Beta

In one sentence: Beta says how much a stock tends to move when the whole market moves: 2 means it moves about twice as far as the market, and 0.5 means about half as far.

$\beta = \frac{\operatorname{Cov}(R_i, R_m)}{\operatorname{Var}(R_m)}$

Read it out loud: “Beta equals the covariance of R i and R m, divided by the variance of R m. Covariance measures how two sets of returns move together. Variance measures how spread out the market returns are. Beta has no unit. It is a plain number that compares the stock with the market.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

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 average of 3, 9 and 12 is 24 divided by 3, which is 8.
Deviation from the average: A deviation is how far one value sits from the average: the value minus the average. A positive deviation is above average and a negative one is below. The deviations of any list always add up to zero, which is why they must be squared or multiplied before they can be added to measure spread.
Variance: The variance is the average of the squared deviations, and it says how spread out a list is. Squaring makes every deviation positive, so a value above the average and a value below it both add to the spread. Its unit is the unit of the values, squared.
Covariance: The covariance of two lists is the average of the products of their deviations. When both lists are above average in the same period, the product is positive, and when one is above while the other is below, the product is negative. A positive covariance means the two tend to move together.
Sigma and the sum: The big Greek letter sigma, $\sum$ , means add up. The index under it, here $t = 1$ , says where to start, and the letter on top, $n$ , says where to stop. Sum notation is a short way to write a long addition, so $\sum_{t=1}^{n}$ means add over all $n$ periods.
Population and sample formulas: A population formula divides by $n$ , the number of values in the list. A sample formula divides by $n - 1$ instead. This page uses the population version for both covariance and variance, so both are divided by $n$ .

What each symbol means

Symbol	Meaning
$\beta$	Beta, a plain number with no unit. It says how strongly the stock’s returns follow the market’s returns. A beta of 1 means the same size of move, a beta above 1 means bigger moves, and a beta below 1 means smaller moves.
$\operatorname{Cov}(R_i, R_m)$	The covariance between the stock’s returns $R_i$ and the market’s returns $R_m$ . It is positive when the two tend to move in the same direction. Its unit is the return unit squared, such as percent squared.
$\operatorname{Var}(R_m)$	The variance of the market’s returns $R_m$ : the average squared distance of each market return from the average market return. It is never negative, and its unit is the return unit squared.
$R_i$	The return of the stock in each period, such as each month, written as a rate. The subscript $i$ names this stock.
$R_m$	The return of the market in the same periods, often measured with a broad stock index. The subscript $m$ stands for market.

The formula piece by piece

$R_i$: The stock’s return in each period, one number per month for example. It is the first list of inputs.
$R_m$: The market’s return in the same periods, lined up month by month with the stock’s returns. It is the second list of inputs.
$\operatorname{Cov}(R_i, R_m)$: The covariance: the average of the products of the stock’s deviations and the market’s deviations. It is the top of the fraction and says how much the two move together.
$\operatorname{Var}(R_m)$: The variance of the market’s returns: the average of the squared deviations of the market from its average. It is the bottom of the fraction and says how much the market itself moves.
$\beta = \frac{\operatorname{Cov}(R_i, R_m)}{\operatorname{Var}(R_m)}$: Beta equals the covariance of the stock and the market divided by the variance of the market, which compares the stock’s movement with the market’s movement.
$\operatorname{Cov}(R_i, R_m) = \frac{1}{n} \sum_{t=1}^{n} (R_{i,t} - \bar{R}_i)(R_{m,t} - \bar{R}_m)$: The covariance written out: for each period $t$ , multiply the stock’s deviation by the market’s deviation, add over all $n$ periods, and divide by $n$ .
$\operatorname{Var}(R_m) = \frac{1}{n} \sum_{t=1}^{n} (R_{m,t} - \bar{R}_m)^2$: The variance written out: for each period $t$ , square the market’s deviation, add over all $n$ periods, and divide by $n$ .

What the formula is doing, and why

Beta compares two lists of returns: the stock’s and the market’s, measured over the same periods. What goes in: one stock return and one market return for each period. What comes out: one plain number, beta. A beta of 1 means the stock moves with the same size as the market. A beta of 2 means that, on average, a 1% move in the market goes with a 2% move in the stock. A beta below 0 means the stock tends to move against the market.

Why covariance on top? Covariance counts how much the two lists move together. If the stock rises when the market rises and falls when the market falls, the covariance is large and positive. If the two move with no pattern between them, the covariance is near zero. The top of the fraction therefore measures the part of the stock’s movement that goes along with the market.

Why divide by the variance of the market? Dividing by the market’s own spread puts the answer on a market scale, so beta says how many market moves the stock makes. The market compared with itself has covariance equal to its variance, so its beta is exactly 1. If every stock return doubles while the market stays the same, the covariance doubles and so does beta. Beta is also the slope of the best straight line through a chart of stock returns against market returns.

Population formulas. The covariance and variance on this page divide by $n$ , the number of periods. That is the population version. Some textbooks and software divide by $n - 1$ instead, which is the sample version. If both are computed the same way, the top and bottom change by the same factor and beta is unchanged. Mixing the two versions shrinks beta by the factor $(n - 1)/n$ .

