CAPM: Tangential risk free rate and efficient frontier in E-sigma space

[George Philip]

I was going through the consequence of the additional assumption for CAPM in Chapter 8 pg 5 and I was hoping someone could explain the second consequence namely:

2. If in addition they are all subject to the same risk-free rate of interest, the efficient frontier collapses to the straight line in E -sigma space which passes through the risk-free rate of return on the E-axis and is tangential to the efficient frontier for risky securities.

I understand as to why the risk-free rate of interest is a straight line in the E-sigma space as proved in Chapter 6 but could someone explain as to why this straight line will always be tangential to the Efficient frontier?

There is a similar question in pg 6 answering this and I understood why the risk-free straight line can't pass below but I was not able to understand why it can't pass above the efficient frontier

 

[CapitalActuary]

It doesn’t really make sense for the capital market line to pass above the efficient frontier. It gives the risk-return profile of the risk free asset + a risky portfolio. You can’t get a risky portfolio above the efficient frontier, by definition, so clearly the best you can do is picking one on the efficient frontier.

In CAPM you assume all investors are following this same logic, so they all do the same and hold a combination of the risk free asset and the same risky portfolio right on the efficient frontier. The only way for everyone to do this is if the risky portfolio is market cap weighted.