Three-fund portfolio returns and variance drain

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longinvest
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Three-fund portfolio returns and variance drain

Post by longinvest » Thu Apr 03, 2014 6:52 am

I just read the latest Bogleheads blog entry which discusses Three-fund portfolio returns. In addition to compound returns (CAGR) and standard deviation, it reports on the variance drain (compound return minus arithmetic return) of different asset allocations (80/20, 60/40,40/60, and 20/80).

In the measured period, a 80/20 portfolio lost up to -1.31% to variance drain, while a 40/60 lost much less (up to -0.26%). Why isn't this variance drain discussed more often?
Historical returns

The following tables give return data for three-fund portfolios assuming investment in Vanguard investor share index funds. Keep in mind that past returns are no guarantee of future returns, but the history reveals how each portfolio allocation has performed over both the 2000 -2002 and 2008 bear markets and ensuing recoveries.

...
Compound returns

The tables below give 3-year, 5-year, 10-year, 15-year, and 17-year compound returns and volatility statistics for each three-fund portfolio allocation.

...
Variance drain

A higher variance of returns results in a fund having a compound return lower than its average arithmetic return. Variance drain (compound return minus arithmetic return) measures the amount of return lost due to variance. The table below gives results over the 1997 to 2013 period.

...
Added on May 20, 2018:

The answer to my question is that variance drain is irrelevant because we shouldn't consider arithmetic returns in the first place.

Here's a summary I wrote later in this topic:
longinvest wrote:
Sun May 20, 2018 2:04 pm
As the original poster of this topic, let me try to summarize what I've learned, so far:
  • One should never think in terms of average arithmetic returns. One should think in terms of compound returns (CAGR) instead when looking at past returns.
  • Future returns are unknown.
  • The total bond market is less volatile than the total stock market because it contains a significant amount of short-term bonds (giving it an overall intermediate duration) and, therefore, can't fluctuate as much in value as stocks, thanks to mathematics.
  • International stocks are riskier than domestic ones (higher political risk, harder to enforce legal rights, currency conversion costs and spreads, etc).
  • The Three-Fund Portfolio promoted by author and Bogleheads forum co-founder Taylor Larimore is an excellent approach to diversify one's exposure to various investment risks and dampen stock volatility with bonds at rock bottom cost.
  • Seeking higher returns by trying to bet on future asset correlations, as promoted by author Larry Swedroe (who has an awful track record), is a form of speculation, mostly driven by greed (e.g. the hope to beat the returns of a simple three-fund portfolio using a portfolio of high-cost actively managed funds of similar volatility). (See viewtopic.php?p=3936798#p3936832 for supporting facts).
Last edited by longinvest on Sun May 20, 2018 4:20 pm, edited 2 times in total.
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Re: Three-fund portfolio returns and variance drain

Post by 1210sda » Thu Apr 03, 2014 7:31 am

The difference between the geometric mean (compound return) and the arithmetic Mean (average) is due to the standard deviation.
If the standard deviation is zero, then the two means are equal. The larger the Standard deviation, the greater the difference in the two returns.

Another way of measuring risk.

Is this not correct ??

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Re: Three-fund portfolio returns and variance drain

Post by longinvest » Thu Apr 03, 2014 7:42 am

It seems correct to me. But the way I see it is: Your expected return is not what you expect. (That would have been a terrific thread title). Does this make sense?
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Re: Three-fund portfolio returns and variance drain

Post by midareff » Thu Apr 03, 2014 8:24 am

Thanks for posting this... I had been meaning to post a question about statistics that may be available for volatilities performance drag.

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Re: Three-fund portfolio returns and variance drain

Post by dbr » Thu Apr 03, 2014 8:28 am

There is certainly common discussion of the fact that arithmetic average is not the correct way to estimate the average return over a period of compounding. I suppose that most people would not call error made in doing arithmetic incorrectly a "drain," and that may be why you don't see the discussion in those terms very often.

In the context of taking the arithmetic average annual return as what you "should" get from your investment if there were not variance, the idea ought to be part of the discussion for why one would try to optimize a portfolio by reducing risk at same return (arithmetic). Maybe people who write in that context just assume everyone knows what is going on.

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Re: Three-fund portfolio returns and variance drain

Post by sweeneyastray » Thu Apr 03, 2014 9:12 am

Here are a few examples to illustrate the effect.

A. You have $100 invested in something. The value goes up 10% one year, then down 10% the next year. How much is the investment worth?
Answer: $99. You lost 1% of your original investment over the course of two years. Where did that extra dollar go? Nowhere. Step by step: after one year, your investment went to $100 + 10% of $100 = $100 + $10 = $110.
The next year, your investment went down 10%. That is, it went to $110 - 10% of $110 = $110 - $11 = $99.
The 10% decrease was off of a higher amount, $110, so in dollar terms, that decrease was bigger ($11) than the increase (10% of $100, or $10).
Notice that the "arithmetic mean" of the two % changes is 0% (+10 and -10 arithmetically average to 0). But I'd say that that arithmetic mean has no meaning, so calculating a variance drain from it isn't particularly helpful.

B. Switch the order of the increase and decrease. Does it matter?
No. Drop 10% the first year, you're down to $100 - $10 = $90. Then go up 10%. Your final amount is $90 + 10% of $90 = $90 + $9 = $99. This time, the decrease is only $10, but your increase is even smaller: $9.

The end result is the same, no matter the order of returns (as long as you just look at the final number). This is because what you're really doing when you add a percentage is multiplying by a factor.
Add 10% = multiply by 100% + 10% = multiply by 110% = multiply by 1.1
Add 25% = multiply by 1.25
Subtract 10% = multiply by 90% = multiply by 0.9

And multiplying by 1.1, then by 0.9 (as in case A above) is the same as multiplying by 0.9, then by 1.1 (as in this case).

