Volatility is critical to risk measurement. Generally, volatility refers to standard deviation, which is a dispersion measure. Greater dispersion implies greater risk, which implies higher odds of price erosion or portfolio loss - this is key information for any investor. Volatility can be used on its own, as in "the hedge fund portfolio exhibited a monthly volatility of 5%," but the term is also used in conjunction with return measures, as, for example, in the denominator of the Sharpe ratio. Volatility is also a key input in parametric value at risk (VAR), where portfolio exposure is a function of volatility. In this article, we'll show you how calculate historical volatility to determine the future risk of your investments. (For more insight, read *The Uses And Limits Of Volatility*.)

**Tutorial:** Option Volatility

Volatility is easily the most common risk measure, despite its imperfections, which include the fact that upside price movements are considered just as "risky" as downside movements. We often estimate future volatility by looking at historical volatility. To calculate historical volatility, we need to take two steps:

1. Compute a series of periodic returns (e.g. daily returns)

2. Choose a weighting scheme (e.g. unweighted scheme)

A daily periodic stock return (denoted below as u_{i}) is the return from yesterday to today. Note that if there was a dividend, we would add it to today's stock price. The following formula is used to calculate this percentage:

$\begin{aligned}&u_i=\frac{S_i-S_{i-1}}{S_{i-1}}\\&\textbf{where:}\\&u_i=\text{daily periodic stock return}\end{aligned}$

In regard to stock prices, however, this simple percentage change is not as helpful as the continuously compounded return. The reason for this is that we can't reliably add together the simple percentage change numbers over multiple periods, but the continuously compounded return can be scaled over a longer time frame. This is technically called being "time consistent." For stock price volatility, therefore, it is preferable to compute the continuously compounded return by using the following formula:

$u_i=ln\bigg(\frac{S_i}{S_{i-1}}\bigg)$

In the example below, we pulled a sample of Google's (NYSE:GOOG) daily closing stock prices. The stock closed at $373.36 on August 25, 2006; the prior day's close was $373.73. The continuous periodic return is therefore -0.126%, which equals the natural log (ln) of the ratio [373.26 / 373.73].

Next, we move to the second step: selecting the weighting scheme. This includes a decision on the length (or size) of our historical sample. Do we want to measure daily volatility over the last (trailing) 30 days, 360 days, or perhaps three years?

In our example, we will choose an unweighted 30-day average. In other words, we are estimating average daily volatility over the last 30 days. This is calculated with the help of the formula for sample variance:

$\begin{aligned}&\sigma^2_n=\frac{1}{m-1}\sum^m_{i=1}(u_{n-i}-\bar{u})^2\\&\textbf{where:}\\&\sigma^2_n=\text{variance rate per day}\\&m=\text{most recent }m\text{ observations}\\&\bar u=\text{the mean/average of all daily returns } (u_i)\end{aligned}$

We can tell this is a formula for a sample variance because the summation is divided by (m-1) instead of (m). You might expect an (m) in the denominator because that would effectively average the series. If it were an (m), this would produce the population variance. Population variance claims to have all of the data points in the entire population, but when it comes to measuring volatility, we never believe that. Any historical sample is merely a subset of a larger "unknown" population. So technically, we should use the sample variance, which uses (m-1) in the denominator and produces an "unbiased estimate", to create a slightly higher variance to capture our uncertainty.

Our sample is a 30-day snapshot drawn from a larger unknown (and perhaps unknowable) population. If we open MS Excel, select the thirty day range of periodic returns (i.e., the series: -0.126%, 0.080%, -1.293% and so on for thirty days), and apply the function =VARA(), we are executing the formula above. In Google's case, we get about 0.0198%. This number represents the *sample daily variance* over a 30-day period. We take the square root of the variance to get the standard deviation. In Google's case, the square root of 0.0198% is about 1.4068% - Google's historical *daily* volatility.

It's OK to make two simplifying assumptions about the variance formula above. First, we could assume that the average daily return is close enough to zero that we can treat it as such. That simplifies the summation to a sum of squared returns. Second, we can replace (m-1) with (m). This replaces the "unbiased estimator" with a "maximum likelihood estimate".

This simplifies the above to the following equation:

$\begin{aligned}\text{variance}=\sigma^2_n=\frac{1}{m}\sum^m_{i=1}u^2_{n-i}\end{aligned}$

Again, these are ease-of-use simplifications often made by professionals in practice. If the periods are short enough (e.g., daily returns), this formula is an acceptable alternative. In other words, the above formula is straightforward: the variance is the average of the squared returns. In the Google series above, this formula produces a variance that is virtually identical (+0.0198%). As before, don't forget to take the square root of the variance to get the volatility.

The reason this is an unweighted scheme is that we averaged each daily return in the 30-day series: each day contributes an equal weight toward the average. This is common but not particularly accurate. In practice, we often want to give more weight to more recent variances and/or returns. More advanced schemes, therefore, include weighting schemes (e.g., the GARCH model, exponentially weighted moving average) that assign greater weights to more recent data

**Conclusion**

Because finding the future risk of an instrument or portfolio can be difficult, we often measure historical volatility and assume that "past is prologue". Historical volatility is standard deviation, as in "the stock's annualized standard deviation was 12%". We compute this by taking a sample of returns, such as 30 days, 252 trading days (in a year), three years or even 10 years. In selecting a sample size we face a classic trade-off between the recent and the robust: we want more data but to get it, we need to go back farther in time, which may lead to the collection of data that may be irrelevant to the future. In other words, historical volatility does not provide a perfect measure, but it can help you get a better sense of the risk profile of your investments.

Check out David Harper's movie tutorial, *Historical Volatility - Simple, Unweighted Average*, to learn more on this topic.