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Question for the math inclined - he has simply added these rates, should he multiply each component? IE: (1+a/100)(1+b/100)etc…-1?
You're right. Further, this fundamental flaw permeates the column, so I wouldn't look at any of the numbers. (I didn't even read much past the second paragraph because it was apparent that Rekenthaler was trying to turn a multiplicative question into an additive one.)
Consider how he calculates the contribution of multiple expansion. He calculates the multiple expansion at roughly 2.66 (31.5/11.8). He tries to back out this factor by saying that if we divide it out, we would have the remaining S&P growth. Dividing 100% by 2.66% leaves 37% for "everything else". So, obviously (and obviously wrong), multiple expansion accounts for 63% of the S&P growth.
This is so wrong that it left me scratching my head about where he got 63% for quite awhile until I saw what his simplistic calculation was. BTW, the 31.5 figure he used was wrong also. He erroneously transcribed the 30.5 figure that one gets from the data he cited.
I think I can illustrate the problem simply:
Consider a 1x1 square. Its area is 1. Suppose its height doubles (100% growth) and its length triples (200% growth). Its area is now 6 (500% growth).
Would you say that 50% of the growth was due to the height doubling? After all, had the height not doubled, the area would have grown just half as much (to 3 instead of 6). According to his reasoning, the trebling of the length thus must likewise account for half the growth.
It's nonsense because it ignores the cross terms - the effect that one factor growing has on the other factor growing. The (1+a/100) effect.
He starts with junk arithmetic. From there we're off to the races. It only gets worse after that, I assume (not having the stomach for the rest).
One item is MSF's lengthy objection to the pie-chart math, which was the column's straw man. MSF says that math is bogus, my column says that math is bogus. We agree.
The other item is RBRT's question. Should I have multiplied thefour components, not added them? Yes, I should have. I opted for simplicity by adding the components so that I wouldn't have to write a distracting paragraph about why they did not sum to the total return. Had I multiplied the components instead, the numbers would have been a few basis points lower, the order of importance the same, and the conclusion unchanged.
In hindsight, I should have done the calculations the right way and made the explanation.
As Rbrt pointed out, the effects of various factors are multiplicative, not additive. It is an error to try to hammer the question into an additive one. This mistake permeates the column. Not only in the "straw men" as John put it, but in the "real" answer.
Percentages assume that the effect of returns from different factors are additive (so that they can be summed, presumably to 100%). That's an erroneous assumption, and it's why I gave my geometric (square) example.
Try to allocate the factor weightings (for height and length) in that example yourself. We assume that height of the square doubles (100% rate of increase) and length triples (200% rate of increase) in a year. Thus the area expands sixfold (500% increase).
What portion of the 500% increase is due to the 100% increase in height, and what portion is due to the 200% increase in length? And remember that the two portions have to sum to 100%.
It's an absurd question. Yet that's the calculation being performed in the table above where factor weightings are assigned. Further, one doesn't need factor weightings at all to answer the original question posed: if one factor (e.g. inflation) were excluded, what would the return be?
You can see how to answer this from my square example. If we exclude the increase in height, then the area triples in size. So the "return", excluding height growth, is 200%. Clean, simple, answers the question, and doesn't go into the weeds with additive percentage factor weights.
Finally, to illustrate just how deeply this conflating of multiplicative and additive effects goes, we can look at the hypothetical in the column where inflation runs at 51% and nominal return runs at 50%/year. With a 50% rate of return, a dollar invested would be worth $92,924,336.24. That's multiplicative (1+50%) x (1+ 50%) x ...
The gain, an additive quantity, would be a dollar less, $92,924,335.24, i.e. an increase of 9,292,433,524%. But what is written is: "investors would have gained 9,292,433,624%". Significant? Not numerically, but conceptually.
I did not know that jrekenthaler follows this board and posts/responds here. That is a pleasant surprise.
I remember even fund managers responding on M* forums when those fourms were vibrant during mid 2000s. I vividly remember Charles Ober of PRNEX responding to question there.
Graphical image to illustrate why one can't decompose factors additively:
Start with 1x1 square (black). If you double the height, you add the yellow square (1 new square). If you triple the length, you add the blue squares (2 new squares). If you do both, and only if you do both, do you add the green (mix of yellow and blue) squares (2 new squares).
We can say that doubling the height doubles the number of squares. That's true whether the length is one (and only the yellow square is added), or the length is three (and the yellow and green squares are added). So doubling the height truly adds 100% to the area.
Likewise, we can say that tripling the length triples the number of squares. That's true whether the height is one (and only the blue squares are added), or the height it two (and the blue and green squares are added). So tripling the length truly adds 200% to the area.
