### Average vs. compound return

We all know the story: Since 1945, the average annual stock market return has been 12.3% (dividends included, not adjusted for inflation). Ten thousand dollars invested in 1945 would be worth \$10k*1.12367 =\$24.7M in 2012.

Alas, this is not true! There are many errors in this sales pitch:

1. The dollar amounts are not adjusted for inflation.

2. Average investor cannot wait 67 years. Short term returns are important!

3. Long term investors don’t get to keep the average return.

Here we discuss the fallacy of average return in more detail. To keep the discussion concrete, we start with an example. Table 1 shows a stock market return over 5 year period. On year 1, the stock market is up by a modest 1.23% followed by two bad years and finally two very good years. The average return for the five years is 7.46%. The total compound return at the bottom row, however is 34.42% and not 43% from the average return (1.07465 = 1.43).

The rate of return that the investor should be interested is the geometric average of the return or the compound annual growth rate (CAGR). In our example, the CAGR is 6.09% which correctly gives the correct compound return of 34.4% (1.0695 = 1.344).

#### Geek notes:

The compound annual growth rate CAGR is approximately related to the average return rAVE by For example, for SP500 the average return is rAVE = 12.3% and the standard deviation is σ = 17% giving average compound rate of 11.9%. While this is still healthy, the total for the post war period drops to \$10k*1.11967 =\$10.3M in 2012. This is less than half from the sales pitch!