Beta and CAPM: The Math Behind Expected Return

Every discounted cash flow model ends with a number that decides most of the valuation: the discount rate. For equity, that number usually comes from the Capital Asset Pricing Model, and the CAPM usually comes down to one input - beta. Beta is quoted everywhere, on every finance site, and it is almost never explained as what it is: the slope of a line fitted to past returns. Understanding that sentence, and its arithmetic, turns beta from a magic number into something you can check, question, and occasionally correct.

This guide derives beta from a six-month return series by hand, puts it into the CAPM, and then covers the places where the model quietly misleads. All figures are arithmetic on stated assumptions, not forecasts, and nothing here is a recommendation to buy or sell any security.

What Beta Actually Measures

Beta is the sensitivity of one asset's return to moves in the market's return. A beta of 1.15 means that when the market's return rises one percentage point above the risk-free rate, this asset's return tends to rise 1.15 points above it; when the market falls one point below, the asset tends to fall 1.15. A beta of 0.6 means the same asset moves 60 cents for every dollar the market moves. A beta near zero describes an asset that barely responds to the market at all.

Formally, beta is the covariance of the asset's returns with the market's returns, divided by the variance of the market's returns:

beta = Cov(asset, market) / Var(market)

Dividing by variance normalises the number, which is why beta is unitless and comparable across assets. It is also why beta depends on the index you choose as "the market": measured against a broad total-market index, a stock can have a beta of 1.1; measured against a sector index, the same stock can have a beta of 0.8. Both numbers are correct. They answer different questions.

Computing Beta From Six Months of Returns

Take a simplified example: six monthly returns for a broad index and for one stock, expressed in percent. Six observations is far too few for a real estimate - professionals use 60 monthly returns, five years - but the arithmetic is identical and short enough to follow.

The index averages 0.833% a month and the stock averages 1.350%. Deviations are each observation minus its own mean. Summing the products of the paired deviations gives 39.35, and summing the squared index deviations gives 34.33. Divide each by n−1 (five) for the sample covariance and variance:

• Covariance = 39.35 / 5 = 7.87
• Variance of the index = 34.33 / 5 = 6.867
• Beta = 7.87 / 6.867 = 1.146

The index's standard deviation here is 2.62% a month, an unusually calm six months for an equity index - real data typically runs 3.5% to 4.5%. That is exactly why the sample matters: a calm period produces a low variance and, all else equal, a less stable beta.

The correlation between the two series is about 0.955, so the R-squared is roughly 0.91: the index explains 91% of this stock's monthly variation. That is high for a single stock - typical values for individual companies against a broad index sit between 0.2 and 0.5 - and it is a reminder that a six-month window can flatter a relationship that will not hold up over five years.

The CAPM Itself

The model states that the expected return on an asset equals the risk-free rate plus beta times the market risk premium:

E(R) = Rf + beta × (E(Rm) − Rf)

Three inputs, and the disagreements are all in the second and third. The risk-free rate is usually a government bond yield matched to the horizon - a 10-year Treasury yield for a long-horizon valuation, a 3-month bill for a one-year hurdle. The market risk premium - the expected excess return of equities over that risk-free rate - is not observable. Long-run historical estimates for US equities cluster between 4% and 6%; forward-looking estimates built from dividend yields and growth tend to run lower. Whichever you use, state it, because a one-point change in the premium moves every valuation in the model.

Check the arithmetic on the worked example: 4.2 + 1.146 × 5.0 = 4.2 + 5.73 = 9.93%. The premium earned over the risk-free rate - 5.73 points - is exactly beta times the market premium, which is the entire content of the model.

Portfolio Beta Is a Weighted Average

Betas add linearly when weighted by market value, which makes portfolio-level CAPM easy. A portfolio of three holdings, weighted 40%, 35%, and 25%, with betas of 0.6, 1.1, and 1.6:

• 0.40 × 0.6 = 0.240
• 0.35 × 1.1 = 0.385
• 0.25 × 1.6 = 0.400
• Portfolio beta = 0.240 + 0.385 + 0.400 = 1.025

At the same 4.2% risk-free rate and 5.0% premium, the portfolio's expected return is 4.2 + 1.025 × 5.0 = 9.33%. Note what this does not mean: the portfolio is not as risky as a single stock with beta 1.025, because the unsystematic risk of the three holdings partly cancels. CAPM prices only the risk that cannot be diversified away, which is the model's central claim and its most contested one.

