To construct the minimum variance portfolio with two assets, we need the expected returns, variances, and correlation between the assets. Let’s denote:
- Asset 1: expected return r1, variance σ12
- Asset 2: expected return r2, variance σ22
- Correlation: ρ12
We want to determine the portfolio weights w1 and w2 such that the portfolio variance σP2 (the risk measure) is minimized. Note that w1 + w2 = 1, so we can express w2 = 1 − w1. The portfolio variance is given by:
σP2 = w12σ12 + w22σ22 + 2w1w2ρ12σ1σ2
Substituting w2 = 1 − w1, we get:
σP2 = w12σ12 + (1 − w1)2σ22 + 2w1(1 − w1)ρ12σ1σ2
To minimize the portfolio variance, we differentiate σP2 with respect to w1 and set the derivative to 0:
$$\frac{\partial \sigma_P^2}{\partial w_1} = 2 w_1 \sigma_1^2 - 2(1 - w_1) \sigma_2^2 + 2 (1 - 2w_1) \rho_{12} \sigma_1 \sigma_2 = 0$$
Solving this equation for w1, we find the optimal weight for asset 1:
$$w_1^* = \frac{\sigma_2^2 - \rho_{12} \sigma_1 \sigma_2}{\sigma_1^2 + \sigma_2^2 - 2 \rho_{12} \sigma_1 \sigma_2}$$
The optimal weight for asset 2 can be found using w2* = 1 − w1*:
$$w_2^* = \frac{\sigma_1^2 - \rho_{12} \sigma_1 \sigma_2}{\sigma_1^2 + \sigma_2^2 - 2 \rho_{12} \sigma_1 \sigma_2}$$
These weights, w1* and w2*, will result in the minimum variance portfolio for the given assets.
Now let’s consider an example with specific values for r1, r2, σ12, σ22, and ρ12. Suppose:
- Asset 1: r1 = 10%, σ12 = 0.04
- Asset 2: r2 = 15%, σ22 = 0.09
- Correlation: ρ12 = 0.5
Using the formulas derived above, we can find the optimal weights:
$$w_1^* = \frac{0.09 - 0.5 \cdot \sqrt{0.04} \cdot \sqrt{0.09}}{0.04 + 0.09 - 2 \cdot 0.5 \cdot \sqrt{0.04} \cdot \sqrt{0.09}} \approx 0.456$$
$$w_2^* = \frac{0.04 - 0.5 \cdot \sqrt{0.04} \cdot \sqrt{0.09}}{0.04 + 0.09 - 2 \cdot 0.5 \cdot \sqrt{0.04} \cdot \sqrt{0.09}} \approx 0.544$$
So the minimum variance portfolio would consist of approximately 45.6% of asset 1 and 54.4% of asset 2.