utility function and CAPM in portfolio theory

I am trying to connect some dots in my understanding between 2 concepts.

Utility function: I can see there there are different utility functions and I can draw them at different levels until I find one that is touching the efficient frontier, call it point A. That will define a portfolio that is optimal for a given utility function.

CAPM line: if I introduce a risk free asset to my portfolio I can draw a straight line between the return of that asset and touch the efficient frontier. Then, this line will have a well known equation and will define an attainable optimal portfolio I can achieve, call it B, and also the one I can get if I lend/borrow using risk free asset.

How those points A and B related. Are they the same in a particular case?

Please have a look at this image, which I have copied from here:

Here, the point M is the tangency portfolio of the capital market line.

As you can see, the investor A (left hand side) can attain higher utility when the risk free asset becomes available: He can "jump" from the efficient frontier (w/o risk-free investment) onto any point on the CML (both leftmost points in the graph).

In any case, the investor's optimal risky portfolio will be exactly M, the same holds for the other investor. BUT their respective total investment mix (risk-free vs. risky) is, of course, different.

HTH?

• To summarize: the curved frontier is attainable by combining risky assets, once you add lending/borrowng at a riskfree rate the attainable set is the tangent line. In the CAPM we further assume that every investor finds his place on that same line (they all have same info about means and covariances but perhaps different utilities), then the line is called the CML capital market line and the tangency point is called the market portfolio. Mar 10, 2022 at 12:49
• @noob2 +1. I don't want to rob you of the "+1"s for that comment; else I'd blatantly copy it into my answer. Mar 10, 2022 at 13:49
• Nice discussion. Other points for OP to keep in mind: (1) As Richard Roll explained back in the 1980s, the sole testable implication of the CAPM theory is that the market portfolio lies on the efficient frontier (2) The CAPM doesn't work. Market beta doesn't explain cross-sectional variation in expected returns (while other variables arguably do). Mar 10, 2022 at 19:19
• @Kermittfrog: thanks for your answer. thus, we plot the efficient frontier, we plot the CAPM line and then we plot various utility to find a point on it for each investor. Is mean-variance utility a particular case of a utility function. Where would that utility be? Or this is a case where there is no risk-free assets included and thus it lands exactly where CAPM touches EF(M)? Curious what is so special about this utility function in this context? Mar 10, 2022 at 19:42