# 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.