Is the Raddr Methodology Analytically Valid?

Research on Safe Withdrawal Rates

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Is the Raddr Methodology Analytically Valid?

Post by hocus2004 »

I believe that we have established beyond any reasonable doubt that the conventional SWR methodology is analytically invalid for purposes of determining SWRs (while valid for purposes of determining historical surviving withdrawal rates). My view is that both the Bernstein methodology (using the Gordon Equation to account for the effect of valuation) and the JWR1945 methodology (using statistical analysis tools to account for the effect of valuation) are analytically valid for purposes of determining SWRs (do you agree, JWR1945?). I have wondered from time to time how to categorize methodologies using Monte Carlo runs, in particular the approach used by raddr in the research he posted at the FIRE board.

My inclination is to say that the raddr approach is a third analytically valid methodology for purposes of determining SWRs (I of course understand that raddr does not agree with me re the analytic invalidty of the conventional methodology). But I am not clear enough on what is involved in a Monte Carlo analysis to say for sure.

Does anyone have thoughts on this question?
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Post by JWR1945 »

Raddr's Monte Carlo model is the best that I have seen. Gummy's is next best. Then follows everything else.

Raddr's approach introduces one of the effects of valuations indirectly. He limits the scatter (or volatility or standard deviation) of market gains and losses according to the Reversion to the Mean. That is, he reduces the scatter after N years to less than 1/(the square root of N) times the scatter in a single year. Valuations are the cause. Mean reversion is the effect.

Gummy paid attention to the statistical probability distributions of the stock market and inflation when he developed his Monte Carlo model. More can be done. For example, Crestmont found that the stock market's statistics can be estimated much better by using two different (but still lognormal) distributions for up and down markets instead of using a single distribution.

Monte Carlo models require a mean and standard deviation (or variance) as inputs at a minimum. These inputs are related to valuations.

The biggest weakness of the Monte Carlo models that I have seen is that they do not incorporate valuations directly, but rely on user specified values instead. John Bogle's version of the Gordon Equation can provide the critical link: the expected return of the market is the investment return (equal to the dividend yield plus the growth rate of earnings) plus the speculative return (which takes multiple expansion and contraction into account). William Bernstein's version of this equation differs slightly.

This is a major source of error. A Monte Carlo model transfers the problem of predicting stock market returns from the historical record to the discretion of the person making use of the model. It does not tell you how closely this person's estimates correspond to the historical data.

Users of Monte Carlo models can fall into the trap of thinking that they have good statistics just because they can make a large number of runs. One can take as many runs as he wants to reach any level of precision with a Monte Carlo model. In this sense, there are no statistical limitations with Monte Carlo models. To the extent that a model introduces the right sources of randomness in the right places, it will have the correct statistics after processing. This can never be done perfectly. It may be good enough.

To the extent that a Monte Carlo model fails to introduce the right sources of randomness at the right places, it can produce large errors. The accuracy of Monte Carlo results is determined by assumptions hidden within models, not by the number of runs.

The fidelity of Monte Carlo model results is often equivalent to those of Historical Sequence calculations. The two approaches presume different things as to how future results will be similar to historical experience. For example, a Monte Carlo model is likely to have several sequences with four or five down years in a row. After all, one typically looks at 1000 or 10000 (possible) sequences. The Historical Sequence method does not consider such sequences directly since there are no such sequences in the historical record.

Compare all of this with The New Tool, which does not incorporate valuations directly. Monte Carlo models and The New Tool are similar. The New Tool introduces valuations indirectly. It requires that you specify the total return of the overall market for a specified number of years. You introduce valuations when you estimate the total return of the market.

We must qualify the results of all Monte Carlo models that I have seen (including Raddr's and Gummy's) for purposes of determining Safe Withdrawal Rates. They are very similar to The New Tool in that they translate estimates of stock market returns into Safe Withdrawal Rates. They do not automatically take valuations into account. Nor do they take into account that valuations and dividend yields are both outside of the historical range. All of that is external.

I consider my method sufficient to provide valid answers for purposes of determining Safe Withdrawal Rates. I have included the relevant factors. You do not have to estimate market behavior.

Pay the greatest attention to my emphasis on common sense and cause and effect relationships. Observe, for example, that I have limited my statistical precision to 90% (two-sided) confidence limits. I do so to make it clear that the statistics are only rough approximations.

