Capital Asset Pricing Model (CAPM)

Table of contents

CAPM Overview

  • The expected return on any asset $i$ is the risk-free interest rate, $R_F$, plus a risk premium, which is the asset's market beta, $\beta_{iM}$, times the premium per unit of beta risk, $E(R_M) - R_f$
  • $E(R_i) = R_f + \beta_{iM}(E(R_m) - R_f)$
    • Which can be rewritten to be expressed in terms of excess return of the asset
      • $E(R_i) - R_f = \beta_{iM}(E(R_m) - R_f)$
    • $\beta_{iM}$ for asset $i$ is the covariance of its return with the market return divided by the variance of the market return
    • (Market Beta) $\beta_{iM} = \frac{\text{cov}(R_i,R_M)}{\sigma^2(R_M)}$
    • Market beta is a risk premium
    • Assets with a market beta of zero must have an expected return equal to the risk free rate because they doesn't contribute to the variance in the market return

CAPM assumptions

  • Expected returns for each asset is linearly related to their market betas
  • Expected return on market portfolio > expected return on assets with market beta of 0
    • Market betas for all assets are positive
  • Assets uncorreleated with the market have expected returns equal $R_f$
  • The beta premium is equal to $E(R_m) - R_f$

Testing CAPM on market data

  • Regress average asset returns on estimates of asset betas
  • Model predicts the intercept is the riskfree rate
  • Coefficient on beta is $E(R_M)-R_f$

Issues that arise

  • Tough to estimate market beta for an individual asset, can vary depending on time range
    • Estimate beta for portfolios of assets instead
  • Residuals from the model for one asset are not independent from another's residuals
    • Caused by variation due to an attribute such as industry or location
In [1]:
import pandas as pd
import edhec_risk_kit as erk
import numpy as np
import statsmodels.api as sm
In [2]:
ind_rets = erk.get_ind_file('returns', n_inds=49)
fff = erk.get_fff_returns()

CAPM Example

What is the market beta for Healthcare industry stocks from 1970 to 2015?

In [3]:
hlth_excess = ind_rets.loc['1970':'2015', ['Hlth']] - fff.loc['1970':'2015', ['RF']].values
mkt_excess = fff.loc['1970':'2015', ['Mkt-RF']]
exp_var = mkt_excess.copy()
exp_var['Constant'] = 1
lm = sm.OLS(hlth_excess, exp_var).fit()
lm.summary()
Out[3]:
OLS Regression Results
Dep. Variable: Hlth R-squared: 0.390
Model: OLS Adj. R-squared: 0.389
Method: Least Squares F-statistic: 352.2
Date: Fri, 13 Mar 2020 Prob (F-statistic): 4.27e-61
Time: 15:18:56 Log-Likelihood: 732.28
No. Observations: 552 AIC: -1461.
Df Residuals: 550 BIC: -1452.
Df Model: 1
Covariance Type: nonrobust
coef std err t P>|t| [0.025 0.975]
Mkt-RF 1.1222 0.060 18.766 0.000 1.005 1.240
Constant -6.651e-06 0.003 -0.002 0.998 -0.005 0.005
Omnibus: 49.539 Durbin-Watson: 1.830
Prob(Omnibus): 0.000 Jarque-Bera (JB): 255.035
Skew: -0.107 Prob(JB): 4.17e-56
Kurtosis: 6.323 Cond. No. 21.8


Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Explore the coefficients in CAPM

The market beta for healthcare stocks is the coefficient on the Mkt-RF variable in the model. In this case, it is 1.1222. This tells us that the expected excess return for healthcare stocks is 1.1222 times the excess return of the market

Which industry had the highest market beta from 1960 to 2015?

  • Consider all 49 industries
In [4]:
industries = list(ind_rets.columns)
betas = []
for ind in industries:
    ind_excess = ind_rets.loc['1970':'2015 ', [ind]] - fff.loc['1970':'2015', ['RF']].values
    lm = sm.OLS(ind_excess, exp_var).fit()
    betas.append(lm.params['Mkt-RF'])

beta_df = pd.DataFrame({'Beta':betas},
                        index = industries)
In [5]:
print(beta_df['Beta'].idxmax(), 'had a market beta of', np.round(beta_df['Beta'].max(),2))

Softw had a market beta of 1.59

Lowest market beta?

In [6]:
print(beta_df['Beta'].idxmin(), 'had a market beta of', np.round(beta_df['Beta'].min(),2))

Util had a market beta of 0.52