We start this lesson by discussing what is meant by **(stock) return volatility**. Then, we explain the **return volatility formula**. Finally, a simple **return volatility calculator** is provided for your convenience.

#### Contents

## What is (stock) return volatility?

Imagine an investor who bought shares of a stock three years ago. According to the investor’s calculations, her annual returns over this period were 7%, 2%, and −3%. Therefore, her average realized return* R*_{A} is:

*R*_{A} = (7% + 2% − 3%) / 3 = 2%

While this figure gives the investor an idea about the average yearly performance of her investment, it doesn’t tell her anything about its riskiness. For that, we rely on return volatility, which captures how risky (or volatile) the returns are.

The stock return volatility is computed as the standard deviation of returns, which is the square root of the variance of returns. Variance and standard deviation are two important statistical measures, which are also used in many areas other than finance (e.g., healthcare, quality control, construction, weather forecasting, etc.). Intuitively, these are measures of dispersion, quantifying how spread out the returns are.

## Return volatility formula

In order to compute realized return volatility, we begin with the variance of realized returns, or simply the **realized variance**. The realized variance is the sum of squared deviations from the average realized return *R*_{A}:

Realized variance = *∑* (*R _{t}* −

*R*

_{A})

^{2}/ Twhere *∑* is the summation symbol, *R _{t}* is the return in period

*t*, and

*T*is the number of periods (e.g., months, years, etc.).

Then, we can write the return volatility formula as the squared root of the realized variance, which is of course the same as the standard deviation of realized returns:

**Technical note**: If the realized volatility will be used as an estimate for future volatility, we divide by *T−1* instead of *T* to obtain an unbiased estimator.

If we return to our earlier example, we calculated the investor’s average realized return as *R*_{A} = 2%. Now, we can compute the realized variance as follows:

Realized variance = [(7% − 2%)* ^{2}* + (2% − 2%)

*+ (−3% − 2%)*

^{2 }*] / 3= 0.00167*

^{2}Finally, we can easily calculate realized volatility by taking the squared root of realized variance:

√(0.00167) = 0.0408 = 4.08%

Now, our investor has a more complete understanding of the stock’s past performance: In the past three years, the stock had an average return of 2% with a volatility of 4%. This helps the investor compare the performance of this stock with similar investment opportunities she had three years ago to determine whether the stock performed relatively well.

## Return volatility calculator

Instructions for using the realized return volatility calculator:

- Enter return observations as percentage points (e.g., enter 10 for 10%).
- You can enter up to 12 observations.
- When you are entering, say, 5 observations, use the first five fields: Return 1, Return 2, …, and Return 5.
- Two figures are reported at the bottom. If you’re going to use the realized volatility as an estimate for future volatility, you can rely on the first figure, which divides the sum of squared deviations by
*T-1*. If you’re simply interested in the realized volatility, you can focus on the second figure (*T*).

##### summary

In finance, we often say that risk and return are the two sides of a coin. Therefore, metrics such as arithmetic average return and geometric average return are helpful when evaluating the past performance of an investment but are not sufficient on their own without proper consideration of risk.

In this post, we have introduced return volatility as a useful measure of risk and have derived the return volatility formula, relying on the statistical measures of variance and standard deviation.

Further reading

Andersen et al. (2001) ‘The distribution of realized stock return volatility‘, *Journal of Financial Economics*, Vol. 61 (1), pp. 43-76.

##### What is next?

This lesson is part of our free course on investments.

**Next lesson**: We will move on to forward-looking measures of investment performance and will introduce the concept of expected return.**Previous lesson**: We discussed how geometric average return can be used to assess investment performance and how it differs from the arithmetic average return.

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