“Arithmetic Average” and “Straight-Line Projections” — frequently applied to investment gains, losses and marketing by Wall Street investment advisors. These “results” are not indicative of “actual” or “real” returns.

Perhaps you’ve wondered why the returns on your investments (and mutual funds in particular) don’t seem to match the claims made by fund managers. You might be puzzled at why your investments don’t provide the same “rate of return”, on average, as the market or a specific index (i.e. S&P, Dow, etc.).

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

First, 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, sum the 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 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!

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 are simple arithmetic averages, but one is irregular (possibly “real”), 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 (the most recent 14 years) – The Real Deal

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 “real return”)**

Total % = 71.2

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

*“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.

This average, 5.08%, might also be applied to the results of the investment, $156.60, with a statement of “It’s 5.08%!” Yes, inaccurate…we’re applying an erroneous average percentage gain to a (lower) real dollar result.

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.*

**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 money, we should get the same results, right? $156.60?

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**

Yikes! So now, we have to reconcile the fact that we saw a *real return result (in the S&P moves above, in years 2000 – 2014) of $156.60, *and using the resultant *“arithmetic average” **of percentage change,* over another 14 years, *an **expected result of $200.10.*

A point of fact:

*This is a *“*straight-line*” *projection based on an arithmetic average *of percentage gains/losses from some historical index year-end levels.** ****As we see from the real 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! Well, maybe if we use the arithmetic average…**

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.*

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

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”.

The point? 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 sales and purchases (timing…) and you will probably not get what you hope for.