Portfolio theory and mathematical models/Tobin's theorem

Tobin's theorem is one of the modern portfolio theories, which was developed from Markowitz theory adding the concept of risk free assets. This is also known as Separation theorem.

Tobin's (separation) theorem
where
 * $$r_f$$: the interest rate of risk-free assets

The investors who don't take risks at all would invest all own money to the risk-free asset, and the investors who will take risks as much as possible would not have risk-free asset at all. The majority investors who like middle risk and riddle return would have both risk-free assets and tangency portfolio.

In case we make a portfolio with both risk-free assets and risk assets, if all investors act risk-aversively like this, we would certainly choose the combination of risk-free assets and tangency portfolio, which is the conclusion that Torbin reached.



If we suppose a indifference curve (of a investor), which is tangent to the line which comes from $$r_f$$, the following is defined( the indifference curve is salient to the line):

'''Risk free assets : the whole Risk assets = line segment between the tangency point of the indifferent curve and the tangency portfolio : line segment between the tangency point of the indifferent curve and the $$r_f$$ '''

What is "separated"?
1. The optimum allocation ratio within the risk assets

and

2. The optimum allocation ratio between the risk-free assets and the whole risk assets

are separated.