Consider an economy with two risky assets: Asset A has an expected return of 15 percent and a standard deviation of 20 percent. Asset B has an expected return of 8 percent and a standard deviation of 12 percent. The correlation coefficient between assets A and B is -1. What is the standard deviation of the minimum variance portfolio?

Question

BlackTom AI · Accepted Answer

With correlation coefficient -1, the two assets are perfectly negatively correlated, so it is possible to offset one asset’s risk exactly with the other’s and achieve a portfolio with zero volatility.

Relevant concepts:
- Portfolio variance for two assets: σp^2 = w^2σA^2 + (1−w)^2σB^2 + 2w(1−w)Cov(A,B).
- Covariance = ρ·σA·σB. For ρ = −1, Cov = −σAσB and the variance reduces to a perfect square: σp^2 = (wσA − (1−w)σB)^2.
- Setting this expression to zero yields the weights that produce zero variance.

Step-by-step reasoning and calculations:
1. Given σA = 0.20, σB = 0.12 and ρ = −1, compute Cov:
   Cov = ρ·σA·σB = −1 · 0.20 · 0.12 = −0.024.
2. Write variance:
   σp^2 = w^2(0.20^2) + (1−w)^2(0.12^2) + 2w(1−w)(−0.024).
   Because ρ = −1 this simplifies algebraically to:
   σp^2 = (w·0.20 − (1−w)·0.12)^2.
3. Set σp^2 = 0 → w·0.20 = (1−w)·0.12.
   Solve: w(0.20 + 0.12) = 0.12 → w = 0.12 / 0.32 = 0.375 (37.5% in A, 62.5% in B).
4. Plugging these weights yields σp^2 = 0 → σp = 0.

Why the correct answer is correct:
Perfect negative correlation allows exact cancellation of risk when weights satisfy wσA = (1−w)σB, producing zero variance and hence zero standard deviation.

Why the other choices are incorrect:
- 12% is simply σB (all invested in B), not the minimum-variance combination.
- 16% might be confused with σA (20%) scaled or some naive weighted average; it is not the minimum.
- 5.5% appears to be an erroneous calculation (no algebraic result for ρ = −1 yields that value).

Conclusion: The standard deviation of the minimum-variance portfolio is zero, matching the provided answer "zero."

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