The Capital Asset Pricing Model

Investment Theory

Lorenzo Naranjo

Fall 2026

From Beta Pricing to the CAPM

  • Beta pricing holds for any mean-variance efficient portfolio Q: \operatorname{E}(r_{A}) = (1 - \beta_{A}) r_{f} + \beta_{A} \operatorname{E}(r_{Q}).
  • It does not say which portfolio is efficient.
  • Building Q directly needs \pmb{w}_{Q} \propto \pmb{\Sigma}^{-1}(\pmb{\mu} - r_{f}\mathbf{1}), and both inputs are hard to estimate.
  • The CAPM supplies the missing link: in equilibrium, Q is the market portfolio M.
    • Market weights need no estimates of expected returns or covariances.
    • In practice, broad value-weighted indexes proxy for M.

The Assumptions

  • Investors evaluate portfolios by their means and variances.
  • They share the same expected returns and covariance matrix (homogeneous expectations).
  • They trade the same assets without frictions and borrow or lend at a common risk-free rate.
  • Markets clear.

Two-Fund Separation

  • An investor with risk aversion A holds weight w^{*} = \frac{\mu_{Q} - r_{f}}{A \sigma_{Q}^{2}} in the tangency portfolio Q.
  • With homogeneous expectations, every investor faces the same frontier and picks the same Q.
  • More risk-averse investors lend and hold less of Q; less risk-averse investors borrow and hold more.
  • Either way, they all hold the risky assets in Q’s proportions.

Two-Fund Separation in a Picture

A capital allocation line from the risk-free rate through the tangency portfolio Q, with three points marked on it representing three investors with different weights in Q, one close to the risk-free rate, one near Q, and one beyond Q on the leveraged portion of the line. A dashed parabola is tangent to the line at each investor's point, representing that investor's indifference curve, with the most risk-averse investor's curve being the most sharply curved.

Figure 1: Three investors hold Q in different amounts, each at the point where an indifference curve (dashed) is tangent to the CAL.

Aggregation and Market Clearing

  • Let q_{j} be asset j’s weight in Q and W_{i} investor i’s wealth. Aggregate demand for asset j is q_{j} \sum_{i} w_{i}^{*} W_{i}.
    • The sum is the same for every asset, so aggregate risky demand has Q’s composition.
  • Market clearing: investors must hold the outstanding supply, whose weights are V_{j}/\sum_{k} V_{k}, the weights of the market portfolio M.
  • Therefore Q = M.
    • The conclusion concerns relative weights within the risky portfolio; the risk-free asset need not be in zero net supply.

When the Argument Fails

  • If investors disagree about expected returns or covariances, face different borrowing rates, or cannot trade the same assets, they need not choose the same tangency portfolio.
  • Two-fund separation then fails at the aggregate level.
  • Market clearing no longer implies that M is mean-variance efficient.

Property 1 (The Capital Asset Pricing Model) If the assumptions of the CAPM hold, the market portfolio M is mean-variance efficient. Substituting Q = M into the beta pricing formula, the expected excess return of any asset A is proportional to its market beta: \operatorname{E}(r_{A}) - r_{f} = \beta_{A} \bigl[\operatorname{E}(r_{M}) - r_{f}\bigr], \tag{1} where \beta_{A} = \frac{\operatorname{Cov}(r_{A}, r_{M})}{\operatorname{V}(r_{M})}.

Reading the CAPM

  • The market risk premium \operatorname{E}(r_{M}) - r_{f} is the expected return investors require for bearing one unit of market risk.
  • An asset with \beta_{A} = 1 has the same expected excess return as the market; one with \beta_{A} < 1 has a smaller one.
  • Expected returns are determined entirely by exposure to market risk.
    • Idiosyncratic volatility does not enter because it can be diversified away.

Example 1 (Computing a Stock Beta) Suppose that the correlation between stock A and the market is 0.6. If the standard deviation of A returns is 40% per year, and the standard deviation of the market is 20% per year, the beta of A is \begin{aligned} \beta_{A} & = \frac{\operatorname{Cov}(r_{A}, r_{M})}{\operatorname{V}(r_{M})} = \frac{\sigma_{A} \sigma_{M} \rho_{A,M}}{\sigma_{M}^{2}} \\ & = \frac{\sigma_{A} \rho_{A,M}}{\sigma_{M}} = \frac{0.4 \times 0.6}{0.2} = 1.2. \end{aligned}

Example 2 (Computing an Expected Return) If the risk-free rate is 5%, \beta_{DELL} = 1.3, and \operatorname{E}(r_{M}) = 14\%, then the CAPM predicts that: \operatorname{E}(r_{DELL}) = 0.05 + 1.3 \times (0.14 - 0.05) = 16.7\%. The 16.7% is Dell’s expected annual total return, including both price changes and distributions such as dividends. It is an average required return, not a guarantee of the return that will be realized next year.

Systematic and Firm-Specific Risk

  • Under the CAPM, the return of any asset can be decomposed into a systematic and a firm-specific part, r_{A} - r_{f} = \beta_{A} (r_{M} - r_{f}) + \varepsilon_{A}, \tag{2} where \operatorname{E}(\varepsilon_{A}) = 0 and \operatorname{Cov}(\varepsilon_{A}, r_{M}) = 0.
  • Because the two parts are uncorrelated, the total variance splits as \sigma_{A}^{2} = \beta_{A}^{2} \sigma_{M}^{2} + \sigma^{2}(\varepsilon_{A}). \tag{3}
  • Systematic variance \beta_{A}^{2} \sigma_{M}^{2} remains in a diversified portfolio; firm-specific variance \sigma^{2}(\varepsilon_{A}) can be diversified away.
  • Only \beta_{A} determines expected return.

Example 3 (Computing the Residual Risk) Suppose that stock A has a standard deviation of returns of 40% per year and a market beta of 1.1. The standard deviation of the market return is 20%.

This means that the residual variance is \sigma^{2}(\varepsilon_{A}) = \sigma_{A}^{2} - \beta_{A}^{2} \sigma_{M}^{2} = 0.40^{2} - 1.1^{2} \times 0.20^{2} = 0.1116. Therefore, the standard deviation of the firm-specific risk is \sqrt{0.1116} = 33.41\%.

Interpreting Empirical Evidence

  • The CAPM is a statement about expected returns, which cannot be observed directly.
    • Realized returns vary around their expectations, so a stock can beat or miss the CAPM prediction in a given year without violating the model.
    • Tests use many realized returns to estimate whether average returns line up with beta.
  • A persistent deviation has several possible explanations:
    • The asset is mispriced.
    • The CAPM’s assumptions fail, so the market portfolio is not mean-variance efficient.
    • The index used in the test is a poor proxy for the theoretical market portfolio, which cannot be observed (Roll, 1977).

The Joint-Hypothesis Problem

  • Every empirical test of market efficiency is also a test of the asset-pricing model used to define the correct expected return.
  • Rejecting the CAPM relation therefore rejects the combined hypothesis, not necessarily market efficiency alone.