The Capital Asset Pricing Model (CAPM)

The CAPM is the first and most influential answer to the question what determines an asset's expected return? Its verdict is startling in its economy: only one thing is priced, an asset's covariance with the market. Idiosyncratic risk earns nothing because it can be diversified away. The model is empirically flawed, yet it remains the scaffolding for beta, the cost of capital, performance attribution, and every multi-factor model that followed. You cannot understand Factor Risk Models without first understanding what CAPM got right and what it got wrong.

From the frontier to equilibrium

Start where the The Efficient Frontier ended: with a risk-free asset, every investor holds the same tangency portfolio, levered up or down. If all investors share the same estimates of $(\mu,\Sigma)$, the homogeneous-expectations assumption, then in aggregate the tangency portfolio they collectively hold must be the market portfolio $M$, the cap-weighted basket of all assets (markets clear; someone owns every share). So the market portfolio is mean-variance efficient. That single fact prices everything.

Deriving the security market line

Consider mixing a fraction $\alpha$ of asset $i$ with $(1-\alpha)$ of the market. The mix has

$\mu(\alpha) = \alpha\mu_i + (1-\alpha)\mu_M, \qquad \sigma(\alpha) = \sqrt{\alpha^2\sigma_i^2 + 2\alpha(1-\alpha)\sigma_{iM} + (1-\alpha)^2\sigma_M^2}.$

Because $M$ is efficient, the curve traced by these mixes must be tangent to the capital market line at $\alpha = 0$, otherwise you could improve on $M$, contradicting its efficiency. Setting the slope $d\mu/d\sigma$ at $\alpha=0$ equal to the CML slope $(\mu_M - r_f)/\sigma_M$ and simplifying (the algebra hinges on $d\sigma/d\alpha|_0 = (\sigma_{iM}-\sigma_M^2)/\sigma_M$) yields the CAPM pricing equation:

$\boxed{\;\mathbb{E}[R_i] - r_f = \beta_i\big(\mathbb{E}[R_M] - r_f\big), \qquad \beta_i = \frac{\operatorname{Cov}(R_i, R_M)}{\operatorname{Var}(R_M)}.\;}$

Expected excess return is linear in beta. Plotted with $\beta$ on the x-axis, this line is the security market line (SML): intercept $r_f$, slope equal to the market risk premium $\mathbb{E}[R_M]-r_f$. Every correctly priced asset lies on it.

What beta means

$\beta_i$ is the regression slope of asset $i$'s excess returns on the market's, exactly the Ordinary Least Squares (OLS) slope $\operatorname{Cov}(R_i,R_M)/\operatorname{Var}(R_M)$ (see Beta (β)). Decompose total risk:

$\sigma_i^2 = \underbrace{\beta_i^2\sigma_M^2}_{\text{systematic}} + \underbrace{\sigma_{\varepsilon_i}^2}_{\text{idiosyncratic}}.$

CAPM's core message is that only the systematic part is compensated. The idiosyncratic term $\sigma_{\varepsilon_i}^2$ is diversifiable and earns zero premium; bearing it is uncompensated risk. Assets above the SML have positive Alpha (α) (cheap, higher return than beta justifies); below the SML, negative alpha.

Assumptions

CAPM is a chain of idealizations: (1) investors are mean-variance optimizers over a single period; (2) homogeneous expectations, everyone agrees on $\mu,\Sigma$; (3) frictionless markets, no taxes, no transaction costs, unlimited borrowing and lending at a single $r_f$; (4) all assets are traded and infinitely divisible; (5) no investor is large enough to move prices. Each is false in reality, and relaxing them motivates the model's successors, the zero-beta CAPM (Black 1972) drops the risk-free asset, the ICAPM and consumption-CAPM add state variables and intertemporal hedging.

Worked example

A stock has $\beta = 1.3$. The risk-free rate is $r_f = 3\%$ and the equity risk premium is $\mathbb{E}[R_M]-r_f = 5\%$. CAPM's required return is

$\mathbb{E}[R_i] = 3\% + 1.3\times 5\% = 9.5\%.$

Suppose the stock actually delivers $11\%$. The $1.5\%$ excess over the CAPM benchmark is Jensen's alpha, the point of a factor regression $R_i - r_f = \alpha + \beta(R_M - r_f) + \varepsilon$ is to test whether $\alpha$ is statistically distinguishable from zero. If the stock's total volatility is $30\%$ and the market's is $18\%$, then systematic vol is $\beta\sigma_M = 1.3\times 18\% = 23.4\%$, leaving idiosyncratic vol $\sqrt{0.30^2 - 0.234^2} \approx 18.8\%$, over which CAPM says the investor earns nothing.

Empirical failures

CAPM is one of the most-tested and most-rejected models in finance:

• The SML is too flat. Empirically, high-beta stocks earn less than CAPM predicts and low-beta stocks earn more, the "low-volatility anomaly," exploited by betting-against-beta strategies. The realized slope is well below the market premium.
• Size and value. Small-cap and high book-to-market stocks earn returns far above their betas (Fama–French 1992), motivating the three- and five-factor models. Beta alone explains little of the cross-section.
• Momentum. Past winners keep winning, a return pattern orthogonal to beta and unexplained by CAPM (see Momentum).
• Roll's critique. The true market portfolio (all wealth: equities, bonds, real estate, human capital) is unobservable. Every test uses a proxy, so CAPM is arguably untestable, a rejection may just mean the proxy is inefficient.

These failures did not discard CAPM; they generalized it. The insight "expected return equals exposures times factor premia" survives, CAPM is simply the one-factor special case of the arbitrage pricing theory and modern Factor Risk Models.

Failure modes in practice

• Unstable beta. Estimated betas drift across regimes and depend on the window and benchmark; shrink them toward 1 (Blume/Vasicek adjustment).
• Wrong benchmark. Beta to the S&P is meaningless for a market-neutral book; match the benchmark to the exposure.
• Single factor. Real returns load on many factors; a one-factor alpha is often just an unmodeled factor exposure in disguise.

In interviews

Know the punchline cold, $\mathbb{E}[R_i]-r_f = \beta_i(\mathbb{E}[R_M]-r_f)$, and be able to justify it: the market is the tangency portfolio, so nothing can beat its Sharpe, which forces every asset onto the SML. Explain the systematic/idiosyncratic split and why only systematic risk is priced (idiosyncratic risk is diversifiable). Expect the question "why does CAPM fail empirically?", name the flat SML, size/value/momentum, and Roll's critique. A sharp follow-up connects it forward: "how is CAPM related to multi-factor models?", it is the one-factor case. See Alpha (α), Beta (β), and Sharpe Ratio.