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I am a bit confused about one assumption of the CAPM. My professor said that in the CAPM model all investors share the same utility function and the same degrees of risk aversion. Then as a final consequence all investors will choose a portfolio composed by a portion of the market portfolio and a portion of the risk-free security according to its preferences. (Each investors could have different portions of risk-free security and risky asset)

As far as I am concerned, I agree with the first condition, but I don't understand how is it possible that all investors have same degree of risk aversion. If so, why investors would choose different portion of risk-free security and risky asset?

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That is only an assumption. Indeed, you should always keep in mind the difference between Asset Allocation and Capital Allocation.

You can see Asset Allocation as the first step of making investments, where you want to decide which securities will provide you with meaningful risk-return characteristics. Of course, Markowitz allocation is the stronghold here and the risk-return characteristics is mean-variance.

In a second step, after you've gotten the efficient risk-return combinations, then comes into play the individual degree of risk aversion. In fact, here each investor would choose a different combination of securities that will deliver a different risk-return profile.

From the first step also comes Merton's 1971 idea of Two funds separation theorem, when the market portfolio and the risk free asset exists and returns come from a geometric Brownian motion.

If you see the Asset Allocation/Capital Allocation separation it should be clearer why you want to keep the assumption of sharing the same degree of risk aversion in the first step, because you want to evaluate the investment opportunities objectively (Asset Allocation) to then allocate capital (capital allocation) subjectively only in a second moment.

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Your intuition is correct, the CAPM does not assume every individual has the same degree of risk aversion. From page 280 of Bodie, Kane, and Marcus Investments (8th ed) (Chapter 9 The Capital Asset Pricing Model) "The thrust of these assumptions [of the basic CAPM] is that we try to ensure that individuals are alike as possible, with the notable exceptions of initial wealth and risk aversion."

The basic idea of the CAPM assumptions is everyone is a Markowitz mean-variance optimizer (which does not require homogeneous degrees of risk-aversion), and everyone has homogeneous expectations, which means everyone will hold the market portfolio, which leads to the CAPM. As a note, the major contribution of Markowitz portfolio optimization was that it showed an investor's optimal risky portfolio was independent of their degree of risk aversion. The CAPM does not then add the risk aversion back in as an assumption.

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