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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.

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  • $\begingroup$ It would be helpful to give a definition of the variables you use in your equation. What is $\pi_\mu$ and $\pi$? $\endgroup$ – muffin1974 Jun 5 '15 at 8:26
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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 $r_f$.

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