From CAPM to Factors

For decades, finance had one tidy answer to the most important question in investing — why does this stock earn what it earns? The Capital Asset Pricing Model said: one number. Your stock’s expected return is entirely explained by how much it swings with the overall market. Defensive stock, low return. Wild stock, high return. One dial, called beta, sets the price of every asset on Earth. Elegant, Nobel-winning, and — as the data eventually screamed — not quite true.

This lesson is the autopsy. We’ll restate CAPM as the world’s simplest factor model, define precisely what a “factor” and a “risk premium” actually are, then walk through the four famous anomalies — size, value, momentum, and low-beta — that one stubborn beta could never explain. We end on the question that powers this entire course: when a stock beats its CAPM prediction, is that skill, a reward for hidden risk, or just a mistake the market hasn’t fixed yet? Spoiler: most of the time, alpha is hidden beta.

CAPM, rewritten as a one-factor model

Analogy. Imagine the entire stock market is a single tide. CAPM claims every boat — every stock — rises and falls only because of that one tide; the only thing distinguishing boats is how much each one bobs. A heavy barge (low beta) barely moves; a dinghy (high beta) lurches with every wave. Know the tide and know the boat’s bobbiness, and you know everything about its expected voyage. There are, CAPM insists, no other currents.

Definition. The Capital Asset Pricing Model says an asset’s expected return in excess of the risk-free rate equals its beta times the market’s expected excess return:

$E[R_i] - R_f = \beta_i \,\big(E[R_m] - R_f\big).$

Here $R_f$ is the risk-free rate (what cash earns safely), $E[R_m] - R_f$ is the market risk premium (the extra you expect for holding stocks over cash), and $\beta_i$ is the asset’s sensitivity to the market. Read it as: take the one reward the market offers, and scale it by how much of that risk you’re carrying.

The testable, empirical form. To actually check CAPM against data, you run a time-series regression of the asset’s excess returns on the market’s excess returns:

$R_i - R_f = \alpha_i + \beta_i \,(R_m - R_f) + \varepsilon_i.$

The slope is $\beta_i$, the noise $\varepsilon_i$ is the diversifiable wiggle CAPM says doesn’t matter, and the intercept $\alpha_i$ is alpha — the average return left over after the market explains what it can. CAPM makes one brutally sharp prediction: for every asset, $\alpha_i = 0$. Expected return is entirely market beta. Any persistent, non-zero alpha is CAPM admitting it missed something.

Beta, precisely. Beta isn’t a vibe; it’s a covariance ratio:

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

It measures how much the asset moves together with the market, normalized by how much the market moves on its own. A beta of 1 means it swings exactly with the market; 0.5, half as hard; 1.5, half again as hard.

The scatter below is that regression. Each dot is one period — market excess return on the X axis, the stock’s excess return on the Y axis. The fitted line’s slope is beta and its intercept is alpha. CAPM’s claim, in one image, is that the line should pass through the origin: $\alpha = 0$.

What exactly is a factor?

Analogy. Think of a portfolio’s return the way a nutritionist thinks of a meal. The total calories are the headline number, but what actually matters is the breakdown into macronutrients — protein, carbs, fat. A factor is a macronutrient of returns: a distinct, systematic source of reward that lots of assets share. CAPM is the claim that returns have exactly one macronutrient (the market). Factor models say: no, there are several, and a stock’s expected return depends on how much of each it contains.

Definition — factor. A factor is a systematic, non-diversifiable common source of return shared across many assets. “Systematic” means it affects whole swaths of the market at once, so you can’t diversify it away by holding more stocks. In modern practice a factor is usually itself a tradable long–short portfolio: you go long the assets with lots of the characteristic and short the assets with little of it, and the return of that portfolio is the factor. The market factor, for instance, is just “long stocks, short cash.”

Definition — factor loading (exposure). A loading is how much of a given factor an asset carries — the regression slope on that factor. It’s CAPM’s $\beta$, generalized: instead of one slope on one market, you get a slope on each factor. A stock might load +0.3 on the market, +0.8 on the value factor, and −0.2 on momentum. Loadings are the “how much of each macronutrient” numbers.