Have fun.

John R.
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Post by unclemick »

I'm not qualified to comment on Raddr's analysis methodology.

But I've been following with great interest his progress on trying to determine the suitiblity of 'recent investment products' in the commodities area for a 'joe average slice and dicer'. The tongue in cheek is mine - but the quest is serious. In the past - commodities were not easy to utilize in a small investors portfolio.

Tangentially - favorable comment on his analysis methods gives me added confidence that his analysis of various commodity type vehicles may be on to something.

Never bought commodity futures(pork bellies, corn, wheat, etc.) - but a past trail of gold/gold stock/timberland/Vanguard PM, oil/gas/copper/metal stocks, real estate, etc has been/is/can be in varying degrees cumbersome.

A better way is of great interest to me.
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Post by JWR1945 »

unclemick
Tangentially - favorable comment on his analysis methods gives me added confidence that his analysis of various commodity type vehicles may be on to something.
..
A better way is of great interest to me.
Warning! Warning! Warning!

Danger! Danger! Danger!

Shame on you! Go out and buy (or borrow) a copy of David Dreman's blockbuster Contrarian Investment Strategies: The Next Generation. Read Chapter 15: Small Stocks, Nasdaq and Other Market Pitfalls. There are all sorts of traps.

Remember the tremendous advantage that was claimed for small stocks? Most of the time, they would drift along with the rest of the market. At other times, they would soar. Or so it was claimed.

The flaw was that no one could buy the small stocks at the prices and quantities that the academics assumed. There were tremendous spreads between bid and ask prices: many times 100% or more. The academic models priced all purchases and sales midway between the bid and asked prices. The number of shares available sometimes was only 100 shares in a transaction and 10000 shares per year.

You are talking about an entirely new product when you talk about commodity index funds. Remember emerging market funds when they first came out? They saturated illiquid markets with purchase orders. Brokers collected commissions. Fund managers collected fees. Investors got creamed.

Commodity index funds will absorb a lot of cash. This will change commodity markets in a fundamental way for many years to come. The old index numbers are highly unlikely characterize this new market successfully.

When was the last time that Wall Street came up with a good story [i.e., engineered a new financial product] that turned out well? For their customers, not their salesmen.

Have fun.

John R.

P.S. While you are at it, read what David Dreman and John Bogle have to say about the Efficient Frontier. HINT: their comments are other than favorable.
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Post by unclemick »

Hmmmm - point well taken. Ease of use can distort the the risk premium. Bernstein sort of addresses the issue (IPO example) in his Fall, Efficient Frontier.

Not Dreman, but vaguely remember some of Bogle's comments about the efficient frontier/MVO SD and his heartburn with the facile use of ETF's. Also have seen some small comments about hedge funds distorting the oil futures markets.

Still interested, but caution is warrented. Fir trees in Oregon, Pine in Mississippi - I'll let my dinky timberland holdings continue to grow - heh,heh.

Don't expect to go overboard like the 70's and 80's - but with inflation predicted on the horizon - expect a lot of media noise in this area in coming years. Gold and real estate or multi asset class investing never made me rich back then either. DCA with De Gaul and the Norwegian widow got me to ER.

Will watch from the sidelines for now.

Balanced index in reserve with dividend stocks as an income booster to defined pension.

BTY - the timberland in Oregon morphed into a land development LLC(due to Spotted Owls) - about 4% of last years income - bought in 1967 - it really got hosed over the years relative to inflation. One of my before Bogle mis adventures.
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Post by JWR1945 »

Neither John Bogle nor David Dreman have good things to say about the efficient frontier.

This is taken from my post From Chapter 8
http://nofeeboards.com/boards/viewtopic.php?t=2572
JWR1945 wrote:From Chapter 8The really interesting part of this chapter, however, is how completely John Bogle discredits the notion of using the efficient frontier. At the efficient frontier, an allocation provides the greatest increase in the overall return relative to risk (as measured by volatility). If you are able to locate the efficient frontier successfully, you are better off reaching a higher level of return by leveraging your investments at that particular allocation than by changing allocations.

