Capital Asset Pricing Model





Kerry Back

What is the CAPM?

  • The Capital Asset Pricing Model (CAPM) is a theory from the 1960s. Its discoverer won the econ Nobel prize.
  • The intuition is:
    • Market risk is the biggest risk that a diversified investor faces.
    • The risk of each asset should be measured in terms of how much it contributes to market risk.
    • The risk premium of each asset should depend (linearly) on this measure of risk.

CAPM Formula

The CAPM formula is

\[\bar{r}_i - r_f = \beta_i(\bar{r}_m-r_f)\]

where \(r_m\) = market return and \(\beta_i\) is the slope of the regression line of \(r_i-r_f\) on \(r_m-r_f\).

Cost of equity capital

  • The CAPM is widely used to estimate expected returns to compute discount factors for corporate investment projects.
    • The return shareholders expect is \(r_f + \beta_i(\bar{r}_m - r_f)\).
    • This is the required return on equity capital for corporate projects.
  • The CAPM does not work well enough to use in an investments context.

Foundation of the theory

  • The hypothesis of the CAPM is that the market is a frontier portfolio, with borrowing rate = savings rate.
    • The rationale is that each investor chooses a frontier portfolio, which is the tangency portfolio with borrowing or saving.
    • So, the market is the tangency portfolio.
  • In the usual application in the U.S., the “market portfolio” is the U.S. stock market.

Deriving the CAPM

  • The equation for a frontier portfolio is \(kCw = \bar{r} - r_f1_n\).
  • If we imagine that we know \(C\) and \(w\), then this is an equation for the risk premia \(\bar{r}-r_f1_n\).
  • Specifying \(w\) = market, this equation is the CAPM.
  • The equation says that the marginal benefit of an asset (its risk premium) is proportional to its marginal contribution to the risk of the market portfolio.
  • This marginal contribution is measured by beta.

Beta

  • Beta is the slope in the regression:

\[r-r_{f}=\alpha + \beta(r_m-r_f)+e\]

  • where \(r_{m}\) = market return and \(e\) = zero-mean risk.
  • Beta answers this question:
    • if the market is up 1%, how much do we expect the asset to be up, all else equal?
    • e.g., \(\beta\) = 2 \(\Rightarrow\) we expect the asset to be up 2%.

The CAPM and the data

  • The CAPM doesn’t fit the data very well.
  • A simple example is industry returns. Average returns are mostly unrelated to betas.