How Utility-Maximizing Portfolios Relate to the Capital Market Line

Summary

The document explains how an investor’s preferred portfolio changes when a risk-free asset becomes available. Without that asset, the investor selects a point on the efficient frontier where an indifference curve for their utility function is tangent to the frontier. Adding a risk-free asset creates a capital market line (CML), tangent to the frontier at portfolio M.

With access to the CML, investors with different risk preferences choose different combinations of the risk-free asset and M. Their optimal risky holdings are therefore the same tangency portfolio, while their overall allocations differ. The explanation is conceptual and refers to a diagram, but the diagram itself is not included here. It does not derive the CML or address assumptions such as borrowing and lending at the risk-free rate, so its conclusion should be understood within the standard portfolio-theory framework.

Key ideas

• Without a risk-free asset, an investor’s utility function identifies an optimal point on the efficient frontier.
• Introducing a risk-free asset creates a capital market line tangent to the efficient frontier.
• The tangency portfolio is the optimal risky portfolio shared by investors with different risk preferences.
• Investors express their different risk preferences through the proportions allocated to the risky portfolio and the risk-free asset.

Tags

# utility function and CAPM in portfolio theory


# 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?

## Answer by Kermittfrog (score 7, accepted)

https://quant.stackexchange.com/a/70119

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?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.