John Bogle showed that the efficient frontier was unusable. He showed that it produced extreme changes in allocations based on changes in risk so small as not to be meaningful. He showed that normal changes in the market caused vast changes in the efficient frontier from one decade to the next. Normal changes swamp out any advantage that an optimal allocation is supposed to provide.
David Dreman wrote a scalding review of the Efficient Market Hypothesis and Modern Portfolio Theory in Chapter 14 What is Risk? and Appendix A (as well as in other chapters) in his blockbuster Contrarian Investment Strategies: The Next Generation.

Here are some relevant comments from page 299 in Chapter 14:
The lack of correlation between risk and return was not the only problem troubling academic researchers. More basic was the failure of volatility measures to remain constant over time, which is central to both the efficient market hypothesis and modern portfolio theory. Although beta is the most widely used of all volatility measures, a beta that can accurately predict future volatility has eluded researchers since the beginning. The original betas constructed by Sharpe, Lintner, and Mossin were shown to have no predictive power, that is, the volatility in one period had little or no correlation with that in the next. A stock could pass from violent fluctuations to lamb-like docility.[Emphasis added.]
Have fun.

John R.
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Post by hocus2004 »

"The biggest weakness of the Monte Carlo models that I have seen is that they do not incorporate valuations directly, but rely on user specified values instead."

Thanks for that detailed and helpful response, JWR1945.

I agree with you that it is better to incorporate valuations directly. The root idea in SWR analysis is to look at what happened in the past to form assessments of the various likelihoods of various scenarios playing out in the future. The logically consistent thing to do when considering the effect of changes in valuations is to look at how changes in valuations have affected long-term returns in the past.

Still, I don't view it as entirely unreasonable to want to explore long-term return possibilities other than those suggested by the historical data. There are some investors who believe that, for any of a number of reasons, stocks are likely to perform in the future in ways different from how they have performed in the past. For example, my understanding of the argument put forward in the book "Dow 36,000" (I have only read reviews of the book, not the book itself) is that the claims in that book are rooted in the premise that the equity premium is going to disappear in the near future. This would cause returns in the early years of a retirement beginning today to be better than what you would come to expect from relying on an analytsis of the historical data.

It appears to me that an investor who wanted to determine the SWR that applies in the event that the "Dow 36,000" scenario plays out could employ a Monte Carlo analysis to do so. It's not the SWR that I would use in my planning. But so long as the investor making use of the number is aware that the premise on which it is based is not in accord with the historical data, I see no harm in a study being done reporting on the results obtained.

The great danger in the conventional methodology studies is that the assumption re the effect of valuatons being employed is both far-fetched and hidden. There are a large number of community members who have indicated a belief that the conventional methodology numbers are reasonable ones, that they provide a useful "rule of thumb." That is not so, of course. The conventional methodology provides a rule of thumb that is not too far off the mark from what the historical data says in times of moderate valuation. At times of low valuation, the conventional studies generate numbers so far off the mark on the low side as to be "highly misleading" (Bernstein's phrase). At times of high valuation, the conventional studies generate numbers so far off the mark on the high side as to be "highly misleading."

The greatest danger lies in the fact that the assumptions are never stated. An investor making use of a Monte Carlo analysis often needs to specify what effect he expects changes in valuation levels to have. That means that he needs to stop and think about the question. The process of doing so underlines for him the significance of the valuation question. Investors using Monte Carlo studies are effectively put on notice that the accuracy of results that the studies generate depend on the accuracy of the assumptions re valuation entered into "the black box."

This is not so with the conventional methodology studies. I am only familiar with the REHP study, but my understanding is that all conventional studies suffer from this flaw. The REHP study nowhere makes the reader aware of how unlikely it is that the core assumption of the study--that in the furture stocks will perform in a way that they never have before and that for the first time in history changes in valuation will have zero effect on long-term returns--will play out in the real world. Raddr once had an analysis posted which, using standard deviation analysis, determined that the odds of this assumption playing out were 1 in 100. (I linked that analysis in a post I put to the Early Retirement Forum, and raddr deleted the reference to the odds from the write-up, but he still indicates that it is a highly far-fetched assumption that drives the conventional studies.)

There is all the difference in the world between a withdrawal rate being "100 percent safe" and a withdrawal rate having a 1 in 100 chance of being 100 percent safe. The danger of the conventional studies is that they mislead readers into thinking that the withdrawal rates they generate are in some real-world sense safe (investors who have not devoted much study to historical returns are often not aware of the significance of the historical correlation between the valuation level that applies at the beginning of an historical sequence and the long-term return that applies for that sequence).

