Let's analyze this problem step by step.

1. Sketching the Situation in the $(\sigma_V, \mu_V)$ Plane:

   • Given that we have $n$ risky assets that are uncorrelated, each with a mean return $m_*$ and variance $\sigma_i^2$, we aim to represent the expected return $\mu_V$ and the variance $\sigma_V^2$ of a portfolio composed of these assets.
   • Since each asset has the same mean return $m_*$, the expected return of any weighted portfolio of these assets will also be $m_*$ (assuming all weights sum up to 1).
   • In the $(\sigma_V, \mu_V)$ plane, the line representing the expected return will be horizontal at $m_*$.
   • The variance (and thus the standard deviation $\sigma_V$) will depend on the allocation of weights across assets. Minimum variance would correspond to the lowest achievable portfolio variance for this fixed mean $m_*$, creating a point on this line at the minimum portfolio standard deviation.

2. Finding the Minimum Variance Point and Associated Weights:

   • To achieve the minimum variance portfolio, we want to minimize the portfolio variance $\sigma_V^2$.
   • For a portfolio with weights $w_i$, the variance $\sigma_V^2$ of the portfolio is given by: $\sigma_V^2 = \sum_{i=1}^n w_i^2 \sigma_i^2$
   • Given that the assets are uncorrelated, we aim to minimize this expression subject to the constraint $\sum_{i=1}^n w_i = 1$.
   • Using the Lagrange multiplier method or known formulas, the minimum variance weights for uncorrelated assets are: $w_i = \frac{\frac{1}{\sigma_i^2}}{\sum_{j=1}^n \frac{1}{\sigma_j^2}}$
   • Substituting these weights back into the expression for $\sigma_V^2$ yields the minimum variance for the portfolio: $\sigma_V^2 = \left( \sum_{i=1}^n \frac{1}{\sigma_i^2} \right)^{-1} = \bar{\sigma}^2$
   • Thus, the minimum variance point on the $(\sigma_V, \mu_V)$ plane has coordinates $(\bar{\sigma}, m_*)$, where $\bar{\sigma}^2$ is defined as: $\bar{\sigma}^2 = \left( \frac{1}{n} \sum_{i=1}^n \frac{1}{\sigma_i^2} \right)^{-1}$

In summary:

• The minimum variance point on the $(\sigma_V, \mu_V)$ plane is $(\bar{\sigma}, m_*)$, with $\bar{\sigma}$ being the square root of the harmonic mean of the variances.
• The optimal weights for each asset to achieve this minimum variance are proportional to the inverse of each asset's variance.

Would you like further details on any part of the derivation, or do you have questions about the specific steps?

Here are some related questions you might find helpful:

1. How does the minimum variance portfolio change if the assets are not uncorrelated?
2. Can we extend this approach if assets have different expected returns instead of a common mean?
3. What is the impact of introducing a risk-free asset to this portfolio optimization problem?
4. How would correlation between assets affect the optimal portfolio weights?
5. Could we explore the effect of constraints on the weights (e.g., no short selling)?

Tip: When constructing portfolios, remember that minimizing variance alone may not achieve an optimal risk-return trade-off; balancing variance and return (like in mean-variance optimization) often yields better results.