Risk Aversion - Quantum Resources Site

Abstract¶

A brief overview of risk aversion, including measures of risk aversion and it’s meaning.

Keywords:UtilityRiskRisk Aversion.¶

Risk aversion can be thought of as an agents attitude toward risk. More formally, it models the tendency of people to prefer outcomes will low uncertainty to those with high uncertainty, even if the average payback from the higher uncertainty outcome is equal to or larger than the payback from the low uncertainty outcome.

People will often agree to take an option that has low uncertainty and lower payoff, then high uncertainty but high payoff. For example, one could put their money in a bank account with reliable but low interest returns. Or, one could put their money into a stock that is volatile, but has a potentially high return.

Risk aversion is encoded into the curvature of the utility function:

If $u(w)$ is concave then the agent being modelled is risk-adverse.
If $u(w)$ is linear then the agent being modelled is risk-neutral.
If $u(w)$ is convex then the agent being modelled is risk-seeking.

The higher the curvature of the utility function, the more the agent is either risk averse/risk seeking.

Whilst utility functions introduce the notion of risk, the concept of risk aversion allows one to quantify it.

Risk Aversion Measures¶

Relative Risk Aversion

Absolute Risk Aversion

Let $u_{R}(w)$ be some utility function.

Relative Risk Aversion (RRA) of $u_{R}(w)$ is given by

\textrm{RRA} u_{R}(w) \coloneqq \frac{-w u_{R}''(w)}{u_{R}'(w)},

(1)

where $u_{R}''(w)$ and $u_{R}'(w)$ are the second and first derivatives of the utility function with respect to $w$ .

Constant Risk Aversion¶

Isoelastic utility functions are functions that are unique in having constant RRA. This means that an agents attitude toward risk would not change if both the potential upsides and downsides were multiplied by some factor (meaning the ratio of upside to downside is constant).

Exponential utility are functions that are unique in having constant ARA. This means that an agents attitude toward risk would not change if both the potential upsides and downsides were linearly scaled by some factor. More formally, this means an agents risk is independent of his liabilities (if the total amount of wealth an agent has changes, there attitude towards risk does not).

Constant ARA is considered to be unrealistic, with people taking less risk if the downside is losing a larger portion of there limited wealth, and more risk if th

The difference between two agents with these risk profiles can be demonstrated through the following couple of examples.