C. Amplify the up and down. Say you start with $100, then you go up 50%, then down 50%. (Still an "arithmetic average" of 0%.)

Start: $100
End of first year: $100 + 50% of $100 = $150.
End of second year: $150 - 50% of $150 = $75.

The actual endpoint is much lower than in cases A and B. This is your "variance drain": case C, with its 50% swings up and down, leads to a worse position than cases A or B, with their 10% swings up and down.

But that's simply a result of the fact that the percent changes are computed from different bases. And the bigger the swings, the bigger the differences in those bases, year to year.

You should never arithmetically average these sorts of sequential percent changes. For that matter, you should never just add percents either, unless you're sure they're being computed from the same base.
At small percentages, the effect is small. Increasing by 1%, then by 2% is only a very slight bit larger than increasing by 1 + 2 = 3%. But the fact that the error is small with low percentages means that it's easy to overapply this shortcut.

(First post! I have been reading the forum for a while, but I teach percentage math for the GMAT, GRE, SAT, etc., so this is finally an area in which I feel I can contribute!)

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Re: Three-fund portfolio returns and variance drain

Post by 1210sda » Thu Apr 03, 2014 9:13 am

dbr wrote: try to optimize a portfolio by reducing risk at same return (arithmetic).
This is part of the efficient frontier, along with trying to increase return at the same risk.

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Re: Three-fund portfolio returns and variance drain

Post by dbr » Thu Apr 03, 2014 9:44 am

1210sda wrote:
dbr wrote: try to optimize a portfolio by reducing risk at same return (arithmetic).
This is part of the efficient frontier, along with trying to increase return at the same risk.

1210
And the OP's question is why one does one not hear the words "variance drain" commonly used in discussion of this topic.

Note there was some discussion a while back about the question of whether the return plotted on the y-axis in most efficient frontier charts is the arithmetic average return for a year or the compound annualized average return and I think I read that it is the average return that is used and therefore we would still have to include variance drag if we wanted to project a compound return.

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Re: Three-fund portfolio returns and variance drain

Post by 1210sda » Thu Apr 03, 2014 9:53 am

dbr wrote:
dbr wrote:
Note there was some discussion a while back about the question of whether the return plotted on the y-axis in most efficient frontier charts is the arithmetic average return for a year or the compound annualized average return and I think I read that it is the average return that is used and therefore we would still have to include variance drag if we wanted to project a compound return.
I agree.

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Re: Three-fund portfolio returns and variance drain

Post by steve_14 » Thu Apr 03, 2014 11:10 am

longinvest wrote:In the measured period, a 80/20 portfolio lost up to -1.31% to variance drain, while a 40/60 lost much less (up to -0.26%). Why isn't this variance drain discussed more often?
Because it's just an invented mathematical construct. The greater the volatility, the greater the difference between the real (geometric) and arithemetic (the average of the geo returns)...but so what?

This random fact has no bearing on economic returns, and should only be of interest to high school math students. That blog entry shouldn't have muddied the waters by even mentioning it.

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Re: Three-fund portfolio returns and variance drain

Post by dbr » Thu Apr 03, 2014 12:44 pm

steve_14 wrote:
longinvest wrote:In the measured period, a 80/20 portfolio lost up to -1.31% to variance drain, while a 40/60 lost much less (up to -0.26%). Why isn't this variance drain discussed more often?
Because it's just an invented mathematical construct. The greater the volatility, the greater the difference between the real (geometric) and arithemetic (the average of the geo returns)...but so what?

This random fact has no bearing on economic returns, and should only be of interest to high school math students. That blog entry shouldn't have muddied the waters by even mentioning it.
I agree that in that blog article the subject should never have been mentioned.

Wasn't there another thread recently where there was a discussion of how fund returns are reported and it was agreed that by SEC rule multi-year returns are compound and not arithmetic returns? I am not sure how arithmetic returns enter the discussion except for people doing efficient frontier analysis, as previously discussed.

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Re: Three-fund portfolio returns and variance drain

Post by 1210sda » Thu Apr 03, 2014 1:19 pm

Unfortunately, both returns are used by some investors.

If investors are not aware of the difference between arithmetic and geometric, they can be using the wrong one and not know why their return projections are so generous (or the reverse). Even well known folks such as Dave Ramsey can mistakenly say the market has returned close to 12% over the long term, when the reality is that CAGR is more like 10%.

For this reason, knowing how or why the two return numbers are different can help.

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Re: Three-fund portfolio returns and variance drain

Post by LadyGeek » Thu Apr 03, 2014 3:41 pm

FYI - This thread is linked from the blog (take a look at the bottom of the article): Three-fund portfolio returns | Bogleheads® Blog

What would really help bring the point home is to expand the wiki article; it's not much more than a definition: Variance drain

If anyone has a suggestion, post here. I'll also take a crack at it.

sweeneyastray - Welcome!
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Re: Three-fund portfolio returns and variance drain

Post by steve_14 » Thu Apr 03, 2014 4:00 pm

dbr - Not sure, didn't see that thread.

1210sda - Good point, there should be a wiki noting the difference. Oh wait

LadyGeek - I've noticed Bogleheads are suckers for catchy two word ideas. I'd resist that urge here and get rid of that page. Nothing is being "drained". Replace it with a page called "Understanding the difference between arithmetic and geometric returns", with an emphasis that price volatility has nothing to do with economic returns.

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Re: Three-fund portfolio returns and variance drain

Post by 1210sda » Fri Apr 04, 2014 9:12 am

While the term "Variance Drain" is new to me, the concept is not.

