Two fund separation when there's a risky asset?

Question

I am currently reading a book which begins its portfolio theory section with the case with $n$ risky assets where it proves that 2-fund separation applies (any minimum variance portfolio is a linear combination of two minimum variance portfolios with distinct returns). It then moves on to the case where there's a riskfree asset as well, and claims at one point that each agent will hold a mix of the tangent portfolio and the riskfree asset.

Why? A result like this would require reproving 2-fund-separation in this new situation as well, but the author doesn't mention it at all. Is there then a different way to see this result, or is it just 2fundseparation that the author just forgot to mention still holds?

Answer 1

The two fund separation theorem still hold. If you have N risky asset with your efficient frontier and add the risk free asset, as result you achieve another efficient frontier that boil down in a straight line (the CML). You can see also this short explanation

Tangency portfolio and CML - Why does it have the highest sharpe ratio?

In other the two-fund separation theorem is easy to understand, and demonstrate, in N risky plus 1 riskless setting. In N risky assets setting is sensibly more complicated. If you understood the second case ... the first is already grasped.