# On Exponential Utility and Conditional Value-at-Risk as Risk-Averse Performance Criteria

arXiv: https://arxiv.org/abs/2108.01771 · HTML: https://mlsanity.com/paper/2108.01771/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Aug 2021.

## Built on by (most cited first, 3 of 3)

- [Risk-averse autonomous systems: A brief history and recent developments from the perspective of optimal control](https://mlsanity.com/paper/2109.08947/) (2021)
- [Risk-averse Total-reward MDPs with ERM and EVaR](https://mlsanity.com/paper/2408.17286/) (2024)
- [Risk-Sensitive and Robust Model-Based Reinforcement Learning and Planning](https://mlsanity.com/paper/2304.00573/) (2023)

## Abstract (arXiv)

The standard approach to risk-averse control is to use the Exponential Utility (EU) functional, which has been studied for several decades. Like other risk-averse utility functionals, EU encodes risk aversion through an increasing convex mapping varphi of objective costs to subjective costs. An objective cost is a realization y of a random variable Y. In contrast, a subjective cost is a realization varphi(y) of a random variable varphi(Y) that has been transformed to measure preferences about the outcomes. For EU, the transformation is varphi(y) = exp(frac-θ2y), and under certain conditions, the quantity varphi-1(E(varphi(Y))) can be approximated by a linear combination of the mean and variance of Y. More recently, there has been growing interest in risk-averse control using the Conditional Value-at-Risk (CVaR) functional. In contrast to the EU functional, the CVaR of a random variable Y concerns a fraction of its possible realizations. If Y is a continuous random variable with finite E(|Y|), then the CVaR of Y at level α is the expectation of Y in the α · 100 % worst cases. Here, we study the applications of risk-averse functionals to controller synthesis and safety analysis through the development of numerical examples, with emphasis on EU and CVaR. Our contribution is to examine the decision-theoretic, mathematical, and computational trade-offs that arise when using EU and CVaR for optimal control and safety analysis. We are hopeful that this work will advance the interpretability and elucidate the potential benefits of risk-averse control technology.