There are many ways to look at risk, volatility being one of them. Some will say that volatility is not risk, but if it has a direct impact on my return, I consider it risk.

If one has a 100% investment in an SP500 index fund AND, there were no volatility, that investment would probably have generated a CAGR (over the long term) of roughly 12%, the same as the average return. Since there IS volatility, (standard deviation was about 20%), the CAGR is reduced to roughly 10%.

If the investment has lower volatility than as described above (bonds, balanced portfolio, etc. ) then the drain is lower. Variance Drain is directly related to volatility......the greater the volatility, the greater the drain and vice versa.

Those who have a hard time understanding standard deviation, may be better able to relate to the reduction of their return from Average to Compound...i.e. Variance Drain.

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Re: Three-fund portfolio returns and variance drain

Post by dbr » Fri Apr 04, 2014 9:27 am

1210sda wrote:
There are many ways to look at risk, volatility being one of them. Some will say that volatility is not risk, but if it has a direct impact on my return, I consider it risk.
Almost all of the worries that people have that they call risk in investing originate in the volatility of investments. People who try to say that volatility is not risk usually mean that some measure or another of volatility, such as standard deviation per se, does not explicitly report the thing they are worried about. Variance drain is an example. The phenomenon exists, but to calculate it you have to take the standard deviation and put it in another formula to see what the effect is. You also have to worry about whether that formula you are using is only valid assuming certain distributions for the returns.

I think the best example of risk originating in volatility is in the study of safe withdrawal rates from portfolios. In that context volatility appears under the name "sequence of returns." But at the time recognizing that issue compared to planning withdrawals as a simple amortization was a big deal. In that case also there is the distinction as to whether one should use Monte Carlo methods assuming certain distributions or use the historic returns method which sort of samples actual historical behavior of assets, no matter how indescribable.

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Re: Three-fund portfolio returns and variance drain

Post by 1210sda » Fri Apr 04, 2014 9:51 am

Thank you, DBR.

As Barry Barnitz showed in his blog post, the measure of Variance Drain is Arithmetic Return minus Compound Return (although I believe he had it reversed in the blog...Compound Return minus Arithmetic Return...:-) ).

My question is, How is "sequence of return" quantified ? Is the reduction applied to the Arithmetic Mean or the Geometric Mean ? My initial reaction is to say from the Geometric mean. But by how much.

Another thought....the sequence of returns drag is accidental. It depends on when you start withdrawals from your portfolio and what history does thereafter. Variance Drag is more intentional....it depends on your AA which we control. Does this make sense to you?

Look forward to hearing from you. I value your input.

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Re: Three-fund portfolio returns and variance drain

Post by dbr » Fri Apr 04, 2014 10:06 am

1210sda wrote:
My question is, How is "sequence of return" quantified ? Is the reduction applied to the Arithmetic Mean or the Geometric Mean ? My initial reaction is to say from the Geometric mean. But by how much.

Another thought....the sequence of returns drag is accidental. It depends on when you start withdrawals from your portfolio and what history does thereafter. Variance Drag is more intentional....it depends on your AA which we control. Does this make sense to you?
It makes total sense. Sequence of returns is first and foremost a phenomenon of historical chance, luck, as Jim Otar puts it. A quantification is the wide range in safe withdrawal rates depending on the year of retirement. That number varied, it seems to me, from around 4% in 1966 to as high as 8% for retirements beginning in 1982. I don't think there is a mathematical formula that expresses that as a correction to returns, although one could find out what rate of compound average return just runs you out of money at any given withdrawal rate. Another quantification would be the variability of final wealth when withdrawing from a portfolio. FireCalc, for example, will give you a spread sheet containing this result whenever you run the model.

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Re: Three-fund portfolio returns and variance drain

Post by midareff » Sat Apr 05, 2014 10:42 am

sweeneyastray wrote:Here are a few examples to illustrate the effect.

A. You have $100 invested in something. The value goes up 10% one year, then down 10% the next year. How much is the investment worth?
Answer: $99. You lost 1% of your original investment over the course of two years. Where did that extra dollar go? Nowhere. Step by step: after one year, your investment went to $100 + 10% of $100 = $100 + $10 = $110.
The next year, your investment went down 10%. That is, it went to $110 - 10% of $110 = $110 - $11 = $99.
The 10% decrease was off of a higher amount, $110, so in dollar terms, that decrease was bigger ($11) than the increase (10% of $100, or $10).
Notice that the "arithmetic mean" of the two % changes is 0% (+10 and -10 arithmetically average to 0). But I'd say that that arithmetic mean has no meaning, so calculating a variance drain from it isn't particularly helpful.

B. Switch the order of the increase and decrease. Does it matter?
No. Drop 10% the first year, you're down to $100 - $10 = $90. Then go up 10%. Your final amount is $90 + 10% of $90 = $90 + $9 = $99. This time, the decrease is only $10, but your increase is even smaller: $9.

The end result is the same, no matter the order of returns (as long as you just look at the final number). This is because what you're really doing when you add a percentage is multiplying by a factor.
Add 10% = multiply by 100% + 10% = multiply by 110% = multiply by 1.1
Add 25% = multiply by 1.25
Subtract 10% = multiply by 90% = multiply by 0.9

And multiplying by 1.1, then by 0.9 (as in case A above) is the same as multiplying by 0.9, then by 1.1 (as in this case).

C. Amplify the up and down. Say you start with $100, then you go up 50%, then down 50%. (Still an "arithmetic average" of 0%.)

Start: $100
End of first year: $100 + 50% of $100 = $150.
End of second year: $150 - 50% of $150 = $75.

The actual endpoint is much lower than in cases A and B. This is your "variance drain": case C, with its 50% swings up and down, leads to a worse position than cases A or B, with their 10% swings up and down.

