Show that there exists a fully invested portfolio such that the covariance between their returns is zero

Question

Background Information:

I came across this question in chapter 2 of Active portfolio Management by Grinold and Kahn. It pertains to the efficient frontier which is displayed below:

Question:

If $T$ is fully invested and efficient and $T\neq C$, prove there exists a fully invested portfoli $T^{*}$ such that $Cov(r_T, r_{T^{*}}) = 0$

We have that $T$ is fully invested on the efficient frontier so $h_{T}^{T}e = 1$ since we want to find a $T^{*}$ that is also fully invested and efficient but $T^{*}\neq C$ it seems to me by the picture above that the only way we can have $Cov(r_T, r_{T^{*}}) = 0$ then $T^{*} = Q$?

Additional thoughts:

Since $T$ is efficient and fully invested it must be a linear combination of $h_C$ and $h_Q$ thus $$h_T = c_1 h_C + c_2 h_Q$$

Then since $h_T^{T}e = 1$,

$$\left(c_1h_C + c_2 h_Q\right)^{T}e = 1 \ \Rightarrow \ c_1 + c_2 = 1$$

Like wise since $T^{*}$ is fully invested so $h_{T^*}^{T}e = 1$. Now, $r_T = h_{T}^{T} r = \left(c_1h_C + c_2 h_Q\right)^{T}r$ and $r_{T^{*}} = h_{T^{*}}^{T}r$ So, \begin{align*} Cov(r_T,r_{T^{*}}) &= Cov((c_1h_{C}^{T} + c_2 h_{Q}^{T})r,h_{T^{*}}^{T}r)\\ &= c_1 c_2 Cov(h_C^{T}r + h_Q^{T}r, h_{T^{*}}^{T}r)\\ &= c_1 c_2\left(E[(h_C^T r + h_Q^{T})h_{T^{*}}^{T}r] - E[h_C^{T}r + h_{Q}^{T}r]E[h_{T^{*}}^{T}r]\right)\\ &= c_1 c_2 \left(E[h_{C}^{T}r h_{T^{*}}r] + E[h_{Q}^{T}r h_{T^{*}}^T r] - E[h_C^{T}r]E[h_{T^{*}}^T r] - E[h_{Q}^{T}r] E[h_{T^{*}}^T r]\right)\\ &= 0 \end{align*}

I am not sure if this is correct although. I don't really know exactly how to approach this problem, any suggestions are greatly appreciated.

Answer 1

There is a procedure for finding $T^*$ starting from the portfolio $T$ on the efficient frontier, such that $cov(T^*,T)=0$:

From the point $T$ draw a line thorough the point $C$ (which represents the global minimum variance portfolio or GMVP) until it intersects the Y axis at a point $R_z$. Draw a horizontal line from this point until it intersect the parabola at a point we will call $T^*$. This is the desired portfolio having zero covariance with $T$.

Clearly if T and C coincides the procedure fails: the line TC is not defined. We know that the GMVP has the same non-zero covariance with every portfolio on the parabola, so this case is not solvable.

The detailed proof can be found for example on page 6,7,8 of this paper

https://eclass.aueb.gr/modules/document/file.php/INF258/ConstMalliarisPortfolioChapter.pdf