You can't just add up squares created by increasing the height alone (the yellow square) and the squares added by increasing the length alone (the blue squares) and figure that you've got everything covered. Area is computed by multiplying height and length, not by adding them. If all you do is add the yellow and blue squares, you miss the green squares, which exist only because both height and length increased.
It's absurd to suggest that the green squares (or parts thereof) are due solely to changes in height or changes in length (exclusive "or").
@msf, in showing how wrong additive decomposition of factors is, are you also making a point about this conclusion?
>> Had I multiplied the components instead, the numbers would have been a few basis points lower, the order of importance the same, and the conclusion unchanged.
(I believe I recall that JR once said in the past that he reads only MFO items brought to his attention, not as a general follower.)
As I recall, JR was granted membership by his request, using his current name; at 3:20 pm on August 27. Not that this in and of itself, is of consequence.
@msf, in showing how wrong additive decomposition of factors is, are you also making a point about this conclusion?
A couple, at least.
That his relying on people's intuition that the factor weightings should sum to 100% is wrong. So his factor weighting column must be wrong. And thus the way he calculated it must be wrong.
That the assertion that this resulted in being off by merely a few basis points is sleight of hand. The misdirection comes about in switching from the time frame used in most of the column, 45 years, 3 months (542/12 years), to a single year toward the end. Calling that single year "annualized" enhanced the illusion, as it makes it seem that the factor weighting (percentages) are the same over time.
That's not true. Over time, the discrepancies compound just like the returns. What may be a rounding of a few basis points over one year becomes a large deviation over decades.
Here's a simple example to illustrate. Take two factors, one that adds 2% to the return per year and one that adds 3%. Call them inflation and earnings if it helps make things more concrete.
Over one year (i.e. annualized), the increase in value is: (1+ 2%) x (1 + 3%) - 1 = 5.06% ≈ (2% + 3%) = 5% (give or take 6 basis points)
Using addition rather than multiplication isn't significant at first. But over 45 years, 3 months we get, first using the correct (multiplicative) annual return of 5.06% and then the rounded (additive) return of 5%:
(1 + 5.06%) ^ (542/12) - 1 = 8.2951 = 829.51% increase in nominal value (1 + 5%) ^ (542/12) = 8.0584 = 805.84% increase in value
That's a difference of 2,367 basis points. Not just "a few basis points". Swept under the rug by redirecting attention from a decades long time span to a time span of a single year.
You're welcome to attempt to calculate your own version of what the four factors contributed, over that same time period. What you will find is -
1) Once you have determined each of the four factors, and you attempt to multiply them, they will not equal the S&P 500's total return. Inconsistencies among your data sources do not permit such accuracy.
2) Thus, if you wish to complete the effort, you will be forced to make alter your results into order to make the numbers fit. These assumptions will introduce variances that will likely be as large as the difference betweent the multiplicative and additive approaches.
3) Shifting the time period that is studied by a year or two will have the same effect of changing the underlying factor results.
None of these items obscure the general pattern. Tinker with them all, and inflation still comes out as the largest factor, followed by profits, then dividends, then multiple expansion. But depending upon your choices, the individual numbers will vary.
It's an exercise with a margin of error, but that margin is not larger than the differences among the findings, assuming that the time period studied doesn't greatly change. (Except perhaps the role of profits vs. dividends.)
I have not read the article for nuances but looking at the table posted in this thread, why are somewhat duplicative profits and dividends stated as separate factors? I would have thought corporate profits would subsume dividends. If they are combined into one factor, would inflation and multiple expansion have been allocated higher returns?
The line item in the table is not well labeled. If you read the text, you'll see that this item is growth in profits, not profits.
Company profits are distributed to investors in two ways that are summed: 1. Dividends 2. Stock appreciation
For simplicity, assume no multiple expansion and no inflation.
Whatever a company earns is either distributed to investors as dividends or plowed back into the company to grow profits. With a constant multiplier, the growth in those profits results in the same percentage growth in the price of stock. Thus in theory the investor receives all the profits one way or another.
Of course this assumes that the company makes good use of the money. If it's known to squander cash (or just sit on it), multiples will contract, since the cash isn't being used to generate future profits.
JR writes in essence that minor discrepancies among multiple sources of data are on the same order of magnitude as the difference between multiplicative and additive calculations. So don't sweat it.
I'll decompose that because I disagree with two out of three parts:
1. Data sources will be inconsistent. While true, that's a problem that can be circumvented by using a single data source containing all the necessary data.
