A riddle about "attributing" gains.

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nisiprius
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A riddle about "attributing" gains.

Post by nisiprius » Mon May 05, 2014 11:24 am

Once upon a time there was a married couple named Capers and Divinity, known to friends as "Cappy" and "Divvy." They invested the family finances according to an unusual plan. During the first six months of each year, Cappy had total charge of the portfolio; during the second six Divvy did. As it happened, wonderful to say, every single year each of them earned precisely 3.5265% over the course of that half-year, so that their total earnings for the year were 7.1774% every year, and their money precisely doubled every ten years. As a result, their starting $100,000 has grown sixteenfold, to $1.6 million.

After forty years of marital and financial bliss, the couple are about to live happily after after in retirement... but suddenly, in an ill-advised moment, it occurs to Divvy to wonder about their comparative contributions to the final result. She approaches Cappy with some scribbling on the back of an envelope and says:

"Our savings grew by $1.5 million. Now, if we do not count my investments, but only yours, it is clear that with only your investments, our savings would only have doubled once every twenty years, and we would only have $400,000. Obviously, then only $400,000 of the $1,600,000 can be attributed to your investments, and the other $1,200,000 are attributable to mine. Therefore, I am responsible for 75% of our gains."

Cappy retorts "But I can do that calculation exactly the other way around. If we only count your investments, it is clear that with your investments alone, our savings would only have grown to $400,000. Therefore, it is my investments should be credited with adding the $1,200,000 and my investments that are responsible for 75% of the gains."

Now, let's revise the story slightly. Let's suppose that Cappy is actually responsible for 7.33% annual growth, and Divvy for 4.08% on top of that, and that we extend time period is extended from 40 years to about 73...

Now read the Wall Street Journal Article, "Test Your Smarts...on Dividend Investing," question 7:

What percentage of the S&P 500's total return since 1940 can be attributed to dividends?

A. 20% B. 35% C. 60% D. 90%

ANSWER: D. Dividends are a big reason why investors make money in stocks, research has found. "Over the long term, dividends have been the main contributors to total return in equity investments," Ian Mortimer and Matthew Page, co-managers of Guinness Global Equity Income, wrote in a recent paper. A key contributor to this phenomenon: dividend reinvestment. The two found that a hypothetical $100 invested in the S&P 500 index in late 1940 would have turned into $17,470 by the end of 2013 if dividends weren't reinvested. But reinvesting them would have produced $265,851.
And I personally believe "B" is the correct answer. And I look forward to a spirited and intricately tangled discussion.
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Re: A riddle about "attributing" gains.

Post by speedbump101 » Mon May 05, 2014 11:41 am

Jack Bogle's quote on this:

"John Bogle, writing on the website IndexUniverse.com, writes the following: “An investment of $10,000 in the S&P 500 Index at its 1926 inception with all dividends reinvested would by the end of September 2007 have grown to approximately $33,100,000 (10.4% compounded). [Using the S&P 90 Stock Index before the 1957 debut of the S&P 500.] If dividends had not been reinvested, the value of that investment would have been just over $1,200,000 (6.1% compounded) – an amazing gap of $32 million. Over the past 81 years, then, reinvested dividend income accounted for approximately 95% of the compound long-term return earned by the companies in the S&P 500.” In other words, the reinvestment of dividends accounted for almost all of stocks’ long-term total return."

http://bonds.about.com/od/Investing-For ... Return.htm

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Re: A riddle about "attributing" gains.

Post by Leeraar » Mon May 05, 2014 12:08 pm

Isn't this like investment expenses? Let's say the total return is 7.5%, dividends are 2.5%.

After 40 years, 1.075^40=18.0, 1.05^40=7.0

Subtract out the original investment, you have (17-6)/17, dividends account for 65% of the gain.

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Re: A riddle about "attributing" gains.

Post by Rodc » Mon May 05, 2014 12:10 pm

Very interesting.

I'm going to vote for B as well.

The logic should be internally consistent, for example who goes first cannot matter, and clearly the accounting used in the OP or in the follow up on Bogle fail this test of reasonableness.
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Re: A riddle about "attributing" gains.

Post by ogd » Mon May 05, 2014 12:18 pm

Nice post, nisiprius. The misuse of those "dividend share" numbers always cracks me up.

The number doesn't answer the hypothetical "what if companies hadn't paid dividends", it only answers this one: "what if dividends had been paid, but you spent them all", or perhaps even more accurately (because spending presumably would have bought you something), "what if dividends had been paid, but they were stolen from your mailbox every quarter". It is, in that way, rather useless. A useful hypothetical, if trivially so, is another one: "what if someone tried to gauge the performance of stocks by looking at price charts alone". Yup, that one happens all the time, beginning with the concept of "lost decade".

