Derivation of the Arrow Pratt Risk Aversion Coefficient.pdf - Derivation of the Arrow Pratt Risk Aversion Coefficient By Dale Nesbitt Draft 1 Enclosed

Derivation of the Arrow Pratt Risk Aversion Coefficient.pdf...

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Derivation of the Arrow Pratt Risk Aversion Coefficient By Dale Nesbitt Draft 1 Enclosed herein is the derivation of the Arrow-Pratt risk aversion coefficient and its reciprocal the risk tolerance. 1 Derivation of Indifference Probability from Any Given u-Curve This section harks back to Lecture 6 and the rules of actional though. Consider the deal for a more preferred prize A and a less preferred price B. If we have any u-curve u(.), the expected utility is indicated in the following figure p ( ) u A ( ) u C 1 p u = We know that the u-curve u can be subjected to an affine transformation without affecting the answer. In particular, the affine transformed u-curve v(.) will give the same answer as the original u curve u(.). The affine transformed u-curve is ( ) ( ) v x a bu x = + We want to set the constants a and b so that v(C)=0 and v(A)=1. In particular, we want to set a and b so that ( ) ( ) ( ) ( ) v A a bu A 1 v C a bu C 0 = + = = + = This is exactly what we did in Lecture 6 with the best and worst prizes. We manipulate the second of these two equations ( ) ( ) ( ) v C a bu C 0 a bu C = + = ⇒ =− Substitute this into the transformed u-curve v to write it ( ) ( ) ( ) ( ) ( ) v x bu C bu x b u x u C = + =
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Our intention to have v(A)=1 implies that ( ) ( ) ( ) ( ) ( ) v A 1 b u A u C 1 b u A u C = = =
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