Shiller's data contains: a. CPI-U by month - so one can calculate inflation over any period of time b. Earnings (E) by month - so one can calculate earnings growth over any period of time c. S&P price (P) by month - so one can calculate P/E by month and thus multiple expansion over any period of time d. Dividends (D) by month, so one can calculate D/P (dividend rate) over any period of time.
(JR adds another source of discrepancy; for earnings growth he cites a source for the entire economy, not for the S&P 500.)
2. The order of magnitude of these discrepancies is roughly the same as that of percentage differences when comparing multiplicative and additive results.
That is, if one factor adds 3%/year and another factor adds 2%/year, we can combine them by adding and get a 5%/year gain, or we can multiple them to get (1+3%) x (1 + 2%) - 1 = 5.06%. It's just a small difference and comparable to differences introduced by using multiple data sources. Granted.
3. The percentages (which may be off slightly due to #2) are unchanged over time. For example, if one factor accounts for 2% of the gain in a year and another factor accounts for 3% of the gain in a year, the same will be true over 45 years. That is, compounding the returns doesn't change the factor impacts (here 40%:60%).
This is wrong, and this is the main point I've been driving at.
Consider inflation and multiplier expansion. For simplicity, we'll assume that no divs are paid and earnings don't grow.
Annualized rates are: Inflation 3.51% and multiple expansion 2.19% (from original column) To allocate weights, we just take each value's percentage of the sum total.
Inflation accounts for 3.51/5.70 = 62% of the total, and multiple expansion accounts for 2.19/5.70 = 38% of the total.
After 45 years, inflation has increased the price by (1.0351)^45 - 1 = 372%, and multiple expansion has increase the price by (1.0219)^45 - 1 = 165%
Over 45 years, inflation accounts for 372/537 = 69% of the total, and multiple expansion accounts for 165/537 = 31% of the total.
One cannot allocate percentage impacts by using annualized figures. Compounded over time, the larger factors loom ever larger while the smaller factors play less and less of a role. The reason for this is compounding, i.e. multiplying, rather than adding returns.
Worth a mention: None of these items obscure the general pattern. Tinker with them all, and inflation still comes out as the largest factor, followed by profits, then dividends, then multiple expansion
He looked at a 45 year period. But when we take the longer view, more than double that time, i.e. a century, the general pattern shifts. Dividends are more important than inflation.
Over 45+ years, annualized inflation is 3.51% and average dividend rate is 2.75% (I get 2.77%). Over a century, annualized inflation is 2.77% and average dividend rate is 3.83%.
I won't bore you with the formulas, but I will make it easy for you to reproduce and check. Using Shiller's spreadsheet, here are the four expressions needed:
Jan 1976 - March 2021: inflation =POWER(E1811 / E1268,1/45.25)-1 av div rate =SUMPRODUCT(C1269:C1811,1/B1269:B1811)/(45.25 * 12)
July 1921 - June 2021: inflation =POWER(E1814/E614,12/1200) av div rate =SUMPRODUCT(C615:C1814,1/B615:B1814)/(100 * 12)
Comments
Question for the math inclined - he has simply added these rates, should he multiply each component? IE: (1+a/100)(1+b/100)etc…-1?
You're right. Further, this fundamental flaw permeates the column, so I wouldn't look at any of the numbers. (I didn't even read much past the second paragraph because it was apparent that Rekenthaler was trying to turn a multiplicative question into an additive one.)
Consider how he calculates the contribution of multiple expansion. He calculates the multiple expansion at roughly 2.66 (31.5/11.8). He tries to back out this factor by saying that if we divide it out, we would have the remaining S&P growth. Dividing 100% by 2.66% leaves 37% for "everything else". So, obviously (and obviously wrong), multiple expansion accounts for 63% of the S&P growth.
This is so wrong that it left me scratching my head about where he got 63% for quite awhile until I saw what his simplistic calculation was. BTW, the 31.5 figure he used was wrong also. He erroneously transcribed the 30.5 figure that one gets from the data he cited.
I think I can illustrate the problem simply:
Consider a 1x1 square. Its area is 1. Suppose its height doubles (100% growth) and its length triples (200% growth). Its area is now 6 (500% growth).
Would you say that 50% of the growth was due to the height doubling? After all, had the height not doubled, the area would have grown just half as much (to 3 instead of 6). According to his reasoning, the trebling of the length thus must likewise account for half the growth.
It's nonsense because it ignores the cross terms - the effect that one factor growing has on the other factor growing. The (1+a/100) effect.
He starts with junk arithmetic. From there we're off to the races. It only gets worse after that, I assume (not having the stomach for the rest).
One item is MSF's lengthy objection to the pie-chart math, which was the column's straw man. MSF says that math is bogus, my column says that math is bogus. We agree.