The truth is, we don't know what would have happened if companies hadn't paid any dividends. Perhaps the misallocation of capital reinvestment would have been so huge that stocks would have done significantly worse. Looking at dividend-less companies doesn't seem to bear that conclusion, but you could argue that those companies self-selected for good reinvestment opportunities. But -- isn't this the relevant question after all? Invest in dividend payers or in nonpayers or a mix thereof? We know the answer to this one -- you can do either and you'll be just fine.

Until the taxes come in, that is. I'm curious about another number, which is "what percentage of stock performance has been lost to dividend taxes by a taxable investor"? Because that share of dividends truly fits the description of having been paid, but "taken" (to use a neutral term) from your mailbox every quarter. What say you, WSJ?

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Re: A riddle about "attributing" gains.

Post by sscritic » Mon May 05, 2014 12:20 pm

You left out their friend Spendy. The end result in the WSJ combines the efforts of Cappy, Divvy, and Spendy.

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Re: A riddle about "attributing" gains.

Post by nisiprius » Mon May 05, 2014 12:22 pm

speedbump101 wrote:Jack Bogle's quote on this:

"John Bogle, writing on the website IndexUniverse.com, writes the following: “An investment of $10,000 in the S&P 500 Index at its 1926 inception with all dividends reinvested would by the end of September 2007 have grown to approximately $33,100,000 (10.4% compounded). [Using the S&P 90 Stock Index before the 1957 debut of the S&P 500.] If dividends had not been reinvested, the value of that investment would have been just over $1,200,000 (6.1% compounded) – an amazing gap of $32 million. Over the past 81 years, then, reinvested dividend income accounted for approximately 95% of the compound long-term return earned by the companies in the S&P 500.” In other words, the reinvestment of dividends accounted for almost all of stocks’ long-term total return."

http://bonds.about.com/od/Investing-For ... Return.htm

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Re: A riddle about "attributing" gains.

Post by nisiprius » Mon May 05, 2014 12:23 pm

sscritic wrote:You left out their friend Spendy. The end result in the WSJ combines the efforts of Cappy, Divvy, and Spendy.
You aren't allowed to introduce that into the discussion unless you can find an actual name whose nickname could be "Spendy."
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Re: A riddle about "attributing" gains.

Post by ogd » Mon May 05, 2014 12:27 pm

nisiprius wrote:
sscritic wrote:You left out their friend Spendy. The end result in the WSJ combines the efforts of Cappy, Divvy, and Spendy.
You aren't allowed to introduce that into the discussion unless you can find an actual name whose nickname could be "Spendy."

Spandex? Spandau? Spenedict?

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Re: A riddle about "attributing" gains.

Post by VictoriaF » Mon May 05, 2014 12:30 pm

ogd wrote:
nisiprius wrote:
sscritic wrote:You left out their friend Spendy. The end result in the WSJ combines the efforts of Cappy, Divvy, and Spendy.
You aren't allowed to introduce that into the discussion unless you can find an actual name whose nickname could be "Spendy."

Spandex? Spandau? Spenedict?


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Re: A riddle about "attributing" gains.

Post by RyeWhiskey » Mon May 05, 2014 12:31 pm

ogd wrote:
nisiprius wrote:
sscritic wrote:You left out their friend Spendy. The end result in the WSJ combines the efforts of Cappy, Divvy, and Spendy.
You aren't allowed to introduce that into the discussion unless you can find an actual name whose nickname could be "Spendy."

Spandex? Spandau? Spenedict?


Esperanza. Now let the metaphors continue.
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Re: A riddle about "attributing" gains.

Post by Leeraar » Mon May 05, 2014 12:32 pm

nisiprius wrote:
sscritic wrote:You left out their friend Spendy. The end result in the WSJ combines the efforts of Cappy, Divvy, and Spendy.
You aren't allowed to introduce that into the discussion unless you can find an actual name whose nickname could be "Spendy."


In the South African archives I find George Spender, George Spendlove, and Alice Spendelow.

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Re: A riddle about "attributing" gains.

Post by Rob Bertram » Mon May 05, 2014 12:41 pm

I agree with Nisiprius and Rodc

The return explained by dividends is 4.08 / (7.33 + 4.08) which is about 35%

Since returns are compounded, they can't be easily separated mathematically.
X: $100,000 @ 7.33% grows to $1,693,733 after 40 years.
Y: $100,000 @ 4.08% grows to $495,098 after 40 years.
Z: $100,000 @ 11.45% grows to $7,641,927 after 40 years.

