The initial price of volatility

We'll start this blog with some theory about volatility.
A short excerpt form the original interesting article on Estopedia, called "The Uses And Limits Of Volatility" by David Harper,CFA, FRM (Contact Author | Biography)

______________________________________________________________

One of the theoretical properties of volatility may or may not surprise you: it erodes returns.

This is due to the key assumption of the random walk idea: that returns are expressed in percentages. Imagine you start with $100 and then gain 10% to get $110. Then you lose 10%, which nets you $99 ($110 x 90% = $99). Then you gain 10% again, to net $108.90 ($99 x 110% = $108.9). Finally, you lose 10% to net $98.01. It may be counter-intuitive, but your principal is slowly eroding even though your average gain is 0%!

If, for example, you expect an average annual gain of 10% per year (i.e. arithmetic average), it turns out that your long-run expected gain is something less than 10% per year. In fact, it will be reduced by about half the variance (where variance is the standard deviation squared). In the pure hypothetical below, we start with $100 and then imagine five years of volatility to end with $157:


The average annual returns over the five years was 10% (15% + 0% + 20% - 5% + 20% = 50% ÷ 5 = 10%), but the compound annual growth rate (CAGR, or geometric return) is a more accurate measure of the realized gain, and it was only 9.49%. Volatility eroded the result, and the difference is about half the variance of 1.1%. These results aren't from a historical example, but in terms of expectations, given a standard deviation of σ (variance is the square of standard deviation, σ^2) and an expected average gain of μ, the expected annualized return is approximately μ - ( σ^2 ÷ 2).
___________________________________________________________________

The 'hidden secret' in this article is in the last line.

We may define the Volatility Rate Loss (VRL) here as the difference between the Average Annual growth Rate (AAGR) and the (real) Compound Annual Growth rate (CAGR).

VRL = AAGR - CAGR

Let's calculate the VRL for several volatilities:

Volatility	VRL
σ	( σ^2) / 2
3%	0,03%
5%	0,13%
10%	0,50%
15%	1,13%
20%	2,00%
25%	3,13%
30%	4,50%
35%	6,13%
40%	8,00%

From this simple table, and given the fact that actual volatility of several main indexes (shares) are varying around 20-30% and that volatilities of 40 (or more) of investments in individual companies are no exception, it's clear that we can not neglect the influence of VRL.

For example if we take an investment with an average return of 10% and a volatility ( σ ) of 20%, the average return (of 10%) will show 2% too high.

Another way of putting it (in this last case) is that when you decide to step over from a non-volatile investment to a volatility risk type of investment, your investment should return on average 2% higher to give the same return as your non-volatile investment. In a way VRL is the initial price you pay for volatility sec.

So, when judging returns, don't look at average returns, but always look at the Compound Annual Growth Rate.