The other item is RBRT's question. Should I have multiplied thefour components, not added them? Yes, I should have. I opted for simplicity by adding the components so that I wouldn't have to write a distracting paragraph about why they did not sum to the total return. Had I multiplied the components instead, the numbers would have been a few basis points lower, the order of importance the same, and the conclusion unchanged.
In hindsight, I should have done the calculations the right way and made the explanation.
Percentages assume that the effect of returns from different factors are additive (so that they can be summed, presumably to 100%). That's an erroneous assumption, and it's why I gave my geometric (square) example.
Try to allocate the factor weightings (for height and length) in that example yourself. We assume that height of the square doubles (100% rate of increase) and length triples (200% rate of increase) in a year. Thus the area expands sixfold (500% increase).
What portion of the 500% increase is due to the 100% increase in height, and what portion is due to the 200% increase in length? And remember that the two portions have to sum to 100%.
It's an absurd question. Yet that's the calculation being performed in the table above where factor weightings are assigned. Further, one doesn't need factor weightings at all to answer the original question posed: if one factor (e.g. inflation) were excluded, what would the return be?
You can see how to answer this from my square example. If we exclude the increase in height, then the area triples in size. So the "return", excluding height growth, is 200%. Clean, simple, answers the question, and doesn't go into the weeds with additive percentage factor weights.
Finally, to illustrate just how deeply this conflating of multiplicative and additive effects goes, we can look at the hypothetical in the column where inflation runs at 51% and nominal return runs at 50%/year. With a 50% rate of return, a dollar invested would be worth $92,924,336.24. That's multiplicative (1+50%) x (1+ 50%) x ...
The gain, an additive quantity, would be a dollar less, $92,924,335.24, i.e. an increase of 9,292,433,524%. But what is written is: "investors would have gained 9,292,433,624%". Significant? Not numerically, but conceptually.
I remember even fund managers responding on M* forums when those fourms were vibrant during mid 2000s. I vividly remember Charles Ober of PRNEX responding to question there.
Stay Kool, Derf
Start with 1x1 square (black).
If you double the height, you add the yellow square (1 new square).
If you triple the length, you add the blue squares (2 new squares).
If you do both, and only if you do both, do you add the green (mix of yellow and blue) squares (2 new squares).
We can say that doubling the height doubles the number of squares. That's true whether the length is one (and only the yellow square is added), or the length is three (and the yellow and green squares are added). So doubling the height truly adds 100% to the area.
Likewise, we can say that tripling the length triples the number of squares. That's true whether the height is one (and only the blue squares are added), or the height it two (and the blue and green squares are added). So tripling the length truly adds 200% to the area.
You can't just add up squares created by increasing the height alone (the yellow square) and the squares added by increasing the length alone (the blue squares) and figure that you've got everything covered. Area is computed by multiplying height and length, not by adding them. If all you do is add the yellow and blue squares, you miss the green squares, which exist only because both height and length increased.
It's absurd to suggest that the green squares (or parts thereof) are due solely to changes in height or changes in length (exclusive "or").
>> Had I multiplied the components instead, the numbers would have been a few basis points lower, the order of importance the same, and the conclusion unchanged.
(I believe I recall that JR once said in the past that he reads only MFO items brought to his attention, not as a general follower.)
That his relying on people's intuition that the factor weightings should sum to 100% is wrong. So his factor weighting column must be wrong. And thus the way he calculated it must be wrong.
That the assertion that this resulted in being off by merely a few basis points is sleight of hand. The misdirection comes about in switching from the time frame used in most of the column, 45 years, 3 months (542/12 years), to a single year toward the end. Calling that single year "annualized" enhanced the illusion, as it makes it seem that the factor weighting (percentages) are the same over time.
That's not true. Over time, the discrepancies compound just like the returns. What may be a rounding of a few basis points over one year becomes a large deviation over decades.
Here's a simple example to illustrate. Take two factors, one that adds 2% to the return per year and one that adds 3%. Call them inflation and earnings if it helps make things more concrete.
Over one year (i.e. annualized), the increase in value is:
(1+ 2%) x (1 + 3%) - 1 = 5.06% ≈ (2% + 3%) = 5% (give or take 6 basis points)
Using addition rather than multiplication isn't significant at first. But over 45 years, 3 months we get, first using the correct (multiplicative) annual return of 5.06% and then the rounded (additive) return of 5%:
(1 + 5.06%) ^ (542/12) - 1 = 8.2951 = 829.51% increase in nominal value
(1 + 5%) ^ (542/12) = 8.0584 = 805.84% increase in value
That's a difference of 2,367 basis points. Not just "a few basis points". Swept under the rug by redirecting attention from a decades long time span to a time span of a single year.