But that's simply a result of the fact that the percent changes are computed from different bases. And the bigger the swings, the bigger the differences in those bases, year to year.

You should never arithmetically average these sorts of sequential percent changes. For that matter, you should never just add percents either, unless you're sure they're being computed from the same base.
At small percentages, the effect is small. Increasing by 1%, then by 2% is only a very slight bit larger than increasing by 1 + 2 = 3%. But the fact that the error is small with low percentages means that it's easy to overapply this shortcut.

(First post! I have been reading the forum for a while, but I teach percentage math for the GMAT, GRE, SAT, etc., so this is finally an area in which I feel I can contribute!)

It isn't (in my mind anyway) about the year to year impact, it is about the impact of the daily volatility accumulated over a year.

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Re: Three-fund portfolio returns and variance drain

Post by steve_14 » Sat Apr 05, 2014 11:33 am

As an example, you buy shares in Joe's Donuts. Sales double and the stock price rises to reflect that. Day to day volatility, or "variance drain" has no bearing on this. It doesn't change the market value of the company.

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Re: Three-fund portfolio returns and variance drain

Post by 1210sda » Sat Apr 05, 2014 11:43 am

steve_14 wrote:As an example, you buy shares in Joe's Donuts. Sales double and the stock price rises to reflect that. Day to day volatility, or "variance drain" has no bearing on this. It doesn't change the market value of the company.
Let's say stock price goes from $100 to $200 to reflect the sales doubling. (100% gain)

In year two, sales are cut in half and stock price drops to reflect that. Stock is now $100 (50% loss)

Arithmetic Average is (100%-50%)/2 = 25% over the two years

Geometric Average is 0% (from $100 to $100 over the two years)

In this example, the Arithmetic Average had a 25% Variance Drain.

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Re: Three-fund portfolio returns and variance drain

Post by dbr » Sat Apr 05, 2014 11:59 am

1210sda wrote:
steve_14 wrote:As an example, you buy shares in Joe's Donuts. Sales double and the stock price rises to reflect that. Day to day volatility, or "variance drain" has no bearing on this. It doesn't change the market value of the company.
Let's say stock price goes from $100 to $200 to reflect the sales doubling. (100% gain)

In year two, sales are cut in half and stock price drops to reflect that. Stock is now $100 (50% loss)

Arithmetic Average is (100%-50%)/2 = 25% over the two years

Geometric Average is 0% (from $100 to $100 over the two years)

In this example, the Arithmetic Average had a 25% Variance Drain.

1210
That's a good illustration. And the point here is that this whole discussion is just about understanding the right way of doing arithmetic when talking about how to characterize returns for several periods when we know the return in each of the periods. Calling this "drain" implies some kind of a problem or a danger when we are really just talking about how to do numbers. The only danger is choosing the wrong computation for what it is we want to know. Since most people want to know the expected result at the end of several periods from applying successive returns, one period after the other, the answer is that one wants to know the compound or geometric average. To say that that using the wrong arithmetic means we have ignored a "drain" seems bizarre to me.

Why would one want to know the arithmetic average over several periods? The answer to that is we would do this if we are trying to estimate the properties of the distribution from which each annual return is thought to be a random sample. To estimate those properties we look at a set of returns and use the arithmetic average and the standard deviation of that set of returns as an estimate of the average and standard deviation of the underlying distribution. This is standard analysis of an assumed statistical model. Also, this is why efficient frontier analysis uses arithmetic average as the return on the vertical axis. If you are looking at such information and you want to project multi-year returns, you are supposed to know that you need to generate a compound average from those numbers, taking into account variance.

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Re: Three-fund portfolio returns and variance drain

Post by 1210sda » Sat Apr 05, 2014 12:34 pm

Thanks again, DBR.

Maybe instead of "drain", it could be called "differential" or some such term.

I think it's important to know the difference between the two calculations so that when it's misused by some (Ramsey, for example) we won't be as easily misled....at least that's my hope.

I agree that "you are supposed to know...." which one to use, but not everybody does.

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Re: Three-fund portfolio returns and variance drain

Post by steve_14 » Sun Apr 06, 2014 12:24 am

1210sda wrote:
steve_14 wrote:As an example, you buy shares in Joe's Donuts. Sales double and the stock price rises to reflect that. Day to day volatility, or "variance drain" has no bearing on this. It doesn't change the market value of the company.
Let's say stock price goes from $100 to $200 to reflect the sales doubling. (100% gain)

In year two, sales are cut in half and stock price drops to reflect that. Stock is now $100 (50% loss)

Arithmetic Average is (100%-50%)/2 = 25% over the two years

Geometric Average is 0% (from $100 to $100 over the two years)

In this example, the Arithmetic Average had a 25% Variance Drain.
Right, and clearly one of these is the actual return, and one is just fun with numbers. My point is, let's not confuse people by subtracting them, and implying money was drained away somehow. I do agree, however that's it's important people understand the difference between the means generally.

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Re: Three-fund portfolio returns and variance drain

Post by Random Walker » Sun Apr 06, 2014 1:46 am

I feel this is a topic that is not discussed here. I remember getting a feel for it first from Ferri's All About Asset Allocation. Portfolio volatility will always cause the compounded return of a portfolio to fall short of the average weighted return of the portfolio components. And one goal in portfolio design should be to bring the compounded return closer to the average annual return. This is not just a mathematical construct. It is real money. The more weakly, negatively, less than perfectly positively correlated components in a portfolio, the more efficient the portfolio will be. This is why a simple portfolio with only a few funds may well not be optimal. In fact it is quite possible that it is worthwhile to have a somewhat higher weighted portfolio expense ratio for the sake of having more components that are weakly correlated. I certainly appreciate that costs are certain while returns, correlations, volatilities can vary. But nonetheless there are strong potential advantages to multiasset class investing. Gibson's Asset Allocation, Balancing Financial Risk summarizes this very well. In fact the most important takeaways from the book are summarized in one graph and a few pages of text that can be found in a short essay on the internet. What matters is overall portfolio volatility. It is a drag on returns. Adding volatile non correlated or weakly correlated asset classes to a portfolio can actually dampen portfolio volatility and increase portfolio efficiency.

