L13: Cost of equity

Cost of equity using CAPM

Under the CAPM assumptions, we have:

\[E[R_{i}] = R_{f} + \beta_i (E[R_{m}] - R_{f}) \]

Therefore, to estimate the cost of equity for any firm \(i\), using the CAPM we need:

  • The current risk free rate
    • It is best to try to match the horizon of the risk-free rate with the horizon over which the cost of equity applies
      • Example: if you want to calculate the cost of equity over the next year, use the yield on a 1-year T-bill as the risk-free rate
      • Example: if you want to calculate the cost of equity over the next 10 years, use the yield on a 10-year T-bill as the risk-free rate
  • The firm’s market beta
    • Regress past excess returns of the firm on past excess returns on the market
  • An estimate for the market risk premium
    • Take an average of excess returns on the market (use at least 20 years of data)

The Fama-French three factor model

The Fama-French three factor model adds two more systematic risk factors in addition to the market return from the single-factor model:

\[R_{i,t} - R_{f,t} = \alpha_i + \beta_{i,m} (R_{m,t} - R_{f,t}) + \beta_{i,smb} R_{smb} + \beta_{i,hml} R_{hml} + \epsilon_{i,t}\]

where: - \(R_{smb}\) is the return on a portfolio that is long on the smallest firms in the economy and short on the largest firms - \(R_{hml}\) is the return on a portfolio that is long on the firms with highest B/M ratios and short on the firms with the lowest B/M ratios - B/M = book value of equity divided by market value of equity - High B/M firms are called “value” firms - Low B/M firms are called “growth” firms

Cost of equity using the Fama-French three factor model

Under the Fama-French model, we have:

\[E[R_{i}] = R_{f} + \beta_{i,m} (E[R_{m}] - R_{f}) + \beta_{i,smb} E[R_{smb}] + \beta_{i,hml} E[R_{hml}] \]

Therefore, to estimate the cost of equity for any firm \(i\), using the Fama-French factor we also need

  • The firm’s market, SMB, and HML betas
    • Regress past excess returns of the firm on past excess returns on the market and the returns on the SMB and HML factors
  • An estimate for the SMB and HML risk premia
    • Take an average of the returns of the SMB and HML factors (use at least 20 years of data)

Application

Calculate the cost of equity for AAPL using both the CAPM and the Fama-French three factor model. Use use monthly data since 2010 for AAPL.

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Application

Recalculate the risk premia of all three risk factors using only data since 1990. Now recalculate the cost of equity of AAPL using these new risk-premia estimates.

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