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FRM Part I · FRM Exam Part I · Modern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM)

In the single-index model, the excess return of stock i is written as R_i = alpha_i + beta_i * R_M + e_i, where all returns are in excess of the risk-free rate. Which statement about the residual term e_i is correct under the model's assumptions?

The residual in the single-index model has zero mean and is uncorrelated with the market return and with other stocks' residuals. It represents firm-specific risk, so all co-movement between stocks arises through their common exposure to the market index.

  1. AIt has zero expected value and is uncorrelated with the market excess return and with the residuals of other stocksCorrect
  2. BIt has zero expected value and is perfectly correlated with the residuals of stocks in the same industry
  3. CIt equals the market risk premium multiplied by beta
  4. DIt is the part of return explained by market movements and is therefore systematic

Explanation

The single-index model assumes the residual is firm-specific noise with mean zero, uncorrelated with the market factor and with other firms' residuals. This is what lets all covariance between stocks come through the market. Correlated residuals would require additional factors, so that option contradicts the model.

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