Yesterday I was kind of burned out on the dialogue with Doc, and thought it was taking us too far off topic, so kept my last reply short. Now I've got my mojo back, and have realized that this probably is an important topic to clarify; i.e., what do we mean by "return" or "returns" in the context of this thread, and how is it calculated?
Doc seems to have two issues. One is related to terminology, and one is related to the subject of whether or not a particular return calculation "assumes" reinvestment of dividends.
Regarding terminology, as I pointed out before, I use the term "annualized returns" on the charts, e.g., in the OP. A quick Google search shows that this is a widely used and well understood term. If we want to hold up Vanguard as representative of "the industry", you can do this search to see if Vanguard uses the term:
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"annualized return" site:Vanguard.com
This returns many links, primarily to research papers they've done. Take a look at any of them, and you'll find the term used. I selected their most recent "Vanguard's economic and investment outlook" (December 2014), and found the term used nine times. So I think we can safely say that this is an "industry standard" term.
I'm using the term exactly the way Vanguard uses it.
On the performance pages/tabs on their website, they use the term "Average annual returns" instead, but it means exactly the same thing. They use this term because the SEC requires it. I happen to think "annualized returns" is better, since as I said before, it leads to less confusion as to whether it's an arithmetic mean or geometric mean being shown. If you read academic papers or finance textbooks, or even many investment articles on the web, you'll see that arithmetic mean and geometric mean are both widely used, so it's important to distinguish between them.
As far as I can tell, "annualized return" is used exclusively to indicate the geometric mean of annual returns when discussing multi-year periods (it's also used to indicate annualized returns for periods shorter than one year). In my opinion, the SEC made a mistake in using "annual" instead of "annualized".
Speaking of the SEC, that's a good segue into the topic of reinvested dividends that Doc seems so concerned about. He mentioned that I insist on including them in my calculations. Well I don't insist on it, but the SEC does insist on it, specifically with respect to calculating "average annual total return". Everyone can read it for themselves here:
Final Rule: Disclosure of Mutual Fund After-Tax Returns (S7-09-00), in the section titled "Item 21. Calculation of Performance Data".
After showing the formula that must be used, which is exactly the simple version of the future value formula I showed in an earlier reply, but using different symbols, they provide this instruction:
SEC wrote:2. Assume all distributions by the Fund are reinvested at the price stated in the prospectus (including any sales load imposed upon reinvestment of dividends) on the reinvestment dates during the period.
So yes, reinvestment of distributions is assumed in the average annual total return values provided by Vanguard, which are the values I use to calculate 5-year and 10-year (or any subsequent N-year) returns.
Further, the geometric mean of annual returns, which is the mathematical term for the calculation of annual[ized] average [total] return, "assumes" that the entire cumulative return from previous years is invested in the subsequent year; i.e., no distributions are taken from the fund--they are all reinvested. So if we have annual total returns r1, r2, and r3, the
geometric mean cumulative total return, expressed as a percentage, is calculated as:
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Cumulative Total Return % = (1+r1) * (1+r2) * (1+r3) - 1
EDIT: And the geometric mean is calculated as:
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GeoMean = [ (1+r1) * (1+r2) * (1+r3) ] ^ (1/3) - 1
Another common term used for this, Compound Annual Growth Rate, or CAGR, makes it explicit that this is a
compound return. Compounding implies reinvestment of the previous returns, whether they came from capital appreciation or dividends. Since I use Vanguard's "total return" values, which incorporates both capital return and income return, I don't have to worry about what part of the return comes from dividends and what comes from capital appreciation. And I don't have to worry about cash flows, because there are no cash flows in the calculation--just annual returns.
If I wanted to do a calculation based on cash flows, instead of looking at annual returns, I'd look at cumulative total return over the period of interest, in which case there would be two cash flows: the original investment, and the final value (SEC uses the terms "initial payment" and "ending redeemable value", or "ERV"). I guess I could make it more complicated, and break out the distributions from the capital appreciation, and use an IRR calculation with more cash flows, but why would I go to the trouble of doing that when I can get the same value with a much simpler calculation?
Now, I think I know where Doc is coming from with the whole "you don't have to assume reinvestment of dividends" thing. He's probably read the same paper that I've read that explains that a common misconception is that a present value calculation or internal rate of return calculation assumes that the cash flows (e.g., dividends) are reinvested. These calculations don't assume anything about what happens with the cash flows, but just describe the relationship between the present value of cash flows and a constant discount rate.
I think the misunderstanding arises because a common statement is something like this (from "Investments", by Bodie, Kane and Marcus"):
The yield to maturity can be interpreted as the compound rate of return over the life of the bond under the assumption that all bond coupons can be reinvested at an interest rate equal to the bond's yield to maturity. Yield to maturity is widely accepted as a proxy for average return.
(Highlights mine)
I highlighted the first phrase because it's import to note that "can be interpreted as" does not say that the formula for yield to maturity assumes anything. It's just a way to think about it--it's a mental model that helps people understand the concept. Doc may feel that it's misleading, but I don't. I highlighted "proxy" to emphasize that YTM is not the same thing as average [annual[ized]] return.
As I've shown before, an IRR calculation will give you the same result whether you assume reinvested dividends or not, but as I also showed, this does not mean that they both necessarily result in the same "ending redeemable value". As an investor, I'm more interested in the ERV than the IRR.
Kevin
If I make a calculation error, #Cruncher probably will let me know.