Volatility is the most common measure of risk, but it comes in several flavors. In a previous article, we showed how to calculate simple historical volatility. (To read this article, see 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).

Historical Vs. Implied Volatility
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 The Uses And Limits Of Volatility.)

CT-EWMA1.gif

If we focus on just the three historical approaches (on the left above), they have two steps in common:

  1. Calculate the series of periodic returns
  2. Apply a weighting scheme

First, we calculate the periodic return. That's typically a series of daily returns where each return is expressed in continually compounded terms. For each day, we take the natural log of the ratio of stock prices (i.e., price today divided by price yesterday, and so on).

CT-EWMA2.gif

This produces a series of daily returns, from ui to ui-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:

CT-EWMA3.gif

Notice that this sums each of the periodic returns, then divides that total by the number of days or observations (m). So, it's really just an average of the squared periodic returns. Put another way, each squared return is given an equal weight. So if alpha (a) is a weighting factor (specifically, a = 1/m), then a simple variance looks something like this:

CT-EWMA4.gif

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:

CT-EWMA5.gif

For example, RiskMetricsTM, 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.

CT-EWMA6new.gif

Simple volatility effectively weighs each and every periodic return by 0.196% as shown in Column O (we had two years of daily stock price data. That is 509 daily returns and 1/509 = 0.196%). But notice that Column P assigns a weight of 6%, then 5.64%, then 5.3% and so on. That's the only difference between simple variance and EWMA.

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:

CT-EWMA7.gif

Recursive means that today's variance references (i.e. is a function of the prior day's variance). You can find this formula in the spreadsheet also, and it produces the exact same result as the longhand calculation! It says: Today's variance (under EWMA) equals yesterday's variance (weighted by lambda) plus yesterday's squared return (weighed by one minus lambda). Notice how we are just adding two terms together: yesterday's weighted variance and yesterdays' weighted, squared return.

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.)

Related Articles
  1. Economics

    The Problem With Today’s Headline Economic Data

    Headwinds have kept the U.S. growth more moderate than in the past–including leverage levels and an aging population—and the latest GDP revisions prove it.
  2. Economics

    Explaining the Participation Rate

    The participation rate is the percentage of civilians who are either employed or unemployed and looking for a job.
  3. Investing Basics

    Calculating the Margin of Safety

    Buying below the margin of safety minimizes the risk to the investor.
  4. Fundamental Analysis

    Is India the Next Emerging Markets Superstar?

    With a shift towards manufacturing and services, India could be the next emerging market superstar. Here, we provide a detailed breakdown of its GDP.
  5. Mutual Funds & ETFs

    ETF Analysis: PowerShares S&P 500 Downside Hedged

    Find out about the PowerShares S&P 500 Downside Hedged ETF, and learn detailed information about characteristics, suitability and recommendations of it.
  6. Mutual Funds & ETFs

    ETF Analysis: Guggenheim Enhanced Short Dur

    Find out about the Guggenheim Enhanced Short Duration ETF, and learn detailed information about this fund that focuses on fixed-income securities.
  7. Mutual Funds & ETFs

    ETF Analysis: iShares Morningstar Small-Cap Value

    Find out about the Shares Morningstar Small-Cap Value ETF, and learn detailed information about this exchange-traded fund that focuses on small-cap equities.
  8. Mutual Funds & ETFs

    ETF Analysis: iShares MSCI KLD 400 Social

    Find out about the iShares MSCI KLD 400 Social exchange-traded fund, and learn detailed information about its characteristics, suitability and recommendations.
  9. Mutual Funds & ETFs

    ETF Analysis: iShares Agency Bond

    Find out about the iShares Agency Bond exchange-traded fund, and explore detailed analysis of the ETF that tracks U.S. government agency securities.
  10. Mutual Funds & ETFs

    ETF Analysis: Guggenheim BulletShrs 2018 HY CorpBd

    Find out about the Guggenheim BulletShares 2018 High Yield Corporate Bond ETF, and get information about this ETF that focuses on high-yield corporate bonds.
RELATED TERMS
  1. Principal-Agent Problem

    The principal-agent problem develops when a principal creates ...
  2. Exchange-Traded Fund (ETF)

    A security that tracks an index, a commodity or a basket of assets ...
  3. Discount Bond

    A bond that is issued for less than its par (or face) value, ...
  4. Compound Annual Growth Rate - CAGR

    The Compound Annual Growth Rate (CAGR) is the mean annual growth ...
  5. Internal Rate Of Return - IRR

    A metric used in capital budgeting measuring the profitability ...
  6. Return On Investment - ROI

    A performance measure used to evaluate the efficiency of an investment ...
RELATED FAQS
  1. Is my IRA/Roth IRA FDIC-Insured?

    The Federal Deposit Insurance Corporation, or FDIC, is a government-run agency that provides protection against losses if ... Read Full Answer >>
  2. What assumptions are made when conducting a t-test?

    The common assumptions made when doing a t-test include those regarding the scale of measurement, random sampling, normality ... Read Full Answer >>
  3. What does a high turnover ratio signify for an investment fund?

    If an investment fund has a high turnover ratio, it indicates it replaces most or all of its holdings over a one-year period. ... Read Full Answer >>
  4. What is the utility function and how is it calculated?

    In economics, utility function is an important concept that measures preferences over a set of goods and services. Utility ... Read Full Answer >>
  5. Does index trading increase market vulnerability?

    The rise of index trading may increase the overall vulnerability of the stock market due to increased correlations between ... Read Full Answer >>
  6. What is the difference between passive and active asset management?

    Asset management utilizes two main investment strategies that can be used to generate returns: active asset management and ... Read Full Answer >>

You May Also Like

Trading Center
×

You are using adblocking software

Want access to all of Investopedia? Add us to your “whitelist”
so you'll never miss a feature!