8% isn’t always 8%…in fact, rarely! Arithmetic Averages are Deceiving

“Arithmetic Average” and “Straight-Line Projections” — frequently applied to investment gains, losses and marketing by Investment Advisors or financial sales people.  These “results” are not indicative of “actual” returns.

Perhaps you’ve wondered why the returns on your investments don’t seem to match the claims made by advisors.

These investments might be in a 401(k), an IRA, an investment account at a Wall Street firm, Variable Life Insurance, etc.  Let’s take a look at how some of these claims of achievement come to be.

   Let’s try a little game:

   In the first year of investing, you’re up 60%.  In the second, down 50%.  So, what’s your return, or average return?

   Up 60, then down 50…so, 60 – 50 = 10 (“net”), divided by 2 (years) = 5% per year, true?      Certainly that’s how people tend to look at it.

   Now let’s try that with money:

   $100 invested, up 60% in year one = $160.  Second year, $160 goes down 50% = $80.     $80!  Not a “gain” of 5%/yr, but a net loss!

   The point?  Advisors, investors, most people tend to discuss or exclaim results as      percentages, and averages, but these are based on market changes.  Just a little misleading, right? 

Now, for more depth, we’ll break the analysis and explanation into 4 parts:

Part 1, Averages.

Let’s imagine that we want to invest some money, and that we are told an average return of 8% is possible.  Now, our first thoughts might be that 8%, over years, would be… Well, what would it be?

$100 invested, and the gains/losses are as follows:

100    x     8% gain  = 108

108    x     8% gain  = 116.64

116.64 x   8% gain  = 125.97

125.97 x   8% gain  = 136.05

136.05 x  8% gain   = 146.93

So, to get the average gain, sum the gains and losses (here expressed as percentage changes):

+8

+8

+8

+8

+8

____

+40, now divide by the five years…

40/5 = 8% avg.  This example result?  146.93 with the straight-line 8% average.

(Is this how the stock market behaves?)

The advisor’s statement: “The investment returned an average of 8% over five years!”

Average is average, right?

Let’s try it again, but with what looks like market volatility:

Same $100 invested, but with gains/losses as follows:

100    x    16% gain   = 116

116    x     0% (no loss, no gain) = 116

116    x     8% gain     = 125.28

125.28 x -24% loss   = 95.21

95.21   x 40% gain   = 133

Again, to get the average, sum the percentage changes:

+16

    0

  +8

-24

+40

_____

+40, now divide by the five years…

40/5 = 8% avg.      This result?  133 with an irregular 8% average, but the same advisor statement!  “8% average!”

Part 2, Actual Returns.

Let’s review the dollar results from the averages above:

Different results, 146.93 and 133.     Hey, what’s up with that?

Both “8% average” according to the advisor….why the difference in investment results?        Both reflect arithmetic averages, but one is irregular (possibly “actual”), the other “straight-line”.               

Now, let’s look at the Standard & Poor’s index, and the market “moves”:

Real Returns vs. Straight-Line Projections

Standard & Poor’s index levels*, year-end 2000 – year-end 2014

Year          Index level        % Change       $100 investment is now:

12/31/00     1320.28                                                $100.00

12/31/01      1148.08                       -13%                  $87.00

12/31/02       879.82                       -23%                  $66.99

12/31/03       1111.92                  +26.4%                  $84.67

12/31/04      1211.92                        +9%                  $92.29

12/30/05     1248.29                     +2.9%                $ 94.97

12/29/06     1418.30                   +13.6%                 $107.88

12/31/07      1468.36                     +3.5%                $111.66

12/31/08       903.25                    -38.5%                 $68.67

12/31/09       1115.10                   +23.5%                 $84.80

12/31/10       1257.64                    +12.7%                $95.57

12/30/11      1257.60                       -0-                      $95.57

12/31/12      1426.19                     +13.5%                $108.47

12/31/13     1848.36                    +29.6%                 $140.58

12/31/14     2058.90                    +11.4%               $156.60 (this is the “actual                                                                                    _____                                              return”)

                                        Total % = 71.2

However…

“Arithmetic Average” of percentage gain = 71.2% / 14 yrs = 5.08% per year.   It’s just adding the (percentage) gains and losses, and dividing by the number of years.

So, how is this used?

The market changed an average of 5.08% during the timeframe above.  Many use that as “the market returned an average of 5.08%”.