Definition — risk premium (factor premium). A factor premium is the expected excess return you earn per unit of loading on that factor — the reward the market pays for carrying that particular risk. The beautiful simplification for a tradable factor: its premium is just the mean return of the factor portfolio itself. If the long–short value portfolio has averaged 4% a year, the value premium is roughly 4%, and your expected reward is your value loading times that 4%.

Priced vs. unpriced. Not every wiggle earns a reward. Idiosyncratic risk — the part of a stock’s movement unique to that one company ($\varepsilon$ in the regression) — is diversifiable: hold enough names and it washes out. Because you can shed it for free, the market refuses to pay you for it. Only non-diversifiable (systematic) risk is priced. This is the deep logic of all asset pricing: you are paid for risk you cannot escape, never for risk you simply chose not to diversify.

The cracks: anomalies CAPM can’t explain

An anomaly is a pattern in average returns that CAPM swears shouldn’t exist — a way to reliably earn more (or less) than your beta predicts. By the 1980s and 1990s, the anomalies were piling up faster than CAPM could explain them away. Four became legendary.

Size (Banz, 1981). Sort stocks by market capitalization and the small ones have historically out-earned what their betas justify. Two firms with identical beta, the smaller one beats the larger — a flat contradiction of “beta is everything.” The long–short portfolio “small minus big” (SMB) captured this premium.

Value (Fama–French, 1992). Cheap stocks — high book-to-market ratio, meaning their accounting book value is large relative to their stock price — have beaten expensive “growth” stocks (low book-to-market, priced for dazzling futures). The long–short portfolio “high minus low” book-to-market (HML) earned a premium that beta couldn’t touch. Boring, cheap, unloved stocks quietly won.

Momentum (Jegadeesh–Titman, 1993). Stocks that won over the past 12 months tend to keep winning over the next few months; past losers keep losing. Buy recent winners, short recent losers, and you earn a premium — one that’s awkwardly hard to pin on any obvious risk story. Momentum is the anomaly that refuses to behave.

Low-beta (Black–Jensen–Scholes, 1972; Frazzini–Pedersen, 2014). The realized Security Market Line is much flatter than CAPM predicts. Low-beta stocks earn more per unit of risk than high-beta stocks — the exact opposite of “more beta, more reward.” Betting against beta (long low-beta, short high-beta, leverage-adjusted) has been a durable strategy.

The hammer blow. Fama and French’s 1992 paper showed that once you control for size and book-to-market, market beta has almost no explanatory power for the cross-section of average returns. The one number CAPM said was everything turned out to explain almost nothing on its own. Beta wasn’t wrong so much as woefully incomplete.

The flat SML

Analogy. CAPM promises a steep, fair staircase: every extra step of risk (beta) lifts you to a higher expected return. The low-beta anomaly says the real staircase is a gentle ramp — and near the high-risk end it barely rises at all. You climbed into the riskiest, most exciting stocks expecting a big reward and got handed a sloped sidewalk.

Definition. The low-beta anomaly is the empirical finding that realized average returns rise much less steeply with beta than CAPM’s Security Market Line predicts. Low-beta stocks deliver higher risk-adjusted returns than high-beta stocks; the realized line is flatter than the theoretical one, sometimes nearly flat. Leverage-constrained investors who can’t borrow tend to overpay for high-beta stocks (chasing return without leverage), bidding their prices up and their future returns down — Frazzini and Pedersen’s mechanism for why the line flattens.

The chart below plots the theoretical CAPM line. Now notice the example assets: the low-beta stock sits above the line (it earned more than CAPM said it should) while the high-beta stock sits below (it earned less). Connect the dots in your head and you get the flat empirical line — rising far less than the steep CAPM one beneath it.

Risk or mistake? Two stories for one premium

So a value stock beats its CAPM prediction year after year. Why? There are exactly two grown-up explanations, and the entire factor-investing debate — including whether your strategy will still work after you read about it — turns on which one is true.

