*Using Volatility To Gauge Future Risk*.) We used Google's actual stock price data in order to compute daily volatility based on 30 days of stock data. In this article, we will improve on simple volatility and discuss the exponentially weighted moving average (EWMA).

**First, let's put this metric into a bit of perspective. There are two broad approaches: historical and implied (or implicit) volatility. The historical approach assumes that past is prologue; we measure history in the hope that it is predictive. Implied volatility, on the other hand, ignores history; it solves for the volatility implied by market prices. It hopes that the market knows best and that the market price contains, even if implicitly, a consensus estimate of volatility. (For related reading, see**

Historical Vs. Implied Volatility

Historical Vs. Implied Volatility

*.)*

*The Uses And Limits Of Volatility***Calculate the series of periodic returns****Apply a weighting scheme**

_{i}to u

_{i-m}, depending on how many days (m = days) we are measuring.

That gets us to the second step: This is where the three approaches differ. In the previous article (

*Using Volatility To Gauge Future Risk*), we showed that under a couple of acceptable simplifications, the simple variance is the average of the squared returns:

**The EWMA Improves on Simple Variance**

The weakness of this approach is that all returns earn the same weight. Yesterday's (very recent) return has no more influence on the variance than last month's return. This problem is fixed by using the exponentially weighted moving average (EWMA), in which more recent returns have greater weight on the variance.

The exponentially weighted moving average (EWMA) introduces lambda, which is called the smoothing parameter. Lambda must be less than one. Under that condition, instead of equal weights, each squared return is weighted by a multiplier as follows:

^{TM, }a financial risk management company, tends to use a lambda of 0.94, or 94%. In this case, the first (most recent) squared periodic return is weighted by (1-0.94)(.94)

^{0}= 6%. The next squared return is simply a lambda-multiple of the prior weight; in this case 6% multiplied by 94% = 5.64%. And the third prior day's weight equals (1-0.94)(0.94)

^{2}= 5.30%.

That's the meaning of "exponential" in EWMA: each weight is a constant multiplier (i.e. lambda, which must be less than one) of the prior day's weight. This ensures a variance that is weighted or biased toward more recent data. (To learn more, check out the Excel Worksheet for Google's Volatility.) The difference between simply volatility and EWMA for Google is shown below.

Remember: After we sum the entire series (in Column Q) we have the variance, which is the square of the standard deviation. If we want volatility, we need to remember to take the square root of that variance.

What's the difference in daily volatility between the variance and EWMA in Google's case? It's significant: The simple variance gave us a daily volatility of 2.4% but the EWMA gave a daily volatility of only 1.4% (see the spreadsheet for details). Apparently, Google's volatility settled down more recently; therefore, a simple variance might be artificially high.

**Today's Variance Is a Function of Pior Day's Variance**

You'll notice we needed to compute a long series of exponentially declining weights. We won't do the math here, but one of the best features of the EWMA is that the entire series conveniently reduces to a recursive formula:

Even so, lambda is our smoothing parameter. A higher lambda (e.g., like RiskMetric's 94%) indicates slower decay in the series - in relative terms, we are going to have more data points in the series and they are going to "fall off" more slowly. On the other hand, if we reduce the lambda, we indicate higher decay: the weights fall off more quickly and, as a direct result of the rapid decay, fewer data points are used. (In the spreadsheet, lambda is an input, so you can experiment with its sensitivity).

**Summary**

Volatility is the instantaneous standard deviation of a stock and the most common risk metric. It is also the square root of variance. We can measure variance historically or implicitly (implied volatility). When measuring historically, the easiest method is simple variance. But the weakness with simple variance is all returns get the same weight. So we face a classic trade-off: we always want more data but the more data we have the more our calculation is diluted by distant (less relevant) data. The exponentially weighted moving average (EWMA) improves on simple variance by assigning weights to the periodic returns. By doing this, we can both use a large sample size but also give greater weight to more recent returns.

(To view a movie tutorial on this topic, visit the Bionic Turtle.)