Your words above make me feel a bit more comfortable about saying that Monte Carlo analyses can be analytically valid, and that the raddr methodology indeed is analytically invalid. My hope is that, once a consensus forms that there is no constructive purpose served by the production of future conventional methodology studies, there will be a number of community members using a number of different analytically valid approaches. We learn from making use of as many approaches as can be reasonably justified. The dangers we have discovered in the conventional methodlogy studies do not appear to be as serious a factor in connection with Monte Carlo studies.

It is my hope that our discussions will cause SWR analyts to be more careful in their handling of the valuation question than they have been in the past. I think it would be extremely helpful if analysts using Monte Carlo approaches in which the researcher decides on the long-term return being assumed would spell out in some detail the justification for the assumption employed. The most important thing is that the reader of the study be put on notice of the weaknesses of any approach using long-term return assumptions not in accord with those obtained by making reference to what the historical data reveals on this question.
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Post by JWR1945 »

hocus2004
The great danger in the conventional methodology studies is that the assumption re the effect of valuations being employed is both far-fetched and hidden. There are a large number of community members who have indicated a belief that the conventional methodology numbers are reasonable ones, that they provide a useful "rule of thumb." That is not so, of course.
What happened is that we were able to bring Safe Withdrawal Rates back up to the 4% previously claimed. It took a tremendous amount of effort. Without changes, the old rule of thumb is highly misleading and dangerous. The rule of thumb should be stated like this: It is possible to achieve the Safe Withdrawal Rates that were indicated by the Conventional Methodology even at today's valuations, but not by using the original approach.

Have fun.

John R.
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Post by JWR1945 »

hocus2004
The greatest danger lies in the fact that the assumptions are never stated. An investor making use of a Monte Carlo analysis often needs to specify what effect he expects changes in valuation levels to have. That means that he needs to stop and think about the question. The process of doing so underlines for him the significance of the valuation question. Investors using Monte Carlo studies are effectively put on notice that the accuracy of results that the studies generate depend on the accuracy of the assumptions re valuation entered into "the black box."
Unfortunately, a typical Monte Carlo model includes default values of the market's behavior that users incorrectly assume to apply today. There is no explanation as to how to come up with reasonable inputs to the model.

Have fun.

John R.
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Post by JWR1945 »

hocus2004
Raddr once had an analysis posted which, using standard deviation analysis, determined that the odds of this assumption playing out were 1 in 100. (I linked that analysis in a post I put to the Early Retirement Forum, and raddr deleted the reference to the odds from the write-up, but he still indicates that it is a highly far-fetched assumption that drives the conventional studies.)
IIRC, the original number was closer to 1 in 700 (or something else that is close to 1000).

Raddr erred when he backed away from his original estimate. He cited his reason as overlapping data. This caused him to reduce the number of degrees of freedom (i.e., the effective number of independent data points) in his calculations.

The correct number of degrees of freedom is much closer to what he used in his original calculations.

Here are the details.

There are only two prices that determine the total return of the market when there are no withdrawals: the original price and the final price. None of the intermediate prices enter into the calculation.

All of the original prices can be treated as totally independent. That makes most, but not all, of the final prices dependent (i.e., they are the same as one of the initial prices, but from a different historical sequence). Some of the final prices are independent because any sequence starting from the same date would be a partial sequence. For example, a 30-year sequence that begins in 1980 is not yet complete, but the 30-year sequence that ends in 1980 [which would be the 1950 sequence] is already complete.

[Intermediate price levels have a slight influence in determining how likely a final price will be. This has a minimal effect.]

I realize that this is a technical nit. But I consider it important to determine the power of statistical tests accurately. Only in economics and financial studies have I seen it considered acceptable when one fails to reveal differences that actually exist.

Have fun.

John R.
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Post by JWR1945 »

Your words above make me feel a bit more comfortable about saying that Monte Carlo analyses can be analytically valid, and that the raddr methodology indeed is analytically invalid.
Is this an editing error?

Have fun.

John R.
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Post by hocus2004 »

"Is this an editing error? "

Yes. Thanks for catching that.

My view is that the raddr methodology is analytically valid.
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Post by hocus2004 »

"Unfortunately, a typical Monte Carlo model includes default values of the market's behavior that users incorrectly assume to apply today. There is no explanation as to how to come up with reasonable inputs to the model."