X + Y != Z

So my answer is purely on using math consistently. I think that the discussion about creative accounting won't be very lively. There is, of course, the deeper question of tradeoffs of how companies apply their cash. Do they pay dividends, do they re-invest and grow their business? Do stocks that pay dividends have lower appreciation? This, I think will be move lively.

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Re: A riddle about "attributing" gains.

Post by Dale_G » Mon May 05, 2014 12:45 pm

The "late 1940" confuses the issue because the compounding period is not precisely defined. My quick calculation based on 73 years and uniform returns gives 7.33% capital gains and 4.08% dividends for a total of 11.4% annual return. As far as I am concerned dividends accounted for 4.08/11.4 = 36.6%. So B.

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Re: A riddle about "attributing" gains.

Post by Methedras » Mon May 05, 2014 12:52 pm

Rob Bertram wrote:I agree with Nisiprius and Rodc

The return explained by dividends is 4.08 / (7.33 + 4.08) which is about 35%

Since returns are compounded, they can't be easily separated mathematically.
X: $100,000 @ 7.33% grows to $1,693,733 after 40 years.
Y: $100,000 @ 4.08% grows to $495,098 after 40 years.
Z: $100,000 @ 11.45% grows to $7,641,927 after 40 years.

X + Y != Z

So my answer is purely on using math consistently. I think that the discussion about creative accounting won't be very lively. There is, of course, the deeper question of tradeoffs of how companies apply their cash. Do they pay dividends, do they re-invest and grow their business? Do stocks that pay dividends have lower appreciation? This, I think will be move lively.


The Bogleheads group is a relatively savvy and diligent group of investors. Typically, we also understand the concept on nonlinearity, and why returns simply can't be divided up this way.

However, I believe the average person will not understand the nonlinear nature of compound returns, and will quickly buy in to the compelling tale about dividend investing.

Thanks to nisiprius for bringing up this point, as it is very important. Unfortunately, I think most of us are already convinced!

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Re: A riddle about "attributing" gains.

Post by House Blend » Mon May 05, 2014 12:55 pm

nisiprius wrote:Now read the Wall Street Journal Article, "Test Your Smarts...on Dividend Investing," question 7:

What percentage of the S&P 500's total return since 1940 can be attributed to dividends?

A. 20% B. 35% C. 60% D. 90%

I stopped reading your post at this point.

Given that the title of the article is "Test Your Smarts...on Dividend Investing," the bias involved is immediately on display, and so we can guess the kind of back of the envelope calculation they are basing this on.

Ignoring your numbers (that would be too much of a clue), I will suppose
the S&P500 returns 10%/year, and 3%/year in dividends. Doing this for 73 years,

No dividends: (1.07)^73 = 139.6
With dividends: (1.10)^74 = 1051.2

The 139.6 is about 13% of 1051.2, so I'm going to predict without reading the article that they think D is the "smart" answer.

And I'm going to agree with you that it is a rather naive calculation, and more importantly, an idiotic way to justify "Dividend Investing".

And now having read the "answer", I have to say it was even worse than I expected. (So if I don't reinvest the dividends, what am I supposed to do with them? Burn them?) If the companies didn't pay dividends does that mean that they would simply be less profitable, or would they simply grow more?

Equally invalid would be to wonder what would happen if the S&P 500 only paid dividends over that same period, and had no capital growth. I see that

(1.03)^73 = 8.65 (less than 1% of 1051.2),

so in a parallel universe, where the WSJ publishes an article titled "Test Your Smarts on Growth Investing", we would see a multiple choice question on what percentage of the S&P 500's total return since 1940 can be attributed to capital growth, and the "smart" answer would be 99+%.

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Re: A riddle about "attributing" gains.

Post by ThePrune » Mon May 05, 2014 1:09 pm

I also noticed this particular question in todays WSJ and was astounded at their suggested correct answer.
House Blend wrote:Equally invalid would be to wonder what would happen if the S&P 500 only paid dividends over that same period, and had no capital growth. I see that

(1.03)^73 = 8.65 (less than 1% of 1051.2),

so in a parallel universe, where the WSJ publishes an article titled "Test Your Smarts on Growth Investing", we would see a multiple choice question on what percentage of the S&P 500's total return since 1940 can be attributed to capital growth, and the "smart" answer would be 99+%.

