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Lecture%2014%20-%20Choice%20Under%20Uncertainty%20II

Lecture%2014%20-%20Choice%20Under%20Uncertainty%20II - FIN...

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1 FIN 501 Financial Economics Lecture 14: Choice Under Uncertainty II Professor Nolan Miller
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2 Announcements Problem Set #4 is due today by 3pm. Problem Set #5 is available on Compass.  Due Oct. 19.
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3 Last time Leftovers from Capital Markets Lotteries Expectations Expected Utility
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4 This time Risk Aversion Certainty Equivalents Insurance Portfolio Choice Normal-Exponential Model and Mean-Variance  Utility Efficient Frontier in Mean-Variance Space Link to CAPM
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5 Risk aversion and utility for money If a DM is risk averse, this implies  something about the shape of u(w), the  DM’s utility for money. In particular, a risk averse DM will have  a strictly concave u(w). u’(w) > 0 u’’(w) < 0 This corresponds to decreasing marginal  utility of wealth. An additional $1 is worth more to me if I  am poor than if I am rich. “more concave”   more risk averse. w u(w) Risk averse
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6 Example Suppose u(w) = ln(w). This function is concave.  u’’(w) = -1/w 2 . You start with wealth $100. You face a lottery where you win $50 or lose $50 with equal probability. EV =  ½  (150) +  ½  (50) = 100 . Is the expected utility  with the lottery higher or lower than the utility of its  expected value, u(100)?
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7 Example, cont’d How do EU and EV compare? Utility of $100 for sure is ln(100)=4.61 Utility of ln(50) and ln(150) EU of the lottery is halfway between  ln(50 and ln(150). EU =  ½  ln(150) +  ½  ln(50)  4.46 By a fact of geometry, EU is also the  height where a line between (50,ln(50))  and (150,ln(150)) crosses EV! w u(w) 100 50 150 ln(100)=4.61 ln(50)=3.91 ln(150)=5.01 EU= 4.46
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8 Example, cont’d Note that ln(100) > EU. So, this DM prefers $100 for  sure to the lottery on $50 and  $150 This is true for any lottery and  concave function Next slides do same  graphical exercise for  Utility = u(w) Lottery = equal chance  to win, lose L w u(w) 100 50 150 ln(100)=4.61 ln(50)=3.91 ln(150)=5.01 EU= 4.46
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9 Example Suppose u(w) is strictly increasing and strictly concave. You start with wealth $w*. You face a lottery where you win $L or lose $L with equal probability. EV =  ½  (w *  + L) +  ½  (w *  - L) = w* . Is the expected utility  with the lottery higher or lower than its expected  value, w * ?
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10 Example, cont’d EU =  ½  ln(w *  + L) +  ½  ln(w *  - L) How do EU and EV compare? Utility of w *  for sure is u(w * ) Utility of u(w * -L), u(w * +L) EU of the lottery is halfway between  u(w * -L), u(w * +L).
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