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

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 $T^{*} = Q$?

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.

• There has to be some assumptions (forgive me for being a bit picky). You need to have at least 2 orthogonal components in the portfolio (after decomposition). Then you can think about this problem as finding a vector that is orthogonal to a existing vector. on a 2-d plane, you obviously have a unique orthogonal vector. In a 3-d or more space, the result is not unique anymore. With this intuition, I think the proof would be easy to construct. – Will Gu Dec 10 '16 at 0:16
• Maybe I am blind - but what is $r_P$ or the corresponding portfolio $P$? – LocalVolatility Dec 10 '16 at 0:46
• The portfolio $P$ as defined by what my class has discussed to be us determined by a holding vector for $N$ available assets: $$h = h_P = \begin{pmatrix} h_1\\ \vdots\\ h_N \end{pmatrix}$$ – Wolfy Dec 10 '16 at 2:11
• You can also think if it as a benchmark portfolio like the S&P 500 I believe – Wolfy Dec 10 '16 at 2:13
• Any portfolio is defined via a weight vector. So what is special about P? It seems to matter here. – LocalVolatility Dec 10 '16 at 12:46

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 $T$.