This is a great way of presenting the difficulty of "decomposing" the contributions of investment factors that combine via geometric growth. From my perspective their are THREE contributions to growth: pure capital growth contribution, pure dividend reinvestment contribution, and a combined (convolution of) capital & dividend growth contribution.

Maybe the best mutual funds are actually "Growth and Income" :wink:
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Re: A riddle about "attributing" gains.

Post by Methedras » Mon May 05, 2014 2:28 pm

This is a great way of presenting the difficulty of "decomposing" the contributions of investment factors...


For the engineers, scientists, and general number geeks on the forum (and there are quite a few of us), the relevant theoretical background can be found in the superposition principle, http://en.wikipedia.org/wiki/Superposition_principle.

Essentially, a system must be linear if you are going to be able to apply two transformations in any order and get the same result. Since we know investment growth is geometric, this WSJ analysis can be known to be false without event doing the math. The numerical results, as have been shown, are even more damning, as they show the gross magnitude of error that can be found due to this creative accounting of dividend gains.

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Re: A riddle about "attributing" gains.

Post by dbr » Mon May 05, 2014 3:21 pm

Methedras wrote:
This is a great way of presenting the difficulty of "decomposing" the contributions of investment factors...


For the engineers, scientists, and general number geeks on the forum (and there are quite a few of us), the relevant theoretical background can be found in the superposition principle, http://en.wikipedia.org/wiki/Superposition_principle.

Essentially, a system must be linear if you are going to be able to apply two transformations in any order and get the same result. Since we know investment growth is geometric, this WSJ analysis can be known to be false without event doing the math. The numerical results, as have been shown, are even more damning, as they show the gross magnitude of error that can be found due to this creative accounting of dividend gains.


Yep, and we can even do some simple arithmetic to see what is happening in the case of compound growth.

Begin Year One: assets = A, growth is c% from capital growth and d% from dividends. We will calculate the evolution from each.
End Year One: assets = (A) + c * A + d * A (So far so good; the growth is in proportion to the proportion of c and d. The () is around the beginning value and rest is return.)
End Year Two: assets = (A+c*A+d*A) + c*A + c*c*A + c*d*A + d*A + d*c*A + d*d*A = A*(1 + (2c+c*c) + (2d + d*d) + (2c*d))

Now what is to notice in the above is that 2c + c*c is the gain and compounding of capital gains, and 2d + d*d is the gain and compounding of dividends. But already there is this other thing, 2c*d, there where capital gains and dividends are compounding each other. Think about that.

I won't write out the End Year Three formula, but what is going to happen is that there will be more and more terms that mix up products of c and d. So when this is happening trying to say that the growth "comes" a certain amount from dividends and a certain amount from capital growth is a tricky sort of statement to make.

To me this is a lesson that people should not "talk" about what numbers do but simply observe and report what numbers do. The original fallacy is in thinking the words "dividends account for" are actually meaningful.

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Re: A riddle about "attributing" gains.

Post by sscritic » Mon May 05, 2014 3:27 pm

You mean [(1 + c + d) ^ n] - 1 has cross product terms for n greater than one? Who knew.

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Re: A riddle about "attributing" gains.

Post by Oicuryy » Mon May 05, 2014 3:30 pm

Here is a 2012 version of the Mortimer and Page paper.
http://www.gafunds.com/wp-content/uploa ... Matter.pdf

Look at Figure 2. The price return line (blue) looks to have climbed about two grid lines and the total return line (red) climbed a little over three. Price provided about two-thirds of the total return and reinvested dividends provided about one-third.

"B" is the correct answer.

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Re: A riddle about "attributing" gains.

Post by JoMoney » Mon May 05, 2014 3:31 pm

The question was:
What percentage of the S&P 500's total return since 1940 can be attributed to dividends?

Not "should be attributed" or "is attributed", just "can be" ... and anyone who's "Smart on Dividend Investing" would be quick-witted enough to lay this story on thick before the math nerds come around. If you want to sell a dividend strategy you have to be smart about all the potential dividend stories.

Or maybe it's just in the definition of Smart: sharp stinging pain. mental pain or suffering. "You mean I have to pay taxes on these dividends immediately? And then buy more of this growing company at full market mark-up to book-value? Ouch! That smarts!"
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Re: A riddle about "attributing" gains.

Post by dratkinson » Mon May 05, 2014 4:17 pm

If it pleases the court.

I call your attention to this topic from Nov 2013,

"Dividends are Different?"
viewtopic.php?f=10&t=118228

in which the senior investors argued that gains and dividends (withdrawals) are exactly the same.

So you senior investors are not allowed to change your story now. :annoyed


For riddle purposes, as each spouse earned exactly the same return during their half year, ... that's the answer, each is responsible for half the return. The math is just for obfuscation.

