CAPM


Kevin Crotty
BUSI 448: Investments

Where are we?

Last time:

  • Market Model Regressions
  • Alphas and Betas
  • Estimating Covariance Matrix

Today:

  • CAPM and the Market Model Regression
  • CAPM: Theory
  • CAPM: Practice

CAPM and the Market Model Regression

What is the CAPM?

  • The Capital Asset Pricing Model (CAPM) is a theory from the 1960s. Its discoverer won the Nobel prize in economics.
  • 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.

Capital Asset Pricing Model

\[E[r_i - r_f] = \beta_i \cdot E[r_m-r_f]\]

  • \(E[r_m-r_f]\) is the market risk premium
  • \(\beta_i = \frac{\text{cov}(r_i, r_m)}{\text{var}(r_m)}\)

Empirically, we estimate a market model regression:

\[ r_{i,t} - r_{f,t} = \alpha_i + \beta_i (r_{m,t} - r_{f,t}) + \varepsilon_{i,t} \]

  • What differs between the top and bottom equations?

Theory: CAPM

CAPM Assumptions

  • Investors have identical beliefs about the same universe of asset returns
  • Investors have mean-variance preferences
  • Single period investment horizon

CAPM Assumptions

  • Frictionless borrowing and lending
    • borrowing rate = savings rate
  • Frictionless trading
    • no transactions costs & no taxation & shorting allowed
  • Perfect competition: investors are price-takers

Equilibrium

  • All investors view the market portfolio as the tangency portfolio.
  • The capital allocation line with respect to the market portfolio is called the capital market line
  • Investors save or borrow at the risk-free rate to locate on the CML according to their risk aversion.
  • Prices will adjust so that the marginal benefit of an asset (its risk premium) is proportional to its marginal contribution to the risk of the market portfolio.

Deriving the CAPM (1/3)

Recall that the tangency portfolio in a frictionless setting satisfies:

\[\begin{align*} \sum_{i=1}^N \text{cov}[r_1,r_i] w_i &= \delta (E[r_1] - r_f) \\ \sum_{i=1}^N \text{cov}[r_2,r_i] w_i &= \delta (E[r_2] - r_f) \\ & \vdots \\ \sum_{i=1}^N \text{cov}[r_N,r_i] w_i &= \delta (E[r_N] - r_f) \end{align*}\] where \(\delta\) is a constant (it is a Lagrange multiplier from the optimization problem)

  • The LHS terms are the contributions of each asset to overall portfolio risk.
  • The RHS terms are proportional to each asset’s risk premium.

Deriving the CAPM (2/3)

  • Previously: we solved the system for weights
  • CAPM: solve for expected returns using market weights

For asset \(j\):

\[ \sum_{i=1}^N \text{cov}[r_j,r_i] w_i = \delta (E[r_j] - r_f) \]

Rearrange and use the fact that \(r_m = \sum_i w_i r_i\) to get:

\[ E[r_j - r_f] = \delta^{-1} \text{cov}[r_j,r_m] \]

Deriving the CAPM (3/3)

Using the definition of beta:

\[ E[r_j - r_f] = \delta^{-1} \beta_j \text{var}[r_m]\,. \]

Now aggregate this at market weights:

\[ \sum_j w_j \cdot E[r_j - r_f] = \delta^{-1}\text{var}[r_m] \sum_j w_j \cdot \beta_j \]

This implies \(\delta = \text{var}[r_m] / E[r_m - r_f]\), so we arrive at the CAPM formula:

\[ E[r_j - r_f] = \beta_j E[r_m - r_f] . \]

Intuition of the equilibium

  • The marginal benefit of an asset (its risk premium) is proportional to its marginal contribution to the risk of the market portfolio
  • The marginal contribution to risk is measured by beta.

What if this weren’t the case?

  • If an asset’s reward to risk contribution ratio is higher than ratios for other assets, what would you do?
    • Hold the asset at a greater weight, reducing weights in others.
    • But purchasing would push price up and return down until all investments had the same reward-to-risk-contribution ratio.

Practice: CAPM

CAPM and Corporate Finance

  • The CAPM is widely used to estimate expected returns to compute discount factors for corporate investment projects.
    • The return shareholders expect is \(r_f + \beta_i \cdot E[r_m-r_f]\).
    • This is the required return on equity capital for corporate projects.
    • aka cost of equity capital

Website

CAPM and Investments

  • The CAPM is somewhat less useful in an investments context.
    • What are the inputs for \(r_f\) and \(E[r_m-r_f]\)?
    • Estimating inputs can be too noisy
    • Doesn’t describe the cross-section of equity returns well

Estimating the market risk premium

  • Empirically, this is challenging.

  • An additional complication: the MRP is likely time-varying.

Historical average market risk premium

  • One option is to use the realized average: \[\frac{1}{T}\sum_t (r_{m,t}-r_{f,t})\] as an estimate of the expected market risk premium
  • Sample means are noisy estimates of population means
    • Need a large \(T\) sample
    • Precision of estimate doesn’t improve with sampling data more frequently.

Precision of historical average

  • Standard error = SD/\(\sqrt{T}\)
    • Annual SD of market return of 20%:
Years of Data Standard Error of Estimates
5 8.94%
10 6.32%
25 4.00%
50 2.83%
100 2.00%

Visualization

Security market line

The security market line is the visual representation of the CAPM and the cross-section of expected returns

The CAPM and cross-sectional data

  • The CAPM doesn’t fit realized returns in the cross-section of stocks very well.

  • Theoretically, the slope of the SML should be:

    • \(E[r_m-r_f]\)

  • Empirically, the slope is much flatter than the realized market risk premium.

Industry returns example

  • A simple example is industry returns.
  • Average returns are mostly unrelated to betas.

Website example

In-class notebook version

Let’s look at what this webpage is doing.

For next time: Predictability in the Cross-Section of Returns





BUSI 448