Asset Beta Versus Equity Beta

The beta you see quoted is an equity beta - it reflects both the business and the debt. To compare companies with different capital structures, or to value a division, you need the asset beta (unlevered beta) and then relever it for the target structure. The standard formula, with a tax rate t and debt-to-equity ratio D/E:

beta_equity = beta_asset × [1 + (1 − t) × D/E]

Take an asset beta of 0.9, a D/E of 0.5, and a 25% tax rate: 0.9 × [1 + 0.75 × 0.5] = 0.9 × 1.375 = 1.2375. Feeding that into CAPM at the same rates gives 4.2 + 1.2375 × 5.0 = 10.39%, about 46 basis points more than the unlevered version. That gap is the compensation shareholders demand for financial leverage, and ignoring it is one of the most common errors in comparable-company analysis - a peer group with a different debt load will show a different beta for reasons that have nothing to do with business risk.

Where Beta Fails

Beta is a single-factor, backward-looking regression, and it inherits every weakness of that description:

• It is unstable. Individual-company betas estimated on rolling five-year windows routinely move by 0.2 to 0.4. A 1.2 today might have been 0.85 three years ago without any change in the business.
• It is index-dependent. Measured against the S&P 500, a mid-cap stock shows a different beta than against the Russell 2000. Neither is wrong; they answer different questions.
• It assumes symmetric risk. Beta treats upside and downside volatility identically, which is not how investors experience risk. Downside-deviation measures such as the Sortino ratio address this, as covered in the Sharpe ratio walkthrough.
• It misses other priced factors. Size, value, profitability, and momentum all explain return differences that beta does not. Multifactor models exist precisely because single-factor CAPM leaves too much unexplained.
• It is measured with error in thin markets. A stock that trades a few thousand shares a day produces stale prices, which biases beta downward - the returns look smooth because they are not updating.

Reading a Quoted Beta

When a screener shows a beta of 0.92 next to a ticker, four things are usually hidden behind that number: the index it was measured against, the window (five years of monthly data is the most common default, two years of weekly data the second), whether it is an equity or asset beta, and when the window ended. A figure computed through a market crash can differ by 0.3 from the same calculation run a year later, because the crash dominates the variance in the denominator.

The practical response is not to distrust the number but to bracket it. Take the quoted beta, add and subtract 0.15, and see whether your conclusion survives both ends. If a valuation only works at the low end, the valuation is a leveraged bet on a regression coefficient - a thin foundation for a decision about a business. Where possible, cross-check against a peer median computed on the same window and index; when the two disagree sharply, the difference usually points to a capital-structure or business-mix change that the quoted figure has not caught up with yet.

A Practical Checklist

When you need a defensible beta rather than a quoted one, work through five steps. Choose the index that matches the opportunity set, and say which one. Use 60 monthly returns where available, because five years is the convention and shorter windows are noisier. Unlever the peer betas before averaging, then relever to your target capital structure. Sanity-check the result against the sector - a software company with a beta of 0.4 deserves a second look at your data. Finally, run the valuation at beta ± 0.2 and report the range: if a 10% change in the discount rate flips the conclusion, the conclusion was never about the business.

For how beta fits into a portfolio rather than a single valuation, the asset allocation guide covers how blended betas translate into expected drawdowns, and the bond price explainer covers the duration side of the same question. If your CAPM output feeds a multiple-based cross-check, the P/E ratio breakdown shows where the two methods agree and where they diverge.

Disclaimer

This article is educational. It explains arithmetic and a widely used model; it does not recommend any security, fund, or allocation, and it is not personalised financial advice. Expected returns produced by CAPM are assumptions dressed as numbers - the model is a tool for organising your reasoning, not a prediction. Consult a licensed adviser about your own situation, and treat every input you cannot observe, especially the equity risk premium, as a range rather than a point.