# Volatility (Standard Deviation of Returns)

> Volatility says how far an investment's returns usually land above and below their average, so a bigger number means a bumpier ride.

Page: https://formulabro.yunusemrevurgun.com/finance/volatility/  
Category: Finance  
Published: 2026-10-23  
Updated: 2026-10-23

## Formula

$$
\sigma = \sqrt{\frac{1}{N - 1} \sum_{i=1}^{N} (R_i - \bar{R})^2}
$$

Read it out loud: "Sigma equals the square root of one over N minus one, times the sum, for i from 1 to N, of R i minus R bar, all squared. Sigma is the volatility, the typical size of the ups and downs of the returns. R i is the return in period i, R bar is the average return, and N is the number of returns in the list."

## Before you start

- **Average (mean) and the bar**: The **average**, or **mean**, is the sum of a list of numbers divided by how many numbers there are. A bar over a letter, as in $\bar{R}$, means the average of that letter's values. In the example the four returns add to 12 and there are four of them, so the average is 3.
- **Deviation and squares**: A **deviation** is a value minus the average, so it is negative when the value is below the average. Squaring a deviation makes it positive whichever way it points, so $(-3)^2 = 9$ and $3^2 = 9$. Squares are used so that up moves and down moves both count toward the spread.
- **Standard deviation**: The **standard deviation** is the typical distance of values from their average. It is the square root of the variance, so it is in the same units as the values. A small standard deviation means the values stay close to the average, and a large one means they spread out.
- **Sigma and the sum**: The Greek letter $\sigma$, sigma, is the usual symbol for standard deviation. The big Greek letter $\sum$ means add up. Under it, $i = 1$ says where to start, and $N$ on top says where to stop, so $\sum_{i=1}^{N}$ means add over all $N$ values.
- **Square root**: The **square root** of a number is the value that, multiplied by itself, gives the number back. The square root of $\frac{20}{3}$ is about 2.58, because $2.58 \times 2.58 \approx 6.67$. A calculator has a square-root function for finding it.
- **Annualize**: To **annualize** a number means to convert a per-month or per-day figure into a per-year figure. Monthly volatility multiplied by the square root of 12 gives yearly volatility, because independent monthly moves add up over a year in a way that grows with the square root of time, not with time itself.

More notation help: https://formulabro.yunusemrevurgun.com/math-basics/

## What each symbol means

- $\sigma$: The **volatility**: the sample standard deviation of the returns, the typical size of their ups and downs. It is in the same unit as the returns and per period, so if the returns are in percent per month, sigma is in percent per month.
- $R_i$: The **return in period $i$**, one number for each month or year in the list, written as a rate or in percent.
- $\bar{R}$: The **average return**: the sum of all the returns divided by $N$. The bar over the R means average.
- $N$: The **number of returns** in the list, a count with no unit. A list of four months of returns has $N = 4$.
- $i$: The **counter** for the sum. It runs over the periods, from 1 up to $N$, and has no unit.

## The formula piece by piece

- $R_i - \bar{R}$: The **deviation** of one return from the average: how far the return in period $i$ sits above or below the typical value.
- $(R_i - \bar{R})^2$: The **squared deviation**, which is never negative, so an above-average and a below-average return both count toward the spread.
- $\sum_{i=1}^{N} (R_i - \bar{R})^2$: The **sum of squared deviations** over all $N$ returns. It is the total spread of the list, before it is averaged.
- $\frac{1}{N - 1} \sum_{i=1}^{N} (R_i - \bar{R})^2$: The **sample variance**: the sum of squared deviations divided by one fewer than the count, because the average came from the same returns.
- $\sigma = \sqrt{\frac{1}{N - 1} \sum_{i=1}^{N} (R_i - \bar{R})^2}$: The volatility equals the square root of the sample variance of the returns, which is the typical size of their ups and downs in the same units as the returns.
- $\sigma_{\text{annual}} = \sqrt{12}\, \sigma_{\text{monthly}}$: The yearly volatility equals the square root of 12 times the monthly volatility, when the monthly moves are independent. For daily returns the same idea uses the square root of the number of trading days in a year.

## What the formula is doing, and why

Volatility answers one question: **how far do returns usually land from their average?** You feed in a list of returns for equal periods, such as months. **What comes out** is one number, $\sigma$, in the same units as the returns and per period. A volatility of 2.58 for monthly returns written in percent means a typical month lands about 2.58 percentage points above or below the average.

**Why measure from the average?** Volatility is about how far returns move around their centre, not how high they are. A fund that returns 5% every month never varies, so its volatility is zero, even though 5% a month is a high return. The formula therefore subtracts the average $\bar{R}$ from each return. The resulting deviations always add up to zero, which is why they cannot be added directly.

**Why square, average and take the square root?** If we added the deviations directly, the positive and negative ones would cancel. Squaring makes each deviation count. Averaging the squares gives the variance, but its units are squared, so the square root brings the answer back to the units of the returns. The factor $\frac{1}{N - 1}$ divides by one fewer than the count, because the average was worked out from the same returns, which uses up one piece of information. This is Bessel's correction, and it gives a fairer estimate of the spread when we only have a sample.

