## Arithmetic Returns vs. Geometric Returns

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jyun
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### Arithmetic Returns vs. Geometric Returns

At the age of 40, I have been investing for about 5 years. Initially, I focused on individual stocks, but now I hold 100% in the S&P 500. Recently, I came across an interesting topic about Arithmetic Returns vs. Geometric Returns (the idea that high returns with high volatility are risky). I found an intriguing article stating that even with an average annual return of 25%, high volatility could result in the principal remaining the same as it was 5 years ago. I find this concept difficult to understand. Has anyone in the Bogleheads community experienced something similar? Additionally, what advice can I share with friends who only pursue high returns?

https://financetrain.com/arithmetic-ret ... ic-returns
Random Walker
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### Re: Arithmetic Returns vs. Geometric Returns

The issue is termed “volatility drag” or “variance drain”. Because of volatility, the compounded or geometric mean return is always less than the arithmetic or simple mean return. An equation approximates the relationship: Ari Mean -0.5* (SD^2) = Geo Mean.

Here are some examples I keep in my head. It takes a 100% gain to break even after a 50% loss. It takes a 50% gain to break even after a 33% loss. It takes a 33% gain to break even after a 25% loss. Most obvious example is start with \$100. Earn 100% year 1 and lose 50% year 2. After two years you have \$100. Your compounded return is 0% but your simple average return is (100-50)/2= 25%. We eat our compounded returns, but we create portfolios based on forward looking expected returns which are simple means. If two portfolios have the same average return but different volatilities, the one with lower volatility will have less volatility drag, a higher compounded return, accumulate more money. Volatility drag is a real cost. The reason it occurs is that after significant losses, there is less money present to experience the subsequent future higher returns.

Dave
esteen
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### Re: Arithmetic Returns vs. Geometric Returns

Random Walker wrote: Mon Jun 03, 2024 9:49 pm Here are some examples I keep in my head. It takes a 100% gain to break even after a 50% loss. It takes a 50% gain to break even after a 33% loss. It takes a 33% gain to break even after a 25% loss. Most obvious example is start with \$100. Earn 100% year 1 and lose 50% year 2. After two years you have \$100. Your compounded return is 0% but your simple average return is (100-50)/2= 25%.
These are exactly the kinds of examples I give folks for understanding arithmetic vs geometric means.

At the risk of beating the dead horse... I had a 50% loss followed by a 50% gain:

\$100 x 0.5 = \$50
\$50 x 1.5 = \$75

My arithmetic mean over the two periods is -50%+50% = 0%... But I ended up with 25% less money!
This post is for entertainment or information only, and should not be construed as professional financial advice. | | "Invest your money passively and your time actively" -Michael LeBoeuf
bh1
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### Re: Arithmetic Returns vs. Geometric Returns

Using arithmetic mean on most financial data is almost always wrong. Not different, just wrong.

https://www.skewray.com/articles/change ... are-ratios
eigenperson
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### Re: Arithmetic Returns vs. Geometric Returns

It's wrong in some contexts. Using geometric returns can also be wrong. You have to know what the two numbers mean.

Say you have a security that either has a +100% or a -55% return each year, both options being equally likely. This has a negative geometric return. If you buy and hold the security for 100 years you will likely end with less than you started with. However, the security has a positive expected return. If you buy it, on average you will make money.

This is not a contradiction. If you buy \$100 of the security and hold it for 100 years, the mean value of your ending portfolio is \$65 billion, but the median value is \$0.51.

A more reasonable strategy is to make the security only a portion of your portfolio, and periodically rebalance. With that strategy, such a security can make you a lot of money with low risk. It would be a mistake to forgo this security entirely just because it has a negative geometric return.
rkhusky
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### Re: Arithmetic Returns vs. Geometric Returns

Arithmetic mean is about the same as geometric mean for small changes, (1+x)(1+y) ~= 1 + x + y, when xy << x or y.
Last edited by rkhusky on Tue Jun 04, 2024 7:05 am, edited 1 time in total.
nisiprius
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### Re: Arithmetic Returns vs. Geometric Returns

Arithmetic means are wrong when averaging across successive time periods, where the return at the end of each time period becomes the starting value for the next. They are right when averaging among multiple assets over the same period of time. That's because across time returns multiply ("compound"), but over the same period of time they add.

Over a single year, a portfolio's return is the weighted arithmetic average of all of the returns of its constituents. It's simple math. If I have \$10,000 in asset A and \$10,000 in asset B, and over a year asset A has a return of 10% and asset B has a return of 5%, at the end of the year I'll have \$11,000 + \$10,500 = \$21,500. So \$20,000 has grown by \$1,500 and the portfolio return is \$1,500/\$20,000 = 0.075 = 7.5%, which is the ordinary arithmetic mean of 10% and 5%.

That's why arithmetic averages of asset performance numbers legitimately get calculated and presented, and become available for unscrupulous or ignorant commentators to misapply.
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gatorking
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### Re: Arithmetic Returns vs. Geometric Returns

Random Walker wrote: Mon Jun 03, 2024 9:49 pm Here are some examples I keep in my head. It takes a 100% gain to break even after a 50% loss.
Dave
The Geometric returns equivalent statement would be:
It takes 69 consecutive 1% losses to go from 100 to ~50 and then 69 consecutive 1% gains to get back to ~100.
dbr
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### Re: Arithmetic Returns vs. Geometric Returns

nisiprius wrote: Tue Jun 04, 2024 7:03 am Arithmetic means are wrong when averaging across successive time periods, where the return at the end of each time period becomes the starting value for the next. They are right when averaging among multiple assets over the same period of time. That's because across time returns multiply ("compound"), but over the same period of time they add.

Over a single year, a portfolio's return is the weighted arithmetic average of all of the returns of its constituents. It's simple math. If I have \$10,000 in asset A and \$10,000 in asset B, and over a year asset A has a return of 10% and asset B has a return of 5%, at the end of the year I'll have \$11,000 + \$10,500 = \$21,500. So \$20,000 has grown by \$1,500 and the portfolio return is \$1,500/\$20,000 = 0.075 = 7.5%, which is the ordinary arithmetic mean of 10% and 5%.

