Hyperbolic Absolute Risk Aversion

What is Hyperbolic Absolute Risk Aversion?

Hyperbolic Absolute Risk Aversion (HARA) is a property of certain utility functions that makes the inverse of an individual's level of risk aversion (their risk tolerance) a linear function of their total wealth. It is generally assumed that this also means a positive relationship, i.e. that risk aversion decreases as total wealth increases. HARA is used in financial modeling to conveniently model investors' choices to hold risk free or risky assets in their portfolios, though this is not necessarily true for all HARA utility functions. 

Key Takeaways

  • Hyperbolic Absolute Risk Aversion (HARA) describes a family of utility functions where individuals' tolerance for risk is proportional to their wealth level. 
  • HARA utility functions provide a convenient and mathematically tractable tool for modeling investor choice between risky and risk-free assets. 
  • HARA does not necessarily represent an accurate picture of how people actually make choices with respect to risk, but provides a simple way to understand how they can be modeled.

Understanding Hyperbolic Absolute Risk Aversion

ARA is a means of measuring risk avoidance via a convenient mathematical equation. If all investors are assumed to have similar utility functions, then the equation predicts that each investor holds the available basket of risky assets in the same proportions as all others, and that investors differ from each other in their portfolio behavior only with regard to the fraction of their portfolios held in the risk-free asset rather than in the basket of risky assets. Hyperbolic absolute risk aversion is part of the family of utility functions originally proposed by John von Neumann and Oskar Morgenstern in the 1940s. Like their other theorems, HARA assumes that investors are rational, which is expressed as a desire to maximize final payouts while mitigating risk.

Similar to other mathematical utility and optimization methods, HARA provides a framework for economists and analysts to model different investor behaviors as well as assess the impact of various decisions. What's more, HARA can be used on a wide array of financial and non-financial problems. As with most mathematical methods, hyperbolic absolute risk aversion works best when one's investment objectives are clearly defined.

What makes HARA unique is that it assumes that an investor holds either the risk-free asset (in the U.S. this typically is short-term Treasuries), or else the basket of all available risky assets in varying allocation proportions. Thus, somebody who is extremely risk averse under the hyperbolic absolute risk aversion framework holds 100% in the risk-free asset. At the other end of the spectrum, a completely risk-seeking person invests 100% in the basket of all risky assets. Those with risk aversion levels in between will have more or less risky assets, with a greater proportion assigned to those with more risk tolerance. Furthermore, the increase in the risky asset given a person's increasing risk tolerance in relation to their utility function will be linear in fashion under HARA (under the assumption that the person is rational and also has a linear utility function).

HARA assumptions for risk tolerance can be incorporated with the capital asset pricing model when using a representative utility function that is the same for all investors and only varies with changes in wealth.

Like most financial models, the HARA framework is not meant to be an accurate depiction of reality and how people really allocate to risky assets. Rather, it is meant as a simplification to help better understand a far more complex world.

Article Sources
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  1. John von Neumann, Oskar Morgenstern. "Theory of Games and Economic Behavior." Princeton University Press, 1944.

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