# Calculating Volatility: A Simplified Approach

Many investors have experienced abnormal levels of investment returns volatility during various periods of the market cycle. While volatility can sometimes be higher than expected, it can also be argued that the way volatility is typically measured contributes to the problem of stocks appearing unexpectedly and inexplicably volatile.

The purpose of this article is to discuss the problems associated with the traditional measurement of volatility and to explain a more intuitive approach that investors can use to help them assess the extent of risk.

## Traditional measure of volatility

Most investors know that standard deviation is the typical statistic used to measure volatility. The standard deviation is simply defined as the square root of the mean variance sound data mean. Although this statistic is relatively easy to calculate, the assumptions behind its interpretation are more complex, which in turn raises concerns about its accuracy. As a result, there is a certain level of skepticism about its validity as a risk measurement.

For standard deviation to be an accurate measure of risk, investment return data must be assumed to follow a normal distribution. In graphical terms, a normal distribution of data will be plotted on a graph in a way that looks like a bell curve. If this standard is true, then approximately 68% of the expected results should fall within ± 1 standard deviation from the investment expected return95% must be within ± 2 standard deviations and 99.7% must be within ± 3 standard deviations.

For example, from 1979 to 2009, the three-year rolling average annualized return of the S&P 500 Index was about 9.5% and its standard deviation was about 10%. Given these baseline performance metrics, one would expect that 68% of the time, the expected performance of the S&P 500 Index would fall within a range of -0.5% and 19.5% (9 .5% ± 10%).

Unfortunately, there are three main reasons why investment performance data may not be normally distributed. First, investment performance is generally skewed, which means that return distributions are generally skewed. As a result, investors tend to experience periods of abnormally high and low performance. Second, investment performance typically exhibits a property known as flattening, which means that the performance of the investments has an abnormally high number of positive and/or negative performance periods. Taken together, these issues distort the appearance of the bell-shaped curve and skew the accuracy of the standard deviation as a measure of risk.

In addition to asymmetry and flattening, a problem known as heteroscedasticity is also a cause for concern. Heteroscedasticity simply means that the variance of the sample investment return data is not constant over time. Therefore, the standard deviation tends to fluctuate depending on the length of the time period used to perform the calculation or the time period selected for performing the calculation.

Like skewness and kurtosis, the ramifications of heteroscedasticity will make standard deviation an unreliable measure of risk. Taken collectively, these three issues can cause investors to misunderstand the potential volatility of their investments and lead them to potentially take on significantly more risk than intended.

## A simplified measure of volatility

Fortunately, there is a much simpler and more accurate way to measure and examine risk, through a process known as the historical method. To use this method, investors simply need to graph the historical performance of their investments, generating a graph known as a histogram.

A histogram is a graph that plots the proportion of observations that fall within a multitude of category ranges. For example, in the chart below, the three-year rolling annualized average performance of the S&P 500 Index for the period June 1, 1979 to June 1, 2009 has been constructed. The vertical axis represents the magnitude of the performance of the S&P 500 index and the horizontal axis represents the frequency with which the S&P 500 index recorded such performance.

As the chart illustrates, using a histogram allows investors to determine the percentage of time an investment’s performance is within, above, or below a given range. For example, 16% of the performance observations of the S&P 500 index achieved a return between 9% and 11.7%. In terms of performance below or above a threshold, it can also be determined that the S&P 500 index suffered a loss greater than or equal to 1.1%, 16% of the time, and a performance greater than 24.8%, 7 .7% of the time.

## Comparison of methods

Using the historical method via a histogram has three main advantages over using the standard deviation. First, the historical method does not require investment returns to be normally distributed. Second, the impact of skewness and flattening is explicitly captured in the histogram, which provides investors with the information needed to mitigate unexpected volatility surprises. Third, investors can examine the magnitude of the gains and losses incurred.

The only downside to the historical method is that the histogram, like the use of standard deviation, suffers from the potential impact of heteroscedasticity. However, this should come as no surprise, as investors should understand that past performance is not indicative of future returns. Regardless, even with this caveat, the historical method is still an excellent benchmark measure of investment risk and should be used by investors to gauge the magnitude and frequency of their potential gains and losses. associated with their investment opportunities.

## Application of the methodology

How do investors generate a bar chart to help them examine the risk attributes of their investments?

One recommendation is to request investment performance information from the investment management businesses. However, the necessary information can also be obtained by collecting monthly data the last price of the investment asset, usually found from various sources, and then manually calculating the performance of the investment.

After performance information has been collected or calculated manually, a histogram can be constructed by importing the data into a software package, such as Microsoft Excel, and using the software’s additional data analysis function. Using this methodology, investors should be able to easily generate a histogram, which in turn should help them gauge the true volatility of their investment opportunities.

## The essential

Concretely, the use of a histogram should allow investors to examine the risk of their investments in a way that will help them assess the amount of money they stand to gain or lose on a annual basis. Given this kind of real-world applicability, investors should be less surprised when markets fluctuate significantly, and as a result should feel much more satisfied with their investment exposure in all economic environments.