The Mean–Variance Frontier with a Risk-Free Asset

Summary

The document sets up a mean–variance portfolio problem with multiple risky assets and one risk-free asset. It defines the risky-asset mean vector and covariance matrix, assumes the covariance matrix is invertible, and introduces three scalar quantities formed from these inputs. The question gives formulas for the minimum-variance portfolio’s mean and variance and asks for a relation connecting portfolio variance to its mean return.

The proposed relation describes a quadratic mean–variance frontier centered at the minimum-variance portfolio. Its curvature is expressed through the minimum-variance portfolio and a second target portfolio. However, the document contains only the problem statement: it provides no proof, derivation, or evidence that the formula holds under the stated setup. It is useful as a prompt about portfolio construction and efficient-frontier geometry, but readers must supply the argument and clarify the target portfolio notation themselves.

Key ideas

• The setup combines risky assets with a risk-free asset in a mean–variance portfolio problem.
• The risky-asset covariance matrix is assumed to be invertible.
• The stated minimum-variance portfolio has mean equal to the ratio of two defined scalars and variance equal to the reciprocal of another.
• The question proposes a quadratic relationship between portfolio variance and mean return around the minimum-variance portfolio.
• The document asks for a proof but does not include one.

Tags

# Relation between mean and variance of a portfolio in modern portfolio theory:


# Relation between mean and variance of a portfolio in modern portfolio theory:












I hope that this is the right place to ask my question!

Let a market with $N\ge1$ risky assets and denote by $(R_i,i=1,\cdots, N)$ their returns and $R$ the vector of these $N$ returns. In addition, there is a riskfree asset with the return $R_f$.

Denote by $\mathcal{E}$ the mean of $R$ and $\Lambda$ the covariance matrix of $R$. We suppose that $\Lambda$ is invertible.

Finally, let $a=~^t\text{1l}\;\Lambda^{-1}\;\text{1l},\;b=~^t\text{1l}\;\Lambda^{-1}\;\mathcal{E} \text{ and } c=~^t\mathcal{E}\;\Lambda^{-1}\;\mathcal{E}$.

An investor looks for a portfolio with a maximum mean return for a constant variance return.

The market portfolio has the renturn $\mu_M=\frac{b.R_f-c}{a.Rf-b}$ and the variance $\sigma_M^2=\frac{a.\mu_M^2-2b.\mu_M+c}{ac-b^2}$.

Here is the question: Prove that the return $\mu$ and variance $\sigma$ of an investor's portfolio satisfies $\sigma^2=\sigma_0^2+(\sigma_T^2-\sigma_0^2).\Big(\frac{\mu-\mu_0}{\mu_T-\mu_0}\Big)^2$ where $\mu_0$ and $\sigma_0^2$ the mean return and the variance of the portfolio with minimum variance (we can show that $\mu_0=\frac{b}{a}$ and $\sigma_0^2=\frac{1}{a}$).

Thank you in advance for any suggestion!

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