All researchers are under an obligation to explain the reasons for their assumptions, in my view. I don't think that the early SWR analysts were trying to hide anything. My sense is that they simply were not aware of the correlation between valuation levels at the starting point of a returns sequence and the long-term return that applies for that sequence.

The big breakthrough of our SWR discussions was to prove the existence of this correlation. (Our work supports earlier showings of this correlation that were put forward by Robert Shiller in a non-SWR context). Now that the existence of the correlation is clear, I am not able to see any justification for failing to address it in some way.

My inclination would be to refer to a Monte Carlo study that plugged in an unrealistic long-term returns assumption without setting forth an justification for it or explanation of it as analytically invalid. My inclination is to refer to those that do include justifications or explanations as analytically valid.
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Post by hocus2004 »

"IRC, the original number was closer to 1 in 700"

Here is a link to the thread where raddr said that the odds of the long-term returns assumption on which the intercst SWR claims are based coming to pass are 1 in 740.

http://www.nofeeboards.com/boards/viewt ... ght=#p9014

Ataloss sent Bernstein an e-mail asking whether he agreed with me that the conventional SWR methodology is analytically invalid.

Bernstein: I *do* like the trinity study--it's groundbreaking work, it's just that the
embedded return is all wrong going forward . . ..the trinity study?? fuggedaboudit. the methodology is sound, but remember, there's a 7% real equity return embedded into it. who in the peanut gallery wants bet their retirement on that? "

raddr: "In fact, looking back at my Gordon equation thread, a 7% real return going forward would be about 3 SD's above the mean for 40-60 years going forward. IOW, the chances that we will see 7% returns long term are about 1 in 740 or 0.135% - if the future is anything like the past."
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Post by JWR1945 »

hocus2004
My inclination would be to refer to a Monte Carlo study that plugged in an unrealistic long-term returns assumption without setting forth an justification for it or explanation of it as analytically invalid. My inclination is to refer to those that do include justifications or explanations as analytically valid.
Yes, but understand that this describes the Monte Carlo model at a typical website that sells investments such as mutual funds.
Bernstein: I *do* like the trinity study--it's groundbreaking work, it's just that the
embedded return is all wrong going forward . . ..the trinity study?? fuggedaboudit. the methodology is sound, but remember, there's a 7% real equity return embedded into it. who in the peanut gallery wants bet their retirement on that? "
It is easy to get confused on this point. Remember that I had difficulty reconciling John Bogle's 50-year long-term real return for the overall stock market, the Gordon Equation and estimates of what will happen in the next 20 to 30 years.

For the market to have a real return of 6.5% to 7.0% from 1980 to 2030 (which is 50 years), it will have to underperform sharply from 2004 to 2030. That is, if the two pieces (split between an early period and a later period) always end up in the same place after 50 years and the first part has been a roaring bull, the second part has to be subdued.

The Gordon Equation gives us a number that fits this hypothesis of a subdued market going forward (from 2004 to 2030) because of today's low dividend yields, BUT its numbers are supposed to be for the very long-term (50 or more years), not for just a decade or two.

I resolved this inconsistency only when I realized that the Gordon Equation's discount rate is not the same as the investment return unless future dividends are reinvested in stocks that are (essentially) identical to those available today. It is the availability or lack of availability of suitable investments in the future that cause the Gordon Model to start to err after a decade or so. The long-term (real) return of the overall market will revert back toward 6.5% to 7.0% because stock shares purchased in the future will have different (higher) dividend yields than what is available today.

This is a subtle and difficult point that neither John Bogle nor William Bernstein helped to clear up.

Have fun.

John R.
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Post by hocus2004 »

"I resolved this inconsistency..."

Would you be willing to describe in a bit more detail the nature of the inconsistency that you are referring to here? I'm not looking (at this point) for your resolution of the inconsistency. The part that I don't understand is the problem you are trying to address with your resolution.

Are you disagreeing with Bernstein that there is a 7 percent long-term return assumption embedded into the conventional methodology studies?

Is the problem that different people are using different definitions of "long-term?" For purposes of the Bernstein statement above, "long-term" means the 30-year period covered by the conventional studies, right?

"For the market to have a real return of 6.5% to 7.0% from 1980 to 2030 (which is 50 years), it will have to underperform sharply from 2004 to 2030. That is, if the two pieces (split between an early period and a later period) always end up in the same place after 50 years and the first part has been a roaring bull, the second part has to be subdued. "

This makes sense to me.

