Arithmetic Mean vs Geometric Mean – Why are Financial Institutions Allowed to Report Returns the Way they Do?

Arithmetic Mean

Arithmetic Mean, more commonly referred to as mean or average, is the summation of observation divided by the number of observations.  Or in plain English:

(1+2+3+4)/4

= 2.5

In finance, we are typically concerned with average rates of return over periods of time.  Returns of the stock market indices, mutual funds, ETFs and individual stocks for 1, 3, 5, and 10 year periods typically.  Calculating the average is real simple.  However it may not be the best form of calculating an average due to outlying data.  Basically arithmetic mean understates the effect a large single data point can have on the average.

Using arithmetic mean (which is used by mutual funds, ETFs, etc.) will lead investors to believing they made more money over time than they actually did.

What does that gibberish from the last sentence mean.  Lets use an example with a 5 year sample of returns for the S&P500 from the years 2006-2010.  I am using this 5 year time frame to include a negative year.

2006= +15.79

2007= +5.49

2008= -37.00

2009= +26.46

2010= +15.06

So in calculating the arithmetic mean we simply add up our returns: 15.79+5.49-37.00+26.46+15.06= 25.8

Next we divide the summation of our observation from above, by the number of occurrences (5).  25.8/5= 5.16%

So any reasonable investor would think  $100 invested for 5 years at an average return of 5.16%  would mean you have $128.60 at the end of 2010.  But is this necessarily correct?

Lets look a bit deeper and see if this is correct, by using $100 invested at the start of 2006 and figuring the value we would have at the end of each year.

At the end of 2006, our $100 investment would be worth $115.79 (calculated 100*1.1579)

At the end of 2007, our $115.79 investment would be worth $122.14 (calculated 115.79*1.0549)

At the end of 2008, our $122.14 investment would be worth $76.95 (calculated 122.14*(1-0.37)).  Oh no, we are in the hole now.

At the end of 2009, our $76.95 investment would be worth $97.31 (calculated 76.95*1.2646)

At the end of 2010, our $97.31 investment would be worth $111.96 (calculated 97.31*1.1506)

Hold the fuck up!  Above we showed an average 5 year return on the S&P500 of 5.16% and applying that average return on $100 we showed a closing value of $128.60.  However when we calculate what $100 actually invested in the S&P500 would be worth, we come up with $111.96.  $111.96 vs $128.60 are not even close with a difference of $16.64 overstated.  Don’t think it’s a big deal.  Imaging $100,000 or some other large dollar amount, and the difference is more substantial.  Is there a better means of calculating mean? (See what I did there)

Geometric Mean

Geometric mean is a way of calculating an average which uses the product (multiplication) of the observations as opposed to arithmetic using the summation (addition).  In other words, geometric mean is the nth root of the product of n observations.

Calculating Geometric Mean for returns by hand is actually quite simple:

{[(1+r)(1+r)…]^(1/n)} -1

where r is the return for each observation point

and n is the total number of observations

 Lets use our data points from the S&P 500 again from above.

2006= +15.79

2007= +5.49

2008= -37.00

2009= +26.46

2010= +15.06

Using geometric mean, we arrive at an average of:

{[(1.1579)*(1.0549)*(0.63)*(1.2646)*(1.1506)]^(1/5)}-1

{[1.11969682]^(.20)}-1

(1.02286917)-1

= 0.02286917 or 2.28%

Geometric mean better handles the effect of outliers, but also negative returns.  Here we come up with a geometric mean of 2.28% vs the arithmetic mean of 5.16% for the same sets of data.  In this case, the arithmetic mean is more than twice as high as the geometric mean.

So, now lets apply our geometric mean of 2.28% to the $100 initial investment and see where we are after 5 years and compare to arithmetic.

Using the 2.28% geometric mean on an initial $100 investment, after 5 years we have a closing value of $111.93.  Compared to the arithmetic mean ending 5 year value of $128.60.  A difference between the 2 averages of $16.67 or  what I’d say is statistically significant when it comes to money.

More interesting is comparing the 5 year ending value of the geometric mean of $111.93 to the actual ending value of investing in the S&P500 from above of $111.96.  A difference of only $0.03 understated. Compare that to the difference of using arithmetic mean at $16.64 overstated ($128.60-$111.96).

Why Does This Matter?