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Re: Three-fund portfolio returns and variance drain

Post by grayfox » Sun Apr 06, 2014 4:13 am

Average is just the measure of central tendency. This wikipedia article, Average, lists about a dozen different types of "averages", or measures of central tendency. Even though each is trying to nail down the same idea, i.e. central tendency, they all have a different calculation and give a different answer.

I suppose you could calculate the difference between each average-type statistic and give it a descriptive name. [BTW, I have heard volatility drag rather than volatility drain. "Drain" sounds like your money is going into the sewer. "Drag" sounds like some kind of head wind that is slowing you down.]

Then there is Statistical dispersion which is how much results can vary. Wikipedia lists half a dozen measures of statistical dispersion.

But then we can ask, the central tendency and statistical dispersion of what?

If we are investing and holding for, say, 30 years, we care about the final outcome.
So what we want to know about is the central tendency and dispersion of the Total Return over the 30-year holding period.

What difference does it make what's the average and dispersion of daily, monthly, annual or bi-annual returns?
All you should care about is the result at the end, i.e. Total Return over the 30-year holding period.

Case 1. Risk Free Asset
In Apr-2014, put $32,565 into 2043 Nov 15 STRIPS.
In Nov-2043, end up with $100,000 +/-0.00
Total Return = 100,000 / 32,565 = 3.07x
No dispersion of final outcome.

Case 2. Risky Asset
In Apr-2014, put $32,565 into Vanguard Total Stock Market (VTSMX)
In Nov-2043 end up with ???
There is a range of outcomes. We can guestimate the range if we like.
Let's make a guess that there is 95% chance of TR being between 1.0x and 10.0x
So 95% chance you will end up with as little as $32,656 or as much as $320,656.
2.5% chance you will have a loss. 2.5% chance > $320,656

Case 3. Even More Risk
In Apr-2014, put $32,565 into Vanguard Emerging Markets (VEIEX)
In Nov-2043 end up with ???.
Let's make a guess that there is 95% chance of TR being between 0.5x and 20.0x
So 95% chance you will end up with as little as $16,328 or as much as $656,120.

:idea: The risk in cases 2 and 3 is that there is a range of outcomes, and some of the outcomes are worse than the No Risk case. In some case, you can even have net loss, i.e. TR < 1.

Observe I did not mention anything about the annual return, arithmetic mean, geometric mean, variance, standard deviation, volatility drain or whatever.
Those are numbers you can calculate, but not necessary.

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Re: Three-fund portfolio returns and variance drain

Post by sweeneyastray » Sun Apr 06, 2014 7:02 am

dbr wrote:
1210sda wrote:
steve_14 wrote:As an example, you buy shares in Joe's Donuts. Sales double and the stock price rises to reflect that. Day to day volatility, or "variance drain" has no bearing on this. It doesn't change the market value of the company.
Let's say stock price goes from $100 to $200 to reflect the sales doubling. (100% gain)

In year two, sales are cut in half and stock price drops to reflect that. Stock is now $100 (50% loss)

Arithmetic Average is (100%-50%)/2 = 25% over the two years

Geometric Average is 0% (from $100 to $100 over the two years)

In this example, the Arithmetic Average had a 25% Variance Drain.

1210
That's a good illustration. And the point here is that this whole discussion is just about understanding the right way of doing arithmetic when talking about how to characterize returns for several periods when we know the return in each of the periods. Calling this "drain" implies some kind of a problem or a danger when we are really just talking about how to do numbers. The only danger is choosing the wrong computation for what it is we want to know. Since most people want to know the expected result at the end of several periods from applying successive returns, one period after the other, the answer is that one wants to know the compound or geometric average. To say that that using the wrong arithmetic means we have ignored a "drain" seems bizarre to me.

Why would one want to know the arithmetic average over several periods? The answer to that is we would do this if we are trying to estimate the properties of the distribution from which each annual return is thought to be a random sample. To estimate those properties we look at a set of returns and use the arithmetic average and the standard deviation of that set of returns as an estimate of the average and standard deviation of the underlying distribution. This is standard analysis of an assumed statistical model. Also, this is why efficient frontier analysis uses arithmetic average as the return on the vertical axis. If you are looking at such information and you want to project multi-year returns, you are supposed to know that you need to generate a compound average from those numbers, taking into account variance.
Right, this arithmetic average of annual returns is used in efficient frontier analysis. But the assumption that annual returns are drawn from a stable distribution that happens to be normal has always seemed really weird to me. Why are annual returns distributed normally? What underlying law causes that to be so? For instance, why not daily returns? If daily returns are distributed normally, then annual returns would be log-normally distributed. And then you follow the chain of logic down to the tiniest tick you can, and finally you realize there isn't a theoretical justification at any time frame.

I was a physics major, so this lack of true theoretical foundation for MPT/CAPM/efficient frontier analysis bothered the heck out of me when I first learned about it in business school. And none of my finance professors could explain it to me. Or why standard deviation of returns is the preferred measure of risk in that analysis. Or, for that matter, why another Nobel Prize was handed out for applying decades-old physics to option pricing, another place where the assumption of normality is not only unjustified, it's positively dangerous. Economics and finance theory want to be harder sciences than they ever can be, and so fall into these kinds of misleading errors, in my opinion.

