Mean variance framework

In the mean variance framework what is the mathematical expression for the market portfolio?

This is the question I was asked in an interview.

What I know about the market portfolio is the following:

  1. it is the portfolio of all the risky assets available in the market with weights equal to the proportion of their market capitalisation.

  2. it is the point at which the ray from the risk free asset is tangent to the efficient frontier.

  3. it is the point on the efficient frontier so that the slope of the CAL or the sharpe ratio (Rm-Rf)/σm is maximum.

But I don’t know how to express the market portfolio mathematically. So it will be great if someone can please help.

market portflio is where beta is equal to 1.

CAPM equation = R§ = Risk-free + Beta(Market Return - Risk-free)

So if your beta is equal to 1 then your Return on Portfolio is equal to the Return on the Market because the Risk-free on both sides cancel and you’re left with R§ = Market Return