How do you calculate the minimum variance portfolio weights for two risky assets?

Question

AcadiFi · Accepted Answer

The minimum variance portfolio (MVP) is the portfolio on the efficient frontier with the **lowest possible volatility**. For two risky assets, there is an elegant closed-form solution.

**The Formula**

For two assets A and B with standard deviations sigma_A and sigma_B, and correlation rho_AB:

**w_A = (sigma_B^2 - rho_AB * sigma_A * sigma_B) / (sigma_A^2 + sigma_B^2 - 2 * rho_AB * sigma_A * sigma_B)**

**w_B = 1 - w_A**

**Worked Example — Kensington Two-Asset Portfolio:**

Asset A (Oakmont Growth Fund): Expected return = 12%, sigma = 20%
Asset B (Silverline Bond Fund): Expected return = 6%, sigma = 10%
Correlation: rho_AB = 0.30

**Step-by-step calculation:**

| Component | Calculation | Value |
|---|---|---|
| sigma_A^2 | 0.20^2 | 0.0400 |
| sigma_B^2 | 0.10^2 | 0.0100 |
| rho * sigma_A * sigma_B | 0.30 × 0.20 × 0.10 | 0.0060 |
| Numerator | 0.0100 − 0.0060 | 0.0040 |
| Denominator | 0.0400 + 0.0100 − 2(0.0060) | 0.0380 |
| **w_A** | 0.0040 / 0.0380 | **10.53%** |
| **w_B** | 1 − 0.1053 | **89.47%** |

**MVP Portfolio Statistics:**

| Metric | Calculation | Value |
|---|---|---|
| Expected return | 0.1053(12%) + 0.8947(6%) | 6.63% |
| Portfolio variance | (0.1053)^2(0.04) + (0.8947)^2(0.01) + 2(0.1053)(0.8947)(0.006) | 0.00927 |
| **Portfolio sigma** | sqrt(0.00927) | **9.63%** |

Notice the MVP has lower volatility (9.63%) than even the less volatile asset B (10%). This is diversification at work.

**How Correlation Affects MVP Weights:**

| Correlation | w_A | w_B | Portfolio Sigma |
|---|---|---|---|
| +1.00 | 0% (corner) | 100% | 10.00% |
| +0.30 | 10.53% | 89.47% | 9.63% |
| 0.00 | 20.00% | 80.00% | 8.94% |
| −0.50 | 28.57% | 71.43% | 7.56% |
| −1.00 | 33.33% | 66.67% | 0.00% |

As correlation decreases, the MVP allocates **more** to the riskier asset because the diversification benefit of adding it increases.

**Key Exam Points:**

1. The MVP formula requires only **variances and correlation** — expected returns are irrelevant for finding minimum variance weights.
2. When rho = +1, the MVP puts 100% in the lower-volatility asset (no diversification benefit).
3. When rho = −1, a **zero-variance** portfolio exists — this is the theoretical maximum diversification.
4. The MVP always lies on the efficient frontier's **leftmost point** (lowest sigma).
5. MVP weights can be **negative** (short selling) if correlation is high and volatilities are similar.

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How do you calculate the minimum variance portfolio weights for two risky assets?

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