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Re: Three-fund portfolio returns and variance drain

Post by grayfox » Sun Apr 06, 2014 7:34 am

I found this chart that shows the histogram of 20-year returns for S&P 500. X-axis is annualized real total return.

Image

S&P 500 Composite 20-year total real returns

It doesn't look like a run-of-the-mill normal distribution. There's a lot in the middle from about 6-10, but then there are secondary peaks around 2-4, and 11-13. Multimodal, like maybe there was some regime switching in the past.

I think this chart shows that you should plan for a range of possible outcomes. During the same period as the chart, you could have put your money in a 20-year Treasury and probably gotten 2% real return with low risk. This chart shows that, with stocks, it's possible to do worse than that. There are some points down around 0% and at least one is negative. Thus the risk being related to the dispersion.

BTW I don't believe MPT requires that the distribution of returns has to be normal. For example, you could model returns as student-t distribution or something else.

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Re: Three-fund portfolio returns and variance drain

Post by sweeneyastray » Sun Apr 06, 2014 8:26 am

Great graph, grayfox! And good point about the assumption of normality. It's more the assumption of stability that I question in all these cases. Until someone has the grand unified theory of why these returns should follow any particular distribution over time, with constant parameters, I'll content myself with the point you just made: plan on a range of returns from equities.

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Re: Three-fund portfolio returns and variance drain

Post by grabiner » Sun Apr 06, 2014 8:52 am

grayfox wrote:I found this chart that shows the histogram of 20-year returns for S&P 500. X-axis is annualized real total return.

Image

S&P 500 Composite 20-year total real returns

It doesn't look like a run-of-the-mill normal distribution. There's a lot in the middle from about 6-10, but then there are secondary peaks around 2-4, and 11-13. Multimodal, like maybe there was some regime switching in the past.
The main reason it isn't normal is that you have overlapping periods; normal distributions assume independent observations. The highest 20-year returns are from 1979-1998, and given those returns, you would expect 1978-1997 and 1980-1999 to have returns almost as high, since they share 19 years. Similarly, the poor returns of 1929-1948 share 19 years with poor returns from 1928-1947 or 1930-1949.

Returns for non-overlapping periods should be closer to normal, but there are only seven non-overlapping 20-year periods available, and they cover such a large range that the nature of the stock market may have changed over time. (There is no reason to expect the stock market to have the same risk level in 1990-2009 that it had in 1870-1899.)
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Re: Three-fund portfolio returns and variance drain

Post by dbr » Sun Apr 06, 2014 9:05 am

sweeneyastray wrote:Great graph, grayfox! And good point about the assumption of normality. It's more the assumption of stability that I question in all these cases. Until someone has the grand unified theory of why these returns should follow any particular distribution over time, with constant parameters, I'll content myself with the point you just made: plan on a range of returns from equities.
Both of the points you bring up about what the distribution actually is and whether it is stable are often discussed but perhaps not so often actually implemented. You could look up Benoit Mandelbrot's attempt to apply fractal models to this problem, an effort which I think he even gave up in the long run. Larry Swedroe often discusses the "Larry" portfolio which is aimed at reducing tail risk of low outcomes by combining a small allocation of high returning volatile small cap stocks with lots of bonds. This is an explicit practical acknowledgement of the non-normality of investment returns.

My guess is that normality is often elected as an assumption for the two reasons that it is a two parameter distribution completely defined by mean and standard deviation, and that it is mathematically tractable for projecting multi-year evolution from repeated one year samples. Hence the infamous sqrt(t) pronouncement about risk and time. (But see here: http://www.norstad.org/finance/risk-and-time.html ). Having also a physics and some applied statistics background, one is certainly accustomed to various possibilities for the distribution most appropriate to a certain phenomenon and to whether that distribution is forced by real characteristics of the phenomenon or is just a convenient model that fits. I remember being struck in school by how the assumption that all nuclei of a given atomic number and mass are identical and that the decay of any one nucleus is random, independent, and exactly the same for all of them immediately generates the phenomenon of exponential decay and that the large number of nuclei in a sample means a Poisson distribution will describe the statistics. In that case all you need to find out is the rate of decays, which is also a measure of the aforesaid probability of one nucleus decaying in a unit of time. Unfortunately no one has yet discovered anything that simple and elegant in investing.

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Re: Three-fund portfolio returns and variance drain

Post by dbr » Sun Apr 06, 2014 9:07 am

grabiner wrote:
Returns for non-overlapping periods should be closer to normal, but there are only seven non-overlapping 20-year periods available, and they cover such a large range that the nature of the stock market may have changed over time. (There is no reason to expect the stock market to have the same risk level in 1990-2009 that it had in 1870-1899.)
Aren't there histograms of non-overlapping one year returns?

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Re: Three-fund portfolio returns and variance drain

Post by longinvest » Sun Apr 06, 2014 9:20 am

According to this paper (http://papers.ssrn.com/sol3/papers.cfm? ... id=2027471), variance drain can be quantified through a simple equation. The theory does not rely on any specific distribution of returns.
The well-known approximation of the difference between the arithmetic average and geometric average returns as one-half of the variance of the underlying returns is reexamined using Jensen's Inequality. The "defect" in Jensen's Inequality, is given an exlicit formula in terms of the variance following some ideas put forward by Holder. A new form of the AM-GM Inequality follows and is is applied to financial returns. Both exact, and approximate relations between the arithmetic average, geometric average, and variance of returns are discussed. The mathematical formulation of these relations are free of distributional assumptions governing the underlying returns process.
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Re: Three-fund portfolio returns and variance drain

Post by dbr » Sun Apr 06, 2014 9:33 am

longinvest wrote:According to this paper (http://papers.ssrn.com/sol3/papers.cfm? ... id=2027471), variance drain can be quantified through a simple equation. The theory does not rely on any specific distribution of returns.
The well-known approximation of the difference between the arithmetic average and geometric average returns as one-half of the variance of the underlying returns is reexamined using Jensen's Inequality. The "defect" in Jensen's Inequality, is given an exlicit formula in terms of the variance following some ideas put forward by Holder. A new form of the AM-GM Inequality follows and is is applied to financial returns. Both exact, and approximate relations between the arithmetic average, geometric average, and variance of returns are discussed. The mathematical formulation of these relations are free of distributional assumptions governing the underlying returns process.
Thanks. That is an interesting read.