Reminds me of the crooked bellhop riddle. 3 guys rent a hotel room for $30 ($10/each). Later the manager notices they should have paid only $25 (total) so sends the bellhop to their room to return $5 (5 single $1). The bellhop, not having the correct change to give each man, only gives $1 to each, and keeps $2 for himself.

Riddle: The men paid $9 each ($27 total), the bellhop kept $2 (27+2=29), so where did the extra dollar go to account for the original $30 paid by the men?


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Re: A riddle about "attributing" gains.

Post by nisiprius » Mon May 05, 2014 4:39 pm

dratkinson wrote:...In which the senior investor argued that gains and dividends (withdrawals from) are exactly the same.

So you senior investors are not allowed to change your story now. :annoyed

For riddle purposes, as each spouse earned exactly the same return during their half year, ... that's the answer, each is responsible for half the return. The math is just for obfuscation.
Oh, sure. I posed a riddle with transparent obfuscation :) in which the error was relatively clear, so as to point out the similarity to the WSJ's example that has covert obfuscation :)
Reminds me of the crooked bellhop riddle. 3 guys rent a hotel room for $30 ($10/each). Later the manager notices they should have paid only $25 (total) so sends the bellhop to their room to return $5 (5 single $1). The bellhop, not having the correct change to give each man, only gives $1 to each, and keeps $2 for himself.

Riddle: The men paid $9 each ($27 total), the bellhop kept $2 (27+2=29), so where did the extra dollar go to account for the original $30 paid by the men?


Spencer = Spendy.
I've always liked that riddle. Since I wasn't able to figure it out for myself, and only know the explanation because it was explained to me, I won't give the answer. Except to say that it's similar to the demonstration, "I have eleven fingers. One, two, let's put these three aside for the moment, four, five, six, seven, eight, now lets add back the three, nine, ten, eleven."
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Re: A riddle about "attributing" gains.

Post by Phineas J. Whoopee » Mon May 05, 2014 4:49 pm

Somehow I seem to remember a time when The Wall Street Journal was aimed at financial professionals who wouldn't be taken in by that kind of malarkey. Shortly prior to the peak of the last cycle, maybe it was.
PJW

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Re: A riddle about "attributing" gains.

Post by Oicuryy » Mon May 05, 2014 5:50 pm

Gains and losses should be measured in nepers.

From $100,000 to $400,000 is a gain of 1.386 Np. From $100,000 to $1,600,000 is a gain of 2.773 Np. Cappy's gain is half the total gain.

From $100 to $17,470 is a gain of 5.136 Np. From $100 to $265,851 is a gain of 7.886 Np. Price gain is 65% of total gain.

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Re: A riddle about "attributing" gains.

Post by Methedras » Mon May 05, 2014 6:06 pm

Oicuryy wrote:Gains and losses should be measured in nepers.

From $100,000 to $400,000 is a gain of 1.386 Np. From $100,000 to $1,600,000 is a gain of 2.773 Np. Cappy's gain is half the total gain.

From $100 to $17,470 is a gain of 5.136 Np. From $100 to $265,851 is a gain of 7.886 Np. Price gain is 65% of total gain.

Ron


An excellent point. Generally speaking, any logarithmic measure would work equally well for purely geometric growth.

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Re: A riddle about "attributing" gains.

Post by nisiprius » Mon May 05, 2014 6:22 pm

Oicuryy wrote:Gains and losses should be measured in nepers.
Had to look up that newfangled unit--I was brought up on good old American decibels--to say nothing of slugs, horsepower, and BTUs--but in fact nepers makes a lot of sense, because for small gains and losses a centineper is the same thing as a percent. So we should get up a lobbying campaign to convince the SEC to introduce centinepers into the system of financial units.
From $100,000 to $400,000 is a gain of 1.386 Np. From $100,000 to $1,600,000 is a gain of 2.773 Np. Cappy's gain is half the total gain.

From $100 to $17,470 is a gain of 5.136 Np. From $100 to $265,851 is a gain of 7.886 Np. Price gain is 65% of total gain.

Ron
Absolutely. And, of course, the extra attributable to dividends is 265,851/17,470 = 2.722 Np = 35% of total gain.

But could you explain why that's correct, and why "90% attributable to dividends" is incorrect, in a couple of sentences, to a someone with average investing savvy?
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Re: A riddle about "attributing" gains.

Post by Oicuryy » Mon May 05, 2014 7:22 pm

nisiprius wrote:But could you explain why that's correct, and why "90% attributable to dividends" is incorrect, in a couple of sentences, to a someone with average investing savvy?