An everyday picture

Think of a dog on a long leash walking beside its owner. When the owner strides forward, the dog tends to stride forward too, and when the owner stops, the dog stops. If the dog covers about twice the owner’s distance on each walk, its beta is about 2. A dog that barely moves when the owner moves has a beta near 0. Beta is the measure of how far the dog follows the owner’s movement, on average.

Worked example: Beta of a stock from three months of returns

We have three months of returns for one stock and for the market. To keep the numbers small, each return is written as a plain number of percent, so 3 means 3%. The stock returns are 3, 9 and 12. The market returns are 2, 4 and 6. The months are in the same order, so the first stock return and the first market return belong to the same month. We want the beta of the stock, using population formulas that divide by $n = 3$ . All numbers are made up for the example.

Write down the paired returns. Each month gives one stock return and one market return, and they must belong to the same month. Month 1 has stock 3 and market 2, month 2 has stock 9 and market 4, and month 3 has stock 12 and market 6. That gives $n = 3$ periods.
Find the average of each list. The average is the centre of each list, and the deviations will be measured from it. Stock: $(3 + 9 + 12) \div 3 = 8$ . Market: $(2 + 4 + 6) \div 3 = 4$ . So $\bar{R}_i = 8$ and $\bar{R}_m = 4$ , both in percent.
Find each deviation from the average. A deviation is a return minus its average, so a negative deviation means below average. Stock: $3 - 8 = -5$ , $9 - 8 = 1$ and $12 - 8 = 4$ . Market: $2 - 4 = -2$ , $4 - 4 = 0$ and $6 - 4 = 2$ . As a check, each list of deviations adds to zero: $-5 + 1 + 4 = 0$ and $-2 + 0 + 2 = 0$ .
Multiply the paired deviations for the covariance. Covariance asks whether the two move together, so in each month we multiply the stock’s deviation by the market’s deviation: $(-5)(-2) = 10$ , $(1)(0) = 0$ and $(4)(2) = 8$ . These add to 18, and dividing by $n = 3$ gives $\operatorname{Cov}(R_i, R_m) = 18 \div 3 = 6$ . The positive result means the stock and the market tended to move together.
Square the market deviations for the variance. Variance measures how much the market itself moves, and squaring stops negative deviations from cancelling positive ones: $(-2)^2 = 4$ , $0^2 = 0$ and $2^2 = 4$ . These add to 8, and dividing by $n = 3$ gives $\operatorname{Var}(R_m) = 8 \div 3 \approx 2.67$ , in percent squared.
Divide covariance by variance. Beta is the covariance divided by the variance of the market, so $\beta = 6 \div 2.67 \approx 2.25$ . Both numbers are in percent squared, so the units cancel and beta has no unit. Dividing both by 3 gives the same answer as dividing 18 by 8: $\frac{18/3}{8/3} = \frac{18}{8} = 2.25$ .
Read the answer. A beta of 2.25 means that in these three months the stock moved about two and a quarter times as far as the market, in the same direction. Three months is far too few for a real estimate, so this example only shows the arithmetic.

Answer: The beta is about 2.25. In these three made-up months, the stock moved about two and a quarter times as far as the market, in the same direction. Real beta estimates use many more periods.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Dividing by the variance of the stock instead of the variance of the market. The bottom of the fraction is always the variance of the market returns.
Mixing the population and sample formulas, for example dividing the covariance by $n$ and the variance by $n - 1$ . That quietly shrinks beta by the factor $(n - 1)/n$ .
Using prices instead of returns. Beta compares percentage changes, and raw prices give a different and misleading number.
Treating beta as the whole risk of a stock. Beta covers only the part of the risk that moves with the market, so a stock can be risky and still have a low beta.

Where you will meet it

The capital asset pricing model, where beta sets how much market risk premium a stock is expected to earn.
Fund fact sheets and brokerage tools, which often show a beta for each stock or fund.
Hedging a portfolio against the market, where beta tells how much index exposure is needed to offset the market risk.
Estimating the cost of equity for a company in corporate finance.

Questions people ask

Is a high beta good or bad?

Neither by itself. A high beta means bigger swings with the market, which can mean bigger gains when the market rises and bigger losses when it falls. Whether that suits an investor depends on their goals and time horizon.

Why is the beta of the market exactly 1?

The covariance of the market with itself equals its variance. The top and bottom of the fraction are then the same number, so the ratio is 1.

Is beta the same as volatility?

No. Volatility is the spread of one investment’s own returns. Beta compares that investment with the market. A stock can be very volatile yet have a low beta if most of its moves have nothing to do with the market.

How many periods do I need?

The example uses three periods only to show the arithmetic. Real estimates use many more periods, and the result depends on the period and the index chosen.

Capital Asset Pricing Model (CAPM): CAPM says the return you should expect from an investment is a safe rate plus extra return for the market risk it carries, scaled by its beta.
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.
Linear Regression: Linear regression predicts a number by drawing the straight line that sits closest to the data points, so you can read off a guess for any new input.

Sources

Published 2026-10-23. Plain-text copy for AI agents and readers: finance/beta.md.