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Re: Three-fund portfolio returns and variance drain

Post by grabiner » Sun Apr 06, 2014 9:42 am

dbr wrote:
grabiner wrote:
Returns for non-overlapping periods should be closer to normal, but there are only seven non-overlapping 20-year periods available, and they cover such a large range that the nature of the stock market may have changed over time. (There is no reason to expect the stock market to have the same risk level in 1990-2009 that it had in 1870-1899.)
Aren't there histograms of non-overlapping one year returns?
There are, although some of the other reasons for independence may fail, leading to heavier tails than normal. Daily returns are not normally distributed because individual events may move the market by huge amounts; there was one day in 1987 on which the market dropped by more than the standard deviation for a single year. In addition, consecutive days are dependent, particularly with increasing volatility in bear markets; the largest one-day gain was two days after the largest one-day loss (in 1987), and the third-largest one-day loss was one day after the second-largest one-day loss (in 1929). And there are still changes over time; for example, this histogram of nominal (not real) annual returns looks a bit too fat-tailed but six of the 19 extreme values were in 1928-1937.

And here's [url=http://www.jpmorganinstitutional.com/bl ... y_long.pdf]a JP Morgan Paper on the non-normality, showing a fat left tail of monthly returns.
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Re: Three-fund portfolio returns and variance drain

Post by sscritic » Sun Apr 06, 2014 9:42 am

longinvest wrote:According to this paper (http://papers.ssrn.com/sol3/papers.cfm? ... id=2027471), variance drain can be quantified through a simple equation. The theory does not rely on any specific distribution of returns.
The "defect" in Jensen's Inequality, is given an exlicit formula in terms of the variance following some ideas put forward by Holder.
And the author put a comma after Inequality. Can you really trust anyone who would do that? What is the subject of this sentence? And what is an exlicit formula? I wouldn't trust this paper at all.

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Re: Three-fund portfolio returns and variance drain

Post by longinvest » Sun Apr 06, 2014 9:48 am

sscritic wrote:
longinvest wrote:According to this paper (http://papers.ssrn.com/sol3/papers.cfm? ... id=2027471), variance drain can be quantified through a simple equation. The theory does not rely on any specific distribution of returns.
The "defect" in Jensen's Inequality, is given an exlicit formula in terms of the variance following some ideas put forward by Holder.
And the author put a comma after Inequality. Can you really trust anyone who would do that? What is the subject of this sentence? And what is an exlicit formula? I wouldn't trust this paper at all.
I think that a paper should be dismissed on the basis of containing unsound scientific/mathematical proofs, not on the basis of a few grammar or typographical errors. But, that's me...
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Re: Three-fund portfolio returns and variance drain

Post by LadyGeek » Sun Apr 06, 2014 9:49 am

FWIW, here's another perspective on the S&P 500 stock market distribution shape, spanning from 1950 through 1999: Normal or Log-Normal

Source: Gummy-stuff - finiki, the Canadian financial wiki

An update to the wiki article on variance drain is in process...
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Re: Three-fund portfolio returns and variance drain

Post by LadyGeek » Sun Apr 06, 2014 7:40 pm

The wiki has been updated: Variance drain
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Re: Three-fund portfolio returns and variance drain

Post by longinvest » Sun Apr 06, 2014 7:46 pm

LadyGeek wrote:The wiki has been updated: Variance drain
Cool!
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Re: Three-fund portfolio returns and variance drain

Post by chenghs » Mon Jul 24, 2017 1:56 pm

First of all, thanks a lot for all the great insights on investment! I am a newbie here and just started to learn all the great info.

One quick thing: I look into the 'Three Fund Portfolio' google sheet shared by the member. There are some minor errors in the calculation. Wondering who I should reach out to get it updated since many people might use this google sheet. Thanks a lot,

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Re: Three-fund portfolio returns and variance drain

Post by Random Walker » Sat May 19, 2018 11:28 pm

I believe this is a very important topic. When we create a portfolio, we look at the expected returns, volatilities, and correlations of potential portfolio additions. The expected return of an investment is an estimate of an arithmetic mean return. The expected return of a portfolio is a weighted average of the expected returns of the individual portfolio components. The expected SD of a portfolio will be less than the weighted average of the component SDs because of less than perfect correlations. So, looking forward we use arithmetic averages.
But the returns we eat are geometric (compounded) averages. So one goal in portfolio construction should be to minimize the variance drain on a portfolio. If we can create two portfolios with the same expected return, but one has a lower expected SD, we can expect it to have less variance drain and a greater compounded return.
The sticky point is that all the low correlated potential portfolio additions are more expensive than the three fund portfolio. The path to uncorrelated portfolio additions includes tilting to size and value, adding other factors, maybe even alternatives. Each of these potential marginal improvements to portfolio efficiency should lessen variance drain, but at increased marginal cost. If two portfolios have the same arithmetic average return, but one has an SD of 15% and the other has an SD of 11%, the less volatile portfolio should have about 0.5% additional compounded return due to less variance drain. So there are trade offs involved in choosing the lowest cost route. Of course, costs are certain and the additional benefits of improved portfolio efficiency only potential.

