The Arrow Pratt Coefficient

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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 the absolute change in his wealth (an extra €1,000 for instance), but does not consider the proportional change in wealth. An individual will not have the same behavior if we are considering a +0.1% or a +10% increase in wealth.

III.2. Arrow-Pratt Relative Risk Aversion

Decreasing absolute risk aversion means that the amount of wealth an agent is willing to put at risk increases as wealth increases; on the other hand, increasing relative risk aversion implies that the proportion of wealth he is willing to put at risk increases as wealth increases. Let us consider a constant absolute risk aversion; the amount of wealth put at risk remains the same as wealth increases. With constant relative risk aversion, it is the proportion of wealth that remains constant. As Milton and Savage indicated, an individual’s utility function may not have the same kind of curvature everywhere. The agent may switch from risk-aversion, to risk-loving and then back to risk-aversion. The figure below shows such a behavior.

At low income levels, agents are risk-averse, like they are at very high income levels. However, between the two inflection points of the curve, agents are risk-loving. They still gain a high utility from a monetary gain, but they have enough resources to be okay in case they lose. In order to take account the level of wealth in the risk-taking behavior, Arrow and Pratt came with this coefficient : Arrow-Pratt Relative Risk Aversion Coefficient = -w.(U''(w))/(U'(w))

Let us illustrate this measure using the log utility function (proposed by Bernoulli in 1713) : U(w)=ln⁡(w)
Then : U^' (w)=1/w and U^'' (w)=-1/w²
Absolute Risk Aversion Coefficient = -(U''(w))/(U'w)) = w
Relative Risk Aversion Coefficient = 1

III.3. Uses of Arrow-Pratt Risk Measurement Coefficient

This abstract measure can be used in several domains and industries. The first that comes to mind is the financial and investing sector. It can help firms predict...
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