Risk aversion#Measures of risk aversion
Named after Kenneth Arrow and John W. Pratt Arrow -Pratt measure is a measure of the risk aversion of a decision, wherein between the Arrow -Pratt measure of absolute risk aversion ARA (x ) and the relative risk aversion RRA (x ) to distinguish is.
The Arrow -Pratt measure of absolute risk aversion
Is a twice differentiable, monotonically increasing utility function, the Arrow -Pratt measure of absolute risk aversion ARA (x ) for this utility function is defined as follows:
Here, negative values imply risk appetite (risk affinity) and positive values of risk aversion (risk aversion). Finally accepts the level of zero, the decision maker is risk neutral.
A decision maker is risk-averse than another decision if and only if the basis of their utility functions and for their Arrow -Pratt Measures:
The derivation of the measure of absolute risk aversion ARA '(x)
Indicates the change in the risk adjustment with increased income. For example, if all possible income that may result from the decision situation is increased by a constant value, thus allowing a positive value of the derivative of ARA 's statement that the decision will strengthen its risk aversion or risk appetite, depending on the value of ARA, a negative value that he will act more risk averse or shy, and a value of zero indicates that the increase of all possible income does not affect his decision-making behavior.
Utility functions with constant absolute Risikoaversionsmaß ARA (x ) = k and its inverse functions include the functions
The Arrow -Pratt measure of relative risk aversion
The Arrow -Pratt measure of relative risk aversion is calculated as follows:
It corresponds to the utility elasticity of income potential, which expresses a change in the willingness to take risks with changing potential income from the decision. Is the measure of relative risk aversion constant, so the decision is his decision not change in a uniform, linear transformation of all possible income. A linear relative risk aversion is a decreasing or increasing risk aversion by increasing the potential gains, depending on whether the degree of RRA is negative or positive - the derivative of the RRA system is in operation.
Utility functions with constant relative Risikoaversionsmaß and their inverse functions include the functions
In the general case of constant relative risk aversion ( CRRA ) results in the isoelastic utility function.
Other properties
The Arrow -Pratt measures are invariant with respect to a positive linear transformation of the utility function and are therefore suitable for the Neumann / Morgenstern theory.