Dave

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Re: Three-fund portfolio returns and variance drain

Post by dbr » Sun May 20, 2018 9:50 am

^ Good explanation.

I noticed in reading the Wiki that there might already be an ambiguity in the first line where average return is mentioned without the qualifier that the term refers to the arithmetic average. The reason this is important is that the conventional data published for returns of investments are actually CAGR's, that is to say compound returns. I am not sure where one finds a source to look up arithmetic average returns as used by academics doing MPT.

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Re: Three-fund portfolio returns and variance drain

Post by IlliniDave » Sun May 20, 2018 10:00 am

To me it's just a quirk of the math.

If I have $100 and lose half in one year (-50% return) then gain it back the next year (+100% return) my cumulative return is zero. That if I sum the two individual returns and divide by two I get 25% doesn't mean I somehow lost 25% due to "volatility drag". Like was mentioned above, that line of thought is mathematically erroneous.
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Re: Three-fund portfolio returns and variance drain

Post by Random Walker » Sun May 20, 2018 10:34 am

IlliniDave,
I’m no math whiz, but I disagree. The issue is not just looking at past returns as you wrote above. The issue is building efficient portfolios for the future. Looking forward, each potential portfolio addition has an expected return, a simple mean with a potential dispersion of outcomes. And the portfolio will have an expected return based on the components. We know, before the fact, that there will be volatility drag on the portfolio. The portfolio compounded return will be less than the weighted simple average return of the portfolio components. By choosing portfolio components with correlations and volatilities that mix well together, we can minimize that drag. Looking forward, if two portfolios have the same arithmetic mean return but different standard deviations, the compounded return of the less volatile portfolio will be greater.
This effect is a function of the square of the standard deviation! So the drag rises rapidly with increased equity allocations in a portfolio. Anything that can be done to keep expected return constant and minimize portfolio volatility will be beneficial. What a potential portfolio component adds to a portfolio depends on expected return, volatility, correlations, when correlations tend to change, and of course cost.

Dave

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Re: Three-fund portfolio returns and variance drain

Post by Tyler9000 » Sun May 20, 2018 11:08 am

dbr wrote:
Sun May 20, 2018 9:50 am
I noticed in reading the Wiki that there might already be an ambiguity in the first line where average return is mentioned without the qualifier that the term refers to the arithmetic average. The reason this is important is that the conventional data published for returns of investments are actually CAGR's, that is to say compound returns. I am not sure where one finds a source to look up arithmetic average returns as used by academics doing MPT.
Yeah, with the proliferation of articles covering valuations and expected returns I've noticed a trend of authors switching back to arithmetic averages even for their historical analyses. For example, In his book "Your complete guide to factor-based investing" Larry Swedroe exclusively uses arithmetic averages to argue for the historical performance of various investing factors. That's my one complaint in an otherwise excellent book, but it's an important one since ignoring the effects of volatility on real-world compound returns has a significant distorting effect on the resulting numbers.

BTW, if you want to compare the historical arithmetic average to the geometric average for a portfolio, try plugging your asset allocation into these Annual Returns and Long Term Returns calculators. Both are adjusted for inflation, but the first shows the arithmetic average and standard deviation while the second shows the CAGR over the exact same timeframe.
Last edited by Tyler9000 on Sun May 20, 2018 11:31 am, edited 1 time in total.

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Re: Three-fund portfolio returns and variance drain

Post by rbaldini » Sun May 20, 2018 11:26 am

In short: the CAGR (geometric mean return) is what matters to long term return; if you know CAGR, arithmetic mean is irrelevant; so the difference between them ("variance drain") is irrelevant. Hopefully when someone reports a historical return to you, they are reporting the historical CAGR- in which case there is no need to know variance drain. If someone instead reports the arithmetic mean, then to get CAGR you'd have to calculate and subtract this "variance drain" (or just calculate CAGR directly from historical data)... but then you probably shouldn't trust them anyway.

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Re: Three-fund portfolio returns and variance drain

Post by longinvest » Sun May 20, 2018 11:47 am

Dear Random Walker,
Random Walker wrote:
Sat May 19, 2018 11:28 pm
I believe this is a very important topic.
Thank you for reviving this old topic.

I think that my second post, in this topic, summarized it best: "Your expected return is not what you expect."

In other words, Bogleheads should beware the writings of authors who promote active investing and discuss so-called "expected returns" using arithmetic returns. Author Larry Swedroe comes to mind; following his advice based on such expected returns, in the past, would have been detrimental to one's wealth. Here are two links about the track record of author Larry Swedroe:
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Re: Three-fund portfolio returns and variance drain

Post by TJSI » Sun May 20, 2018 12:13 pm

Bah, humbug.

There is no such thing as volatility drag. It is just the use of mathematics to obfuscate what is going on.
If there is a loss due to volatility, it is due to your investment plan. You are buying high and selling low.

As for me, I prefer to get my free lunch at the diversification banquet.

TJSI

(PS: RandomWalker I though your were cured. But no you persist.)

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Re: Three-fund portfolio returns and variance drain

Post by dbr » Sun May 20, 2018 12:21 pm

TJSI wrote:
Sun May 20, 2018 12:13 pm
Bah, humbug.

There is no such thing as volatility drag. It is just the use of mathematics to obfuscate what is going on.
If there is a loss due to volatility, it is due to your investment plan. You are buying high and selling low.

As for me, I prefer to get my free lunch at the diversification banquet.

TJSI

(PS: RandomWalker I though your were cured. But no you persist.)
Now there is some real irony.

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