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Geometric Average vs. Arithmetic Average

When considering investment returns it is the geometric average,  not arithmetic average, that matters. Understanding the difference between arithmetic and geometric average returns will cause you to invest differently and improve your investment profits by taking volatility into account.

Most returns are reported as an arithmetic average because this is the highest average that can be reported. But an arithmetic average is inaccurate unless there is no volatility in returns. The inaccuracy is caused by volatility and the fact that returns are compounded. An investor who understands this can make adjustments that can greatly improve returns.

Lets take a look at the basics first.

## What are Arithmetic and Geometric Sequences?

A sequence is a group of numbers, called terms, arranged in a specific order.

### Arithmetic Sequences

Arithmetic sequences have a common difference. In other words   the same value is added or subtracted from term to term.

Example:  1, 3, 5, 7, 9, 11  (common difference is +2)

or          25, 20, 15, 10, 5   (common difference is -5)

### Geometric Sequences

Geometric sequences have a common ratio. In other words the ratio between each chronological term in the sequence is the same.

Example:  50, 100, 200, 400, 800  (common ratio is 2)

or               800, 400, 200, 100, 50   (common ratio is 1/2)

Investment average returns must be figured as a geometric average in order to be accurate. This is because through compounding each successive term is dependent on the previous outcome. When calculating investment returns the only time an arithmetic average will be accurate is when there is no volatility (i.e. 5% return each period).

## Volatility Lowers Investment Returns

Here is a simple example to illustrate how volatility lowers your investment returns.

Example:  A 50% gain and a 50% loss. It does not matter the order, the results are the same.

Arithmetic Average =  ( .50 + (- .50)) / 2 =  0

Many people mistakenly believe they are at break even. In reality  they have lost 25% of their capital! If an investor starts with \$100 and losses 50% they now have \$50. A 50% increase only brings them back t0 \$75. If the order of the returns are reversed the results are the same. A 50% increase on \$100 brings them to \$150. The 50% decline will then bring their capital down to \$75.

The arithmetic average is not reality! The geometric average (I will show you how to do this shortly) will provide the actual results:

Geometric Average =  [ (1.5 x .50) ^(1/2)] -1  =  <.1339745962> or  -13.4%

Proof:    Year 1:  \$100.0 – 13.4%  = \$86.6

Year 2:   \$86.6 – 13.4% = \$75.0

## Arithmetic and Geometric Averages

Lets say we have 6 year sequence of investment returns as follows: +30%, -20%, +30%,  -20%, +30%, and -20%.

An arithmetic average is simply the sum of all the terms (numbers) divided by the count of that sequence.

Example: (0.30 + (-.20) + 0.30 + (-.20) + 0.30 + (-.20)  / 6  =  .05 or 5.00%

The geometric average of the same numbers is quite different. This is how to figure geometric average with a finance calculator:

1. First add 1 to each number in the sequence. This is to avoid problems with negative numbers.

2. Multiple each number in the sequence.

3. Raise the answer to the power of one divided by the count of the numbers in the sequence.

4. Subtract one from the result.

Now it looks like this:

Example:    [ (1.3  x  .80  x  1.3 x  .80  x  1.3  x  .80 )  ^(1/6) ] -1   =   .019803903 or 1.98%

If you don’t have a finance calculator you can use a Geometric Mean Calculator and just plug in the numbers. This is an easy way to get a your result.

The average investor is often misled by the media and institutions which incorrectly use the arithmetic average return. An investment manager or mutual fund will probably quote the 5.0% return. But is this your real return? NO. The correct calculation, or your actual return, would be 1.98%.

### S&P 500 Returns

Let’s look a the S&P 500 actual returns (including dividends):

Arithmetic Average S&P 500 Total Returns (2000 – 2012)

((-9.2) +  (-11.9) + (-22.1) + 28.7 + 10.9 + 4.9 +15.8 + 5.5 + (-37.0) + 26.5 + 15.1 + 2.1 + 15.8) / 13  =  4.89%

Geometric Average S&P 500 Total Returns (2000 – 2012)

[ .908 x .881 x .779 x 1.287 x 1.109 x 1.049 x 1.158 x 1.055 x .63 x 1.265 x 1.151 x 1.021 x 1.158) ^(1/13)] – 1  =  1.64%

### Retirement Planning

In the real life S&P 500 example the average investor is led to believe he has made 4.89% average on his equity portfolio.  In reality he has only made 1.64%. This is a major reason investors fall short of their expectations!

Lets look at what this does to your retirement planning using a Savings & Investment Calculator:

Example:  To save \$1,000,000 over 40 years an investor needs to invest:

at 4.89% =  \$693.18 / month

at 1.64% = \$1,479.54 / month

In other words, the investor who assumes he will make 4.89 % and invest \$693.18 /month but only makes a 1.64% return will only have \$468,511.78 not the \$1,000,000 planned!

Of course we could argue that the returns the last 13 years were lower than the long term average. However, the concept of volatility and lower returns applies regardless of the assumptions you choose.

## Conclusion

These examples demonstrate that volatility lowers your investment returns. Arithmetic and geometric averages serve different purposes and only geometric averages will accurately reflect compounded investment returns. Arithmetic averages will always over state investment returns unless there is zero volatility.

The greater the volatility the greater the difference will be between arithmetic and geometric averages. When it comes to investment returns and retirement planning it is compounded (geometric) averages that matter.

The above example provides evidence that even small differences in investment returns can make huge differences in results over long periods of time. The consequence is investors need to put additional emphasis on the amount of volatility they are willing to accept.

It may be that you can increase your long term investment returns by taking LESS risk!