Why No Tangency Portfolio Exists When the Risk-Free Rate Equals GMV Return

# How do I show that there is no tangency portfolio?


# How do I show that there is no tangency portfolio?












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 by Bernardo Scarpelli (score 2)

https://quant.stackexchange.com/a/73366

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.

## Answer by Iron Soles (score 2)

https://quant.stackexchange.com/a/77033

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.

## Answer by Stefan Voigt (score 1)

https://quant.stackexchange.com/a/18184

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 $r_f$.