Risk and Utility

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Risk and Utility

• Professor Walter Powell
• Derived from slides of Dr. Youngren

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• In a Reference Lottery, you can
• Vary the probabilities
• Vary the payoffs associated with the risk
• Vary the Certainty Equivalent
• In all cases, you must set all of the other
  values to find the one you want

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Risk - Introduction
Payoff
0.5
30
14.50
Game 1
0.5
- 1
0.5
2,000
50.00
Game 2
0.5
- 1,900
Figure 13.1
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Risk-Averse Utility Function
Minimum utility 0 _at_ -3000 Maximum Utility 1 _at_
4000 ? utility from -3000 to -2000 is greater
than from 3000 to 4000 Economics Law of
diminishing returns
Note the Concave curve - this denotes Risk Averse
- typical for most people
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Different Risk Attitudes
Different Risk Attitudes
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Investing in the Stock Market
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Investing Risk
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Certainty Equivalents and Utility

• A Certainty Equivalent is the amount of money you
  think is equal to a situation that involves risk.
• The Expected (Monetary) Value - EMV - is the
  expected value (in dollars) of the risky
  proposition
• A Risk Premium is defined as
• Risk Premium EMV Certainty Equivalent
• The Expected Utility (EU) of a risky proposition
  is equal to the expected value of the risks in
  terms of utilities, and EU(Risk)
  Utility(Certainty Equivalent)
• Take the expected value of Utilities not the
  Utilities of the Expected Values

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Finding a Certainty Equivalent
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Assessing Using Certainty Equivalents

• In a Reference Lottery, you can
• Vary the probabilities
• Vary the payoffs associated with the risk
• Vary the Certainty Equivalent
• In all cases, you must set all of the other
  values to find the one you want

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Eliciting a Utility Curve
CE 30 U(100) 1 and U(10) 0 therefore,
U(30) 0.5
CE 50 U(100) 1 and U(30)
0.5 therefore, U(50) 0.5(1) 0.5(.5) 0.75
30
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Eliciting a Utility Curve (Cont.)
30
CE 18 U(30) 0.5 and U(10) 0 therefore,
U(18) 0.5(.5) 0.25
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Utility function matching the two assumed bounds
(10 and 100) and the three points that were
elicited (the 25th, 50th, and 75th percentiles)
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The Exponential Utility Function

• We assume the utility function can match an
  exponential curve

• R will affect the shape of the exponential curve,
  making it more or less concave ? more or less
  risk averse, thus
• R is the risk tolerance
• There is an approximation that can be used to
  estimate the risk tolerance

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The Risk Assessment Lottery

• The highest value of Y at which you are willing
  to take the lottery (bet) instead of remaining
  with 0 loss or gain is approximately equal to R

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Alternate CE Calculations
From page 544 of Clemens, we have a lottery with
payoffs and probabilities . The Risk Tolerance
R is assumed to be 900.
Using either method, you must compute
and
to get
So we have alternative calculations
Or if using the exponential ftn
With calculators that have ln functions or Excel,
I think that the more precise answer is about as
easy to calculate.
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Axioms for Expected Utility

• Ordering and transitivity
• Reduction of compound uncertain events
• Continuity
• Substitutability
• Monotonicity
• Invariance
• Finiteness
• Unfortunately, people dont always obey axioms

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Equivalent under Utility Axioms
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Paradoxes

• Risk attitudes can change depending how you pose
  or frame the problem
• this a problem with polls of all kinds
• Utility curves can be different depending upon
  the status quo and where the lottery falls
  relative to the status quo
• People act irrationally (i.e., they do not choose
  the highest expected payoffs) under various
  conditions
• A common inconsistency is the Certainty Effect -
  too much weight on certain vs. uncertain outcomes

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Allais Paradox
Usually, A is preferred to B and D is preferred
to C, but the two together are inconsistent.
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Implications of Paradoxes

• People do not always choose rationally, even when
  utilities are used to elicit personal preferences
  for different choices
• This leads to various discussions among
  professionals about the best ways to elicit
  preferences - some prefer the PE vice the CE
  approach
• At other times, we have to account for other
  bad decision attitudes - valuing sunk costs,
  variations from status quo, unrealistic attitudes
  about risk, etc.