Market Model Regression


Kevin Crotty
BUSI 448: Investments

Where are we?

Last time:

  • Input sensitivity

Today:

  • Market Model Regressions
  • Alphas and Betas
  • Estimating the Covariance Matrix
  • Estimation Error

Single Benchmark Models

Benchmark Regression

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

  • Regress stock excess returns on benchmark excess returns
  • \(\beta_i = \frac{\text{cov}(r_i-r_f, r_b-r_f)}{\text{var}(r_b-r_f)}\)
  • Most common benchmark is a market return
    • CRSP value-weighted market, S&P 500
    • I’ll refer to this as the market model

Understanding the Market Model Regression

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

  • Meaning of \(\alpha\)?
  • Meaning of \(\beta\)?
  • Meaning of \(\varepsilon\)?

Visualization

Meaning of \(\beta\)

Beta answers the question:

if the benchmark is up 1%, how much do we expect the asset to be up, all else equal?

  • If \(\beta\)=2, we expect the asset to be up 2%
  • If \(\beta\)=0.5, we expect the asset to be up 0.5%

Meaning of \(\alpha\)

Alpha answers the question:

if I were holding the market, could I have improved mean-variance efficiency by investing something in the asset?

  • The answer is “yes” if and only if \(\alpha\) > 0
  • If \(\alpha\) < 0, mean-variance efficiency could have been improved by shorting the asset.

Visualization

A warning

  • Alphas with respect to a benchmark regression are backward-looking.
  • We should only use them for forming portfolios if we believe that the alpha will persist!

Estimating Covariances

Number of Parameters

How many parameters do we need to estimate for an \(N\) asset covariance matrix?

\[\begin{equation*} \begin{bmatrix} \text{var}[r_1] & \text{cov}[r_1,r_2] & \dots & \text{cov}[r_1,r_N] \\ \text{cov}[r_2,r_1] & \text{var}[r_2] & \dots & \text{cov}[r_2,r_N] \\ \vdots & \vdots & \ddots & \vdots \\ \text{cov}[r_N,r_1] & \text{cov}[r_N,r_2] & \dots & \text{var}[r_N] \\ \end{bmatrix} \end{equation*}\]

How many variance terms?

\[N\]

How many distinct covariance terms?

\[ \frac{N^2-N}{2} \]

Curse of Dimensionality

N(Assets) N(Parameters)
5 15
10 55
25 325
50 1,275
100 5,050
  • A great deal of estimation risk with 5,000 parameters to estimate!

Market Model-Implied Covariances

Under the market model, what is the covariance of two assets \(i\) and \(j\), \(\text{cov}(r_i,r_j)\)?

\[\text{cov}(\alpha_i + \beta_i (r_{m} - r_{f}) + \varepsilon_{i}, \alpha_j + \beta_j (r_{m} - r_{f}) + \varepsilon_{j})\]

  • The alphas are constant, so we can ignore them.
  • If we are willing to assume that \(\varepsilon_i\) is uncorrelated with \(\varepsilon_j\), the covariance reduces to:

\[ \beta_i \beta_j \text{var}(r_m-r_f) \]

Market Model-Implied Variances

For variance terms, we definitely should not ignore the residual variance:

\[ \text{var}(r_i) = \beta_i^2 \text{var}(r_m) + \text{var}(\varepsilon_i) \]

Alternatively, we can just estimate the stock-specific variance directly.

Reduced parameter dimensionality

  • Pairwise \(\rho\): \(\frac{N^2-N}{2}\) correlations, \(N\) variances
  • Market Model: \(N\) betas, \(N\) variances, \(1\) mkt variance
N(Assets) Pairwise \(\rho\) N(Parameters) Market Model N(Parameters)
5 15 11
10 55 21
50 1,275 101
100 5,050 201

Industry Portfolios

  • Notebook #1: Estimate betas for industry portfolios and calculate market model-implied covariance matrix
  • Notebook #2: Backtest performance of using the market model-implied covariance matrix for industry portfolios

Persistence of \(\beta\) (and \(\alpha\))

Estimation error

  • Alpha and beta are estimates, so will be subject to the usual concerns about estimation error.

Shrinking betas

  • On average, what value should beta have?
  • A simple way to deal with estimation error is to shrink betas towards 1.

\[ \beta_{\text{adjusted}} = 0.67 \cdot \beta_{\text{adjusted}} + 0.33 \cdot 1 \]

  • Many fancier alternatives exist.

Let’s return to notebook #1 and consider how well shrinking betas performs for our industry portfolios.

For next time: “Capital Asset Pricing Model”





BUSI 448