Skip to content
All library documents

Risk Aversion, Portfolio Choice, and Expected Returns

Article Quant Q&A · Author: lkonoplev
Read the original

Summary

The answer separates mean-variance portfolio choice from asset pricing. Mean-variance optimization describes how an investor chooses a portfolio given return distributions and preferences. It does not, by itself, dictate how expected returns relate to risk. That relationship comes from an asset pricing model, so an optimizer could select assets under a model where higher-beta assets offer higher, lower, or no higher expected returns.

It also distinguishes individual risk aversion from impatience. An individual’s risk aversion is represented by the shape of their utility function, while a personal discount rate may represent how they value consumption across time. In an equilibrium asset pricing model, a discount rate can instead reflect the representative investor’s risk preferences. Thus the question of whether a more risk-averse individual earns higher expected returns cannot be answered just by invoking rationality or mean-variance analysis. It depends on the asset pricing assumptions and on what “discount rate” means in the context. The answer is conceptual and does not develop a specific model or provide empirical tests.

Key ideas

  • Mean-variance optimization is a portfolio choice framework, not an asset pricing theory.
  • Expected return and beta relationships depend on the asset pricing model being assumed.
  • Individual risk aversion is encoded in utility preferences, while impatience concerns intertemporal discounting.
  • A market-equilibrium discount rate may represent aggregate risk preferences.
  • Rationality and market efficiency alone do not require one universal risk-return relationship.

Tags

StatisticsPortfolio constructionRisk management
Full text
# Beyond the mean-variance framework, can expected returns be HIGHER for an individual due to a HIGHER risk aversion?


# Beyond the mean-variance framework, can expected returns be HIGHER for an individual due to a HIGHER risk aversion?












In the mean-variance framework, the only way to get a higher expected return is to be exposed to a higher beta, and the more risk-averse an agent, the lower the beta of their portfolio (lending portfolios). However, could it be that a more risk-averse individual has a higher discount rate than a less risk-averse individual (i.e., the more risk-averse individual is less keen to provide financing for a particular venture). Or does the risk-return trade-off need to hold in all models that assume rationality and market efficiency (ie., higher expected returns are only achieved by higher risk exposure, given a certain level of aggregate risk aversion as in the mean-variance framework)?

## Answer by Richard Hardy (score 1)

https://quant.stackexchange.com/a/75246

The mean-variance framework is about optimal portfolio choice given the distribution(s) of asset prices/returns. On the other hand, the risk-return trade-off comes from an asset pricing model that produces the distribution(s). Therefore, the following is not quite right:

> In the mean-variance framework, the only way to get a higher expected return is to be exposed to a higher beta.

This follows from the asset pricing model, not the optimization framework. Hypothetically, if the asset pricing model implied higher risk assets had lower expected returns, exposure to a higher beta would not lead to a higher expected return – whether we use mean-variance optimization or not.

> However, could it be that a more risk-averse individual has a higher discount rate than a less risk-averse individual?

Risk aversion for a particular individual is characterized by their utility function, not the discount rate. The discount rate may characterize impatience, though. This is in the context of portfolio optimization, e.g. the mean-variance framework. Meanwhile, in an asset pricing model the discount rate characterizes the risk aversion of a representative individual in a market equilibrium.

> Or does the risk-return trade-off need to hold in all models that assume rationality and market efficiency?

The trade-off is due to the asset pricing model. Hypothetically, we could introduce an asset pricing model where the trade-off goes the other way (higher risk brings about lower expected return) or there is not trade-off at all.

Shown 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.