# Investors degree of risk aversion in capm model

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?

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.

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.

It is a pity that all textbooks now (2020-04-30) are WRONG on CAPM. Cause there is a Misunderstanding of the market portfolio: The portfolio separation of Tobin (1958) brings the market portfolio front and center. Although investors have different wealth and preferences, investors with mean-variance preferences all hold the same portfolio of risky assets. However, there is a precondition for this assertion in his paper, Tobin (1958) assumes that the mean vector and variance matrix of the returns of risky securities are given.

However, the returns are endogenous in CAPM. It is the result of the equilibrium of the entire market according to the mean-variance criterion.

"Initial wealth and risk aversion" will affected the equilibrium. Supposing that some investor double her initial wealth, or alter her risk aversion, such that she put more fund in risky assets, then the total value of market portfolio will be increased, and the rate of market return will be changed (the total future payoff is fixed). The market portfolio must be affected! For more on CAPM, see CAPM: Absolute Pricing, or Relative Pricing? or Arbitrage Opportunity, Impossible Frontier, and Logical Circularity in CAPM Equilibrium

Edit (2020-05-26): dislikes are anticipated. Young scientist seeks for new truths, old-fashioned scientist defends old ideas

For any investor's wealth and preference (simple trade-off coefficient) in Example 4.1 Abad (2020): If the only change is an increase in wealth of the first investor, such that $$W_{1}=150$$, then $$\mu_{M}^{\,}=\frac{R_{0}D}{R_{0}% x+D}=\frac{2820\,783}{2411\,900}=1.1695$$. If the only change is an increase in preference parameter of the last investor, such that $$G_{4}=\frac{158}{13}$$, then $$\mu_{M}^{\,}=\frac{2586\,801}{2189\,060}=1.1817$$. If the only changes are parameters $$W_{1}=150$$ and $$G_{4}=\frac{158}{13}$$, and all other parameters remain unchanged, then $$\mu_{M}^{\,}=\frac{1857}{1580}=1.1753$$. Which restores the original expected market return, because of these two parameters have opposite effect and are offset completely. For these three cases, the weight vectors of the market portfolio are $$\frac{1}{120\,595} \begin{bmatrix} 15\,214\\ 6100\\ 99\,281 \end{bmatrix} ,\quad\frac{1}{109\,453} \begin{bmatrix} 13\,906\\ 5548\\ 89\,999 \end{bmatrix} ,\text{ and }\frac{1}{79} \begin{bmatrix} 10\\ 4\\ 65 \end{bmatrix}$$ respectively.