That's why arithmetic averages of asset performance numbers legitimately get calculated and presented, and become available for unscrupulous or ignorant commentators to misapply.
Yes, this is a good explanation of what you care about.

It has to do with whether or not you are presenting the returns as compounded over intervals of time or you are using the returns as samples from which to estimated the mean and other statistics of a hypothetical underlying distribution of possible returns or, as in the example combining present returns.

It is basic mathematics that if the outcome arises from a product of successive numbers one would want to know what single number could be put in that product and generate the same result as the actual product. Such a number is called the geometric average. In the second case one is finding what number could be put in place in the actual numbers in a sum and produce the same result. Such a number is called the arithmetic average. There are lots of other kinds of averages.

A quibble is that in compound growth it is not actually the geometric mean of the returns that is calculated. What is calculated is the geometric mean of the growth factors, the number by which each end amount is multiplied to get the next end amount. That number is 1 plus the return. One then extracts a return out of that mean by subtracting one. In that math returns in % are first converted to decimal equivalents.

Another way to express averages as a general concept is to associate with each concept of average a transformation of the data followed by arithmetic average of the transformed data followed by inverse transformation of the data. In the case of geometric average of a set of numbers the transform is take the log, take the arithmetic average, exponentiate. In the case of CAGR of returns in percents the process is divide by 100, add 1, take the log, take the arithmetic average, exponentiate, subtract one, multiply by 100.
Last edited by dbr on Tue Jun 04, 2024 7:26 am, edited 1 time in total.
dbr
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### Re: Arithmetic Returns vs. Geometric Returns

gatorking wrote: Tue Jun 04, 2024 7:12 am^
Random Walker wrote: Mon Jun 03, 2024 9:49 pm Here are some examples I keep in my head. It takes a 100% gain to break even after a 50% loss.
Dave
The Geometric returns equivalent statement would be:
It takes 69 consecutive 1% losses to go from 100 to ~50 and then 69 consecutive 1% gains to get back to ~100.
This is also a useful observation. Since return involves multiplying a previous number by 1+r to get a new number, an example is that to overcome one's wealth being divided by 2 one must then multiply it by two. Or 1/2 is the inverse of 2. The first r is -50% and second is +100% but the math is that multiply by 2 wipes out divide by 2. So .99^69=.499837 and 1.01^69=1.986894.
DaufuskieNate
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### Re: Arithmetic Returns vs. Geometric Returns

Most of the time when expressing multi-year returns the geometric return is appropriate. When doing a monte carlo simulation, the arithmetic return should be used.
dbr
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### Re: Arithmetic Returns vs. Geometric Returns

DaufuskieNate wrote: Tue Jun 04, 2024 8:10 am Most of the time when expressing multi-year returns the geometric return is appropriate. When doing a monte carlo simulation, the arithmetic return should be used.
Exactly. That average is an estimator of the mean of the distribution used in the simulation. The variance of the data is an estimator of the variance of the distribution used in the simulation. The larger the variance for the same mean the less the compound average generated by the simulation will be and the larger the variance of the simulation will be. So volatility drag is real. And the better reporting of investment results should be the CAGR over a time rather than the expected return (arithmetic average) in a year unless you process that average through a compound growth simulation.
Random Walker
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### Re: Arithmetic Returns vs. Geometric Returns

bh1 wrote: Tue Jun 04, 2024 1:10 am Using arithmetic mean on most financial data is almost always wrong. Not different, just wrong.

https://www.skewray.com/articles/change ... are-ratios
I think when deciding whether to add a new asset to a portfolio, we need to think about expected return, volatility, correlations with other portfolio components, costs. The expected return is usually thought of as an arithmetic mean. But we hope the way it mixes in the portfolio positively affects the portfolio compounded return. The expected return of a portfolio I believe is the weighted simple mean of its components. The expected volatility of a portfolio is always less than the weighted mean volatilities of the components because of correlations less than 1.

I suppose someone could model the expected Geo Mean return of a portfolio based on simple mean expected returns, individual SDs, correlations, but that is out of my league.

Dave
Nowizard
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### Re: Arithmetic Returns vs. Geometric Returns

Once the difference is understood, what is the benefit of an arithmetic return? Does use of CAGR or figuring average return for a time period based on dollar amounts in the asset on an annual basis not represent a better figure at the practical level? Are reported returns for stocks or mutual funds not based on CAGR?
BrooklynInvest
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### Re: Arithmetic Returns vs. Geometric Returns

All other things being equal, an average annual return with low vol is better than the same average return with higher vol.

\$100 with exactly 10% return each year for 6 years is \$177

\$100 with alternating returns of zero and 20%, so averaging 10% annual return, is \$172.

\$100 with alternating returns of -10 and 30%, so averaging 10% annual return, is \$160.

Of course, what this means in practice I don't know. Diversification is the one "free lunch" I suppose but rarely are we looking at investment choices that offer similar returns and sustained diversification.
Random Walker
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### Re: Arithmetic Returns vs. Geometric Returns

BrooklynInvest wrote: Tue Jun 04, 2024 8:59 am All other things being equal, an average annual return with low vol is better than the same average return with higher vol.

\$100 with exactly 10% return each year for 6 years is \$177

\$100 with alternating returns of zero and 20%, so averaging 10% annual return, is \$172.

\$100 with alternating returns of -10 and 30%, so averaging 10% annual return, is \$160.

Of course, what this means in practice I don't know. Diversification is the one "free lunch" I suppose but rarely are we looking at investment choices that offer similar returns and sustained diversification.
We can attempt to create more efficient portfolios: portfolios with similar expected return and lower volatility. Diversification across multiple independent sources of risk can help achieve this. For example this is why factor junkies believe in heavy tilts to size and value on the equity side, concomitantly decreasing overall equity allocation, increasing allocation to safe bonds. Also potential reason to consider Alts. Of course all this comes at certain increased cost. The advantages are only potential.

Dave
bh1
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### Re: Arithmetic Returns vs. Geometric Returns

Random Walker wrote: Tue Jun 04, 2024 8:42 am
bh1 wrote: Tue Jun 04, 2024 1:10 am Using arithmetic mean on most financial data is almost always wrong. Not different, just wrong.

https://www.skewray.com/articles/change ... are-ratios
I think when deciding whether to add a new asset to a portfolio, we need to think about expected return, volatility, correlations with other portfolio components, costs. The expected return is usually thought of as an arithmetic mean. But we hope the way it mixes in the portfolio positively affects the portfolio compounded return. The expected return of a portfolio I believe is the weighted simple mean of its components. The expected volatility of a portfolio is always less than the weighted mean volatilities of the components because of correlations less than 1.

