The Great SWR Investigation-Part 2
wanderer:
That stuff on past-future
http://home.golden.net/~pjponzo/past-future.htm
was written months ago, long before I knew NFB existed.
I was motivated by a silly discussion on Morningstar where posters kept saying that the past is of no value in predicting the future, then suggested some optimal stock/bond ratio (as though they were born with this information).
The tutorial has nothing to do with you or NFB or these discussions.
When I read your comment about historically generated stuff, it fit right in ... so I added it. The comment had no "author". It was NOT referring to your financial philosphy. Neither the comment nor the tutorial had anything to do with you. It was a neat sentence that just fit in.
Anyway, I've removed it, since you seem to think that the tutorial had you in mind. It didn't.
This strange game of "you said this" and "that's not what I meant" and "but look at your post on May xx" and "you quoted me out of context" ... that game ain't no fun.
That stuff on past-future
http://home.golden.net/~pjponzo/past-future.htm
was written months ago, long before I knew NFB existed.
I was motivated by a silly discussion on Morningstar where posters kept saying that the past is of no value in predicting the future, then suggested some optimal stock/bond ratio (as though they were born with this information).
The tutorial has nothing to do with you or NFB or these discussions.
When I read your comment about historically generated stuff, it fit right in ... so I added it. The comment had no "author". It was NOT referring to your financial philosphy. Neither the comment nor the tutorial had anything to do with you. It was a neat sentence that just fit in.
Anyway, I've removed it, since you seem to think that the tutorial had you in mind. It didn't.
This strange game of "you said this" and "that's not what I meant" and "but look at your post on May xx" and "you quoted me out of context" ... that game ain't no fun.
JWR1945:
About my question: "Anybuddy recognize the curve (outlining the final portfolio distribution)?", you respond:
Actually, I have no idea what that curve is ... but I'm workin' on it :^)
I have a math-type description of a final buy-and-hold portfolio (which I found when I was studying Ito calculus), but with monthly withdrawals? T'ain't easy (for me, at least).
Once upon a time, here:
http://home.golden.net/~pjponzo/SWR-2.htm#SHIFT
I thought I could just take the annual stock distribution after N=1 year, shift it left by the annual withdrawal, then continue for N=2, 3, ...
Silly me.
About my question: "Anybuddy recognize the curve (outlining the final portfolio distribution)?", you respond:
My best guess is that gummy wants someone to say that it is a lognormal curve and so forth ...
Actually, I have no idea what that curve is ... but I'm workin' on it :^)
I have a math-type description of a final buy-and-hold portfolio (which I found when I was studying Ito calculus), but with monthly withdrawals? T'ain't easy (for me, at least).
Once upon a time, here:
http://home.golden.net/~pjponzo/SWR-2.htm#SHIFT
I thought I could just take the annual stock distribution after N=1 year, shift it left by the annual withdrawal, then continue for N=2, 3, ...
Silly me.
This strange game of "you said this" and "that's not what I meant" and "but look at your post on May xx" and "you quoted me out of context" ... that game ain't no fun
I agree with your comment here, Gummy. My sense is that every single person who posts on this board agrees.
Yet the problem continues to pop up. We need to step back and ask ourselves "Why?" I have been at this long enough now that I think I can provide some insights.
Some posters are coming to this discussion with a preconception that the conventional SWR analysis is more or less right and others do not share that preconception, and still others are somewhere in the middle. This is what is causing the problem.
Never did I make any personal comment in my discussions at the other board. I put forward a post that proved to be highly controversial precisely because it was so on-topic. There are posters on the other board who have expressed the viewpoint that the founder of that board "invented" the concept of early retirement by offering a specific plan for achieving it safely. I was saying that that plan doesn't work in the real world, that it is all spreadsheets and calculations, but that those spreadsheets and calculations don't provide the information needed to craft a plan likely to prove successful in many circumstances.
I am not aiming to "misrepresent" anything that anyone has said. I am trying to point out serious flaws in an analytical methodology. If I am wrong, people should be able to present reasons why I am wrong. I have been at this for over 12 months now, and no one has presented an effective argument against what I am saying yet. The longer this goes on without anyone being able to do that, the more confidence I gain that the conventional methodology really is as flawed as I say it is.
