Entropic risk measure
Encyclopedia
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. Given the connection to utility functions already, it is an obvious choice for the constraints in utility maximization problem
s.
(the risk aversion parameter) is defined as
where
is the relative entropy of Q << P.
for the entropic risk measure is the set of payoffs with positive expected utility. That is
where
is the exponential utility function.
is given by
This is a time consistent risk measure if
is constant through time.
Risk measure
A Risk measure is used to determine the amount of an asset or set of assets to be kept in reserve. The purpose of this reserve is to make the risks taken by financial institutions, such as banks and insurance companies, acceptable to the regulator...
which depends on the risk aversion
Risk aversion
Risk aversion is a concept in psychology, economics, and finance, based on the behavior of humans while exposed to uncertainty....
of the user through the exponential utility
Exponential utility
In economics exponential discounting is a specific form of the discount function, used in the analysis of choice over time...
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. Given the connection to utility functions already, it is an obvious choice for the constraints in utility maximization problem
Utility maximization problem
In microeconomics, the utility maximization problem is the problem consumers face: "how should I spend my money in order to maximize my utility?" It is a type of optimal decision problem.-Basic setup:...
s.
Mathematical definition
The entropic risk measure with parameter (the risk aversion parameter) is defined as
where
is the relative entropy of Q << P.Acceptance set
The acceptance setAcceptance set
In financial mathematics, acceptance sets are a mathematical concept related to risk measures. As the name suggests an acceptance set is a set of acceptable future net worth, which are acceptable to the regulator.-Mathematical Definition:...
for the entropic risk measure is the set of payoffs with positive expected utility. That is
where
is the exponential utility function.Dynamic entropic risk measure
The conditional risk measure associated with dynamic entropic risk with risk aversion parameter is given byThis is a time consistent risk measure if
is constant through time.