Other things equal, diversification is most effective when

Question

BlackTom AI · Accepted Answer

Portfolio variance for two assets is directly affected by the correlation between their returns, so negative correlation gives the greatest diversification benefit.

1) Relevant knowledge points and concepts
- Diversification reduces portfolio risk by combining assets whose returns do not move perfectly together.
- Covariance and correlation measure how returns move together: correlation ρ ranges from -1 to +1.
- Two-asset portfolio variance: Var(P) = w1^2σ1^2 + w2^2σ2^2 + 2w1w2σ1σ2ρ12.
- When ρ is lower (more negative), the third term becomes negative and offsets individual variances, lowering overall variance. If ρ = -1 and weights chosen appropriately, risk can be completely eliminated.

2) Step-by-step reasoning
- Use the two-asset variance formula to see the effect of ρ.
- As ρ decreases from +1 → 0 → -1, the covariance term 2w1w2σ1σ2ρ moves from positive (adding risk) to zero (neutral) to negative (subtracting risk), so portfolio variance falls.

3) Numerical intermediate steps (example)
Assume σ1 = σ2 = 10% = 0.10 and equal weights w1 = w2 = 0.5.
- Var(P) = 0.5^2(0.10)^2 + 0.5^2(0.10)^2 + 2(0.5)(0.5)(0.10)(0.10)ρ
- Simplify: Var(P) = 0.005(1 + ρ)
Evaluate:
- ρ = +1: Var = 0.005(2) = 0.01 → SD = 10% (no diversification benefit)
- ρ = 0: Var = 0.005(1) = 0.005 → SD ≈ 7.07% (reduced risk)
- ρ = -1: Var = 0.005(0) = 0 → SD = 0% (perfect risk elimination)

4) Why the correct answer is correct
Negative correlation makes returns move in opposite directions, so losses in one security are offset by gains in another, reducing portfolio variance. The formula and numeric example show that the more negative ρ is, the lower the portfolio variance — with ρ = -1 producing maximum diversification (perfect hedging).

5) Why other options are incorrect
- Uncorrelated (ρ = 0): provides diversification but less effective than negative correlation.
- Positively correlated (ρ > 0): reduces or eliminates diversification benefits; if ρ = +1 there is no benefit.
- Securities' returns are high: return level does not determine diversification effectiveness; correlation does.
- Positively correlated and high: positive correlation still undermines diversification regardless of return magnitude.

6) Conclusion
Because negative correlation minimizes portfolio variance (and can eliminate risk when perfectly negative), the most effective diversification occurs when securities' returns are negatively correlated. This matches the provided answer.

Other things equal, diversification is most effective when单项选择题

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