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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 DA 5. Risk Aversion 1 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 2 Example • Suppose that it has been determined that U(x) = ln(x+5) for C∗ = 2 and • Let -4.5 ≤ x ≤ 4.5 (in $ million) C∗ = −2 • 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 DA 5. Risk Aversion 3 DA 5. Risk Aversion 4 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. p1 x1 p1 L x1- BP L' pm pm xm π(0) = U(L') = ∑ p i π( xi − BP ) xm- BP i DA 5. Risk Aversion 5 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 π(0-BP) = 0.61 • 1.18 ln(6-BP) – 1.29 + 1.18 ln(5-BP) – 1.29 = 1.22 • ln(6-BP) + ln(5-BP) = 3.22 ⇒ DA 5. Risk Aversion 6 Example (cont’d) • ln[(6-BP) (5-BP)] = 3.22 • (BP)2 – 11(BP) + 30 = exp(3.22) = 25.04 • (BP)2 – 11(BP) + 4.96 = 0 • BP = 0.47 (the other root is 10.53 and is rejected) • EMV = 1x0.5 + 0x0.5 = 0.5 > 0.47 ⇒ risk aversion DA 5. Risk Aversion 7 The Selling Price of a Lottery • SP = CE (certainty equivalent) • We now interpret x = 0 to represent the DM’s total assets except of the lottery. • Utility of present wealth (situation) is π(L). • π(x) = 1.18 ln(x+5) – 1.29 ⇒ π(0) = 0.6091 DA 5. Risk Aversion 8 Example ⇒ • π(x) = 1.18 ln(x+5) – 1.29 π(0) = 0.6091 • L(1, 0; 0.5, 0.5) • π(L) = 0.5 π(1) + 0.5 π(0) = 0.5x0.8243 + 0.5x0.6091 • π(L) = 0.7167 = π(CE) ⇒ ⇒ Using the figure on p. 4 we get • CE ≅ 0.4771 < EMV = 0.5 DA 5. Risk Aversion 9 Risk Aversion • The DM is always willing to sell any lottery for less than its expected monetary value. • L(x1, x2; p1, p2) EMV = p1 x1 + p2 x2 • U(p1 x1 + p2 x2) = U(EMV) > p1 U(x1)+ p2 U(x2) • A DM with a concave utility for money will refuse a monetarily fair bet and is said to be risk averse. DA 5. Risk Aversion 10 Example • L(1, 0; 0.5, 0.5) • EMV = $0.5 M ⇒ • 0.5 π(1) + 0.5 π(0) = 0.7167 ⇒ π(0.5) = 0.6091 SP = $0.4771 • Risk Premium ≡ EMV – CE = 0.5 – 0.4771 ⇒ • RP = $0.0229 M DA 5. Risk Aversion 11 Risk Aversion and Insurance • You own a house worth $500,000. • A fire (p = 10-2 per year) may destroy it completely. • Your utility function is • π(x) = 1.18ln(x+5) – 1.29; C* = $2M, C* = -$2M • How much premium would you be willing to pay? DA 5. Risk Aversion 12 Decision Tree p -x I 1-p NI p -x -$500,000 1-p 0.0 DA 5. Risk Aversion 13 Utilities • π(I) = π(-x) π(NI) = p π(-0.5) + (1- p) π(0.0) • π(-0.5) = 1.18ln(4.5) – 1.29 = 0.4848 • π(0.0) = 0.6091 • π(NI) = 0.4848 x 10-2 + (1- 10-2) x 0.6091 = 0.6078 • π(-x) = 0.6078 ⇒ x = $5,661 DA 5. Risk Aversion 14 Your Perspective • You are willing to pay up to $5,661 to insure your house. • The expected loss is 10-2 x $500,000 = $5,000 < $5,661 ⇒ You are willing to pay more than the expected loss because you are risk averse. DA 5. Risk Aversion 15 The Company’s Perspective • Why would the insurance company agree to insure you? • The company may win $5,661 with probability 0.99 or lose $494,339 with probability 10-2. DA 5. Risk Aversion 16 The Company’s Decision Tree p - 494,339 Accept I 5,661 1-p Do not Accept I p 0.00 1-p 0.00 DA 5. Risk Aversion 17 The Company’s Perspective (2) • The company has $1 billion dollars in assets. Its wealth is either $999,505,661 with probability 10-2, or $1,000,005,661 with probability 0.99. • For such small changes, the company’s utility of money is linear, i.e., the company makes decisions using the EMV. • EMV = 5,661 x 0.99 - 494,339 x 10-2 = $661 DA 5. Risk Aversion 18 The Company’s Perspective (3) • The alternative is for the company to refuse the premium, in which case the EMV is zero. • The company should agree to insure the house. • Note: The company’s overhead expenses have not been factored in. DA 5. Risk Aversion 19 Assessment Using Certainty Equivalents 1. Set U(C*) = 0 and U(C*) = 1 2. Consider the reference lottery L1(C*, C*; 0.5, 0.5) • Its certainty equivalent is derived by solving U(CE1) = 0.5 U(C*) + 0.5 U(C*) = 0.5 DA 5. Risk Aversion 20 Assessment Using Certainty Equivalents (2) Thus, we have found a third point of the utility function, that with utility 0.5. 3. Repeat the process with a new reference lottery L2(C*, CE1; 0.5, 0.5) Solve U(CE2) = 0.5 U(C*) + 0.5 U(CE1) = 0.75 to get a fourth point of the utility function, CE2. DA 5. Risk Aversion 21 Assessment Using Certainty Equivalents (3) 4. Repeat using L3(CE1, C*; 0.5, 0.5) ⇒ U(CE3) = 0.5 U(CE1) + 0.5 U(C*) = 0.25 to get a fifth point, and so on. DA 5. Risk Aversion 22