# How to prove that every point on the capital market line corresponds to a unique portfolio

Prove that every point on the capital market line corresponds to a unique portfolio.

Attempted proof

I know that each point on the capital market line represents a linear combination of the risk free rate and some portfolio. But I am not really sure how to show that we have uniqueness. Any suggestions or references are greatly appreciated.

Note that we can assume the risk-free rate is zero, which is what my professor presented. Perhaps we would need a different proof if the risk-free rate was not zero but I am not sure, as usual I am lost.

• You are right: Two different points have different proportions in the risk free asset, so the composition is unique. – Alex C Nov 24 '16 at 20:54
• @AlexC Then let's add this as an answer rather than a comment. – SRKX Nov 25 '16 at 17:22
• @SRKX I tried to provide an answer to this question, this is roughly what my professor told me to do, not sure if it is complete though yet – Wolfy Dec 14 '16 at 23:28

I believe to prove this rigorously we need to first note that we want to $$\min{h^T V h } \ \ \text{subject to} \ \ h^T f = f_P$$ Using Lagrange multiplier we have the problem
\begin{cases} 2Vh &= \lambda f\\ h^T f &= f_P \end{cases} Thus, $$h = \frac{\lambda}{2}V^{-1}f = c\times h_Q$$ where $c = \frac{\lambda}{2}$ so, every point on the capital market line corresponds to a unique portfolio.