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# da5 - Engineering Risk Benefit Analysis 1.155 2.943 3.577...

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DA 5. Risk Aversion 1 Engineering Risk Benefit Analysis 1.155, 2.943, 3.577, 6.938, 10.816, 13.621, 16.862, 22.82, ESD.72, ESD.721 DA 5. Risk Aversion George E. Apostolakis Massachusetts Institute of Technology Spring 2007

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DA 5. Risk Aversion 2 Calibration of utility functions We can apply a positive linear transformation to a utility function and get an equivalent utility function. π (x) = a U(x) + b a>0 A calibrated utility function is such that π (C * ) = 0 and π (C * ) = 1
DA 5. Risk Aversion 3 Example Suppose that it has been determined that U(x) = ln(x+5) for -4.5 x 4.5 (in \$ million) Let Let π (x) = aln(x+5) + b. Then, 1 = aln(7) + b and 0 = aln(3) + b to obtain a = 1.18 and b = -1.29 2 2 = = C and C

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DA 5. Risk Aversion 4
DA 5. Risk Aversion 5 The Buying Price for a Lottery It is the purchase price at which the DM is indifferent between the alternatives of buying the lottery and not buying it. Let π (x) the DM’s utility function with π (0) the utility of his present assets, i.e., before he buys the lottery. x 1 p 1 x m p m x 1 - BP p 1 p m x m - BP π = = π i i i ' ) BP x ( p ) L ( U ) 0 ( ' L L

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DA 5. Risk Aversion 6 Example π (x) = 1.18 ln(x+5) – 1.29 π (0) = 0.61 L(1, 0; 0.5, 0.5) 0.5 π (1-BP) + 0.5
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