Risk Premium, Risk Aversion, and Portfolio Utility
Summary
The document introduces portfolio risk premium as compensation investors may require for accepting uncertain returns instead of a stable cash flow with the same average return. It explains this preference through risk aversion: an investor may give up some expected gain to obtain more stable outcomes. In utility terms, risk aversion is represented by a concave utility function, with Jensen’s inequality connecting variability in outcomes to lower expected utility.
The answer describes the underlying rationale as a model of human preferences rather than a mathematical proof that variance is universally undesirable. Different investors can have different risk aversion and care about different forms of risk. It mentions the Sharpe ratio as a common way to assess average compensation per unit of volatility. The note does not derive a portfolio pricing model or specify how to estimate a premium; its explanation is introductory, and volatility alone may not capture every risk that matters to an investor.
Key ideas
- A risk premium can compensate investors for accepting uncertain rather than stable returns.
- Risk aversion reflects a preference for stability and is represented by concave utility.
- Jensen’s inequality helps explain why variability can reduce expected utility for a risk-averse investor.
- The desirability of risk depends on investor preferences rather than a universal rule about variance.
- The Sharpe ratio summarizes average return relative to volatility, but does not describe every risk dimension.
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Full text
# What is risk premium of a portfolio? # What is risk premium of a portfolio? I am not an expert in the field. So bear with me if my terminology is bad. I want to understand what risk premium of a portfolio is. I understand that there are different forms of risk. The basic idea seem to stem from the following basic difference. A portfolio which gives a fixed return of 5%, is considered different to a portfolio which gives a mean of 5% but has variance. Since variance is not "desired", the latter needs a premium return to match the former portfolio. - Is this understanding correct? - Is there any mathematical basis in saying that variance is not "desired"? Or is it purely psychological? (based on individual risk aversion, utility functions and so on). - If the reasoning is purely psychological, should we logically factor in this premium when creating a portfolio? ## Answer by Stéphane (score 1, accepted) https://quant.stackexchange.com/a/51441 - This intuition is correct. Formally, we consider that people are risk averse which is just another way of saying that they prefer more stable to less stable cash flows. Another equivalent way of saying this is that they are disposed to sacrifice some gains on average for the added stability. - The fundamental reason is indeed entirely psychological. We approximate the tendency of people to require compensation for accepting more fluctuation in their portfolio through risk aversion. In terms of utility functions, per the inequality of Jensen, that implies a concave utility function. - Normal human beings tend to care about how much and what type of risk they are taking with whatever portion of their wealth they have invested in a given portfolio. This is often done using Sharp ratios: people trying to get a sense of how much on average you'd get paid to take "units" of risk, understood as volatility.
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