If I am right, then people should take the matter more seriously than they have so far. If the conventional methodology is fundamentally flawed, then corrections are needed. It is not a good answer to say "oh, well, nothing is perfect." We all know and accept that nothing is perfect. But there are people out in the world who are using the conventional SWR analysis to plan their retirements. If I am right, some of those retirements are in great danger of going bust. We are under a responsibility to let people know this, in the event that we come to agree that I really am right in what I am saying.
I have zero interest in "winning" the argument in the sense of me being shown to have been right and others being shown to have been wrong. That has never been the point of this. The point is to determine how a SWR analysis should be done for it to provide valid answers to the question "What allocation and what take-put percentage is safe?" It is an important question, and we should all be interested in coming up with the right answer according to what the data reveals, not just any old guess that sounds plausible at some particular moment in time.
The conventional methodology provides a guess that is so far off the mark in some circumstances that it is presents a serious danger. There is a chance that people are going to believe that the numbers generated by the conventional analysis are in the right ballpark and that their retirements are going to go bust as a result. There are circumstances in which the numbers produced by the conventional analysis are not in the same ballpark as the numbers that a valid methodology would produce.
I think that our biggest problem is that many people are being blocked by their preconceptions from even giving serious consideration to the idea that the conventional methodology simply does not do the job. We are complicating things immensely on our efforts to avoid discussion of the real question on the table. The question is--Is the conventional methodology valid? I say no. I have no objections to the idea of others saying "yes." But it seems to me a time-waster for us to even discuss the answer "it doesn't matter." It matters a lot. If the conventional methodology is as flawed as I say, we should refrain from any use of SWR analysis altogether. A tool that produces such widely varying results, depending on whether you look at all the data or only selective bits of it, is a dangerous tool.
We need to try to set our preconceptions aside and consider the root questions. What it is we are trying to obtain from a SWR analysis? Does the conventional methodology provide what we are trying to obtain or does it often generate numbers that mislead us as to what the historical data says rather than inform us as to what it really says?
I say that the purpose is to use data to assess the probabilities of various future possibilities. I say that it is dangerous to try to do this without looking at all the data that bears on the question. I say that the only valid answer is the one based on an examination of as much of the data bearing on the question as is available for our use. We do not possess access to enough data to provide a perfect answer. But we possess access to enough data to provide far more accurate answers than the ones generated by the conventional methodology.
I agree with your comment here, Gummy. My sense is that every single person who posts on this board agrees.
Yet the problem continues to pop up. We need to step back and ask ourselves "Why?" I have been at this long enough now that I think I can provide some insights.
Some posters are coming to this discussion with a preconception that the conventional SWR analysis is more or less right and others do not share that preconception, and still others are somewhere in the middle. This is what is causing the problem.
Never did I make any personal comment in my discussions at the other board. I put forward a post that proved to be highly controversial precisely because it was so on-topic. There are posters on the other board who have expressed the viewpoint that the founder of that board "invented" the concept of early retirement by offering a specific plan for achieving it safely. I was saying that that plan doesn't work in the real world, that it is all spreadsheets and calculations, but that those spreadsheets and calculations don't provide the information needed to craft a plan likely to prove successful in many circumstances.
I am not aiming to "misrepresent" anything that anyone has said. I am trying to point out serious flaws in an analytical methodology. If I am wrong, people should be able to present reasons why I am wrong. I have been at this for over 12 months now, and no one has presented an effective argument against what I am saying yet. The longer this goes on without anyone being able to do that, the more confidence I gain that the conventional methodology really is as flawed as I say it is.
If I am right, then people should take the matter more seriously than they have so far. If the conventional methodology is fundamentally flawed, then corrections are needed. It is not a good answer to say "oh, well, nothing is perfect." We all know and accept that nothing is perfect. But there are people out in the world who are using the conventional SWR analysis to plan their retirements. If I am right, some of those retirements are in great danger of going bust. We are under a responsibility to let people know this, in the event that we come to agree that I really am right in what I am saying.
I have zero interest in "winning" the argument in the sense of me being shown to have been right and others being shown to have been wrong. That has never been the point of this. The point is to determine how a SWR analysis should be done for it to provide valid answers to the question "What allocation and what take-put percentage is safe?" It is an important question, and we should all be interested in coming up with the right answer according to what the data reveals, not just any old guess that sounds plausible at some particular moment in time.
