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 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 introduce return volatility as a useful measure of risk and explain how it is related to the statistical measures of variance and standard deviation.

Learning objectives:

- Define the concepts of variance and standard deviation.
- Understand how return volatility can be used as a measure of risk.

## Consider risk as well as return

Imagine an investor who bought corporate bonds of a company 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* is:

*R* = (7% + 2% — 3%) / 3 = 2%

While this figure gives the investor an idea about the average past performance of her investment, it doesn’t tell anything about the riskiness of the investment. For that, we need to define two other statistical measures: **variance **and **standard deviation**. Both are widely-used measures wherever statistics is involved (e.g., healthcare, quality control, construction, weather forecasting, etc.).

## Return volatility as a measure of investment risk

The **variance of realized returns**, or simply** realized variance** is based on the sum of squared deviations from the average and is formally defined as follows:

Realized variance = *∑* (*R _{t}* —

*R*)

^{2}/ Twhere *∑* is the summation factor and *T *is the number of periods (e.g., months, years, etc.). So, in our example, we have:

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

*+ (—3% — 2%)*

^{2 }*= 0.0016*

^{2}Because realized variance takes the “square” of deviations from the average return, it can’t be readily compared with the latter. To fix that, we should take the square root of the realized variance, which we define as the **standard deviation of realized returns**, or simply **return volatility**:

Return volatility = √(Realized variance)

Then, we have:

Return volatility = √0.0016 = 0.04 = 4%

Now, we have a more complete understanding of the past performance of our investor’s investment in corporate bonds: In the past three years, the corporate bonds yielded an average return of 2% with a volatility of 4%. The investor can now compare this performance with the other investment opportunities she had three years ago and can decide whether her investment performed relatively well or not.

##### What is next?

This post is part of the series on investments. The next post in the series is the expected return. The previous post was the average return.

Feel free to share this post if you enjoyed reading it. Also, we would love to hear your thoughts about this post. Please leave a comment below if you noticed any errors or have any suggestions/questions.