Beta, Alpha & the CAPM

Download the prices

Yahoo shows two price columns. Close is the price as it was quoted that day. Adj Close is rewritten after the fact to account for stock splits and dividends. Here are both for :

See the downloaded table

Turn prices into returns

The CAPM is about returns, not prices. Each period's return is the percent change in Adj Close:

In Excel, with Adj Close in column B starting in row 2, type =B3/B2−1 in C3 and fill down. The most recent :

Subtract the risk-free rate

The CAPM compares excess returns: what the stock and the market earned beyond a riskless Treasury bill.

Fit the line

Each dot is one . Left to right is the market's excess return that ; bottom to top is 's. Try to draw the line that best follows the cloud. Dashed segments are your line's misses.

Excel's =SLOPE(stock excess, market excess) and =INTERCEPT(…) pick the one line with the smallest total squared miss. That is all a regression is.

Where β and α come from

The best-fitting line has an exact formula, so nobody really has to search for it. Write R_i for the stock's excess return and R_m for the market's, in each of the n s, and put a bar on top for an average. Then:

$$\beta =\frac{\mathrm{Cov}(R_i,R_m)}{\mathrm{Var}\left(R_m\right)}=\frac{\sum _{t=1}^n(R_{m,t}-{\bar{R}}_m)(R_{i,t}-{\bar{R}}_i)}{\sum _{t=1}^n{(R_{m,t}-{\bar{R}}_m)}^2}$$

β is the covariance of the stock with the market, divided by the variance of the market. In words: in the s when the market was above its average, how far above its own average was the stock, relative to how far the market moved? Once β is known, the line must pass through the point of averages, which gives α:

$$\alpha ={\bar{R}}_i-\beta \times {\bar{R}}_m$$

And R², the share of the stock's variance explained by the market:

$$R^2=\frac{{\beta }^2\times \mathrm{Var}\left(R_m\right)}{\mathrm{Var}\left(R_i\right)}$$

With the data above:

β, the slope, says how much the stock moves with the market. α, the intercept, is where the line crosses the vertical axis: the stock's average excess return in a when the market did exactly zero. ε, the miss, is each dot's distance from the line: the firm-specific part of that , such as earnings surprises, product news, or lawsuits.

How sure are we about β? A different stretch of history would give a somewhat different line. The reasonable range above is wide for volatile stocks and short windows.

Take one apart

Pick any , or click a dot above. Using Excel's best line, the stock's return splits into three pieces. It starts on the with the biggest firm-specific surprise.

Two kinds of risk

What we are doing: using the line from Step 4 to split the stock's risk into two pieces. Risk here means variance, the average squared distance of returns from their mean. Because every return is market part + α + miss, and the misses are unrelated to the market, the variances add up:

$$\underset{\text{total risk}}{\mathrm{Var}\left(R_i\right)}=\underset{\text{market risk}}{{\beta }^2\times \mathrm{Var}\left(R_m\right)}+\underset{\text{firm-specific risk}}{\mathrm{Var}\left(\epsilon \right)}$$

Step by step, for (all numbers per ):

Now diversify. Suppose you split your money equally across N stocks like this one: same β, but each with its own unrelated firm news. Every stock feels the same market, so the market part stays. The firm-specific parts partly cancel: good news at one firm offsets bad news at another, and their variance shrinks by a factor of N:

$$\mathrm{Var}\left(R_p\right)={\beta }^2\times \mathrm{Var}\left(R_m\right)+\frac{\mathrm{Var}\left(\epsilon \right)}{N}$$

Why this matters for the CAPM. Firm-specific risk is easy to get rid of, so investors are not paid for bearing it. Market risk cannot be diversified away, so that is the risk that has to earn a reward. β measures how much market risk a stock adds, which is why β is the only input in the CAPM.

Is α skill or luck?

The computer invents a fake stock with the same β and the same amount of firm-specific noise, but a true α of exactly zero. It lives through the same market history, and we run the same regression. Any α it shows is pure luck. Repeat many times.

Reading it. A stock with a lot of firm-specific noise can show a large α over a few years by luck alone. And with thousands of stocks out there, some will land far out in the tail by chance. Looking back and picking the winner is easy. Knowing in advance which stock will have a positive α is not. That is why the CAPM treats the expected α as zero.

Check how much your choices matter

There is no single "correct" β. Analysts choose how far back to look and how often to measure returns. Each choice changes every step above.

Compare with the β that Yahoo publishes

Yahoo Finance reports a β for every stock and fund, labeled Beta (5Y Monthly): 5 years of monthly returns against the S&P 500, the same recipe as the ◂ cell when you choose 5 years, monthly and SPY. Open the links and compare. Small differences are normal, because each data provider makes slightly different choices: the exact months, the index with or without dividends, and whether to subtract the risk-free rate.