"its numbers are supposed to be for the very long-term (50 or more years), not for just a decade or two. "

I don't understand the nature of the problem you are making reference to here.

"The long-term (real) return of the overall market will revert back toward 6.5% to 7.0% because stock shares purchased in the future will have different (higher) dividend yields than what is available today. "

Are you saying that the long-term return will be 7 percent from any possible starting point? If that is what you are saying, that's probably the part that I am having a hard time understanding. It's hard for me to understand why that would be so. Shouldn't the long-term return vary according to the valuation level that applies at the starting point?

I am missing at least one link in the logic chain on this one. And my sense is that this is as aspect of the question that I need to understand fully.
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Post by JWR1945 »

"I resolved this inconsistency..."
The discrepancy is that the long-term historical (real) return has been 7% while the Gordon Equation indicates that the long-term (real) return is now 3%. If the Gordon Equation is correct, then the long-term return has changed.

In this case the long-term is 50 years or so.

If we start with an initial balance that grows for N years at an annualized rate of return r (without making any deposits or withdrawals), the final balance satisfies the following equation: [final balance/initial balance] = (1+r)^N.

If the annualized rate of return r is 7%, then the final balance should be 1.07^N times the initial balance. Letting N=50 years represent the long-term, the final balance should be 29.46 times the initial balance. This has been the long-term return of the stock market.

If the annualized rate of return is 3%, then the final balance will be 1.03^N times the initial balance. Again letting N=50 years to represent the long-term, the final balance should be only 4.38 times the initial balance. This is what the Gordon Model predicts.

One way to reconcile these different numbers is to look more closely at what we mean when we speak of a long-term return. Suppose that we are able to purchase stocks at half-price. Then the [final balance/initial balance] = twice as much as the calculations indicate. On the other hand, if you pay twice as much as you should, the [final balance/initial balance] = only one-half of what the calculations indicate.

Now let us see what that does to our calculations.

If the long-term annualized rate of return were 7%, then the final balance would be 1.07^N or 29.65 times the initial balance. But if you purchased stock at half-price, the ratio would be twice as big or 59.30. [Here is where An Illusion of Numbers enters in.] We now use the formula to calculate r, (1+r)^N = 59.30 where N=50 years. Taking the 50th root on both sides of the equals sign, (1+r) = 1.085 or r = 8.5%.

Similarly, if you paid twice as much for the stock, the final balance would only be one-half of 29.65 times the initial balance or 14.825. Solving for r when N=50 years, the rate of return satisfies (1+r) = 1.055 or r = 5.5%.

Continuing, if the long-term annualized rate of return were 3% as predicted by the Gordon Equation, then the final balance would be 1.03^N or 4.38 times the initial balance. If you purchased stock at half-price, the ratio would be twice as big or 8.76. We now use the formula to calculate r, (1+r)^N = 8.76 where N=50 years. Taking the 50th root on both sides of the equals sign, (1+r) = 1.044 or r = 4.4%. If you paid twice as much for the stock, the final balance would only be one-half of 4.38 times the initial balance or 2.19. Solving for r when N=50 years, the rate of return satisfies (1+r) = 1.016 or r = 1.6%.

If the long-term rate of return continues to be 7%, which it has been historically, a reasonable amount of variation in the measured (annualized total) return of the stock market would be (of the order of) 5.5% to 8.5%. If the Gordon Equation is correct, a reasonable amount of variation is 1.6% to 4.4%.

Something has to give. It is the Gordon Equation.

The Gordon Equation is a mathematical theorem with assumptions about an initial dividend yield, dividends that grow (forever) at a constant rate and a constant ratio of what people will pay for prices relative to dividends. Obviously, this is an idealization. The formula is realistic enough to be helpful. The theorem is always true. It is not always applicable.

The thing about the Gordon Equation is that its mathematical calculations become better and better as the number of years increase. That is, if you compare the calculated rate of return (which corresponds to an infinite number of years) to the results after a finite number of years N, the difference between the two becomes smaller and smaller as N becomes larger and larger.

The assumptions behind the Gordon Equation become less and less reliable as the number of years N increases.

The Gordon Equation starts us in the right direction, but it fails later on.