The conventional methodology provides a guess that is so far off the mark in some circumstances that it is presents a serious danger. There is a chance that people are going to believe that the numbers generated by the conventional analysis are in the right ballpark and that their retirements are going to go bust as a result. There are circumstances in which the numbers produced by the conventional analysis are not in the same ballpark as the numbers that a valid methodology would produce.
I think that our biggest problem is that many people are being blocked by their preconceptions from even giving serious consideration to the idea that the conventional methodology simply does not do the job. We are complicating things immensely on our efforts to avoid discussion of the real question on the table. The question is--Is the conventional methodology valid? I say no. I have no objections to the idea of others saying "yes." But it seems to me a time-waster for us to even discuss the answer "it doesn't matter." It matters a lot. If the conventional methodology is as flawed as I say, we should refrain from any use of SWR analysis altogether. A tool that produces such widely varying results, depending on whether you look at all the data or only selective bits of it, is a dangerous tool.
We need to try to set our preconceptions aside and consider the root questions. What it is we are trying to obtain from a SWR analysis? Does the conventional methodology provide what we are trying to obtain or does it often generate numbers that mislead us as to what the historical data says rather than inform us as to what it really says?
I say that the purpose is to use data to assess the probabilities of various future possibilities. I say that it is dangerous to try to do this without looking at all the data that bears on the question. I say that the only valid answer is the one based on an examination of as much of the data bearing on the question as is available for our use. We do not possess access to enough data to provide a perfect answer. But we possess access to enough data to provide far more accurate answers than the ones generated by the conventional methodology.
If the conventional methodology is as flawed as I say, we should refrain from any use of SWR analysis altogether.
What I intended to say with this sentence was the following:
If the conventional methodology is as flawed as I say, and we are not open to making the adjustments needed to make it a valid analytical tool, we should refrain from any use of SWR analysis altogether.
What I intended to say with this sentence was the following:
If the conventional methodology is as flawed as I say, and we are not open to making the adjustments needed to make it a valid analytical tool, we should refrain from any use of SWR analysis altogether.
gummy
My (current) sandbox is here:
http://home.golden.net/~pjponzo/SWR-3.htm
At the bottom is a link to a file you can download. It unZIPs to two spreadsheets. One gives the distribution of portfolios (after 40 years). This "distribution" spreadsheet will do the earlier years too if you just copy/paste the portfolios from the SWR-MC spreadsheet.
Thanks.
You have more confidence in my computing abilities than I do, I think
Anything more intellectually challenging than pressing the F9 key with one finger is likely to overextend my limited grey cell capacity. However I'll try.
Thanks again
My (current) sandbox is here:
http://home.golden.net/~pjponzo/SWR-3.htm
At the bottom is a link to a file you can download. It unZIPs to two spreadsheets. One gives the distribution of portfolios (after 40 years). This "distribution" spreadsheet will do the earlier years too if you just copy/paste the portfolios from the SWR-MC spreadsheet.
Thanks.
You have more confidence in my computing abilities than I do, I think
Anything more intellectually challenging than pressing the F9 key with one finger is likely to overextend my limited grey cell capacity. However I'll try.
Thanks again
KenM
Never try to teach a pig to sing. It wastes your time and annoys the pig.
Never try to teach a pig to sing. It wastes your time and annoys the pig.
I've been following 10,000 investors, each (initially) with a $1.00 portfolio and a Withdrawal Rate of 4%, increasing at 2% per year (that's inflation) and getting a random annual return (a la Monte Carlo, selected from a lognormal distribution, Mean Return = 9%, Standard Deviation = 15%).
I have the distribution of "final" portfolios and the distribution of "final" withdrawal rates (after 40 years).
The pictures are here (at the end):
http://home.golden.net/~pjponzo/SWR-3.htm
Why do I do this?
It's my (perhaps feeble) attempt to understand how the "final", 40-year distributions depend upon the assumptions (Mean Return, Inflation rate, etc. etc.).
I have the distribution of "final" portfolios and the distribution of "final" withdrawal rates (after 40 years).
The pictures are here (at the end):
http://home.golden.net/~pjponzo/SWR-3.htm
Why do I do this?
It's my (perhaps feeble) attempt to understand how the "final", 40-year distributions depend upon the assumptions (Mean Return, Inflation rate, etc. etc.).
gummy
Thanks for the update.
I emphatically agree with the highlighted statements.
Have fun.
John R.