I doubt it.

Does a person with average investing savvy know about continuous compounding? Do they remember Pert? Nepers is just rt.

An ert of 4 from Cappy takes P from $100k to $400k. An ert of 4 from Divvy takes that $400k to $1600k. e is a constant, t is the same for both Cappy and Divvy, so r must be the same for both. They contributed equally to the total growth.

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Re: A riddle about "attributing" gains.

Post by dbr » Mon May 05, 2014 9:37 pm

nisiprius wrote:But could you explain why that's correct, and why "90% attributable to dividends" is incorrect, in a couple of sentences, to a someone with average investing savvy?


It can't be explained except to someone who has a reasonable level of mathematical knowledge. Otherwise it is just blathering words without understanding what one is actually saying. "Attributable to dividends" is not defined mathematically unless the definition is exactly the demonstration presented by, for example, Mr. Bogle, in which case it is true that 90% of returns are attributable to dividends -- by definition of "attributable." However, the conclusion under that definition is not interesting. It is in the category of "not even wrong."

I am not sure what investing savvy is. I think it is entirely possible to be savvy about investing without being knowledgeable about associated mathematical concepts. Mathematics really isn't very necessary to most of what most people do very successfully every day. I say that as a one time math major and person who made a career in fields where mathematics actually was necessary pretty nearly every day.

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Re: A riddle about "attributing" gains.

Post by Epsilon Delta » Tue May 06, 2014 11:46 am

You can challenge the dividend claim without bringing in exponential growth.

Investment gains due to dividends are being defined as a residual -- A half hearted attempt is made to attribute some of the gains to growth and then the claim is made that everything else must be dividends. This is such a common method of misleading that it really should be rejected out of hand, unless you care enough to delve into the argument and make sure it actually takes account of all possibilities.

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Re: A riddle about "attributing" gains.

Post by Leeraar » Wed May 07, 2014 11:15 am

nisiprius wrote:Wall Street Jornal:
What percentage of the S&P 500's total return since 1940 can be attributed to dividends?

The two found that a hypothetical $100 invested in the S&P 500 index in late 1940 would have turned into $17,470 by the end of 2013 if dividends weren't reinvested. But reinvesting them would have produced $265,851.

Over 43 years, to get a multiple of 174.70 requires an annual factor of 1.0733.

To get a multiple of 2658.41 requires a factor of 1.7741 1.1141.

If the dividends are not reinvested, the annual factor is (1-d) * 1.1141 = 1.0733.

Solving, d = 0.0366, the dividend yield is 3.66%. Spin that any way you want.

The correct answer to the question is zero, since dividends have no effect on the total return.

L.
(Edited to fix typo.)
Last edited by Leeraar on Wed May 07, 2014 11:59 am, edited 2 times in total.
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Re: A riddle about "attributing" gains.

Post by dbr » Wed May 07, 2014 11:50 am

Leeraar wrote:
nisiprius wrote:Wall Street Jornal:
What percentage of the S&P 500's total return since 1940 can be attributed to dividends?

The two found that a hypothetical $100 invested in the S&P 500 index in late 1940 would have turned into $17,470 by the end of 2013 if dividends weren't reinvested. But reinvesting them would have produced $265,851.

Over 43 years, to get a multiple of 174.70 requires an annual factor of 1.0733.

To get a multiple of 2658.41 requires a factor of 1.7741.

If the dividends are not reinvested, the annual factor is (1-d) * 1.1141 = 1.0733.

Solving, d = 0.0366, the dividend yield is 3.66%. Spin that any way you want.

The correct answer to the question is zero, since dividends have no effect on the total return.

L.


The WSJ article asked about return and then presented information about gains. Gains and return are two different words that talk about different things. If a newspaper is not even competent to get the meaning of words straight, then what hope is there they can get the math straight.

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Re: A riddle about "attributing" gains.

Post by Higman » Wed May 07, 2014 12:07 pm

I have a minor quibble with those that say Cappy and Divvi contributed equally. While it’s true they each contributed equally as a percentage (3.5625%) during their half year it is Divvi who contributes more in total return. Since Divvi invests in the second half of the year she always has a larger principle to work with than Cappy. Or have I missed something (very possible)?

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Re: A riddle about "attributing" gains.

Post by Rodc » Wed May 07, 2014 12:20 pm

Higman wrote:I have a minor quibble with those that say Cappy and Divvi contributed equally. While it’s true they each contributed equally as a percentage (3.5625%) during their half year it is Divvi who contributes more in total return. Since Divvi invests in the second half of the year she always has a larger principle to work with than Cappy. Or have I missed something (very possible)?



If I double the money over X years from $100K to $200K, and you take over and double it again to $400K in X, of course you "helped" most in dollars, but we both did an equal job from a CAGR point of view. I think this is the point.

Even more to the point is the hidden issue of ignoring that fact that if you have two factors (say dividends and price growth) they don't contribute in a linear (additive) way. You get dividend growth off of the price growth and price growth off the shares bought with reinvested dividends. There is a synergy that the WSJ failed to properly account for.
We live a world with knowledge of the future markets has less than one significant figure. And people will still and always demand answers to three significant digits.

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Re: A riddle about "attributing" gains.

Post by dbr » Wed May 07, 2014 3:10 pm

Rodc wrote:
Higman wrote:I have a minor quibble with those that say Cappy and Divvi contributed equally. While it’s true they each contributed equally as a percentage (3.5625%) during their half year it is Divvi who contributes more in total return. Since Divvi invests in the second half of the year she always has a larger principle to work with than Cappy. Or have I missed something (very possible)?



If I double the money over X years from $100K to $200K, and you take over and double it again to $400K in X, of course you "helped" most in dollars, but we both did an equal job from a CAGR point of view. I think this is the point.

Even more to the point is the hidden issue of ignoring that fact that if you have two factors (say dividends and price growth) they don't contribute in a linear (additive) way. You get dividend growth off of the price growth and price growth off the shares bought with reinvested dividends. There is a synergy that the WSJ failed to properly account for.


Right. I wrote out some expressions in a previous thread that tried to illustrate that, but maybe it was only meaningful to people who already are familiar with the picture.

Consider this: If we multiply something by the sum of c and d twice the result is (c+d)*(c+d) = c2+d2+2cd. So the c2 is attributable to c, and the d2 is attributable to d, but what is 2cd attributable to? That's those mixed terms mentioned above. What you discover is that the idea that the result can be "attributed to" doesn't make any sense. You can use the word in a sentence, but that doesn't say it means anything. It gets twistier though. Suppose we suggest that the dividends are all withdrawn and spent. Then d=0 and the result is only c2, the capital growth. It is a meaningful and correct statement that the loss of gain in that case can be attributed to the fact that d=0. However the exact words "attributed to the fact that d=0" is not the same as saying "attributed to dividends" because there is some necessary detail and precision in that first statement that is not in the second statement.

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Re: A riddle about "attributing" gains.

Post by dratkinson » Wed May 07, 2014 4:17 pm

nisiprius wrote:...it's similar to the demonstration, "I have eleven fingers. One, two, let's put these three aside for the moment, four, five, six, seven, eight, now lets add back the three, nine, ten, eleven."


The way I learned to demonstrate we have eleven fingers is to count down: ten, nine, eight, seven, six... on one hand, plus five on the other makes eleven.
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Re: A riddle about "attributing" gains.

Post by nisiprius » Wed May 07, 2014 8:27 pm

dratkinson wrote:
nisiprius wrote:...it's similar to the demonstration, "I have eleven fingers. One, two, let's put these three aside for the moment, four, five, six, seven, eight, now lets add back the three, nine, ten, eleven."


The way I learned to demonstrate we have eleven fingers is to count down: ten, nine, eight, seven, six... on one hand, plus five on the other makes eleven.
You do that first. That one is good enough to puzzle children, but not usually adults (besides, they've heard it before--when they were kids). You do it the "ten, nine, eight, seven, six" say first, then wait for someone to start to object and then say "Look, let's double-check it by doing it a different way," The second way often baffles adults for a good minute or more by the stopwatch.

It's an example of the magician's technique. A magician never repeats a trick, but may often offer to repeat it and do a trick with a similar effect in a completely different way.

Abbott and Costello had a great demonstration of how 13 x 7 = 28... three different ways.

https://www.youtube.com/watch?v=XnICFjDn97o
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Re: A riddle about "attributing" gains.

Post by dl7848 » Wed May 07, 2014 8:51 pm

dbr wrote:
I am not sure what investing savvy is. I think it is entirely possible to be savvy about investing without being knowledgeable about associated mathematical concepts. Mathematics really isn't very necessary to most of what most people do very successfully every day. I say that as a one time math major and person who made a career in fields where mathematics actually was necessary pretty nearly every day.