It was Ben Solar who pointed out that the Gordon Equation calculates a long-term return, not an intermediate return. When we see it applied, however, it is for the intermediate term.
Are you disagreeing with Bernstein that there is a 7 percent long-term return assumption embedded into the conventional methodology studies?
No. I agree that the long-term (annualized real) return of the stock market is 7%. That is the historical record. Any study that uses historical sequences has this (7%) return embedded in it.
Are you saying that the long-term return will be 7 percent from any possible starting point? If that is what you are saying, that's probably the part that I am having a hard time understanding. It's hard for me to understand why that would be so. Shouldn't the long-term return vary according to the valuation level that applies at the starting point?
The mathematical theorem is that the Nth root of any (fixed) positive number becomes closer and closer to 1 as N increases. This may require an exceedingly large value of N. The net effect is that the rate of return r at prices starting at twice as large and at prices starting at one-half as large as is typical come closer and closer together. THIS EFFECT IS FOR THE CALCULATED RATE OF RETURN r. THE RATIO OF THE FINAL BALANCE TO INITIAL BALANCE DOES NOT CHANGE. I refer to this as An Illusion of Numbers.

The false notion that one's purchase price does not matter in the long-run is based upon looking at the rate of return r instead of the ratio of the final balance to the initial balance.

Have fun.

John R.
Last edited by JWR1945 on Wed Oct 06, 2004 4:46 am, edited 1 time in total.
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Post by hocus2004 »

"Understand that this describes the Monte Carlo model at a typical website that sells investments such as mutual funds. "

That's what creates the opportunity for us to do so much good with the work we are doing here, in my assessment. If most people understood this stuff, it would not be worth putting so much time into getting it straight. It's because so many middle-class investors are being led down the garden path that we have a chance to make a difference.

When I was putting together my plan in the mid-90s, I sweated every detail. I didn't want to make a mistake that was going to cost my family big-time in the years to come. It took me a long time to figure out what the historical data was really saying. I hope that at the end of this I am going to be able to present the realities in a nice little package of a Research Report that saves a lot of aspiring early retirees a lot of time and a lot of headaches.

Time will tell the tale. But it's the opportunity we have to make a difference that makes this worth doing, in my view. And it's the reality that confusion on the points we are exploring is so widespread that makes it possible for us to make a difference.
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Post by hocus2004 »

"The discrepancy is that the long-term historical (real) return has been 7% while the Gordon Equation indicates that the long-term (real) return is now 3%."

Thanks for hanging in there with me, JWR1945. It looks like this is going to be one of those threads that I am going to need to return to a number of times over the course of several months for the points being made to sink into my brain. About all that is clear to me at this point is that the points being made in this thread are points that I need to understand better than I do today.

Can we put aside the Gordon Equation for a bit? My bigger concern is with the more general question of whether expected long-term returns vary with changes in starting-point valuation levels.

Say that we define the term "long term" to mean 50 years. Am I right in thinking that the expected long-term return from a starting point of low valuation should be higher than the long-term return from a starting point of high valuation? The expected return obtained for the time-period from January 1929 to Janaury 1979 is not the same as the expected return for the time-period from January 1930 to January 1980, is it?

I have more questions. But I am thinking that it is best that I hold back on them until I obtain clarification on this most basic point.

I am thrown by your statement that "Any study that uses historical sequences has this (7%) return embedded in it." This seems to suggest that the 7 percent expected return applies regardless of the starting-point valuation level. But then you say that "the false notion that one's purchase price does not matter in the long-run...." which suggests to me that the same return assumption should not always be embedded in an analysis.

In the statement I quote at the top of this post, you indicate that the Gordon Equation's prediction of a 3 percent long-term real return is wrong. Is the correct expectation 7 percent or something between 3 percent and 7 percent?
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Post by JWR1945 »

hocus2004
Say that we define the term "long term" to mean 50 years. Am I right in thinking that the expected long-term return from a starting point of low valuation should be higher than the long-term return from a starting point of high valuation? The expected return obtained for the time-period from January 1929 to January 1979 is not the same as the expected return for the time-period from January 1930 to January 1980, is it?
I got distracted by producing some numbers. I will get back to your questions later, but here is what I have at this moment.

This is complicated a little bit because the 6.5% to 7.0% long-term return assumes that all dividends are reinvested. Fortunately, I can use my calculators to generate this data easily.

