The CAPM

The Capital Asset Pricing Model is one of the most-tested, most-rejected, and most-still-used ideas in finance. It says expected return on any asset is determined by its exposure to a single risk factor — the market portfolio. The empirical evidence is mixed at best, but CAPM remains the textbook bench from which every other asset pricing model is launched.

Assumptions

1. All investors are mean-variance optimisers.
2. Single risk-free rate, available to all, with no spread.
3. Homogeneous expectations: everyone agrees on μ and Σ.
4. No transaction costs, taxes, or short-selling constraints.
5. Asset returns are jointly normal (or quadratic utility).

Under these (manifestly false) assumptions, all investors hold the market portfolio (Sharpe-Lintner-Mossin tangency theorem) and the expected return of any asset is a linear function of its market beta.

The CAPM equation

E[R_i] - r_f = β_i (E[R_M] - r_f)
β_i = Cov(R_i, R_M) / Var(R_M)

β_i is the slope of a regression of asset i's excess returns on the market's excess returns. β = 1 means asset moves one-for-one with the market; β = 0 means the asset is uncorrelated with the market (and earns only the risk-free rate); β > 1 means the asset amplifies market moves.

The Security Market Line (SML)

Plot E[R_i] - r_f against β_i. Under CAPM, all assets should lie on a single straight line (the SML) with slope E[R_M] - r_f. Any asset above the line is undervalued (high excess return for its beta); below the line is overvalued. Active managers explicitly seek SML-deviating opportunities.

Implications

• Idiosyncratic risk is not priced. Only market risk (β) commands an expected return premium.
• Total risk ≠ priced risk: a high-σ stock with low β earns only the risk-free rate in expectation.
• The market portfolio is mean-variance efficient.
• Risk-adjusted alpha (excess return after subtracting β-times-market) is the active-manager's score.

Estimating beta

OLS regression: R_i,t - r_f,t = α_i + β_i (R_M,t - r_f,t) + ε_i,t. Typically use 3-5 years of monthly or 1-2 years of daily returns. Practical issues:

• Beta drifts: a firm's beta changes as its business mix changes.
• Beta-bias correction: Vasicek (1973) and Blume (1971) showed sample betas regress toward 1 over time; Bloomberg's 'adjusted beta' applies a 2/3 weighting to sample beta plus 1/3 to 1.
• Choice of market: in practice, the CAPM 'market' is approximated by a broad equity index. Domestic vs world, market-cap vs equal-weight all give different betas.

Empirical failures

• Low-beta anomaly: low-β stocks earn higher Sharpe than high-β stocks — opposite to CAPM (Frazzini-Pedersen 2014, 'Betting against Beta').
• Size effect (Banz 1981): small-cap stocks earn higher returns than CAPM predicts.
• Value effect (Fama-French 1992): high book-to-market stocks beat low B/M.
• Momentum (Jegadeesh-Titman 1993): past 6-12 month winners outperform losers.

Why CAPM survives anyway

• Pedagogically clean: it's the simplest asset pricing model with content.
• Capital budgeting: many corporations use CAPM to compute cost of equity for project evaluation.
• Benchmark: every alternative pricing model is defined relative to CAPM.
• Approximately right: CAPM beta still explains the largest single chunk of cross-sectional return variation.