I suppose someone could model the expected Geo Mean return of a portfolio based on simple mean expected returns, individual SDs, correlations, but that is out of my league.

Dave
The bolded sentence doesn't tell us who is thinking. I certainly don't compute expected return as an arithmetic mean, unless I get to take the logarithm first. This is why we have to deal with the nasty formulas for APR https://en.wikipedia.org/wiki/Annual_percentage_rate.

If you think about using the arithmetic mean for, say, the last 50 years of DJIA, the earlier data points become insignificant. Does that mean that there was no reason to invest back then?
snackdog
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### Re: Arithmetic Returns vs. Geometric Returns

jyun wrote: Mon Jun 03, 2024 9:29 pm ...high volatility could result in the principal remaining the same as it was 5 years ago. I find this concept difficult to understand...
The principal always remains the same, doesn't it? How would it change?
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Hyperchicken
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### Re: Arithmetic Returns vs. Geometric Returns

How does something go down 90%?

First, it goes down 50%. Then, it goes down the rest 80%.
dbr
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### Re: Arithmetic Returns vs. Geometric Returns

Nowizard wrote: Tue Jun 04, 2024 8:49 am Once the difference is understood, what is the benefit of an arithmetic return? Does use of CAGR or figuring average return for a time period based on dollar amounts in the asset on an annual basis not represent a better figure at the practical level? Are reported returns for stocks or mutual funds not based on CAGR?
One use is that it can be a statistical estimator for the mean of whatever distribution of returns we imagine we are drawing from for return each year. We think of the returns over a few years as being a sample set of the underlying distribution. The standard deviation of that sample set could be an estimator for the standard deviation of that distribution. This is useful if you are attempting some sort of statistical analysis of the supposedly underlying behavior. A thing one might attempt, for example, is a Monte Carlo simulation of the progression in time of the portfolio value. Another thing one might attempt is an optimization of the mean return relative to the volatility, for, example, computing a Sharpe ratio.

As a practical matter the CAGR directly reports how much money you have now compared to what you had to start. And, reported returns for periods of some number of years are the CAGR. The only quibble is that you can compute a CAGR even if there have been contributions or withdrawals, but in that case it won't be a return.
nisiprius
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### Re: Arithmetic Returns vs. Geometric Returns

Here is a real-world example of a fund which lost money over a ten-year period despite having an arithmetic average annual return of +1.12%. It's the Vanguard 500 Index Fund, VFINX, from 2000 through 2009.

This can be verified with a calculator with just a little work. I used annual returns rounded to two places, but I let the spreadsheet retain full precision on the cumulative growth calculations.

For the arithmetic mean, you add up the ten annual returns and divide by 10.

For the growth calculations, you calculate

10000 - 9.06 % = 9094
9094 - 12.03 % = 7999.99

etc.

Although the arithmetic average was +1.12%, a \$10,000 investment became \$9,011.66 after ten years.

The other columns show the calculation of the CAGR. You add 1 to the percentages to make them multiples--a 28.50% return means you are multiplying your asset value by 1.2850. You take the geometric mean of that. And then you subtract 1 to convert it back to an average annual return.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.
Northern Flicker
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### Re: Arithmetic Returns vs. Geometric Returns

It is faulty reasoning. Volatility drag is a myth. Your return is your return. The arithmetic mean of multiyear returns is just an incorrect calculation of annualized return from multiyear returns. It has no bearing on realized returns. The "anomaly" is just a result of using incorrect formulas to convert between annual and compounded return. The BH wiki entry about this also should be removed.

It is an error analogous to computing compound returns by adding annual returns.
Last edited by Northern Flicker on Tue Jun 04, 2024 5:20 pm, edited 3 times in total.
IDpilot
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### Re: Arithmetic Returns vs. Geometric Returns

nisiprius wrote: Tue Jun 04, 2024 3:46 pm Here is a real-world example of a fund which lost money over a ten-year period despite having an arithmetic average annual return of +1.12%. It's the Vanguard 500 Index Fund, VFINX, from 2000 through 2009.

This can be verified with a calculator with just a little work. I used annual returns rounded to two places, but I let the spreadsheet retain full precision on the cumulative growth calculations.

For the arithmetic mean, you add up the ten annual returns and divide by 10.

For the growth calculations, you calculate

10000 - 9.06 % = 9094
9094 - 12.03 % = 7999.99

etc.

Although the arithmetic average was +1.12%, a \$10,000 investment became \$9,011.66 after ten years.

The other columns show the calculation of the CAGR. You add 1 to the percentages to make them multiples--a 28.50% return means you are multiplying your asset value by 1.2850. You take the geometric mean of that. And then you subtract 1 to convert it back to an average annual return.
Think you got your geometric mean return wrong. Looks like you forgot the -1 part of the formula. Your geometric mean return should be -0.0104

Geometric Mean Return = [(1+R1)(1+R2)...(1+Rn)]1/n-1
nisiprius
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### Re: Arithmetic Returns vs. Geometric Returns

It's a terminology problem and generally things get a little sloppy in this kind of discussion. I don't think I'm actually wrong. I don't think usage is perfectly consistent.

The problem is that we usually deal with "return," the percentage increase, R, and we don't (as far as I know) have a good name for the growth factor, (1 + R).

What you call the "geometric mean return" is what I called "CAGR" (compound average growth rate) on my spreadsheet.

It is the CAGR that we are interested in comparing with the arithmetic mean return.

What mutual fund companies present as the "average return" is CAGR.

I notice that MoneyChimp's useful S&P 500 calculator uses the phrase "annualized return (= true CAGR)" rather than "geometric mean return."

Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.
avalpert1
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### Re: Arithmetic Returns vs. Geometric Returns

nisiprius wrote: Tue Jun 04, 2024 6:17 pm What you call the "geometric mean return" is what I called "CAGR" (compound average growth rate) on my spreadsheet.
The 'A' in CAGR stand for 'annual', not 'average'. And this isn't a trivial difference - there is nothing really being averaged when computing CAGR. For investments (other than fixed income with 100% reinvestment) it isn't even measuring a real thing, it is more of a hypothetical construct we use to make comparisons easier even though doing so sometimes leads to sloppy thinking and decisions.

All it really is is saying the realized return over the given timeframe behaved as if it had returned X% every year.
Northern Flicker
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### Re: Arithmetic Returns vs. Geometric Returns

And you won't find volatility/variance in any of the correct formulas for converting between annual and multiyear returns.

CAGR/geometric mean is the annual return that compounds to the same multiyear return as that to which a series of annual returns compounds. That is the correctness criterion that the arithmetic mean of a sequence of annual returns fails to meet.

This wiki page should be deleted. It is based on this point that is made on the web page:
Some authors have observed that the more variable a given asset's return is, the greater the difference between the arithmetic and geometric averages.
That point is irrelevant because the arithmetic mean of a sequence of returns is an incorrectly calculated annual return. The fact that a higher variance of returns leads to a greater inaccuracy in the incorrect result is irrelevant.
IDpilot
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### Re: Arithmetic Returns vs. Geometric Returns

I agree with you that it is a terminology problem.

Looking at your sheet again, what you have in the block you labeled "Geo. Mean" is in fact the Geometric Mean of the growth factors you computed above that value. The Geometric Mean can be computed for any finite set of real numbers, n, by taking the nth root of the product of their values and that is what you did. Since this thread has been discussing the Geometric Mean Return, I just mentally added "Return" when I read it. Since the Geometric Mean Return is the Geometric Mean of the growth factors minus one it jumped out at me.

CAGR general means compounded annual growth rate and represents the mean annualized growth rate for compounding values. I've always seen it computed based on beginning value (BV) and ending value (EV) and not based on shorter period returns. If these values are over an annual, or multiples of a year, for n number of years, then the equation is 100[(EV/BV)1/n-1]. No need to compute growth factors and Geometric Means.
nisiprius
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### Re: Arithmetic Returns vs. Geometric Returns

IDpilot wrote: Tue Jun 04, 2024 7:05 pm ...I've always seen it computed based on beginning value (BV) and ending value (EV) and not based on shorter period returns. If these values are over an annual, or multiples of a year, for n number of years, then the equation is 100[(EV/BV)1/n-1]. No need to compute growth factors and Geometric Means...
In the equation you wrote yourself,

"Geometric Mean Return = [(1+R1)(1+R2)...(1+Rn)]1/n-1"

the terms (1+R1) are the things I called annual growth factors. In the "growth of \$10,000" my spreadsheet is showing the cumulative multiplication, step by step, that you show in your equation.

Having done that, I then display the 1+R's in another column, and apply a GEOMEAN function to them. The GEOMEAN function is exactly what your equation is showing when you multiply them and take the nth root.

You can also, if you like start with a BV and perform the multiplications to arrive at an EV and calculate (EV/BV)1/n and (EV/BV)1/n-1. But it's all the same thing.

I did it the way I did so that I could use an explicit "geometric mean" as part of the calculation. But it's the same calculation.

Anyway, I still feel that I did something useful in presenting a real-world example of a series of returns with a positive arithmetic mean that loses money.

P.S. There's another confusing question here. Do we want to think of the arithmetic mean return as

[(1+R1)+(1+R2)+ ... +(1+Rn)]/n - 1

or as

[(R1)+(R2)+ ... +(Rn)]/n ?

They are the same calculation, but structured mentally in two different ways.

For the arithmetic mean, we don't need to go round Robin Hood's barn adding in all the 1's so that we can subtract them again, and there's something intuitive about saying the arithmetic mean return is the arithmetic mean of the returns. But for the compounding case, there's no escaping adding in the 1's and then subtracting it at the end, if we want to follow the normal financial conventions of what "percent return" means.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.
hnd
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### Re: Arithmetic Returns vs. Geometric Returns

I was talking to a friend a while back and he was talking about the 14% he had averaged over the past like 20 years. I was like thats incredible how do you calculate that with all of your accounts and he said he had a spreadsheet. I always like messing with spreadsheets and asked him to send me a copy. and he did send me an empty one.

He had it like year 1 = X%, year 2 = Y%, year 3 = Z% and so on. then it had the total of all the percent and divided it by the number of years. and I was like oh no. I figured the math ant it was closer to 9%. nothing to be ashamed of but i kind of sent him an email explaining all of it (i did if you had \$100 and gained 100% one year you had \$200. the next year you gain -50% and it goes back to \$100. in real life investment returns you had a 0% return. on your spreadsheet you would have a 25% annual return but have 0 dollars to show for it) he was like that's not how it works and i was like okie dokie. He doesn't talk to me about investing anymore.
Random Walker
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### Re: Arithmetic Returns vs. Geometric Returns

Northern Flicker wrote: Tue Jun 04, 2024 3:57 pm It is faulty reasoning. Volatility drag is a myth. Your return is your return. The arithmetic mean of multiyear returns is just an incorrect calculation of annualized return from multiyear returns. It has no bearing on realized returns. The "anomaly" is just a result of using incorrect formulas to convert between annual and compounded return. The BH wiki entry about this also should be removed.

It is an error analogous to computing compound returns by adding annual returns.
I disagree. I appreciate that the only real money is our compounded return. But we can take advantage of the concept to design more efficient portfolios. Is the concept of volatility drag useful? That’s effectively the same question as “can I improve a portfolio by keeping its expected return (a weighted Ari mean) constant and decreasing its volatility?” Many of us believe that this can indeed be accomplished by diversifying across unique and independent sources of risk and expected return. We evaluate returns looking backwards, but we design portfolios looking forwards.

Dave
Northern Flicker
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### Re: Arithmetic Returns vs. Geometric Returns

Random Walker wrote: Tue Jun 04, 2024 11:50 pm
Northern Flicker wrote: Tue Jun 04, 2024 3:57 pm It is faulty reasoning. Volatility drag is a myth. Your return is your return. The arithmetic mean of multiyear returns is just an incorrect calculation of annualized return from multiyear returns. It has no bearing on realized returns. The "anomaly" is just a result of using incorrect formulas to convert between annual and compounded return. The BH wiki entry about this also should be removed.

