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.
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.
Any point on the efficient frontier is the sum $$ Z +\lambda X, \qquad \lambda\in\mathbb{R},$$ where $Z$ is the minimum variance fully invested portfolio and $X$ is an efficient zero-cost portfolio. By optimality of $Z$, any zero-cost portfolio is uncorrelated with $Z$. This shows that on the efficient frontier one has $$\sigma^2_R = \sigma^2_Z + \sigma^2_{\lambda X} = \sigma^2_Z +\frac{(\mu_R-\mu_Z)^2}{\mathrm{SR}_X^2}.$$
It so happens that if a risk-free asset is added with return $\mu_Z$, then the zero-cost efficient frontier does not expand. This, if you like, is the surprising bit. The old $X$ still has the best Sharpe ratio among all zero-cost portfolios even once the previously unavailable risk-free asset has been added. In the new market, we have the same $\mu_Z$ and $\mathrm{SR}_X$ as before but $\sigma_Z$ has become zero. So the new efficient frontier reads $$ \sigma^2_R = \frac{(\mu_R-\mu_Z)^2}{\mathrm{SR}_X^2}.$$ There is no intersection between the old and new frontier. The new frontier is an asymptote to the old one, so poetically speaking the tangency point is at infinity.
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 $r_f$.