Actually, I have no idea what that curve is ... but I'm workin' on it :^)
I have a math-type description of a final buy-and-hold portfolio (which I found when I was studying Ito calculus), but with monthly withdrawals? T'ain't easy (for me, at least).
Thanks for the update.
I emphatically agree with the highlighted statements.
Have fun.
John R.
I've been following 10,000 investors, each (initially) with a $1.00 portfolio and a Withdrawal Rate of 4%, increasing at 2% per year (that's inflation) and getting a random annual return a la Monte Carlo
I think I've probably brought this up several times in different ways (sorry for being repetitive) but I still don't have it clear in my mind.
If I look at gummy's latest excellently(my word) feeble(his word) sandbox stuff, I see the chart
The point indicated is at 15 years at a 95% safe 9.6% withdrawal. Now is this year 25 of a 40 year period starting with a $1.00 portfolio and the 95% SWR of 9.6% relates to the year 25 current portfolio value? Or does this solely relate to the start of a new 15 year period, with a starting $1.00 portfolio and an SWR of 9.6%? Or does it apply to both?
Even with grey cells in overdrive I still can't work it out
I think I've probably brought this up several times in different ways (sorry for being repetitive) but I still don't have it clear in my mind.
If I look at gummy's latest excellently(my word) feeble(his word) sandbox stuff, I see the chart
The point indicated is at 15 years at a 95% safe 9.6% withdrawal. Now is this year 25 of a 40 year period starting with a $1.00 portfolio and the 95% SWR of 9.6% relates to the year 25 current portfolio value? Or does this solely relate to the start of a new 15 year period, with a starting $1.00 portfolio and an SWR of 9.6%? Or does it apply to both?
Even with grey cells in overdrive I still can't work it out
KenM
Never try to teach a pig to sing. It wastes your time and annoys the pig.
Never try to teach a pig to sing. It wastes your time and annoys the pig.
KenM
You are seeing a work in progress. Right now, we are looking into this circumstance (when the answer is yes):
Now is this year 25 of a 40 year period starting with a $1.00 portfolio and the 95% SWR of 9.6% relates to the year 25 current portfolio value?
It turns out that you have asked some incredibly important and significant questions. Early answers were surprising and gummy is retracing his steps very carefully so that we understand exactly what is going on and to make sure that there are no errors.
When we are able to answer your original questions, we will have advanced our knowledge substantially.
Have fun.
John R.
I think I've probably brought this up several times in different ways (sorry for being repetitive) but I still don't have it clear in my mind.
You are seeing a work in progress. Right now, we are looking into this circumstance (when the answer is yes):
Now is this year 25 of a 40 year period starting with a $1.00 portfolio and the 95% SWR of 9.6% relates to the year 25 current portfolio value?
It turns out that you have asked some incredibly important and significant questions. Early answers were surprising and gummy is retracing his steps very carefully so that we understand exactly what is going on and to make sure that there are no errors.
When we are able to answer your original questions, we will have advanced our knowledge substantially.
Have fun.
John R.
KenM:
With regard to the charts that you displayed, they're inventions(!), but are meant to illustrate the relationship between "safe" withdrawal rates and years, n: "safe" meaning 95% probability of surviving n years.
If Sam has just retired and expects to drop dead in n years, he looks at the left chart.
It says that (for example), to last n = 15 years, he can withdraw 9.6% (initially).
On the other hand, if Sally wants to withdraw x% (initially), how many years can she count on (with 95% probability of surviving)?
Sally would look at the right chart.
It says that (for example), if she withdraws 9.6% (initially), it'll last n = 15 years (with 95% probability of surviving).
So (as you've suggested) they refer to "... the start of a new 15 year period ...".
However, as SWR1945 says: "You are seeing a work in progress"
Amen! It changes day to day ... and anything that's there today may not be, tomorrow %#$^&@!?
Currently, I'm working on a section which deals with the distribution of 1/Portfolios, given the distribution of Portfolios (because the withdrawal rates depend upon withdrawal/Porfolio, hence the reciprocal of the portfolios.)
I read it this morning and it makes no sense &$%@!?
Ah well ... creeping senility ...
Oops. Did I say SWR1945?
Is that a Freudian error?
P.S. I think I should have my blue sidekick (the guy who asks me embarassing questions) ... he should ask Ken's questions :^)
With regard to the charts that you displayed, they're inventions(!), but are meant to illustrate the relationship between "safe" withdrawal rates and years, n: "safe" meaning 95% probability of surviving n years.
