How do I show that there is no tangency portfolio?

Question

Question: Suppose that the risk-free return is equal to the expected return of the global minimum variance portfolio. Show that there is no tangency portfolio.

A hint for the question states: Show there is no $\delta$ and $\lambda$ satisfying

$$\delta\Sigma^{-1}(\mu-R_f\iota)= \lambda\pi_\mu + (1-\lambda)\pi\iota$$

but I'm not sure what to make of it. Any help is appreciated.

Answer 1

Intuitively speaking this statement should be clear, as in case the risk-free rate is equal to the expected return of the global minimum variance portfolio you can just assume that the minimum variance portfolio is just an investment into the risk-free rate. Therefore the intersection between the efficient frontier and the tangent line between $r_f$ and the efficient frontier is at $0$ standard deviation and expected return $r_f$.

Answer 2

Recalling that: \begin{align} \begin{cases} A = \mu'\Sigma^{-1}\mu\\ B = \mu'\Sigma^{-1}\iota\\ C = \iota'\Sigma^{-1}\iota\\ \pi_{gmv} = \frac{1}{C}\Sigma^{-1}\iota\\ \pi_{\mu} = \frac{1}{B}\Sigma^{-1}\mu \end{cases} \end{align} Note that if we pose $R_f = \frac{B}{C}$ . We can assume by way of contradiction that there exists $(\delta, \lambda)$ such that the equation holds,and pre multiply both sides of the equation by $\iota'$. implying $\delta \times 0 = 0 = \lambda \frac{B}{B} + (1- \lambda)\frac{C}{C} = 1$ which is a contradiction.