It is an error analogous to computing compound returns by adding annual returns.
I disagree. I appreciate that the only real money is our compounded return. But we can take advantage of the concept to design more efficient portfolios. Is the concept of volatility drag useful? That’s effectively the same question as “can I improve a portfolio by keeping its expected return (a weighted Ari mean) and decreasing its volatility?
The arithmetic mean of a sequence of returns is not the expected return of an investment. Be careful not to confuse the arithmetic mean of a sequence of returns with the concept of a sample mean from a sample of independent trials being used as an estimator for a parameter, such as for estimating expected return. In that case, when the sample is a sequence of annual returns, the arithmetic mean of return and sample mean of the returns have the same formula, but a different interpretation and use. The variance drain argument is confusing these two concepts. (As an aside, such a sample also is a very poor quality sample from which to compute a sample mean of return to estimate expected return. It is neither random nor independent, leading to substantial bias.)

Given two portfolios with the same expected return, if one has lower volatility, we would prefer it. That has nothing to do with the fact that the arithmetic mean of a multiyear sequence of returns is not the annual return that compounds to the same multiyear return as the multiyear sequence compounds to.
Last edited by Northern Flicker on Wed Jun 05, 2024 2:23 pm, edited 1 time in total.
dbr
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### Re: Arithmetic Returns vs. Geometric Returns

nisiprius wrote: Tue Jun 04, 2024 8:35 pm
IDpilot wrote: Tue Jun 04, 2024 7:05 pm ...I've always seen it computed based on beginning value (BV) and ending value (EV) and not based on shorter period returns. If these values are over an annual, or multiples of a year, for n number of years, then the equation is 100[(EV/BV)1/n-1]. No need to compute growth factors and Geometric Means...
In the equation you wrote yourself,

"Geometric Mean Return = [(1+R1)(1+R2)...(1+Rn)]1/n-1"

the terms (1+R1) are the things I called annual growth factors. In the "growth of \$10,000" my spreadsheet is showing the cumulative multiplication, step by step, that you show in your equation.

Having done that, I then display the 1+R's in another column, and apply a GEOMEAN function to them. The GEOMEAN function is exactly what your equation is showing when you multiply them and take the nth root.

You can also, if you like start with a BV and perform the multiplications to arrive at an EV and calculate (EV/BV)1/n and (EV/BV)1/n-1. But it's all the same thing.

I did it the way I did so that I could use an explicit "geometric mean" as part of the calculation. But it's the same calculation.

Anyway, I still feel that I did something useful in presenting a real-world example of a series of returns with a positive arithmetic mean that loses money.

P.S. There's another confusing question here. Do we want to think of the arithmetic mean return as

[(1+R1)+(1+R2)+ ... +(1+Rn)]/n - 1

or as

[(R1)+(R2)+ ... +(Rn)]/n ?

They are the same calculation, but structured mentally in two different ways.

For the arithmetic mean, we don't need to go round Robin Hood's barn adding in all the 1's so that we can subtract them again, and there's something intuitive about saying the arithmetic mean return is the arithmetic mean of the returns. But for the compounding case, there's no escaping adding in the 1's and then subtracting it at the end, if we want to follow the normal financial conventions of what "percent return" means.
Absolutely. This is an illustration of a general understanding of what it means to find an average. The notion is that given a set of numbers and a function of those numbers then the average would be finding a single number that can replace all the instances of the actual data in the function and get the same value out. If the function is to add the numbers then then average is the arithmetic average. If the function is to multiply all the numbers together you get the geometric average. That does not work if any of the numbers are zero or negative. If the function is to add the inverses of the numbers you get the harmonic average. If the function is to take the square root of the sum of the squares of the numbers you get the root mean square. In the case illustrated above you get the annualized compound average growth rate.
IDpilot
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### Re: Arithmetic Returns vs. Geometric Returns

nisiprius wrote: Tue Jun 04, 2024 8:35 pm

Anyway, I still feel that I did something useful ...
That you did! Especially by triggering or esoterica discussion about the meaning of terms that are so often used in these forums.
dbr
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### Re: Arithmetic Returns vs. Geometric Returns

Northern Flicker wrote: Wed Jun 05, 2024 12:18 am
The arithmetic mean of a sequence of returns is not the expected return of an investment. Be careful not to confuse the arithmetic mean of a sequence of returns with the concept of a sample mean from a sample of independent trials being used as an estimator for a parameter, such as for estimating expected return. In that case, when the sample is a sequence of annual returns, the arithmetic mean of return and sample mean of the returns have the same formula, but a different interpretation and use. The variance drain argument is confusing these two concepts. (As an aside, such a sample also is a very poor quality sample from which to compute a sample mean of return to estimate expected return. It is neither random nor independent, leading to substantial bias.)

Given two portfolios with the same expected return, if one has lower volatility, we would prefer it. That has nothing to do with the fact that the arithmetic mean of a multiyear sequence of returns is not the annual return that compounds to the same multiyear return as the muatiyear sequence compounds to.
I am always a little confused why a sequence of returns in time as it actually is would not more often regarded as a problem in time series analysis rather than as a problem about a statistical distribution. This has to do with secular trends (we certainly are constantly reminded of bull and bear markets), serial correlation, whether any underlying statistics are stationary, and so on. A classic illustration we hear of is in the debate that if we wait long enough then international stocks will outperform US stocks for awhile and over very long times have equal performance. Another debate is whether tilting to some factor or another will persistently produce higher return.

PS One of the most astonishing things to me about the US stock market is the apparent goodness of fit of an exponential trend line in time to the market. This also implies a constant growth rate factor, aka constant return. Fitting a time function to a time series of data is one example of a time series analysis method. The next most astonishing feature of that same data is the occurrence of secular bull and bear markets around the trend. I don't think a frequency analysis (another time series technique) of those secular trends is particularly revealing though. But I am an amateur here. The simplest model apparently commonly chosen is a random walk model, of course.
Hydromod
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### Re: Arithmetic Returns vs. Geometric Returns

nisiprius wrote: Tue Jun 04, 2024 7:03 am Arithmetic means are wrong when averaging across successive time periods, where the return at the end of each time period becomes the starting value for the next. They are right when averaging among multiple assets over the same period of time. That's because across time returns multiply ("compound"), but over the same period of time they add.

