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

An equally weighted portfolio holds n securities, each with beta 1 and residual variance 0.0100, residuals uncorrelated, market variance 0.0225. Which is the portfolio variance for n = 10, and what happens as n grows?

The variance is 0.0235, tending to 0.0225 as n grows. Systematic variance of 0.0225 remains because portfolio beta is 1, while diversified specific variance falls as 0.0100 divided by n, reaching zero in the limit.

  1. A0.0235; variance tends to 0.0225 as n growsCorrect
  2. B0.0325; variance tends to 0 as n grows
  3. C0.0235; variance tends to 0 as n grows
  4. D0.0225; variance tends to 0.0100 as n grows
  5. 0.0010; variance tends to 0.0225 as n grows

Explanation

Portfolio beta = 1, so systematic variance = 0.0225. Residual variance = 0.0100/10 = 0.0010. Total = 0.0235. As n increases the specific component vanishes, leaving the non-diversifiable 0.0225.

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