As diversification increases, the firm-specific risk of a portfolio approaches

Question

BlackTom AI · Accepted Answer

Idiosyncratic (firm-specific) risk can be diversified away, so it approaches 0 as the number of independent holdings grows.

1) Relevant concepts:
- Total risk = systematic (market) risk + firm-specific (idiosyncratic) risk.
- Systematic risk cannot be eliminated by diversification; firm-specific risk can, if idiosyncratic shocks across firms are uncorrelated.
- For an equally weighted portfolio, the idiosyncratic component of variance falls with portfolio size.

2) Step-by-step reasoning:
Model each asset return as Ri = αi + βiRm + εi, where εi is the firm-specific shock with Var(εi) = σe^2 and Cov(εi, εj) = 0 for i ≠ j. For an equally weighted portfolio of n assets, the portfolio idiosyncratic term is the average of the εi’s scaled by 1/n.

3) Calculations (all intermediate steps):
- Portfolio idiosyncratic variance = Var((1/n) Σi εi) = (1/n^2) Σi Var(εi) because cross-terms are zero
- = (1/n^2) × n × σe^2 = σe^2 / n
- As n → ∞, σe^2 / n → 0.

4) Why the correct answer is correct:
The math shows the firm-specific variance declines proportionally to 1/n and therefore tends to zero as diversification (n) increases without bound. That means the only remaining risk in a very large, diversified portfolio is systematic risk, not firm-specific risk.

5) Why other options are incorrect:
- 1: No theoretical basis; idiosyncratic risk scales as σe^2/n, not a constant 1.
- infinity or (n − 1) × n: these grow with n, but firm-specific risk decreases with n, so these are opposite.
- the maximum: diversification reduces idiosyncratic risk, so it cannot approach a maximum value.

6) Conclusion:
Therefore the firm-specific risk of a portfolio approaches 0, which matches the provided correct answer.

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