Over a single year, a portfolio's return is the weighted arithmetic average of all of the returns of its constituents. It's simple math. If I have \$10,000 in asset A and \$10,000 in asset B, and over a year asset A has a return of 10% and asset B has a return of 5%, at the end of the year I'll have \$11,000 + \$10,500 = \$21,500. So \$20,000 has grown by \$1,500 and the portfolio return is \$1,500/\$20,000 = 0.075 = 7.5%, which is the ordinary arithmetic mean of 10% and 5%.

That's why arithmetic averages of asset performance numbers legitimately get calculated and presented, and become available for unscrupulous or ignorant commentators to misapply.
I think that folks should realize that the additional step of periodic portfolio rebalancing is used to increase the geometric returns by harvesting the arithmetic gains. Without rebalancing, you are stuck with adding together the geometric returns of individual assets at the end of the entire investment period. This may or may not be what one wants, depending on the assets.

For the engineers among us, the analogy is a bunch of resistances (electrical, thermal, or fluid) in a mix of parallel and series links. An analysis proceeds by arithmetically summing resistances for components in parallel to get an equivalent resistance and calculating a geometric resistance for components in series.
McQ
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### Re: Arithmetic Returns vs. Geometric Returns

Hmmm...sentiment appears to be running against the arithmetic mean to this point.

Am I correct that the sense of the thread is that standard deviation should also be set aside as uninformative? It was computed against the arithmetic mean last time I looked. Bye bye Sharpe ratio!

Or should we keep the concept of standard deviation of returns, but going forward, compute it against the geometric mean return? Anybody have that formula handy?
You can take the academic out of the classroom by retirement, but you can't ever take the classroom out of his tone, style, and manner of approach.
Northern Flicker
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### Re: Arithmetic Returns vs. Geometric Returns

McQ wrote: Wed Jun 05, 2024 6:06 pm Hmmm...sentiment appears to be running against the arithmetic mean to this point.

Am I correct that the sense of the thread is that standard deviation should also be set aside as uninformative? It was computed against the arithmetic mean last time I looked.
Sample variance (square of standard deviation) is an estimator for the variance of the distribution of the random variable for return. It is computed using the sample mean, which is an estimator for the mean of the random variable for return. Nothing here implies that sample variance is uninformative.

For a sample that is a time series of consecutive annual returns, there is no theoretical reason that any particular estimator for the mean of the distribution of returns should compound to the sample's return over the period. Using the sample mean as an estimator for distribution mean certainly gives that up. What is being called variance drain is nothing more than just this observation about the sample mean estimator.

Yes, if the distribution of return has high variance, the return over some period may fall well short of the expected return. That is what variance is. It only harms return for unfavorable outcomes of the random variable. There also can be favorable outcomes that exceed expected return. I could just as easily talk about a variance boost to returns as talk about variance drain. Both are erroneous concepts.

The entry for "variance drain" should be removed from the BH wiki. It is a mis-interpretation of sample statistics.
rkhusky
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### Re: Arithmetic Returns vs. Geometric Returns

Northern Flicker wrote: Wed Jun 05, 2024 11:08 pm The entry for "variance drain" should be removed from the BH wiki. It is a mis-interpretation of sample statistics.
Plus it seems to confuse geometric return and geometric mean.
IDpilot
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### Re: Arithmetic Returns vs. Geometric Returns

McQ wrote: Wed Jun 05, 2024 6:06 pm Hmmm...sentiment appears to be running against the arithmetic mean to this point.

Am I correct that the sense of the thread is that standard deviation should also be set aside as uninformative? It was computed against the arithmetic mean last time I looked. Bye bye Sharpe ratio!

Or should we keep the concept of standard deviation of returns, but going forward, compute it against the geometric mean return? Anybody have that formula handy?
I would say that the sentiment is running against using the arithmetic mean of annual returns to give one any sense of what the total return one might expect for periods longer than one year. As it should be since that is a gross misunderstanding of the arithmetic mean of annual returns.

If one understands what an arithmetic mean of annual returns, geometric mean and geometric mean of a return series are, they all have their usefulness.
dbr
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### Re: Arithmetic Returns vs. Geometric Returns

IDpilot wrote: Thu Jun 06, 2024 7:48 am
McQ wrote: Wed Jun 05, 2024 6:06 pm Hmmm...sentiment appears to be running against the arithmetic mean to this point.

Am I correct that the sense of the thread is that standard deviation should also be set aside as uninformative? It was computed against the arithmetic mean last time I looked. Bye bye Sharpe ratio!

Or should we keep the concept of standard deviation of returns, but going forward, compute it against the geometric mean return? Anybody have that formula handy?
I would say that the sentiment is running against using the arithmetic mean of annual returns to give one any sense of what the total return one might expect for periods longer than one year. As it should be since that is a gross misunderstanding of the arithmetic mean of annual returns.

If one understands what an arithmetic mean of annual returns, geometric mean and geometric mean of a return series are, they all have their usefulness.
Yes, if the numbers you have are a sequence of periodic (annual) returns for successive periods of time and you want to compute the end point wealth over the beginning wealth the right computation to use for that is to apply the CAGR using the formula for CAGR from a sequence of returns compounded. Similar math applies to dealing with fractional periods. What use one would make of the arithmetic average of those returns is a different and more esoteric discussion.

Note geometric average of those returns, if meant literally, does not apply because return in a period can be zero or negative and if one or more of the numbers in the sequence is zero or negative then the geometric mean of that set of numbers does not exist.

I have no idea how any of this is a matter of sentiment. Math is math. I am neutral on whether or not "variance drag" is a helpful term. It is probably both unnecessary and confusing to bring it up. The usual expression is that the exceptions to this know who they are and don't need to be discussed.
rkhusky
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### Re: Arithmetic Returns vs. Geometric Returns

For clarity, here is how the geometric mean is related to geometric return:

Suppose one has a sequence of returns, x_1 ... x_n, equally spaced in time, with the x_i > -1. You can calculate the geometric return by solving (1+x_1)(1+x_2)...(1+x_n) = (1+x_g)^n, resulting in x_g = ((1+x_1)(1+x_2)...(1+x_n))^(1/n) - 1, i.e. the geometric return is the nth root of the product of the terms (1+x_i) minus one or the geometric mean of the (1+x_i) minus one (or Prod(1+x_i)^(1/n) - 1).