If Sam has just retired and expects to drop dead in n years, he looks at the left chart.
It says that (for example), to last n = 15 years, he can withdraw 9.6% (initially).
On the other hand, if Sally wants to withdraw x% (initially), how many years can she count on (with 95% probability of surviving)?
Sally would look at the right chart.
It says that (for example), if she withdraws 9.6% (initially), it'll last n = 15 years (with 95% probability of surviving).
So (as you've suggested) they refer to "... the start of a new 15 year period ...".
However, as SWR1945 says: "You are seeing a work in progress"
Amen! It changes day to day ... and anything that's there today may not be, tomorrow %#$^&@!?
Currently, I'm working on a section which deals with the distribution of 1/Portfolios, given the distribution of Portfolios (because the withdrawal rates depend upon withdrawal/Porfolio, hence the reciprocal of the portfolios.)
I read it this morning and it makes no sense &$%@!?
Ah well ... creeping senility ...
Oops. Did I say SWR1945?
Is that a Freudian error?
P.S. I think I should have my blue sidekick (the guy who asks me embarassing questions) ... he should ask Ken's questions :^)
P.S.
If'n yer curious to see my misadventures:
http://home.golden.net/~pjponzo/distributions-stuff.htm
... but don't tell nobuddy.
If'n yer curious to see my misadventures:
http://home.golden.net/~pjponzo/distributions-stuff.htm
... but don't tell nobuddy.
gummy -
This strange game of "you said this" and "that's not what I meant" and "but look at your post on May xx" and "you quoted me out of context" ... that game ain't no fun
hocus -
I agree with your comment here, Gummy. My sense is that every single person who posts on this board agrees.
Yet the problem continues to pop up. We need to step back and ask ourselves "Why?" I have been at this long enough now that I think I can provide some insights.
I think gummy liked a quote of mine. Doesn't sound like he attached any attached any significance to the melody - he liked the lyric.
I just wanted to be clear that I wasn't being flippant and I found a certain amount of what I considered to be illogic in the link. Not terribly important to either of us, imo, in the scheme of things. I attempted to remove the post but I don't have the privilege the rest of you do (not es' problem - something with my hookup).
I haven't referenced May anything yet (iinm). periodically a poster asserts that quantitative assurances about the future were not made. Those statements are inaccurate.
This strange game of "you said this" and "that's not what I meant" and "but look at your post on May xx" and "you quoted me out of context" ... that game ain't no fun
hocus -
I agree with your comment here, Gummy. My sense is that every single person who posts on this board agrees.
Yet the problem continues to pop up. We need to step back and ask ourselves "Why?" I have been at this long enough now that I think I can provide some insights.
I think gummy liked a quote of mine. Doesn't sound like he attached any attached any significance to the melody - he liked the lyric.
I just wanted to be clear that I wasn't being flippant and I found a certain amount of what I considered to be illogic in the link. Not terribly important to either of us, imo, in the scheme of things. I attempted to remove the post but I don't have the privilege the rest of you do (not es' problem - something with my hookup).
I haven't referenced May anything yet (iinm). periodically a poster asserts that quantitative assurances about the future were not made. Those statements are inaccurate.
regards,
wanderer
The field has eyes / the wood has ears / I will see / be silent and hear
wanderer
The field has eyes / the wood has ears / I will see / be silent and hear
There's a fella on Morningstar who calls himself Ozark. I have a great deal of respect for his opinions. Here's an interesting post:
http://home.golden.net/~pjponzo/Ozark.htm
P.S. Ozark wasn't talking about anybuddy on this board
http://home.golden.net/~pjponzo/Ozark.htm
P.S. Ozark wasn't talking about anybuddy on this board
gummy wrote: There's a fella on Morningstar who calls himself Ozark. I have a great deal of respect for his opinions. Here's an interesting post:
http://home.golden.net/~pjponzo/Ozark.htm
P.S. Ozark wasn't talking about anybuddy on this board
Greetings!
I liked the post:
There really is an Efficient Frontier. There really is a withdrawal rate that will allow my wife and I to spend all our money during our life times, but never go broke.
But these things are unknown and unknowable, going forward. Such things are only knowable looking backward.