1) Once you have determined each of the four factors, and you attempt to multiply them, they will not equal the S&P 500's total return. Inconsistencies among your data sources do not permit such accuracy.
2) Thus, if you wish to complete the effort, you will be forced to make alter your results into order to make the numbers fit. These assumptions will introduce variances that will likely be as large as the difference betweent the multiplicative and additive approaches.
3) Shifting the time period that is studied by a year or two will have the same effect of changing the underlying factor results.
None of these items obscure the general pattern. Tinker with them all, and inflation still comes out as the largest factor, followed by profits, then dividends, then multiple expansion. But depending upon your choices, the individual numbers will vary.
It's an exercise with a margin of error, but that margin is not larger than the differences among the findings, assuming that the time period studied doesn't greatly change. (Except perhaps the role of profits vs. dividends.)
Company profits are distributed to investors in two ways that are summed:
1. Dividends
2. Stock appreciation
For simplicity, assume no multiple expansion and no inflation.
Whatever a company earns is either distributed to investors as dividends or plowed back into the company to grow profits. With a constant multiplier, the growth in those profits results in the same percentage growth in the price of stock. Thus in theory the investor receives all the profits one way or another.
Of course this assumes that the company makes good use of the money. If it's known to squander cash (or just sit on it), multiples will contract, since the cash isn't being used to generate future profits.
I'll decompose that because I disagree with two out of three parts:
1. Data sources will be inconsistent. While true, that's a problem that can be circumvented by using a single data source containing all the necessary data.
Shiller's data contains:
a. CPI-U by month - so one can calculate inflation over any period of time
b. Earnings (E) by month - so one can calculate earnings growth over any period of time
c. S&P price (P) by month - so one can calculate P/E by month and thus multiple expansion over any period of time
d. Dividends (D) by month, so one can calculate D/P (dividend rate) over any period of time.
(JR adds another source of discrepancy; for earnings growth he cites a source for the entire economy, not for the S&P 500.)
2. The order of magnitude of these discrepancies is roughly the same as that of percentage differences when comparing multiplicative and additive results.
That is, if one factor adds 3%/year and another factor adds 2%/year, we can combine them by adding and get a 5%/year gain, or we can multiple them to get (1+3%) x (1 + 2%) - 1 = 5.06%. It's just a small difference and comparable to differences introduced by using multiple data sources. Granted.
3. The percentages (which may be off slightly due to #2) are unchanged over time. For example, if one factor accounts for 2% of the gain in a year and another factor accounts for 3% of the gain in a year, the same will be true over 45 years. That is, compounding the returns doesn't change the factor impacts (here 40%:60%).
This is wrong, and this is the main point I've been driving at.
Consider inflation and multiplier expansion. For simplicity, we'll assume that no divs are paid and earnings don't grow.
Annualized rates are: Inflation 3.51% and multiple expansion 2.19% (from original column)
To allocate weights, we just take each value's percentage of the sum total.
Inflation accounts for 3.51/5.70 = 62% of the total, and
multiple expansion accounts for 2.19/5.70 = 38% of the total.
After 45 years, inflation has increased the price by (1.0351)^45 - 1 = 372%, and
multiple expansion has increase the price by (1.0219)^45 - 1 = 165%
Over 45 years, inflation accounts for 372/537 = 69% of the total, and
multiple expansion accounts for 165/537 = 31% of the total.
One cannot allocate percentage impacts by using annualized figures. Compounded over time, the larger factors loom ever larger while the smaller factors play less and less of a role. The reason for this is compounding, i.e. multiplying, rather than adding returns.
Worth a mention: None of these items obscure the general pattern. Tinker with them all, and inflation still comes out as the largest factor, followed by profits, then dividends, then multiple expansion
He looked at a 45 year period. But when we take the longer view, more than double that time, i.e. a century, the general pattern shifts. Dividends are more important than inflation.
Over 45+ years, annualized inflation is 3.51% and average dividend rate is 2.75% (I get 2.77%).
Over a century, annualized inflation is 2.77% and average dividend rate is 3.83%.
I won't bore you with the formulas, but I will make it easy for you to reproduce and check. Using Shiller's spreadsheet, here are the four expressions needed:
Jan 1976 - March 2021:
inflation =POWER(E1811 / E1268,1/45.25)-1
av div rate =SUMPRODUCT(C1269:C1811,1/B1269:B1811)/(45.25 * 12)
July 1921 - June 2021:
inflation =POWER(E1814/E614,12/1200)
av div rate =SUMPRODUCT(C615:C1814,1/B615:B1814)/(100 * 12)