**Annualizing.** Returns are often measured per month, but a yearly figure is often wanted. If each month's move is independent of the others, the variances of the months add up over a year. A year has 12 months, so the yearly volatility is $\sqrt{12} \approx 3.46$ times the monthly volatility. For daily returns, a common convention is 252 trading days in a year, which gives a factor of $\sqrt{252}$. The rule assumes the periods are independent, which is a simplification.

## An everyday picture

Think of the speed readings on a drive. A car that cruises at a steady 100 kilometers an hour has no spread around its own average speed, while a car that keeps speeding up and braking has a large spread. Volatility measures the bumpiness of the drive, the spread of the returns around their average, and not how fast the car goes on average.

## Worked example: Volatility of four monthly returns

A made-up fund returned 2%, 4%, 0% and 6% in four consecutive months. We write each return as a plain number of percent, so 2 means 2%. We treat these four months as a sample, so $N = 4$ and we use the $N - 1$ version of the formula. We want the monthly volatility, and then the yearly volatility, assuming the months move independently and a year has 12 months.

1. **Write down the list and count it.** The four monthly returns are 2, 4, 0 and 6 percent, so $R_1 = 2$, $R_2 = 4$, $R_3 = 0$ and $R_4 = 6$. The count is $N = 4$. Writing the list out and counting it first stops a value from being missed in the later steps.
2. **Find the average return.** The average is the sum divided by the count: $\bar{R} = (2 + 4 + 0 + 6) \div 4 = 12 \div 4 = 3$. So the average monthly return is 3%, and that is the centre from which the spread is measured.
3. **Find each deviation from the average.** Subtract 3 from each return: $2 - 3 = -1$, $4 - 3 = 1$, $0 - 3 = -3$ and $6 - 3 = 3$. As a check, the deviations add to zero, because $-1 + 1 - 3 + 3 = 0$. That is always true for deviations from an average.
4. **Square each deviation and add them up.** Squaring makes every deviation positive, so a deviation of minus 3 counts as much as a deviation of plus 3: $(-1)^2 = 1$, $1^2 = 1$, $(-3)^2 = 9$ and $3^2 = 9$. The total is $1 + 1 + 9 + 9 = 20$.
5. **Divide by N minus 1.** The formula divides by $N - 1 = 3$, one fewer than the count, because the average was worked out from these same four numbers. So the sample variance is $\frac{20}{3} \approx 6.67$ in percent squared per month.
6. **Take the square root to get the monthly volatility.** The square root brings the number back to percent: $\sigma = \sqrt{\frac{20}{3}} \approx 2.58$. The monthly volatility is about 2.58% a month, so a typical month lands about 2.58 percentage points away from the 3% average.
7. **Annualize the volatility.** There are 12 months in a year, and if the months move independently their variances add up over the year. So the yearly volatility is $\sigma_{\text{annual}} = \sqrt{12} \times 2.58 \approx 3.46 \times 2.58 \approx 8.94$. Another way to get the same result is $\sqrt{12 \times \frac{20}{3}} = \sqrt{80} \approx 8.94$.

**Answer:** The monthly volatility is about 2.58% a month, and the yearly volatility is about **8.94% a year**. The yearly figure assumes that the months move independently. Four months is far too few for a real estimate, so treat this as a picture of the arithmetic.

## Common mistakes

- Forgetting to subtract the average. Without the subtraction, a steady positive return looks like a large spread, and the answer is wrong.
- Dividing by $N$ instead of $N - 1$ for a list that is a sample. The two versions give different answers, so the formula must be used as written.
- Stopping at the variance. The square root is needed to get the standard deviation, which is in the same units as the returns.
- Annualizing by multiplying by 12 instead of by the square root of 12. Volatility grows with the square root of time, not in direct proportion to it.

## Where you will meet it

- Risk statements for funds, which often quote a volatility calculated from past returns.
- The denominator of the Sharpe ratio, which divides the return above the safe rate by the volatility.
- Option pricing models such as Black-Scholes, where volatility is the key input for the price of an option.
- Risk measures for portfolios, such as value at risk, which need the volatility of the whole mix.

## Questions people ask

### Is volatility the same as risk?

Not exactly. Volatility is a common measure of risk because it captures how much returns can move. It counts upward moves as well as downward ones, and it says nothing directly about the worst case.

### Why divide by N minus 1?

The average is estimated from the same returns, so the list fits its own average a little too closely. Dividing by one fewer than the count corrects for this and gives a fairer estimate. With long lists the difference is tiny.

### Why multiply by the square root of 12?

If each month's move is independent, the variances add over the year. Twelve equal monthly variances add to 12 times one of them, and the square root of that is the square root of 12 times the monthly volatility.

### How many returns do I need?

The example uses four months only to show the arithmetic. A volatility from four numbers is very uncertain. Real estimates use many more returns, and they need a long history.

## Related formulas

- [Sharpe Ratio](https://formulabro.yunusemrevurgun.com/finance/sharpe-ratio/)
- [Beta](https://formulabro.yunusemrevurgun.com/finance/beta/)
- [Portfolio Variance (Two Assets)](https://formulabro.yunusemrevurgun.com/finance/portfolio-variance/)

## Sources

- [Volatility (finance), Wikipedia](https://en.wikipedia.org/wiki/Volatility_(finance))
- [Standard deviation, Wikipedia](https://en.wikipedia.org/wiki/Standard_deviation)
- [Bessel's correction, Wikipedia](https://en.wikipedia.org/wiki/Bessel%27s_correction)

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