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In CAPM, we assume people are risk-averse and people get compensated for the systematic risk they suffer. The assumption that most people are risk-averse makes sense, but why are the rational investors also risk-averse? Consider the following example, suppose investment $i$ has an expected return of $10\%$ and beta coefficient $2$ while another safer investment offers $5\%$ but is virtually risk free. By the weak law of large numbers, investment $i$'s average return rate will be close to $10\%$ if many years passes. In fact, if invested for many years, this investment can be seen as an investment with $10\%$ return but less risky because the distribution for the average return rate has less and less varaince as the number of years increases. So a rational investor should understand it and simply choose the investment with the risk premium. The idea is in the long run, yearly varaince on return diminishes.

Any suggestions or criticisms are appreciated.

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Risk-averse does not means that you avoid taking up any additional risk, or taking up the lowest risk investment. As long as the coefficient of a risk is fairly compensated, rational investors does not mind taking up additional risk. And it depends solely on the risk appetite of each individual investors. However some sources have the wrong idea that risk-averse refers to investors dislike and avoids risk, which isn't really the case. As long as the risk is fairly compensated, why not? – Sky Mar 2 at 15:33
    
@Sky My question is why do we even need to be compensated for taking additional risk. I mean, although you might loose a lot of money, you also got the chance to win a lot of money. My point is risk premium's existence is actuarially unfair. – Kun Mar 2 at 20:28
up vote 5 down vote accepted

Below you find some observations...

In CAPM, we assume people are risk-averse and people get compensated for the systematic risk they suffer. The assumption that most people are risk-averse makes sense, but why are the rational investors also risk-averse?

The "rational investors" prefer high (expected) returns and low volatity. In this sense, the rational investors are risk-averse and ask premiums (higher returns) to take more risks (more volatile investments).

Consider the following example, suppose investment $i$ as an expected return of 10% and beta coefficient 2 while another safer investment offers 5% but is virtually risk free.

Ok...

By the weak law of large numbers, investment $i$'s average return rate will be close to 5% if many years passes.

No. Investment $i$'s annual expected return is $10\%$, with an unspecified annual volatility $\sigma > 0$. Theoretically speaking, after many years the annual return rate is still $10\%$. Indeed $10\%$ is the expected annual return rate.

In fact, if invested for many years, this investment can be seen as an investment with 10% return but less risky because the distribution for the average return rate has less and less varaince as the number of years increases.

Actually, the volatility increases in $\sqrt{T}$ where $T$ is the time. So if the annual volatility for investment $i$ is $\sigma = 2$, the volatility for $T = 4$ (years) is $\sigma_{T=4} = \sigma * \sqrt{4} = 2 * 2 = 4$.

Conversely, the (virtually) risk-free asset has a $5\%$ annual return rate and zero volatility.

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I am not sure i follow your question but there are a few points worth making

  • Investors can and do (based on their utility function) chose the highest return/highest risk investment. The assumption in CAPM about risk adversity is that for the same level of expected return, investors will always choose the investment with less risk.
  • On the long run your point is correct and is included in the definition of expected return. However some investors prefer different level of risk based on their constraints and marginal use for more or less money (utility function). Imagine someone who is retired that needs to pay a fixed amount every year for his expenses. He would rather choose an investment that brings an almost certain (or certain) amount every year that match his expenses rather than one that has a chance of bringing 0 returns or worse :) Even if the expected return is higher. If you have an unlimited or very long investment horizon with no constraints on what you use the returns for and you can sleep at night not worrying about your money then obviously you would prefer a higher risk/higher return proposition. It's all about the investor preference and constraints.

Two final notes on that last point :

  • This is in no way contradicting CAPM (see my first point above)
  • In real life though, you never know the real expected return of an investment nor the exact level of risk so people act on perception of those... Food for thoughts.
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It is a big question. You may be interested in an article by Andrew Lo "The Origin of Risk Aversion" (2014) in which he proposes "an evolutionary explanation of risk aversion"

http://alo.mit.edu/wp-content/uploads/2015/06/The_Origin_of_Risk_Aversion_Final.pdf

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>Caveman 1: Hey, what's that sound? Caveman 2: What, that rustling sound from the long grass? It's probably just the wind, come on, let's go hunter-gathering. Caveman 1: Nah, I think I'm going to hang back here for a bit. Caveman 2: Pfft, whatever, I'm going to come back with a nice fat proto-chicken to eat. Maybe even some eggs. Caveman 2 wanders off into the long grass; a few seconds later a proto-lion jumps out and eats him. Caveman 1 wanders back home and contributes his risk-averse genes to the pool. Just so. An intutive account for the article you suggested. – Kun Mar 16 at 19:06

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