Think about the differences in numbers above using arithmetic vs geometric mean.  When you pull information for returns on mutual funds and ETFs, they are reporting arithmetic mean.  If a return was constant year after year, no big deal.  However, because of variability/volatility of returns and compounding, arithmetic mean will overstate the performance of a mutual fund, which in turn translates to overstating your potential account balance.  This sucks, because in reality your balance will be lower.  Geometric mean offers a better may of stating the returns of mutual funds over time and better reflect the actual dollar balance an investor has.  That’s why this matters.  It affects your money.

So what do the SEC and FINRA actually do to protect investors?  Returns are being misquoted to investors.  Add on top of this the ridiculous fees that many mutual funds have, and investors are fighting even more of an uphill battle with no actual regulators doing a thing to protect them.

Coming soon, we will take another look at average returns and how the timing of returns also makes a big difference on portfolio values.

Follow me on the social medias:

Readers Comments (71)

  1. This is a very nice investment math explanation and it is absolutely horrible that SEC and FINRA are not doing anything to protect investors from such blatant misinformation.

    I particularly liked that you provided an example of calculating geometric mean with negative returns. However, I struggled a bit in calculating it in Excel since GEOMEAN just returns NaN if there is a negative value in a sequence of returns.

    After much searching around I found this geomean calculator: https://www.gigacalculator.com/calculators/geometric-mean-calculator.php which handles negative returns automatically, as well as percentages as input (many of the others I found did not!). I ran your example through it like so: +15.79% +5.49% -37.00% +26.46% +15.06% and it returned 2.2869% as is your result. Maybe it will be helpful to other readers of yours as well.

  2. This suЬmit is your Ԍrilled Salmon in Foil toolkit.

  3. No fuгther nnon permkanent disc area is reգuired.

  4. Effiсient WMA MP3 Converter distribuutes as freeware.

  5. Yօu can add a number of AIϜF information at one time.

  6. I’ve been using frе:ac for ɑbout 5 years now.

  7. Perfoг rerverse and inversion of audi᧐ information.

  8. ᒪoooking for a dedicated audio trіmming
    website?

  9. Εaaѕy methods to convert M4P to MP3 without iTunes?

  10. Selеct which audioo reсordsdata you want tto merge.

  11. Dгag n drop with a vrry clear trying interface.

  12. Еxport vide᧐o in frequent internet video ϲodecs.

  13. Magic_M4A_to_MP3_Converter_dow (@Magic_M4A_to_MP3.

  14. Files must be 50 MB or smaller in measurement.

  15. Now I might also like to hav all of the music iin MP3.

  16. Set the start andd ennd time of the media file.

  17. Because of this it’s obly audio disk format.

  18. Right here the primary setting is thee encoder.

  19. I was extraordinarily happy with the outcomes.

  20. Click Convert to transform your MP3 file(s).

  21. Podporuje i DTS Audio CD, a ripovбnн DVD do MP3.

  22. buy vardenafil vardenafil – purchase vardenafil online vardenafil

  23. Echt schade. Umwandeln. Verbindens. Und jetzt geht’s.

  24. stromectol 2mg – buy stromectol for humans stromectol online pharmacy

  25. My developer is trying to persuade me to move to .net from PHP.

    I have always disliked the idea because of the expenses.

    But he’s tryiong none the less. I’ve been using WordPress on numerous websites for about a year and am anxious about switching to another platform.
    I have heard good things about blogengine.net. Is there a way I can transfer all my wordpress posts into it?

    Any kind of help would be really appreciated!

  26. erectile dysfunction doctor – drugs for ed non prescription ed pills

  27. This is very interesting, You’re a very professional blogger.
    I’ve joined your rss feed and look forward to in search of
    extra of your excellent post. Also, I have shared your site in my social networks

  28. Buut calories aren’t the only factor that issues.

    My homepage … dailyro.com

  29. I know this if off topic but I’m looking into starting my
    own weblog and was wondering what all is required to get set up?
    I’m assuming having a blog like yours would cost a pretty penny?
    I’m not very web savvy so I’m not 100% positive.
    Any tips or advice would be greatly appreciated.
    Appreciate it

  30. cytotec buy online usa – cytotec 400 mcg cytotec mexico

  31. I blog frequently and I genuinely appreciate your information. This great article
    has truly peaked my interest. I’m going to book
    mark your website and keep checking for new details about once per
    week. I subscribed to your RSS feed as well. https://tadalafili.com/

  32. What’s Going down i’m new to this, I stumbled upon this I have discovered It
    absolutely helpful and it has helped me out loads. I’m hoping
    to contribute & help other users like its aided me.
    Great job.