Story 1 — rational / risk-based (Fama–French). The premium is real compensation for systematic risk. Cheap value stocks, the argument goes, are cheap because they’re genuinely scarier — they tend to do badly precisely in bad economic states (recessions, credit crunches) when you can least afford losses. You’re paid extra to hold something that hurts when everything else hurts too. If that’s the truth, the premium is a fair price for risk and it should persist indefinitely, because the risk it compensates never goes away.

Story 2 — behavioral / mispricing (Lakonishok–Shleifer–Vishny, 1994). The premium is an error. Investors systematically overpay for glamorous growth stories and underprice boring value names; their mistakes create the gap. Arbitrageurs who’d normally trade it away face limits to arbitrage (costs, career risk, the danger a cheap stock gets cheaper before it recovers), so the mispricing lingers — but it’s still a mistake. If that’s the truth, the premium should erode once enough investors learn about it and pile in.

Why this is the whole course in miniature. These two stories make opposite predictions about the future:

• If a premium is risk, publishing a famous paper about it changes nothing — risk is risk, and it survives publication.
• If a premium is a mistake, publishing the paper is the beginning of its end — capital floods in, the error gets corrected, and the premium decays.

That’s the test the final lesson of this course will run. And momentum is the awkward case that embarrasses both camps: it’s too strong and too persistent to be pure luck, but nobody has a clean risk story for why winners should keep winning — it smells behavioral, yet it has stubbornly refused to fully decay.

Worked example — alpha is hidden beta

Here’s the single calculation that is the spine of this entire course. Take a value stock and price it with CAPM, then compare to what it actually earned.

Setup. The risk-free rate is 3%. The market risk premium is 6%. Our value stock has a beta of 0.9. What does CAPM say it should return?

$E[R] = R_f + \beta \,(E[R_m] - R_f) = 3\% + 0.9 \times 6\%.$

Computing the beta term: $0.9 \times 6 = 5.4$, so CAPM’s prediction is $3 + 5.4 = 8.4$, i.e. 8.4%. But this stock — like value stocks historically — actually averaged 12%. The leftover is CAPM’s alpha:

$\alpha = 12\% - 8.4\% = 3.6\%.$

The twist that names the course. That +3.6% looks like alpha — magical, unexplained, skill-flavored outperformance. It isn’t. It’s the value premium showing up in a model too crude to see it. The stock didn’t have a brilliant manager; it had a value loading CAPM had no factor for. Add an HML (value) factor to the model and that 3.6% gets reabsorbed as $\text{value loading} \times \text{value premium}$ — and the “alpha” collapses back to roughly zero.

This is the thesis: alpha is hidden beta. Most apparent outperformance isn’t skill at all; it’s uncompensated exposure to a factor your model forgot to include. Build a richer factor model and yesterday’s alpha turns into today’s plainly-explained beta. Genuine, durable, can’t-be-explained-by-any-factor alpha is rare — and that scarcity is the most important fact in active management.

Putting it together

CAPM is the simplest possible factor model: a single factor — the market — whose loading is beta, with the sharp prediction that every asset’s alpha is zero. The data refused to comply. Size, value, momentum, and low-beta are durable patterns in average returns that one beta can’t explain; Fama–French (1992) showed that controlling for size and book-to-market leaves market beta nearly powerless. The fix is to recognize that returns have several macronutrients: a factor is a systematic, undiversifiable source of return (often a tradable long–short portfolio), a loading is how much of it you hold, and a premium is the reward per unit of loading. Idiosyncratic risk is diversifiable, so it’s never priced. Why premia exist splits into a rational/risk story (persists) and a behavioral/mistake story (decays) — the publication test that the final lesson will run. And the spine of it all: when a stock beats its CAPM prediction, that “alpha” is usually hidden beta — exposure to a factor your model forgot.

Next up, we’ll meet the workhorse factor models themselves — Fama–French three-factor and beyond — and start measuring loadings and premia for real, turning “alpha is hidden beta” from a slogan into a calculation.