Capital Asset Pricing Model

Definition

The Capital Asset Pricing Model (CAPM) relates expected excess returns to a single market-wide risk factor: $E[r_i] - r_F = \beta_i(E[r_M] - r_F)$, where $\beta_i = \mathrm{Cov}(r_i, r_M)/\mathrm{Var}(r_M)$ is the security's systematic-risk loading. Developed by Sharpe (1964), Lintner (1965), and Mossin (1966) from Markowitz mean-variance portfolio theory; treated as standard received theory in the wiki sources.

Key Ideas

• CAPM alternative derivation of Black-Scholes: Because the option's instantaneous beta is $\beta_w = (xw_1/w)\beta_S$ (the product of the delta ratio and the stock beta), setting $\alpha_w - r = \beta_w(\alpha - r)$ and substituting into the expected return of the hedged portfolio yields the Black-Scholes Partial Differential Equation (PDE) exactly. Preference parameters cancel because beta scales continuously with delta — no continuous hedging argument is needed. See Black-Scholes Option Pricing (Black-Scholes 1973; Rubinstein 1976).

• Log-CAPM (Rubinstein 1976): Under Constant Proportional Risk Aversion (CPRA) with coefficient $b$ and joint lognormality of security and market returns, $\ln E(r_i) = \ln r_F + b \cdot \mathrm{Cov}(\ln r_i, \ln r_M)$. This is the CAPM in log-return space; $b$ can be estimated as $b = [E(\ln r_M) - \ln r_F] / \mathrm{Var}(\ln r_M)$.

• CPRA pricing kernel (Rubinstein 1976): The Stochastic Discount Factor (SDF) under CPRA is $Y_t = R_{Mt}^{-b}$. This makes CAPM a special case of the general SDF valuation $P_0 = E[X_t Y_t]/R_{Ft}$. See Stochastic Discount Factor.

• Intertemporal CAPM (ICAPM) (Merton 1973): Extends CAPM to a continuous-time setting where investors hedge against shifts in future investment opportunities. Expected returns depend on multiple betas (one per state variable) rather than only the market beta. Provides the preference-based foundation for the Black-Scholes extension to stochastic interest rates. See Merton (1973).

How It Works

In the standard single-period CAPM, all investors hold the mean-variance efficient frontier and the market portfolio is the tangency portfolio. In equilibrium, each asset's excess return is proportional to its covariance with the market. The Log-CAPM version replaces arithmetic returns with log-returns and holds exactly under bivariate lognormality + CPRA. The ICAPM adds state-variable hedging demands that generate additional risk premia beyond the market beta.

Why It Matters

CAPM provides the no-continuous-hedging derivation of Black-Scholes — showing the formula is robust to the specific mechanism used to eliminate arbitrage. The CPRA kernel $Y_t = R_{Mt}^{-b}$ is the discrete-time antecedent of modern SDF-based asset pricing (Hansen-Jagannathan bounds, stochastic discount factor tests). The ICAPM is the preference foundation for affine term structure models and intertemporal risk premia.

Open Questions

• Empirical failure of the static CAPM (Fama-French 1992 anomalies) motivates multifactor extensions; the wiki sources treat CAPM primarily as a theoretical tool rather than an empirical model.
• Time-varying betas via multivariate Generalized Autoregressive Conditional Heteroskedasticity (GARCH) are noted in the Nobel Committee (2003) survey but not developed in detail in the wiki.

Related

• Black-Scholes Option Pricing
• Stochastic Discount Factor
• Rational Option Pricing Bounds
• Equity Premium Puzzle
• Rubinstein (1976)
• Black-Scholes (1973)
• Merton (1973)