# For any efficient portfolio, does there exist another efficient portfolio which has zero correlation with it?

For any portfolio on mean-variance efficient frontier, does there exist a portfolio on the frontier which has zero correlation with it?

I tried to play around with the covariance, by setting correlation equal to zero and then showing that since s.d. are positive, the only way it can be zero is when covariance is zero. And from there, tried to prove that this exists (when the covariance between two portfolios are zero). But no success.

How to prove it?

Let the portfolio be $T.$ Suppose $T$ is not the minimal variance portfolio, $M.$

Consider $$\theta T + (1-\theta) M$$ the covariance of this with $T$ is $$\theta \operatorname{Var}(T) + (1-\theta ) \operatorname{Cov}(T,M).$$ By varying $\theta$ we can get zero (i.e. let $\theta=\frac{\operatorname{Cov}(T,M)}{\operatorname{Cov}(T,M)-\operatorname{Var}(T)})$. This portfolio is a linear combination of frontier portfolios so it is a point on the frontier. Call it $Z.$ However, we still have to discuss which side of the frontier it is on, efficient (max return for a given risk) or inefficient (min return for a given risk). I will show that $Z$ is inefficient rather than efficient:

Note that $T$, $Z$ and $M$ all lie on a straight line in weight space.

By considering the minimal variance combination of $T$ and $Z$ which is $M$ we see that they must lie on different sides of $M$ since the weights in the formula for a minimal variance combination of two assets will be positive.

So $Z$ is on the inefficient frontier not the efficient one.

Note all this assumes $T \neq M.$ (Otherwise $\theta$ is not defined, due to division by zero).

(see my book on portfolio theory for more discussion.)

• Thank you for your answer! So does it mean that there doesnt exist a portfolio with a zero correlation given that this portfolio ends up on the inefficient part of the frontier? – user25910 Dec 29 '16 at 9:51
• Besides, what do you mean by inefficient frontier? – SRKX Dec 29 '16 at 10:14
• the inefficient frontier are the points where the return is minimal for a given variance – Mark Joshi Dec 29 '16 at 11:11