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IAI Actuarial Core Principles · Economic Modelling · Single and multifactor models for investment returns

An analyst uses a single-index model, R_i = a_i + b_i R_M + e_i, to split the risk of a portfolio of Indian equities. Which statement about the model's use in portfolio construction is correct?

Total security variance splits into b squared times market variance plus residual variance. Because residuals are assumed uncorrelated across securities, diversification reduces the specific component, while the systematic part driven by beta stays in the portfolio.

  1. ASpecific risk of a well-diversified portfolio remains large because residuals e_i are perfectly correlated across securities
  2. BTotal variance of a security equals b_i squared times market variance plus the variance of e_i, and diversification mainly reduces the second componentCorrect
  3. CBeta measures the security's specific risk and so can be eliminated by holding many securities
  4. DThe model requires estimating a full covariance matrix of all securities directly, with no simplification
  5. The model assumes that residual terms of different securities are positively correlated with the market return

Explanation

In the single-index model the variance is systematic (b² σ_M²) plus specific variance σ_e². Residuals are assumed uncorrelated across securities, so diversification averages away specific risk but not systematic risk. Beta measures systematic risk, not specific risk.

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