Adjusted β. Very high and very low βs tend to drift back toward 1 over time. So data providers such as Bloomberg report an adjusted β = ⅔ × raw β + ⅓ × 1.

Use adjusted β in Steps 11 and 12

From portfolio theory to the CAPM

What we are doing: connecting everything so far to the Portfolio Optimizer from class, where each investor finds the portfolio with the highest Sharpe ratio. The CAPM is what you get when everyone uses the Optimizer with the same inputs. Four steps:

1. Everyone holds the tangency portfolio. In the Optimizer, each investor picks the portfolio with the highest Sharpe ratio, the tangency portfolio, and then mixes it with T-bills according to their risk aversion. If everyone agrees on expected returns, volatilities and correlations, everyone picks the same tangency portfolio.
2. The tangency portfolio is the market. Every share must be held by someone, so the portfolio everyone holds has to be the whole market, M. That is why this page measures β against SPY or the Fama–French market. The capital allocation line through M is called the capital market line (CML):$$E\left[r_p\right]=r_f+\frac{E\left[r_M\right]-r_f}{{\sigma }_M}\times {\sigma }_p$$
3. The risk that matters is covariance with the market. An investor who holds the market does not care about a stock's own σ. What matters is how much the stock adds to the risk of the portfolio they already hold, which is its covariance with the market. In units of the market's variance, that is β, exactly the slope from Step 4:$${\beta }_i=\frac{\mathrm{Cov}(r_i,r_M)}{\mathrm{Var}\left(r_M\right)}$$
4. Every asset pays the same reward per unit of that risk. Inside the tangency portfolio, each asset must earn the same extra return per unit of covariance with the market. If one paid more, tilting toward it would raise the Sharpe ratio, so the Optimizer would not have been at its maximum. The same condition for every asset gives the security market line (SML), which is the CAPM:$$\frac{E\left[r_i\right]-r_f}{\mathrm{Cov}(r_i,r_M)}=\frac{E\left[r_M\right]-r_f}{\mathrm{Var}\left(r_M\right)}\Rightarrow E\left[r_i\right]=r_f+{\beta }_i\times \left(E\right[r_M]-r_f)$$

So the Optimizer and the CAPM are the same idea at two levels: one investor choosing a portfolio, and the whole market in equilibrium. It also explains α from Step 7: a positive α is a reason to tilt away from the market, because adding that asset would raise your Sharpe ratio.

Total risk or market risk? The same assets, two ways

Each dot is one asset from the top of the page, placed at the return the CAPM requires for it. The risk-free rate and premium come from Step 11 (defaults: T-bill and 6%), and σ and β from the window you selected. Switch the horizontal axis and watch the dots.

Many stocks, one line

One stock only gives us a β. The CAPM makes a claim about all stocks together: average returns should rise in a straight line with β. The line starts at zero for β = 0 (T-bills) and passes through the market at β = 1. This is the security market line.

To test it, researchers sort all U.S. stocks every year by β into ten groups. Each dot is one group, measured from July 1963 to August 2026.

See the numbers

What the picture says. Higher β does come with somewhat higher average returns, which is the core CAPM idea. But the real line is flatter than the CAPM line: low-β stocks did better than predicted, and high-β stocks did worse. It is one of the best-known findings in finance, and part of why later models add factors like SMB and HML.

Gather the three CAPM inputs

What we are doing: getting ready to compute the return investors should require from , per year, going forward. The CAPM formula needs three inputs:

$$E\left[r_i\right]=\underset{\text{today}}{r_f}+\underset{\text{from the past}}{{\beta }_i}\times \underset{\text{market risk premium}}{\left(E\right[r_m]-r_f)}$$

β comes from the past because we cannot observe the future co-movement, so we use history as our best guess. It has no units, so a β estimated from ly data can be used directly with annual rates. The other two inputs describe today, because we want the return investors require from now on.

Input 1: the risk-free rate today

Take the current yield on a U.S. Treasury, as an annual rate. Nothing to convert: FRED already quotes it per year, and the answer we want is per year.

Input 2: the market risk premium

This is how much more than the risk-free rate investors expect to earn from the whole stock market in a typical year: E[r_m] − r_f. It cannot be looked up; it has to be estimated. Three common ways:

Step 12 plugs your three inputs into the formula.

The required return

All assets under the same choices, from lowest to highest required return:

Look at the ranking. Low-β assets such as Coca-Cola, utilities and gold move little with the market, so investors ask for little more than the risk-free rate. High-β assets such as Nvidia and the leveraged funds amplify market swings, so they need a much higher expected return to be worth holding. Negative-β assets, such as the short S&P 500 fund and the volatility fund, need less than the risk-free rate. They pay off when everything else falls, so they work like insurance, and investors accept a low or even negative expected return for that protection. How risky an asset is on its own plays no role. Only β does.