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This note explains the classic Arrow-Pratt measure of risk aversion. The standard reference is Pratt (1964) and Arrow (1965). 1 Measures of risk aversion. Consider an agent with expected utility function E[u(w)], where w is wealth and u : R++ ! R is a utility function with u0 > 0 (increasing) and u00 < 0 (concave).
An agent is risk-averse if, at any wealth level w, he or she dislikes every lottery with an expected payoff of zero: ∀ w , ∀˜ z with E ˜z = 0, Eu(w +˜ z) u(w) .
It is sometimes useful to quantify the degree of risk aversion. There are two important measures of risk aversion. The first one is absolute risk aversion: r ((A. x)=−u x )/u ( , which is also called Arrow-Pratt coefficient of absolute risk aversion. Note that u measures the concavity of the utility function, while u normalizes the concavity ...
The Arrow–Pratt index of relative risk aversion combines the important economic concepts of elasticity and marginal utility. The index is used by many authors writing in relation to utility theory.
Arrow-Pratt measure of relative risk aversion: Where x is the payoff of a given lottery and U(x) the utility derived from that payoff. In this LP we’ve started by seeing what the difference between risk and uncertainty is.
Agent B has the following utility: U(Potatoes) = (Potatoes)0.99 U (P o t a t o e s) = (P o t a t o e s) 0.99 (So degree of RRA=0.99). For this agent, more potatoes is always better. In the deterministic case, the utility function represents preferences over certain outcomes.
The most common and frequently used measure of risk aversion are the Arrow-Pratt measures of absolute and relative risk-aversion. Named after John W. Pratt’s paper “Risk Aversion in the Small and in the Large”, 1964, and Kenneth Arrow ’s “The Theory of Risk Aversion”, 1965, these are the measures:
