**Annualized Standard Deviation**

Unlike implied volatility - which belongs to option pricing theory and is a forward-looking estimate based on a market consensus - regular volatility looks backward. Specifically, it is the annualized standard deviation of historical returns.

Traditional risk frameworks that rely on standard deviation generally assume that returns conform to a normal bell-shaped distribution. Normal distributions give us handy guidelines: about two-thirds of the time (68.3%), returns should fall within one standard deviation (+/-); and 95% of the time, returns should fall within two standard deviations. Two qualities of a normal distribution graph are skinny "tails" and perfect symmetry. Skinny tails imply a very low occurrence (about 0.3% of the time) of returns that are more than three standard deviations away from the average. Symmetry implies that the frequency and magnitude of upside gains is a mirror image of downside losses.

SEE: Volatility's Impact On Market Returns

Consequently, traditional models treat all uncertainty as risk, regardless of direction. As many people have shown, that's a problem if returns are not symmetrical - investors worry about their losses "to the left" of the average, but they do not worry about gains to the right of the average.

We illustrate this quirk below with two fictional stocks. The falling stock (blue line) is utterly without dispersion and therefore produces a volatility of zero, but the rising stock - because it exhibits several upside shocks but not a single drop - produces a volatility (standard deviation) of 10%.

Theoretical Properties

Theoretical Properties

For example, when we calculate the volatility for the S&P 500 index as of Jan. 31, 2004, we get anywhere from 14.7% to 21.1%. Why such a range? Because we must choose both an interval and a historical period. In regard to interval, we could collect a series of monthly, weekly or daily (even intra-daily) returns. And our series of returns can extend back over a historical period of any length, such as three years, five years or 10 years. Below, we've computed the standard deviation of returns for the S&P 500 over a 10-year period, using three different intervals:

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:

**Are Returns Well-Behaved?**

The theoretical framework is no doubt elegant, but it depends on well-behaved returns. Namely, a normal distribution and a random walk (i.e. independence from one period to the next). How does this compare to reality? We collected daily returns over the last 10 years for the S&P 500 and Nasdaq below (about 2,500 daily observations):

These are distributions of separate interval returns, but what does theory say about returns over time? As a test, let's take a look at the actual daily distributions of the S&P 500 above. In this case, the average annual return (over the last 10 years) was about 10.6% and, as discussed, the annualized volatility was 18.1%. Here we perform a hypothetical trial by starting with $100 and holding it over 10 years, but we expose the investment each year to a random outcome that averaged 10.6% with a standard deviation of 18.1%. This trial was done 500 times, making it a so-called Monte Carlo simulation. The final price outcomes of 500 trials are shown below:

SEE: Multivariate Models: The Monte Carlo Analysis

Finally, another finding of our trials is consistent with the "erosion effects" of volatility: if your investment earned exactly the average each year, you would hold about $273 at the end (10.6% compounded over 10 years). But in this experiment, our overall expected gain was closer to $250. In other words, the average (arithmetic) annual gain was 10.6%, but the cumulative (geometric) gain was less.

**The Bottom Line**

Volatility is annualized standard deviation of returns. In the traditional theoretical framework, it not only measures risk, but affects the expectation of long-term (multi-period) returns. As such, it asks us to accept the dubious assumptions that interval returns are normally distributed and independent. If these assumptions are true, high volatility is a double-edged sword: it erodes your expected long-term return (it reduces the arithmetic average to the geometric average), but it also provides you with more chances to make a few big gains.

SEE: Implied Volatility: Buy Low And Sell High