The arithmetic return is the mean of the x_i, which is also the mean of the (1+x_i) minus one (or Mean(x_i) = Mean(1+x_i) - 1).

Given that the arithmetic mean of the (1+x_i) is greater than or equal to the geometric mean of the (1+x_i), the arithmetic mean of the (1+x_i) minus one is greater than or equal to the geometric mean of the (1+x_i) minus one. Hence, the arithmetic return is greater than or equal to the geometric return, or Mean(x_i) = Mean(1+x_i) -1 >= Prod(1+x_i)^(1/n) - 1 (the geometric mean of the x_i doesn't enter).
bh1
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### Re: Arithmetic Returns vs. Geometric Returns

Northern Flicker wrote: Tue Jun 04, 2024 7:04 pm And you won't find volatility/variance in any of the correct formulas for converting between annual and multiyear returns.
Since geometric returns have heavy tails, the variance always will go to infinity as the data set grows larger. The variance (the second moment of the probability distribution of geometric returns) is a really bad measure of volatility, and should never be used. Sadly, many do so anyway, leading to piles of nonsense in the literature. For very risky investments like startups, even the geometric average might not exist (math-speak for going to infinity as the data set adds more points).
Northern Flicker
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### Re: Arithmetic Returns vs. Geometric Returns

Variance of (compounded) long-term return increases over the length time. Volatility is the sample variance computed from a sample of actual returns. It is not the variance of a return that was compounded over all sample data points, so it does not increase without bound as sample size increases.

Volatility as an estimator for the variance of non-compounded shorter term return would seem to affect the rate at which the variance of long-term compounded return increases over time, so the investor should still care about volatility.
Last edited by Northern Flicker on Thu Jun 06, 2024 12:03 pm, edited 3 times in total.
02nz
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### Re: Arithmetic Returns vs. Geometric Returns

Surprised Dave Ramsey hasn't been mentioned yet. He's long used arithmetic returns to advertise unrealistically high returns on mutual funds that pay him commission.
Northern Flicker
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### Re: Arithmetic Returns vs. Geometric Returns

I think a major source of confusion results from using a time series of returns over a historical period as a sample of returns. It fosters the confusion of sample statistics with time value of money calculations done with the data to compute compound returns from individual data points or in reverse, computing a constant annualized return corresponding to a compounded return.

When computing a constant annualized return that compounds to the same return over some period as a time series of varying returns, the geometric mean of the returns is the correct computation.

If estimating the expected return of an investment using an outcome sample, the sample mean is an appropriate estimator. Its use is motivated by its various beneficial properties as an estimator of distribution mean (although some of those properties flow from the central limit theorem, which requires a sufficiently large random sample of independent trials). It is not chosen as an estimator based on any relationship with time value of money calculations.

The geometric mean of sample returns also could in theory be used as an estimator for expected return. If one actually believed that it was a better estimator for expected return than is the sample mean, then using the sample mean to estimate expected return while critiquing it as overstating expected return due to a claim of variance drain is an irrational position. If variance drain actually were a thing that establishes the inaccuracy of sample mean as an estimator for expected return, why would you use the sample mean as the estimator?
Last edited by Northern Flicker on Thu Jun 06, 2024 7:45 pm, edited 1 time in total.
hnd
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### Re: Arithmetic Returns vs. Geometric Returns

02nz wrote: Thu Jun 06, 2024 11:43 am Surprised Dave Ramsey hasn't been mentioned yet. He's long used arithmetic returns to advertise unrealistically high returns on mutual funds that pay him commission.
the best part is that the excel data sheet provided on the link they point to on the Ramsey website gives both the arithmetic mean AND the geometric mean

1928-2023 11.66%
1974-2023 12.54%
2014-2023 12.98

Geometric Average Historical Return
1928-2023 9.80%
1974-2023 11.10%
2014-2023 11.91%
McQ
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### Re: Arithmetic Returns vs. Geometric Returns

Northern Flicker wrote: Thu Jun 06, 2024 12:44 pm I think a major source of confusion results from using a time series of returns over a historical period as a sample of returns. It fosters the confusion of sample statistics with time value of money calculations done with the data to compute compound returns from individual data points or in reverse, computing a constant annualized return corresponding to a compounded return.

When computing a constant annualized return that compounds to the same return over some period as a time series of varying returns, the geometric mean of the returns is the correct computation.

If estimating the expected return of an investment using an outcome sample, the sample mean is an appropriate estimator. Its use is motivated by its various beneficial properties as an estimator of distribution mean (although some of those properties flow from the central limit theorem, which requires a sufficiently large random sample of independent trials). It is not chosen as an estimator based on any relationship with time value of money calculations.

The geometric mean of sample returns also could in theory be used as an estimator for expected return. If one actually believed that it was a better estimator for expected return than is the sample mean, then using the sample mean to estimate expected return while critiquing it as overstating expected return due to a claim of variance drain is an irrational position. If variance drain actually were a thing that establishes the inaccuracy of sample mean as an estimator for expected return, why would you use the sample mean as the estimator?
Alas, it appears that the late Harry Markowitz went to his grave not understanding the arguments made by forum member Northern Flicker, among others here in the thread.

On p. 78 of Risk-Return Analysis, https://www.google.com/books/edition/Ri ... =en&gbpv=0 his magisterial summary of fifty years of work, he writes:

Why Inputs to a Mean-Variance Analysis Must Be Arithmetic Means

That’s the section title.* He continues:

“The inputs to a mean-variance optimizer must be the (estimated forthcoming) expected (that is, arithmetic mean) returns rather than the (estimated forthcoming) geometric mean returns.”

*Later in that section he will shows that the old kluge—that geometric mean can be approximated by the arithmetic mean minus one half the variance—is not bad, but by no means the best approximation out there.
You can take the academic out of the classroom by retirement, but you can't ever take the classroom out of his tone, style, and manner of approach.
Northern Flicker
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### Re: Arithmetic Returns vs. Geometric Returns

McQ wrote: Thu Jun 06, 2024 10:09 pm
Alas, it appears that the late Harry Markowitz went to his grave not understanding the arguments made by forum member Northern Flicker, among others here in the thread.

