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.
HI @sumitmaan19 Three of my videos in our recent P1.T1. playlist might be helpful, esp T1-8 and T1-9. Your (1) is consistent with the definition given by Elton: "First, we have shown that, under the assumptions of the CAPM, the only portfolio of risky assets that any investor will own is the market portfolio. Recall that the market portfolio is a portfolio in which the fraction invested in any asset is equal to the market value of that asset divided by the market value of all risky assets." [Elton, Edwin J.; Gruber, Martin J.; Brown, Stephen J.; Goetzmann, William N.. Modern Portfolio Theory and Investment Analysis, 9th Edition (Page 302). Wiley. Kindle Edition.] ... but that's a description of the market portfolio in equilibrium. In practice, I think we treat its solution as an optimization problem, knowing that we are merely approximating because whatever is our set of investable assets is not the whole market. In the XLS for these, I actually assume a two-asset universe (ridiculously unrealistic)! How to i then define the market portfolio? I solve for the portfolio with the highest Sharpe ratio consistent with your #3 (because the assumptions are returns, volatilities and correlations -- prices do not enter this optimization approach, as opposed to the price-based description under equilibrium); this can be done for a large matrix of n assets. So, in my ridiculously unrealistic 2-asset world the mathematical solution to the market portfolio is a highest-Sharpe-ratio optimization problem, for which I recently found the analytical solution (via Mathematica), see here https://www.bionicturtle.com/forum/threads/market-portfolio-and-derivative-of-weight.9919/post-45560 I hope that's a bit of clarity!
Thanks for the detailed response and that definitely was really helpful particularly the maths. So, theoretically the market portfolio is a portfolio on the efficient frontier in which the fraction invested in any asset is equal to the market value of that asset divided by the market value of all risky assets (or our approximated set of investable assets) and the CML passes through it when we add the risk free asset in the scenario. But mathematically it can simply be considered as an optimization problem (with the assumptions of returns, volatilities and correlations) and solving for the weights for the maximum Sharpe ratio.