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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.

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    $\begingroup$ You are right: Two different points have different proportions in the risk free asset, so the composition is unique. $\endgroup$ – Alex C Nov 24 '16 at 20:54
  • $\begingroup$ @AlexC Then let's add this as an answer rather than a comment. $\endgroup$ – SRKX Nov 25 '16 at 17:22
  • $\begingroup$ @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 $\endgroup$ – Wolfy Dec 14 '16 at 23:28
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You have the right idea:

Two different points A and B on the CML have different proportions in the risk free asset, the point to the left has more in the risk free asset than the point on the right. This means that the composition of the two portfolios is distinct , they are not the same, but rather each is unique.

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  • $\begingroup$ Is this rigorous enough as a proof? $\endgroup$ – Wolfy Dec 11 '16 at 20:35
  • $\begingroup$ any idea not sure if you saw my last comment $\endgroup$ – Wolfy Dec 14 '16 at 17:00
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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.

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