If we had invested $100000 in January 1929 and made no further deposits or withdrawals (and if there were no fees), the real balance in January 1979 would have grown to $1018564. The ratio of the [final balance/initial balance] is 10.18564. From January 1930 to January 1970, the balance would have grown from $100000 to $1133432. The ratio of the [final balance/initial balance] is 11.33432.

Over 50 years, the formula (1+r)^50 = 10.18564 has a solution of (1+r) = 1.0475138 or r = 4.75%.

Over 50 years, the formula (1+r)^50 = 11.33432 has a solution of (1+r) = 1.0497549 or r = 4.98%.

If the long-term returns had been exactly 7.0%, the final balances would have been $2 945 703. [Oops! I should have used a ratio of 29.46 instead of 29.65 in my previous example.] If the long-term returns had been exactly 6.5%, the final balances would have been $2 330 668. The ratio of the final balance to the initial balance would have been 23.31.

You picked two of the worst possible start years when it comes to 50-year returns. The only other years with 50-year annualized real returns below 5.0% were 1892 (with a total return of 4.90%) and 1899 (with a total return of 4.83%).

Here are the actual 50-year annualized real returns of the S&P500 with all dividends reinvested using the calculator (and Professor Shiller's database). I have limited the final year to 1952 since the calculator uses dummy stock market returns after 2002.

Year, 50-Year Annualized Real Return, Final Balance If Starting With $100000.

Code: Select all

1871   5.42    1398519
1872   5.59    1515075
1873   5.94    1791451
1874   5.93    1780685
1875   6.15    1980833
1876   6.32    2137094
1877   6.88    2791395
1878   7.25    3308257
1879   7.54    3784620
1880   6.93    2857353
1881   6.00    1840233
1882   5.34    1345962
1883   5.27    1305115
1884   6.05    1885857
1885   5.84    1709699
1886   6.12    1944707
1887   6.43    2257151
1888   5.80    1677134
1889   5.96    1806104
1890   5.80    1679033
1891   5.77    1654346
1892   4.90    1095142
1893   5.14    1228227
1894   5.61    1534044
1895   5.77    1650368
1896   6.32    2142909
1897   5.60    1522899
1898   5.15    1229449
1899   4.83    1056611
1900   5.45    1417061
1901   5.44    1412473
1902   5.42    1400004
1903   5.71    1607032
1904   6.02    1859062
1905   6.31    2126625
1906   6.42    2247329
1907   6.57    2407594
1908   6.86    2763587
1909   6.93    2850459
1910   6.97    2902838
1911   6.98    2912688
1912   7.23    3279564
1913   7.15    3159410
1914   7.65    3995060
1915   8.08    4861929
1916   7.77    4222831
1917   7.64    3972838
1918   8.62    6255777
1919   8.78    6712274
1920   8.44    5744992

Code: Select all

1921   8.72    6550430
1922   8.49    5871586
1923   8.21    5173310
1924   7.61    3920231
1925   6.43    2256815
1926   6.58    2415487
1927   6.42    2244111
1928   5.44    1415040
1929   4.75    1018564
1930   4.98    1133432
1931   5.53    1474075
1932   6.03    1867018
1933   6.42    2245833
1934   5.86    1725810
1935   6.17    1992009
1936   5.70    1601005
1937   5.69    1590222
1938   6.30    2120477
1939   6.22    2045172
1940   6.46    2283274
1941   6.53    2366025
1942   7.45    3629384
1943   7.32    3421105
1944   7.12    3119705
1945   6.77    2648633
1946   6.72    2579465
1947   7.76    4203849
1948   8.40    5630286
1949   8.77    6690902
1950   8.64    6305366
1951   8.02    4727385
1952   7.37    3506174
The real explanation is going to be along these lines:

(1+r)^50 = [final balance/initial balance]

In these examples, the initial balance is always the same (i.e., $100000) and any particular year's 50-year return varies with the final balance. Since a ratio [final balance/initial balance] is used, we could use different initial balances and keep all of the final balances the same. Variations in the initial balances and the percentage return would be according to the formula. Small changes in the percentage return would correspond to large changes in the ratio [final balance/initial balance] because there are 50 years of compounding. Similarly, large changes in the initial balance (which correspond to large changes in the ratio) correspond to large changes in prices but only result in small changes in the percentage return.

Have fun.

John R.
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