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

An equally weighted portfolio holds N stocks, each with beta 1.0 and residual variance 0.0400. The market variance is 0.0225. Residuals are uncorrelated. What is the smallest N for which the portfolio's residual standard deviation is at most 2.0%?

The minimum is 100 stocks. Portfolio residual variance equals single-stock residual variance divided by N, or 0.04/N. A 2% standard deviation means variance of 0.0004, so N must be at least 100. Market variance does not affect residual risk.

  1. A100Correct
  2. B75
  3. C25
  4. D50

Explanation

Portfolio residual variance = 0.04/N. Requirement: sd at most 0.02 means variance at most 0.0004, so N is at least 0.04/0.0004 = 100. At N=50 variance is 0.0008 (sd 2.83%), too high. The market variance is irrelevant to the residual.

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