We are tasked with finding the weights of a minimum-variance portfolio made up of two uncorrelated assets, A and B, with an expected portfolio return of 12%.

Given:

• Expected return of Asset A ($R_A$) = 0.02
• Standard deviation of Asset A ($\sigma_A$) = 0.10
• Expected return of Asset B ($R_B$) = 0.10
• Standard deviation of Asset B ($\sigma_B$) = 0.25
• Desired portfolio return ($R_p$) = 0.12
• The assets are uncorrelated ($\rho = 0$).

Step 1: Define the portfolio return and weights

The portfolio return $R_p$ is given by the weighted sum of the returns of the two assets: $R_p = w_A R_A + w_B R_B$ where:

• $w_A$ is the weight of Asset A in the portfolio,
• $w_B$ is the weight of Asset B in the portfolio.

We also know that: $w_A + w_B = 1$ Since we aim for a portfolio return of 12%, we substitute $R_p = 0.12$ into the return equation: $0.12 = w_A (0.02) + w_B (0.10)$ We now have two equations:

1. $w_A + w_B = 1$
2. $0.12 = 0.02 w_A + 0.10 w_B$

Step 2: Solve for $w_A$ and $w_B$

From equation (1): $w_B = 1 - w_A$ Substitute this into equation (2): $0.12 = 0.02 w_A + 0.10 (1 - w_A)$ Expand and simplify: $0.12 = 0.02 w_A + 0.10 - 0.10 w_A$ $0.12 - 0.10 = (0.02 - 0.10) w_A$ $0.02 = -0.08 w_A$ Solve for $w_A$: $w_A = \frac{0.02}{-0.08} = -0.25$ So, the weight of Asset A is $w_A = -0.25$.

Now, using $w_B = 1 - w_A$: $w_B = 1 - (-0.25) = 1.25$ So, the weight of Asset B is $w_B = 1.25$.

Step 3: Minimum-variance portfolio

The variance of the portfolio $\sigma_p^2$ is given by: $\sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2$ Since the assets are uncorrelated, the covariance term is 0.

Substitute the values for $w_A$, $w_B$, $\sigma_A$, and $\sigma_B$: $\sigma_p^2 = (-0.25)^2 (0.10)^2 + (1.25)^2 (0.25)^2$ $\sigma_p^2 = (0.0625)(0.01) + (1.5625)(0.0625)$ $\sigma_p^2 = 0.000625 + 0.09765625$ $\sigma_p^2 = 0.09828125$ The standard deviation (volatility) of the portfolio is: $\sigma_p = \sqrt{0.09828125} \approx 0.3136$

Summary of results:

• $w_A = -0.25$
• $w_B = 1.25$
• Portfolio variance $\sigma_p^2 = 0.09828125$
• Portfolio standard deviation $\sigma_p \approx 0.3136$

These weights suggest that the investor should short sell 25% of Asset A and invest 125% in Asset B to achieve a 12% return with minimum variance.

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Would you like more details on any step or have additional questions? Here are 5 related questions to deepen your understanding:

1. How would the portfolio change if the assets were correlated?
2. What if the desired portfolio return was 8% instead of 12%?
3. How does short selling affect the overall portfolio risk?
4. Can the variance of the portfolio be reduced further by adding more assets?
5. What if the standard deviation of Asset B was lower—how would that affect the weights?

Tip: When constructing a portfolio, consider the trade-off between expected return and risk (volatility) to make informed investment decisions.