# Entropic risk measure

Entropic risk measure is one possible risk measure to use in context of risk measurement as an alternative to other risk measures as value-at-risk or expected shortfall. In financial mathematics, the entropic risk measure is a risk measure which depends on the risk aversion of the user through the exponential utility function. This makes it a theoretically interesting measure because it would provide different risk values for different individuals. However, in practice it would be difficult to use since quantifying the risk aversion for an individual is difficult to do. The entropic risk measure is the prime example of a convex risk measure which is not coherent.[1] Given the connection to utility functions already, it is an obvious choice for the constraints in utility maximization problems.

## Mathematical definition

The entropic risk measure with the risk aversion parameter $\theta > 0$ is defined as

$\rho^{\mathrm{ent}}(X) = \frac{1}{\theta}\log\left(\mathbb{E}[e^{-\theta X}]\right) = \sup_{Q \in \mathcal{M}_1} \left\{E^Q[-X] -\frac{1}{\theta}H(Q|P)\right\} \,$[2]

where $H(Q|P) = E\left[\frac{dQ}{dP}\log\frac{dQ}{dP}\right]$ is the relative entropy of Q << P.[3]

## Acceptance set

The acceptance set for the entropic risk measure is the set of payoffs with positive expected utility. That is

$A = \{X \in L^p(\mathcal{F}): E[u(X)] \geq 0\} = \{X \in L^p(\mathcal{F}): E\left[e^{-\theta X}\right] \leq 1\}$

where $u(X)$ is the exponential utility function.[3]

## Dynamic entropic risk measure

The conditional risk measure associated with dynamic entropic risk with risk aversion parameter $\theta$ is given by

$\rho^{\mathrm{ent}}_t(X) = \frac{1}{\theta}\log\left(\mathbb{E}[e^{-\theta X} | \mathcal{F}_t]\right).$

This is a time consistent risk measure if $\theta$ is constant through time.[4]