# Why the market portfolio is the tangency portfolio in the Mean-Variance Optimization model?

I read in an explanation that the tangency portfolio has all securities with weights proportional to their market value because supply equal’s demand. But I can't understand why supply equals demand explains why the “best” portfolio is the combination of all stocks, with weights proportional to their market value.

If anyone could elaborate on that, I would really appreciate.

It's a good question, if only because it's about a bad assertion that conflates related concepts!

The easiest answer to your question is the argument that "if the market isn't tangency, then everyone would do tangency, pushing the market to tangency". Which is just a basic re-hashing of all the standard arguments about market efficiency.

One obvious pushback (to which, I think, you allude) is that this does not follow; but in the absence of reliable evidence of mispricing, I have no better prior for any stock's valuation than its market valuation. Semi-strong efficiency 101, in the jargon. I believe therefore that market equals or approximates tangency, because I have no superior prior.

But this argument cannot neatly reconciled with "supply and demand" ;-) Because supply and demand are both conditioned by price. What is the supply at price X; what is the demand at the same price X. It follows there will be a difference at price equals not-X. X being a function of expected return, for simplicity's sake.

Then ask yourself whether X has to be the same for every stock in the market. This is not true; and it's even axiomatic to standard market theory. This says that if we floated "Ladca & DEM's Speculative Punts in New Crypto Inc" on the NASDAQ tomorrow, our little levered venture would probably have a higher cost of capital than the Dow Jones. You could rationalise this fact, either through a higher WACC fundamental argument or though a technical CAPM higher vol equals higher beta one. In any language, two people who have never met punting i-coins by flipping coins is not like Amazon, Proctor & Gamble, or Bank of America.

Our stock would thus require a higher return to attract demand; and create higher supply at a lower rate of return. Irrespective of our correlation to AMZN, PG, and/or BAC, this at a stroke invalidates the tangency=market argument. Because the tangency portfolio could choose to run a small-cap bias that generates a higher return than an index with concentrated exposure to mega-caps, without this necessarily creating an equal-and-opposite volatility.

It might be higher "risk"; but the "risk" there is not an optimised return slope to vol. The argument you're asking about is, in my opinion, intellectually lazy BS. I wouldn't lose sleep over it.

best, DEM

[ps if you don't believe this, ask yourself how the bond market (as big as the equity market, larger if you include credit) can function with yield curves where volatility is an almost linear function of duration, but the compensation for this risk is linear maybe 1% of the time. Different kinds of volatility can be differently priced; just as different kinds of profitability can be. Drawing straight lines through "return" and "vol" is thus wrong to begin with]

There are two questions in here, 1. why only the tangency portfolio and 2. why the equilibrium (supply/demand) argument - let me give it a shot ...

On 1.:

Consider all efficient portfolio combinations (blue line). It would be rational for all investors - regardless of their risk and return preferences - to choose a combination of risk free asset (black) and the tangency portfolio (green), i.e. the capital market line.

On 2.:

If you now consider a very well diversified portfolio, such as the market portfolio, you are only exposed to systemic risk - as the idiosyncratic risk diversifies away. This gives rise to the Capital Asset Pricing Model (CAPM), where sigma is replaced by beta (a measure for systemic risk).

Under the CAPM, stocks above (below) the red line are considered under (over) valued. This is because all investors together demand a risk premium only for systemic risk, i.e. a premium for the systemic risk contribution a single stock to the market portfolio. The relative value argument should then lead to a shift in supply and demand until the stock is priced in line with the CAPM, i.e. the market is in equilibrium.