On p. 78 of Risk-Return Analysis, https://www.google.com/books/edition/Ri ... =en&gbpv=0 his magisterial summary of fifty years of work, he writes:

Why Inputs to a Mean-Variance Analysis Must Be Arithmetic Means

That’s the section title.* He continues:

“The inputs to a mean-variance optimizer must be the (estimated forthcoming) expected (that is, arithmetic mean) returns rather than the (estimated forthcoming) geometric mean returns.”
That view is 100% aligned with mine. My point is that the use of the term arithmetic mean for sample mean is causing misunderstanding, but a sample mean has the right properties. If you repeatedly sample a distribution, the sample mean is normally distributed in the sampling distribution, and converges on the distribution mean as the number of samples increases. This follows from the central limit theorem if the sample meets the requisite preconditions. The sample mean also is a uniform, minimum-variance, unbiased estimator of distribution mean.

I'm not aware that Markovitz ever advocated for sufficiently large random samples of independent trials for his samples. This in fact is why Modern Portfolio Theory has not been as successful as it might have been. With the biased samples used in backtests, what you get for the efficient portfolio depends on the beginning and ending points of the sample. I think Markovitz probably was aware of this, but did not have a solution for the problem. Generating large random samples of investment returns is a difficult problem.
dbr
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### Re: Arithmetic Returns vs. Geometric Returns

Northern Flicker wrote: Thu Jun 06, 2024 10:31 pm
McQ wrote: Thu Jun 06, 2024 10:09 pm
Alas, it appears that the late Harry Markowitz went to his grave not understanding the arguments made by forum member Northern Flicker, among others here in the thread.

On p. 78 of Risk-Return Analysis, https://www.google.com/books/edition/Ri ... =en&gbpv=0 his magisterial summary of fifty years of work, he writes:

Why Inputs to a Mean-Variance Analysis Must Be Arithmetic Means

That’s the section title.* He continues:

“The inputs to a mean-variance optimizer must be the (estimated forthcoming) expected (that is, arithmetic mean) returns rather than the (estimated forthcoming) geometric mean returns.”
That view is 100% aligned with mine. My point is that the use of the term arithmetic mean for sample mean is causing misunderstanding, but a sample mean has the right properties. If you repeatedly sample a distribution, the sample mean is normally distributed in the sampling distribution, and converges on the distribution mean as the number of samples increases. This follows from the central limit theorem if the sample meets the requisite preconditions. The sample mean also is a uniform, minimum-variance, unbiased estimator of distribution mean.

I'm not aware that Markovitz ever advocated for sufficiently large random samples of independent trials for his samples. This in fact is why Modern Portfolio Theory has not been as successful as it might have been. With the biased samples used in backtests, what you get for the efficient portfolio depends on the beginning and ending points of the sample. I think Markovitz probably was aware of this, but did not have a solution for the problem. Generating large random samples of investment returns is a difficult problem.
What we are sampling and what distribution we are trying to describe is a different discussion from what statistics regarding that data we want to attend to. I doubt anyone is suggesting the best statistic to use is the CAGR of a time series of returns as opposed to the arithmetic mean of whatever data set we can manage to gather.

The different kinds of averages have uses for data representing different situations but if the problem is to characterize a distribution by estimating the moments of that distribution and the first moment is the arithmetic mean then presumably the statistic we use for estimation is going to be the arithmetic mean. What data sets we examine to work that problem is, as described above, a discussion.

When the problem is to describe the compound growth of a time series of growth factors, then one would indeed compute the geometric mean of those factors, the data being the terms 1+r where r is the return. In the first problem we characterize the returns, r, directly and not the growth factors. The reason for that is that the model of interest is a compound growth time series and not that the data is simply a collection of random samples of some property. One of the basic analysis methods for examining a time series is to fit a curve to the data hoping to extract an obvious function of time to describe the series. In the case of investment returns an exponential function of time is often a good fit so it makes sense to try to extract the time constant that should be in that function.

Another exercise that can be done is to transform the data. If we believe we are best off by thinking the data is normally distributed we might go directly to arithmetic mean. If we imagine, as an example that a transform of the data is easier to understand then we might study transformed data, such as the log of the data if we think the distribution is log-normal.

Keep in mind that the concept of generalized average derives whole families of different averages from the arithmetic average by a process of taking the inverse transform of the arithmetic average of a transform of the data. If the transform is the identity you get arithmetic average. If the transform is to square the data you get the root mean square. If the transform is taking the log, you get the geometric average. If it is the inverse you get the harmonic average. If it is the log of one plus the decimal equivalent of the returns normalized to yearly scale you get the CAGR. Interestingly if you consult a mathematical article about generalized mean such as this one: https://en.wikipedia.org/wiki/Generalized_mean the CAGR is not even listed as an example while in financial articles CAGR is pretty much the only mean mentioned albeit misnamed the geometric mean (of the returns).
Northern Flicker
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### Re: Arithmetic Returns vs. Geometric Returns

Suppose the sample from which we will estimate expected return is annual returns from 25 randomly selected years between 1900 and 2023. What would even be the point of computing a geometric mean? It's not like we are going to try to compound returns over a randomly distributed set of years. And we would not try to compound the sample mean/arithmetic mean either. We would just view it as an estimator for the mean of the distribution of return.

Had Markovitz been using such a sample in the derivation McQ quoted, it likely would not have even occurred to him to mention that the geometric mean should not be used because it would have no applicability to a random sample. And as a result, there would be no notion of variance drain even to try to specify for discussion.

We have a well developed framework to understand statistical parameter estimation.
Using a highly biased sample such as a backtest leads to estimates for expected return that do not generalize to future returns or reproduce in a different sample. If trying to use a backtest to estimate the expected return, bias and lack of independence of data points in such samples, and the non-interpretability of a t-statistic are the significant concerns. The fact that the sample mean as an estimator for expected return does not replicate the CAGR of the backtest is not an issue. Viewing that as a defect is an incorrect observation about an estimator for the mean of a distribution, and that would be being raised for what already is a highly flawed experimental design anyway.