  33. Thanks for one’s marvelous posting! I genuinely enjoyed reading it, you may be
    a great author. I will be sure to bookmark your blog and
    definitely will come back at some point. I want to encourage yourself to continue your
    great writing, have a nice holiday weekend!

  34. tadalafil price uk – 100mg cialis best online pharmacy no prescription

  35. cialis dosage https://cialiswithdapoxetine.com/# – cialis pills

  36. Please let me know if you’re looking for a
    author for your weblog. You have some really great articles and
    I think I would be a good asset. If you ever want to take some of
    the load off, I’d absolutely love to write some material
    for your blog in exchange for a link back to mine. Please shoot me
    an email if interested. Kudos!

  37. If you are going for most excellent contents like myself, just pay a
    quick visit this site every day because it gives feature contents, thanks

  38. Hi, Neat post. There is a problem together with your website in internet
    explorer, may check this? IE still is the market chief and a good element of other people will leave out your excellent writing due
    to this problem.

  39. I have to thank you for the efforts you have put in penning this site.
    I’m hoping to check out the same high-grade blog posts from you later on as well.
    In fact, your creative writing abilities has inspired me to
    get my own, personal blog now 😉

  40. Hi there, just became aware of your blog through Google,
    and found that it’s really informative. I’m gonna watch
    out for brussels. I’ll be grateful if you continue this in future.

    Numerous people will be benefited from your writing.
    Cheers!

  41. For most recent news you have to visit internet and on the web I
    found this web page as a best web site for most up-to-date updates.

  42. My coder is trying to convince me to move to .net
    from PHP. I have always disliked the idea because of the costs.
    But he’s tryiong none the less. I’ve been using WordPress on several websites for about a year and
    am anxious about switching to another platform.
    I have heard very good things about blogengine.net. Is there a way I can import all my wordpress content into it?
    Any help would be really appreciated!

  43. Quality articles is the crucial to interest the viewers to pay a visit
    the web site, that’s what this site is providing.

  44. Good blog post. I definitely love this site.
    Continue the good work!

  45. I like this internet site because so much useful stuff
    on here :D.

  46. You really make it seem so easy with your presentation but I find this topic to be actually
    something that I think I would never understand.
    It seems too complicated and very broad for me.
    I am looking forward for your next post, I will try to get
    the hang of it!

  47. Great delivery. Great arguments. Keep up the amazing effort.

  48. I know this website provides quality based articles and extra material,
    is there any other site which gives these kinds of information in quality?

  49. I have been syrfing online morе thаn three һoᥙrs latelу, but
    Ι never discovered any interesting article ⅼike
    ʏоurs. It’s beautiful value enoᥙgh fօr me.

    In my opinion, іf all webmasters and blogggers mаde just right ϲontent material as
    you did, the net will probabⅼy bbe muϲh mire hlpful than еvеr ƅefore.

    Lookk аt my sitee – スロットログイン

  50. Are you a fan of the blogger’s body of labor?

  51. Does your site have a contact page? I’m having a tough time locating
    it but, I’d like to shoot you an e-mail. I’ve got some creative
    ideas for your blog you might be interested in hearing. Either way, great blog and I look
    forward to seeing it improve over time.

    Stop by my web blog: Unknown

  52. I like the helpful information you supply for your articles.
    I’ll bookmark your blog and test once more here frequently.
    I am moderately sure I’ll be informed many new stuff right here!
    Good luck for the following!

  53. Excellent article. I will be going through some of these
    issues as well..

  54. Hi, I would like to subscribe for this blog to obtain latest updates, so where can i
    do it please help out.

  55. We’re empowering absolutely everyone to be a Trailbblazer with our culture of Accomplishment from Anyplace.

    Here is my blog post … 유흥알바

  56. Thank you, I have just been searching for information approximately this topic for a
    long time and yours is the greatest I have discovered till now.
    However, what about the conclusion? Are you certain in regards to the
    source?

  57. Someone necessarily lend a hand to make severely posts I might state.
    That is the first time I frequented your website page and to this point?
    I amazed with the analysis you made to create this particular post
    extraordinary. Excellent task!

Leave a comment

Your email address will not be published.


*


Follow me on the social medias: