# Average Stock Market Returns Aren't Average

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Overly complex post. I'll save other board members some time and summarize the post:

Geometric average returns are lower than arithmetic average returns.
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Overly complex post. I'll save other board members some time and summarize the post:

Geometric average returns are lower than arithmetic average returns.

Sorry that you thought it was such a waste of time. Cliff Asness seemed to like it enough to leave a comment ...

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Overly complex post. I'll save other board members some time and summarize the post:

Geometric average returns are lower than arithmetic average returns.

Bonus points for providing the formula showing how the difference between the two returns relates to volatility.  ;)

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This is the part I think is a useful reminder, especially for long-term buy and hold investors, of which there are many on this board (including me):

"Many people think that uncertainty washes out when you buy and hold for a long period of time. Not so, that is the fallacy of time diversification. Although the average return becomes more certain with more periods you don’t get the average return you get the total payoff and that becomes more uncertain with more periods."

There's a link in the original article that's good, too.

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Other than the arithmetic average versus geometric point, I'm not sure I understand either what he is saying or the use of what he is saying.

e.g., if you take actual returns over the history of the S&P over 25 years, the complete data are (from the distribution of CAGR for rolling 25 year periods from 1871-2013 (where the rolls are performed monthly)):

Minimum - 3.79%

Maximum - 17.08%

Average - 9.36%

Median - 8.71%

There were no negative returns.  This data seems more useful than what the article posits, to me.  Perhaps I am missing something, however?

FYI, there are no negative returns after 15 year rolling periods, and even at 15, the minimum is -0.3%.

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Other than the arithmetic average versus geometric point, I'm not sure I understand either what he is saying or the use of what he is saying.

e.g., if you take actual returns over the history of the S&P over 25 years, the complete data are (from the distribution of CAGR for rolling 25 year periods from 1871-2013 (where the rolls are performed monthly)):

Minimum - 3.79%

Maximum - 17.08%

Average - 9.36%

Median - 8.71%

There were no negative returns.  This data seems more useful than what the article posits, to me.  Perhaps I am missing something, however?

FYI, there are no negative returns after 15 year rolling periods, and even at 15, the minimum is -0.3%.

You're looking at one of the winners globally.  The experience in other countries isn't as good.

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You're looking at one of the winners globally.  The experience in other countries isn't as good.

Do you know of any place I can go to look at other countries, which show the distribution CAGR in the same way as above?  That would be interesting to see.

Moreover, I guess the question turns into, is it likely that the U.S. will not act as it has in the past?  And if so, how likely?

Taking a particular case, the lowest returns was investing at the peak of 1929--how much worse should be expected?

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Overly complex post. I'll save other board members some time and summarize the post:

Geometric average returns are lower than arithmetic average returns.

Bonus points for providing the formula showing how the difference between the two returns relates to volatility.  ;)

Easy :): G = A - V / 2

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Do you know of any place I can go to look at other countries, which show the distribution CAGR in the same way as above?  That would be interesting to see.

Don't know about the "same way" part, but the Credit Suisse Global Investment Returns Yearbook is a good place for long-term international numbers ... https://publications.credit-suisse.com/tasks/render/file/?fileID=0E0A3525-EA60-2750-71CE20B5D14A7818

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What I found interesting was the link between volatility of return and final return. He seems to argue that a volatile return of the same magnitude is less preferable than a stable one over time (obviously) and slightly lower (maybe less obvious). So we've heard value investors like Buffet say that they prefer a lumpy 20% vs a smooth 15% anytime. In this case, what they are saying is that a lumpy higher return is still higher than a gradual lower return. But perhaps implicit in this is that the lumpy return be high enough to compensate for that volatility, otherwise, any reasonable investor would just take the gradual return.

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I think this topic is always confusing in the investment context, because engineers (I am one) bring an often-flawed or incomplete perspective to the discussion, but they bring it with a didactic style that makes the discussion more charged than needed.  Generally, I see this issue raised by people who understand math, but not economics.

1) It is true and obvious (I hope to most reading here), and also helpful to point out that geometric average returns are indeed not the same as arithmetic average returns.  As Heilko said...

2) And indeed, mathematically, the more volatile the returns are by year, the bigger difference there will be between the two return measures.

3) The jump that seems to be made by some above in the thread, is that returns in the stock market by real assets are then somehow path dependent on their volatility... hence the injection of Buffett's lumpy vs. smooth return comments.

I think #3 is the odd jump that leverages math, but doesn't understand economics properly.

Real asset (stocks) returns are not (by and large, exceptions due to capital raising and buybacks we all agree do exist) path dependent.  If next year stocks drop 75%, the ultimate value in 20 years (on aggregate - eg, for an index) won't change... so yes, high volatility will make the arithmetic / geometric return spread increase, but it won't do so by lowering geometric / realized returns, it will actually increase arithmetic average returns to compensate.

Thus, the beautiful and famous quote below has a two edged meaning:

"A 50% loss requires a 100% gain to get back to even."  This quote implies you want to avoid 50% losses at all costs (obviously) because they are nearly impossible to make up, but also inverting the statement is a wonderful reminder that a company / index that goes down 50% for a non fundamental reason must double to reach "fair" value.

Two sides to every coin.

Ben

PS, there are a class of investments that are path dependent, and of course volatility is catastrophic to their returns (3x ETFs).... that hasn't been show to be true to aggregate stocks over time, in many countries, and many market cycles.

PPS, further to the original article's point about "time diversification" (and the linked article from 2000) and the fact that cumulative return differences actually widen with time whereas annualized / geometric returns narrow....  Again, I think this is an engineering mindset "explaining" something for the sake of sounding smart, but missing the whole point.  *Yes* if you hold stocks for 15-20 years, there is a DRAMATICALLY wide range of outcomes (measured cumulatively), but I think we can all agree that the fact that those outcomes (in the US markets, in the last 100 years, disclaimer disclaimer, etc etc) are all positive is a pretty big deal, and also they are all positive with regards to the majority of alternative investments that could be made... this is the point, not that stocks don't have a wide range of outcomes.  Someone once said that risk is permanent loss of capital, not a range of expected outcomes.  In this context, it seems that Time Diversification (is that a phrase?) is real, and works (for stocks - certainly behavior may be different in other countries for various reasons)... At least, to me, that's always been the point.... you can invest in something highly like to give you a positive real return after tax over a long time horizon.

Sorry for the length...

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