# Calculating volatility: a simplified method

Many investors have experienced abnormal levels of investment performance volatility during different periods of the market cycle. Although sometimes volatility may be greater than expected, there is also an example that the usual way of measuring volatility can cause stocks to look unexpected and unexplainable volatility.

The purpose of this article is to discuss issues related to traditional volatility measures and explain more intuitive methods that investors can use to help them assess the magnitude of risk.

## Traditional measure of volatility

Most investors know that standard deviation is a typical statistic used to measure volatility. The standard deviation is simply defined as the square root of the average variance of the data and its 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. Therefore, there is a certain degree of doubt about its effectiveness as an accurate risk measure.

In order to make the standard deviation an accurate measure of risk, it must be assumed that the investment performance data obey a normal distribution. In terms of graphics, the normal distribution of the data will be drawn on the chart in a way that looks like a bell curve. If this criterion is true, then approximately 68% of the expected results should be within ±1 standard deviation of the expected return on investment, 95% should be within ±2 standard deviations, and 99.7% should be within ±3 standard deviations.

For example, from 1979 to 2009, the three-year rolling annualized average performance of the Standard & Poor’s 500 Index was about 9.5%, and its standard deviation was about 10%.Given these benchmark performance parameters, it is expected that the expected performance of the S&P 500 will fall within the range of -0.5% and 19.5% (9.5% ± 10%) 68% of the time.

Unfortunately, there are three main reasons why investment performance data may not be normally distributed. First, investment performance is usually biased, which means that the distribution of returns is usually asymmetric. Therefore, investors often experience periods of unusually high and low performance. Second, investment performance usually exhibits a characteristic called kurtosis, which means that investment performance exhibits an unusually large number of periods of positive and/or negative performance. In short, these issues distort the appearance of the bell curve and distort the accuracy of the standard deviation as a measure of risk.

In addition to skewness and kurtosis, a problem called heteroscedasticity is also a concern. Heteroskedasticity simply means that the variance of the sample investment performance data is not constant over time. Therefore, the standard deviation tends to fluctuate according to the length of the time period for calculation or the length of the time period for calculation.

Like skewness and kurtosis, the consequences of heteroscedasticity will cause standard deviation to become an unreliable risk measure. Taken together, these three issues will cause investors to misunderstand the potential volatility of their investments and lead them to take more risks than expected.

## Simplified measure of volatility

Fortunately, there is a simpler and more accurate way to measure and check risk through a process called the historical method. To use this method, investors only need to plot the historical performance of their investments by generating a chart called a histogram.

A histogram is a chart that plots the proportion of observations that fall within a range of categories. For example, in the figure below, the three-year rolling annual average performance of the Standard & Poor’s 500 Index from June 1, 1979 to June 1, 2009 is constructed. The vertical axis represents the magnitude of the performance of the S&P 500 index, and the horizontal axis represents the frequency of such performance of the S&P 500 index.

S&P 500 index performance histogram.

As shown in the figure, using a histogram allows investors to determine the percentage of time that investment performance is within, above, or below a given range. For example, 16% of the S&P 500 index performance observations have a return rate between 9% and 11.7%. In terms of performance below or above the threshold, it can also be determined that the S&P 500 has experienced a loss greater than or equal to 1.1%, 16% of the time, and 24.8% and 7.7% of the time.

## Comparison method

Compared with using the standard deviation, using the historical method through the histogram has three main advantages. First, the historical method does not require investment performance to follow a normal distribution. Second, the influence of skewness and kurtosis is clearly captured in the histogram, which provides investors with necessary information to mitigate unexpected volatility surprises. Third, investors can check the size of the gains and losses experienced.

The only disadvantage of the historical method is that the histogram, just like using the standard deviation, is potentially affected by heteroscedasticity. However, this is not surprising, because investors should understand that past performance does not represent future returns. In any case, even with this warning, the historical method is still an excellent benchmark for measuring investment risk, and investors should use it to assess the magnitude and frequency of potential gains and losses associated with their investment opportunities.

## Application of Methodology

How can investors generate histograms to help them check the risk attributes of investments?

One suggestion is to obtain investment performance information from investment management companies. However, it is also possible to obtain the necessary information by collecting the monthly closing prices of investment assets (usually found through various sources), and then manually calculating the investment performance.

After collecting or manually calculating performance information, you can construct a histogram by importing the data into a software package (such as Microsoft Excel) and using the software’s additional data analysis functions. By using this method, investors should be able to easily generate histograms, which in turn should help them gauge the true volatility of investment opportunities.

## Bottom line

In fact, the use of histograms should allow investors to check their investment risk in a way that helps them measure the amount they earn or lose each year. Given this real-world applicability, investors should not be surprised when the market fluctuates sharply, so they should be more satisfied with their investment exposure in all economic environments.

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