As diversification increases, the standard deviation of a portfolio approaches

Question

BlackTom AI · Accepted Answer

In a single-factor (CAPM) representation, each asset return can be written as Ri = αi + βiRm + εi, where Rm is the market return and εi is the idiosyncratic (firm-specific) shock with E[εi]=0 and uncorrelated across assets.

1) Relevant knowledge points and concepts
- Total portfolio risk (variance) decomposes into systematic (market) risk and idiosyncratic risk.
- Diversification reduces idiosyncratic risk but cannot eliminate systematic (market) risk.
- For a well-diversified portfolio the idiosyncratic component tends to zero; the remaining variance is the market-driven part.

2) Step-by-step reasoning
- Portfolio return Rp = Σ wi Ri = Σ wi(αi + βiRm + εi) = αp + βpRm + εp, where βp = Σ wi βi and εp = Σ wi εi.
- Portfolio variance Var(Rp) = βp^2 Var(Rm) + Var(εp) + 2βp Cov(Rm, εp). Since εi are uncorrelated with Rm, Cov(Rm, εp)=0, so Var(Rp) = βp^2 Var(Rm) + Var(εp).

3) Intermediate/calculation steps
- If idiosyncratic variances are finite and εi are uncorrelated, Var(εp) = Σ wi^2 Var(εi). As the number of assets increases and weights are spread (e.g., equal weights), Σ wi^2 → 0, so Var(εp) → 0.
- Thus as diversification increases: Var(Rp) → βp^2 Var(Rm). A perfectly diversified portfolio that represents the market has βp = 1, so Var(Rp) → Var(Rm) and the standard deviation → σm (the market portfolio’s standard deviation).

4) Why the correct answer is correct
- Diversification eliminates only idiosyncratic risk; systematic market risk remains. The limit of portfolio standard deviation when fully diversified equals the market portfolio’s standard deviation.

5) Why the other options are incorrect
- 0: would imply elimination of market risk too, which diversification cannot achieve.
- 1: is a unitless number not meaningfully the portfolio SD.
- Infinity / the maximum: diversification reduces risk, so SD does not blow up to infinity or necessarily reach a “maximum.”

6) Conclusion
Therefore, the portfolio standard deviation approaches the standard deviation of the market portfolio, matching the provided correct answer.

As diversification increases, the standard deviation of a portfolio approaches单项选择题

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