This is done partly because the actual calculation of the real return is not performed, however a marketing statement applicable to the investment vehicle’s percentage changes (again, using arithmetic average) is assumed to apply to the dollar results.

Part 3: Projections.

What if we were able to look at the history of the index, i.e. as above, and say “the average returns over fourteen years was 5.08%, so lets use that in projecting the next many years…”    

Using the same starting money, over the same number of years, we should get the same results, right?  $156.60?  And if we get that result, we’d have validation of using the historical average for a reasonable projection.

If we take $100 here (to start) and assume those 5.08% gains annually, we’d get:

after yr 1: 100 x 5.08% gain  = 105.08

              2: 105.08 x 5.08%    = 110.42

              3: 110.42 x 5.08%     = 116.03

              4: 116.03 x 5.08%     = 121.92

              5: 121.92 x 5.08%     = 128.11

              6: 128.11 x 5.08%     = 134.62

              7: 134.62 x 5.08%     = 141.46

              8: 141.46 x 5.08%     = 148.64

              9: 148.64 x 5.08%    = 156.19

            10: 156.19 x 5.08%     = 164.12

            11: 164.12 x 5.08%      = 172.46

            12: 172.46 x 5.08%     = 181.22

            13: 181.22 x 5.08%     = 190.43

            14: 190.43 x 5.08%    = $200.10

Well, we didn’t get validation.

In Part 2, irregular changes, 5.08% average, investment result = $156.60

In Part 3, consistent changes, 5.08% average, investment result = $200.10

A point of fact:

This is a “straight-line” projection based on an arithmetic average of percentage gains/losses from some history.  As we see from the actual returns, the use of an arithmetic average of percentage gains/losses can be wildly misleading.

This projection is actually displaying a “Cumulative Rate of Return” (CROR) in its consistent 5.08% rate, building upon itself, year after year.

Oh, and the (real) Cumulative Rate of Return (CROR) on the $156.60?  3.25%  

Wait!  What?  S&P, over 14 years, up 3 percent?  And taxable?  I’m hearing at least 5% from my broker, my advisor!  Huh…

What is the danger in straight-line projections?  Straight-Line projections are typically used to express what become expectations.  Unfortunately, the securities markets are not consistent, but are volatile.   Straight-line is wishful thinking.

Straight-line projections are used regularly in the financial world for convenience and necessity; after all, how would volatility be projected?  But it’s still wishful thinking.

Actual returns of any investment that has inherent volatility will not perform in line with straight-line projections over time; the longer the timeframe, the greater the potential variation.

Part 4: Friend-to-friend examples.

One more thought on the use of averages and, in particular, percentages:  Gravity wins!

What do we mean by that?

Let’s imagine you have $100, and you invest that money.

In the first year, you gain “10%”.  Your Financial Advisor tells you you’re up 10%.  That should give you $110.

In the second year, your advisor tells you you’re down 10%.

So, up 10%, then down 10%.  “Even”, right?  Back to square one?

$100, up 10% = $110.

$110, down 10% = $99.  Yes, 10% off of that bigger number, 110.

How about the other way?

$100, down 10% = $90.

$90, up 10% = $99. The 10% gain on a smaller number.

Gravity wins.  Not “even”.

Let’s try our opening question again:

In the first year of investing, you’re up 60%.  In the second, down 50%.  So, what’s your return, or average return?

Up 60, then down 50…so, 60 – 50 = 10, divided by 2 (years) = 5% per year, true?  Certainly that’s how people tend to look at it, because it seems arithmetically correct.

Now let’s try that with money:

$100 invested, up 60% in year one = $160. Second year, $160 goes down 50% = $80.  $80!

Clear?  Advisors, investors, most people tend to discuss or exclaim results as percentages, and averages.  Just a little misleading, right?

In the real world, the use of “Arithmetic Averages” can’t be totally, or even largely avoided.  The expectations created are off, to say the least.  Throw in fees (rarely, if ever, incorporated into expressions of results) and taxes, market buys and sells (timing…) and you will probably not get what you hope for.

*S&P History Resource:  http://www.fedprimerate.com/s-and-p-500-history.htm

About L N Himel

Site creator and author has well over 30 years experience in the world of Finance at both the institutional and personal levels. As is often true, his successes, but also the pain of the author's mistakes, have resulted in insights that you, the reader, might find helpful in your efforts to avoid pitfalls and poor results. Associated with Himel Financial Services, at http://www.himelfinancial.com/#intro
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