## Definition

Standard deviation is a metric used in statistics to estimate the extent by which a random variable varies from its mean. In investing, standard deviation of return is used as a measure of risk. The higher its value, the higher the volatility of return of a particular asset and vice versa.

It can be represented as the Greek symbol σ (sigma), as the Latin letter “s,” or as Std (X), where X is a random variable.

## Formula

### Probability Approach

This approach is used when the probability of each economic state can be estimated along with the corresponding expected rate of return on the asset. The formula to calculate the true standard deviation of return on an asset is as follows:

where r_{i} is the rate of return achieved at *i*th outcome, ERR is the expected rate of return, p_{i} is the probability of *i*th outcome, and n is the number of possible outcomes.

### Historical Return Approach

The practice of investing historical returns on an asset is often used to calculate the standard deviation. It is assumed that each point in the data has equal probability.

#### Population Standard Deviation

If a data set represents the entire population, the true standard deviation can be calculated as follows:

where r_{i} is the *i*th value of the rate of return on an asset in a data set, ERR is the expected rate of return or the true mean, and N is the size of a population.

#### Sample Standard Deviation

In practice, the sample data set is often used instead of the entire population. The formula above is transformed to calculate a sample standard deviation:

where r_{i} is the *i*th value of the rate of return on an asset in a sample data set, ERR is the expected rate of return or sample mean, and N is the size of a sample.

## Calculation Examples

### Example 1

A portfolio manager has to estimate two securities. The available information is shown in the table below.

__Step 1__: Compute the expected rate of return.

ERR of Security A = -5%×0.02 + 6%×0.25 + 15%×0.40 + 24%×0.30 + 34%×0.03 = 15.62%

ERR of Security B = -18%×0.02 + 2%×0.25 + 16%×0.40 + 27%×0.30 + 36%×0.03 = 22.14%

__Step 2__: Follow the formula above.

As we can see, Security A has a lower expected return and lower volatility measured by standard deviation. In contrast, Security B has a higher expected return but also a higher volatility of return.

### Example 2

The information about annual returns during the last five years of two mutual funds is shown in the table below.

__Step 1__: Calculate the expected rate of return on each fund.

ERR of Mutual Fund A = (7% + 15% + 2% – 5% + 6%) ÷ 5 = 5%

ERR of Mutual Fund B = (3% – 2% + 12% + 4% + 8%) ÷ 5 = 5%

__Step 2__: Follow the formula of sample standard deviation.

Both mutual funds have an equal expected rate of return of 5%, but Mutual Fund B has a lower standard deviation than Mutual Fund A. Other things being equal, Mutual Fund B should be preferred because of the lower risk.

## Standard Deviation in Excel

Excel offers the following functions:

- STDEV.P to calculate standard deviation based on the entire population
- STDEV.S to estimate standard deviation based on the sample

Let’s estimate standard deviation of a return using data in Example 2.

- Select output cell
**G3**. - Click
button, select**fx****All**category, and select**STDEV.S**function from the list. - In field
**Number1**, select the data range**B3:F3**, leave empty field**Number2**, and press the**OK**button.

## The Bottom Line

As was mentioned above, standard deviation of return is used in investing to measure volatility. The bigger its value, the higher the risk inherent in investing in a particular asset. In contrast, lower values indicate lower risk. If other parameters are equal, the asset with the lowest standard deviation of return is preferred.