# 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!

• A warm welcome to Quant.SE! So what have you tried so far to answer the question? – vonjd Jul 17 '16 at 17:33
• If $R_p$ is the return of the investor's portfolio, we can write $R_p=xR_T+(1-x)R_f$ where $x$ is the proportion of his wealth invested in risky assets. After resolving the optimisation problem, I write $\mu=x\mu_T+(1-x)R_f$ and so $x=\frac{\mu-F_f}{\mu_T-R_f}$. And, I prove $\frac{\mu-R_f}{\sigma}=\frac{\mu_T-R_f}{\sigma_T}$. In particular, for the portfolio with the minimim variance, we have $\frac{\mu-R_f}{\sigma}=\frac{\mu_0-R_f}{\sigma_0}$. Combining these inequalities, I found $(\sigma-\sigma_0)^2=(\sigma_T-\sigma_0)^2(\frac{\mu-\mu_0}{\mu_T-\mu_0})^2$. I wonder that there is a mistake! – Zoro-X Jul 17 '16 at 18:42
• I have two problems in this exercice. You can see the first one in my first post (before editing), and I can discuss it with anyone who is interested. The second one is specified in the current topic. I like to add that the question before this one was to prove that $c=\frac{(\mu_T-\mu_0)^2}{\sigma_T^2-\sigma_0^2}+\frac{\mu_0^2}{\sigma_0^2}$ which is easy. – Zoro-X Jul 17 '16 at 18:55