The Arrow Pratt Coefficient
III. The Arrow-Pratt coefficient
Considering Bernoulli’s proposition that utility matters over wealth for risky behavior, and adding the fact that no two economical agents are alike, we can state that risk aversion can vary very widely across individuals. In this section we examine the coefficient determined by the two economists Kenneth arrow and John Pratt. In order to develop models for dealing with risk in business, economists need precise measurements which can be used in sectors such as investments, bank, insurance and many others.
III.1. Arrow-Pratt Absolute Risk Aversion
If we can specify the link between utility and wealth in a same function, this risk aversion coefficient would measure how much utility an agent gains (or losses) as he gains or losses wealth. To determine this, we would instinctively appeal to the first derivative of the utility function U’(w). However, as utility functions are not unique –we will discuss this in details later, derivative functions are not unique either, and thus it is not possible to compare risk aversion between the utility curves of two different individuals.
Thus, Arrow and Pratt looked at the second derivative of the utility function, which measures how the change in utility itself changes as a function of the wealth level, and divide it by the first derivative to obtain a risk-aversion coefficient.
Arrow-Pratt Absolute Risk Aversion Coefficient = -(U''(w))/(U'(w))
This number shall be positive for risk-averse investors and increase with risk aversion. On the contrary, it will be positive for agents with risk-proclivity (convex utility function, and thus positive second derivative). This coefficient enables comparison between individuals with different utility functions. Thus, for instance, an insurance company can set up a pricing strategy according to the risk aversion and the risk premium of its customers.
Though, this coefficient helps to understand the reaction of an individual according to
Considering Bernoulli’s proposition that utility matters over wealth for risky behavior, and adding the fact that no two economical agents are alike, we can state that risk aversion can vary very widely across individuals. In this section we examine the coefficient determined by the two economists Kenneth arrow and John Pratt. In order to develop models for dealing with risk in business, economists need precise measurements which can be used in sectors such as investments, bank, insurance and many others.
III.1. Arrow-Pratt Absolute Risk Aversion
If we can specify the link between utility and wealth in a same function, this risk aversion coefficient would measure how much utility an agent gains (or losses) as he gains or losses wealth. To determine this, we would instinctively appeal to the first derivative of the utility function U’(w). However, as utility functions are not unique –we will discuss this in details later, derivative functions are not unique either, and thus it is not possible to compare risk aversion between the utility curves of two different individuals.
Thus, Arrow and Pratt looked at the second derivative of the utility function, which measures how the change in utility itself changes as a function of the wealth level, and divide it by the first derivative to obtain a risk-aversion coefficient.
Arrow-Pratt Absolute Risk Aversion Coefficient = -(U''(w))/(U'(w))
This number shall be positive for risk-averse investors and increase with risk aversion. On the contrary, it will be positive for agents with risk-proclivity (convex utility function, and thus positive second derivative). This coefficient enables comparison between individuals with different utility functions. Thus, for instance, an insurance company can set up a pricing strategy according to the risk aversion and the risk premium of its customers.
Though, this coefficient helps to understand the reaction of an individual according to