+1

I'd further say that while many Boglehead investors are math-savvy, hopefully Boglehead investment principles don't require such knowledge and don't drive the non-math types away. I don't think that's the case, though. I think it's just that when you get a predonimance of engineers in a thread, the outcome is a lot of mathematical posts that would confuse the ordinary investor, even when the attempt is to clear up the myths the ordinary investors live by. :D

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Re: A riddle about "attributing" gains.

Post by Waba » Wed May 07, 2014 9:38 pm

Many moons ago in high school I had to explain to my teacher in Economy that a 10% gain followed by a 10% loss leaves you with 99%, and not 100% as she claimed. She entirely failed to grasp that.
I'm little hopeful that many will get this dividend calculation fallacy.

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Re: A riddle about "attributing" gains.

Post by dl7848 » Wed May 07, 2014 10:57 pm

Waba wrote:Many moons ago in high school I had to explain to my teacher in Economy that a 10% gain followed by a 10% loss leaves you with 99%, and not 100% as she claimed. She entirely failed to grasp that.
I'm little hopeful that many will get this dividend calculation fallacy.


I remember hearing the rules that if you lose 50%, it takes 100% to get even, and so on. I thought that was interesting, but not really practical. Looking at a security or index on a chart seems to be more relevant. One can follow visually how much one is losing and how much is needed to get even again.

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Re: A riddle about "attributing" gains.

Post by JoMoney » Thu May 08, 2014 12:18 am

nisiprius wrote:
dratkinson wrote:
nisiprius wrote:...it's similar to the demonstration, "I have eleven fingers. One, two, let's put these three aside for the moment, four, five, six, seven, eight, now lets add back the three, nine, ten, eleven."


The way I learned to demonstrate we have eleven fingers is to count down: ten, nine, eight, seven, six... on one hand, plus five on the other makes eleven.
You do that first. That one is good enough to puzzle children, but not usually adults (besides, they've heard it before--when they were kids). You do it the "ten, nine, eight, seven, six" say first, then wait for someone to start to object and then say "Look, let's double-check it by doing it a different way," The second way often baffles adults for a good minute or more by the stopwatch.

It's an example of the magician's technique. A magician never repeats a trick, but may often offer to repeat it and do a trick with a similar effect in a completely different way.

Abbott and Costello had a great demonstration of how 13 x 7 = 28... three different ways.

https://www.youtube.com/watch?v=XnICFjDn97o

:D
Three men go into a motel. The desk clerk said the room was $30, so each man paid $10 and went to the room. A while later, the desk clerk realized the room was only $25, so he sent the bellboy to the three guys' room with $5. On the way, the bellboy couldn't figure out how to split $5 evenly between three men, so he gave each man $1 and kept the other $2 for himself.

This meant that the 3 men each paid $9 for the room, which is a total of $27. Add to that the $2 the bellboy kept and the total is $29. Where is the other dollar?
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Re: A riddle about "attributing" gains.

Post by Jack » Thu May 08, 2014 12:43 am

Waba wrote:Many moons ago in high school I had to explain to my teacher in Economy that a 10% gain followed by a 10% loss leaves you with 99%, and not 100% as she claimed. She entirely failed to grasp that.
I'm little hopeful that many will get this dividend calculation fallacy.

In economics it is common practice to use the log difference to compute percentage (also known as neper in physics). When you use this log percentage, a 10% loss and a 10% gain are exactly equal. So depending on the context, she may have been right.

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Re: A riddle about "attributing" gains.

Post by Oicuryy » Thu May 08, 2014 9:57 am

nisiprius wrote:But could you explain why that's correct, and why "90% attributable to dividends" is incorrect, in a couple of sentences, to a someone with average investing savvy?

Second try.

90% of the total dollar return is "attributable" to dividends in the sense that it comes from shares purchased by reinvesting all dividends. Much of the total return is due to the price increase of those additional shares. Crude calculations on Shiller's data suggest the sum of the dividend payments equals about 25% of the total dollar return.

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Re: A riddle about "attributing" gains.

Post by dbr » Thu May 08, 2014 3:12 pm

Oicuryy wrote:
nisiprius wrote:But could you explain why that's correct, and why "90% attributable to dividends" is incorrect, in a couple of sentences, to a someone with average investing savvy?

Second try.

90% of the total dollar return is "attributable" to dividends in the sense that it comes from shares purchased by reinvesting all dividends. Much of the total return is due to the price increase of those additional shares. Crude calculations on Shiller's data suggest the sum of the dividend payments equals about 25% of the total dollar return.

Ron


That is actually pretty good. The key insight is the part about the price increase of the additional shares. A missing sentence is the other side of that coin, the dividends paid on increased value of the investment due to capital gain.

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