I agree to a large extent. But I've seen enough to be convinced that good valuation measures like dividend yield or PE-10year do have some predictive ability for long term returns, and that it is prudent to heed what they say. And I do think that it is well worth getting informed on Modern Portfolio Theory and a bit of SWR stuff. Otherwise you are just pulling numbers out of a hat, right?
Ben
"Do not spoil what you have by desiring what you have not; remember that what you now have was once among the things only hoped for." - Epicurus
Ozark also says:
"There's lots of stuff we can learn by studying the past ..."
Amen!
P.S.It was the discussion that followed Ozark's post that prompted:
http://home.golden.net/~pjponzo/past-future.htm
"There's lots of stuff we can learn by studying the past ..."
Amen!
P.S.It was the discussion that followed Ozark's post that prompted:
http://home.golden.net/~pjponzo/past-future.htm
gummy wrote: http://home.golden.net/~pjponzo/past-future.htm
That is a wonderful page. I like this quote a lot:
My point is that you shouldn't place your faith in a mathematical prediction. Use it as a guide. Think about what it's saying. Don't trust your financial future to some mathematical ritual.
BenS
"Do not spoil what you have by desiring what you have not; remember that what you now have was once among the things only hoped for." - Epicurus
This is to let gummy know that I am still interested.
At gummy's last post on this thread, he had established that each term in the Magic Sum formula was lognormal with a mean of -M (if we ignore inflation). M should always be positive, even after accounting for inflation.
The current problem has the form of z1+z2^2+...+zN^N or, actually, it can be written z1+(z1*z2)+(z1*z2*z3)+... where all terms have the same mean. If we look at the expectation and variance of a product, we find that E(x*y) = mean(x)*mean(y) and Var(x*y) = [(mean(x))^2]*Var(y) + [(mean(y))^2]*Var(x) + E([(error of x)^2]*[(error of y)^2)]) provided that all terms are independent. For the terms in the Magic Sum formula, the mean of the k'th term is (-M)^k and the variance always increases. The first term for the (k+1)th product is k*[(-M)^2k]*(SD^2). If the details of my calculations are correct...and I do not guarantee that at this point...the others always add [(-M)^(2*[k-s])*E([error of z1)^2]*[(error of z2)^2]*...*[(error of zs)]) terms for s=1 through s=k.
This would mean that the gMS formula consists of the sum of terms from lognormal distributions and that the variance of each new term is larger than the one just before it. This is related qualitatively to the original thought that the final distribution of portfolios would be a simple shift of a lognormal distribution toward zero as time increases. (Negative balances are identified as portfolio failures.) We see that this is part of the answer, but not all. The final distribution is also increasing its variance with time.
In any event I notice a geometrical sum in the mean of gMS.
I will add that we are talking about distributions without memory. In particular, the correlation terms are all assigned an expectation value of zero.
Have fun.
John R.
At gummy's last post on this thread, he had established that each term in the Magic Sum formula was lognormal with a mean of -M (if we ignore inflation). M should always be positive, even after accounting for inflation.
The current problem has the form of z1+z2^2+...+zN^N or, actually, it can be written z1+(z1*z2)+(z1*z2*z3)+... where all terms have the same mean. If we look at the expectation and variance of a product, we find that E(x*y) = mean(x)*mean(y) and Var(x*y) = [(mean(x))^2]*Var(y) + [(mean(y))^2]*Var(x) + E([(error of x)^2]*[(error of y)^2)]) provided that all terms are independent. For the terms in the Magic Sum formula, the mean of the k'th term is (-M)^k and the variance always increases. The first term for the (k+1)th product is k*[(-M)^2k]*(SD^2). If the details of my calculations are correct...and I do not guarantee that at this point...the others always add [(-M)^(2*[k-s])*E([error of z1)^2]*[(error of z2)^2]*...*[(error of zs)]) terms for s=1 through s=k.
This would mean that the gMS formula consists of the sum of terms from lognormal distributions and that the variance of each new term is larger than the one just before it. This is related qualitatively to the original thought that the final distribution of portfolios would be a simple shift of a lognormal distribution toward zero as time increases. (Negative balances are identified as portfolio failures.) We see that this is part of the answer, but not all. The final distribution is also increasing its variance with time.
In any event I notice a geometrical sum in the mean of gMS.
I will add that we are talking about distributions without memory. In particular, the correlation terms are all assigned an expectation value of zero.
Have fun.
John R.