### Key Points

**Volatility**measures the dispersion of returns of a security, which can be estimated by calculating the**standard deviation**of historical returns.- Based upon this standard deviation measure, and an understanding of a stock’s average historical return we can estimate a stock’s range of expected returns.
- This information can be useful in determining whether a particular stock lies within one’s
**risk**tolerance when making investment decisions. - However, there are many assumptions associated with this analysis, which must be taken into consideration.

### Into to Vol

When we think of a volatile individual, we think of someone that is impulsive, rash, and unpredictable. In finance, we can correlate how risky a stock is with how volatile the company’s stock tends to be. Specifically, we can quantify how much the returns of the company’s stock tend to vary over time. A stock whose returns tend to vary significantly would be classified as having a high volatility; while a stock whose returns do not vary so much would be classified as having a low volatility, by comparison.

Volatility is a statistical measure of the dispersion of returns for a given security or market index. —Investopedia

We can calculate the dispersion of returns (meaning the amount by which returns tend to vary) by using the statistical measure known as standard deviation (**σ),** which aims to determine how much a given return is likely to deviate from its average return. **Table 1** below provides a simple example of this notion.

Over this six-month period, the average monthly return for **Stock A** is 1.0%, and for **Stock B** is 2.0%. From these numbers, we can clearly see that **Stock B** tends to move twice as much in both directions as **Stock A**, so it’s not surprising to learn that the standard deviation (or volatility) of **Stock B** is twice that of **Stock A**, 4.2% versus 2.1%, respectively.

Accordingly, a standard deviation of 2.1% means that **Stock A**‘s monthly return is most likely to be around 1.0% plus/minus 2.1%. Put differently, we are most likely to find **Stock A**‘s monthly return to be between -1.1% and 3.1%, which comes from 1.0% minus 2.1%, and 1.0% plus 2.1%, respectively. Similarly, we are most likely to find **Stock B**‘s monthly return to be between -2.2% and 6.2%. As you can see, a doubling of volatility leads to not only more downside, but also more upside, which is generally how it works in investing—i.e., you in order to enhance gains, you also need to increase your chance of loss.

### Expected Range, Confidence Interval

Statistically, this volatility measure can be quite informative if we are to believe that the returns of a stock are random in nature. When we say a stock’s return behaves randomly, we’re basically saying that we can’t use past information to tell us what the return for the next period will be—just as a roll of a fair dice is random, let’s assume the same for the return of a stock (note: there are many additional nuances to surrounding this assumption, but we’ll dig into this later in this series).

Given this and the information presented in **Table 1** above, we can infer that about 68% of the time the monthly return for **Stock A** will be 1.0% +/- 2.1% (which is the average return plus/minus one standard deviation), or between -1.1% and 3.1%, as shown in **Table 2** below. Further, we can infer that about 95% of the time the monthly return for **Stock A** will be 1.0% +/- 4.2% (which is the average return plus/minus two standard deviations), or between -3.2% and 5.2%. Finally, we can infer that about 99.7% of the time the monthly return for **Stock A** will be between -5.3% and 7.3% (which is the average return plus/minus three standard deviations).

This understanding of each stock’s volatility, and expected range of movement can be helpful in **managing risk**. For example, let’s say you don’t want to lose more than 3.0% in a given month, which stock would you pick?

Any stock can go to zero, obviously. But that’s not very helpful, is it? Thinking it over again, and using **Table 2** as our guide, we can see that in 95% of months, **Stock A** should not fall more than 3.2% or rise more than 5.2%.

Another way of putting this same information is that we are 95% confident that a given month’s return for **Stock A** will fall between -3.2% and 5.2%; i.e., the 95% confidence interval for **Stock A** is -3.2% to 5.2%. Accordingly, this means you should only be down more than -3.2% or above 5.2% in about one in twenty months (or 5% of the time) should you decide to invest in **Stock A**. On the other hand, **Stock B** will be down more than -6.5% or above 10.4% about one in twenty months. Although this large positive return sounds appealing, this large negative return is beyond your risk tolerance. From this perspective, you consider the risk of **Stock A** to be in line with your tolerance, and thus, you decide to invest in **Stock A** as opposed to **Stock B.**

But how can we possibly know all this from just six months of returns for each stock? Seems almost too simple, right? Indeed, there are many assumptions at play here, which should be taken into consideration, so let’s dig deeper into all this in Part 3.