x)?
That's good, now your problem is to solve [A] for the function MC.
>But where do I get the numbers W1 and W2 and so on?
Ask Monte. However I should point out that if all investors started with a \$1.00 portfolio and if we let Pk(m) be the portfolio of investor k at the m-year mark,
then Wk = W (1+i)m / Pk(m) where W (1+i)m is the initial withdrawal, increased by m years of inflation.
We'd want to know the distribution (over k) of Pk(m), the \$1.00 portfolios after m years.
>So ... what's the solution to [A]??
I have no idea, but I'm thinking ...