Risk Aversion, Investor Constraints, and the Limits of Long-Horizon Averaging
Article Quant Q&A · Author: Kun
Summary
The discussion examines why rational investors may dislike risk in the CAPM framework and challenges the claim that holding a risky investment for many years makes it effectively safer. It distinguishes the expected annual return from realized returns: a long horizon does not turn an uncertain annual return into a guaranteed one. The response argues that cumulative return volatility can grow with the square root of time, while averaging observations is a different calculation.
A second answer connects risk preference to utility, cash-flow needs, and investment constraints. An investor who must fund regular expenses may favor a more predictable return even when a riskier asset has a higher expectation. The discussion also notes that expected returns and risk are not known precisely in practice. These are conceptual explanations, not an empirical test, and the long-run volatility point depends on assumptions about return behavior and the measure of return being considered.
Key ideas
- Risk aversion in CAPM reflects a preference for lower risk when expected returns are equal.
- A long holding period does not guarantee an asset’s expected annual return or make its annual risk disappear.
- Cumulative return volatility can increase with time, while the variance of an average is a distinct quantity.
- Investor utility, spending needs, and constraints can make predictable cash flows more valuable.
- Expected return and risk are uncertain in practice, so observed choices also reflect beliefs.
Tags
Full text
# Why are investors risk-averse?
# Why are investors risk-averse?
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.
## Answer by stochazesthai (score 5, accepted)
https://quant.stackexchange.com/a/24598
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.
## Answer by NegativeJo (score 2)
https://quant.stackexchange.com/a/24600
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.
## Answer by Alex C (score 2)
https://quant.stackexchange.com/a/24641
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.pdfShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.