Mean-Variance as Expected Utility

1
Mean-Variance as Expected Utility

• Lecture 4

2
Expected Utility IV(1)

• Expected Utility/Expected Value-Variance

3
Expected Utility IV(2)
4
Expected Utility IV(3)

• Next, we want to complete the square in the
  exponent

5
Expected Utility IV(4)

• Elicitation of Risk Aversion Coefficients
• Equally Likely Certainty Equivalence (ELCE)
• Experience has shown that manipulation of
  probabilities while holding the consequences
  fixed is not very satisfactory.
• Thus, most methods involve changing the monetary
  gamble while keeping the probabilities fixed.
• One way to do this is to hold the gamble at 50/50
  or make the outcomes equally likely.

6
Expected Utility IV(5)

• In this method you cut a gamble by eliciting the
  decision makers certainty equivalence.

7
Expected Utility IV(6)

• Equally Likely but Risky Outcomes (ELRO) method
  is similar, but instead of varying the certainty
  equivalence, you form two lotteries and vary one
  of the payoffs until the decision maker is
  indifferent between the two lotteries.

8
Expected Utility IV(7)

• Other transformations of the Arrow-Pratt risk
  aversion coefficient.
• Following Raskin and Cochran, we turn to the use
  of published risk aversion coefficients.
• Rederiving the Arrow-Pratt measures

9
Expected Utility IV(8)
10
Expected Utility IV(9)

• This association between the risk aversion and
  the level of income then raises the question of
  the change in outcome scale.
• For example, what if the original utility
  function was elicited on a per acre basis, and
  you want to use the results for a whole farm
  exercise?
• Theorem 1 Let r(x)u(x)/u(x). Define a
  transformation of scale on x such that wx/c,
  where c is a constant. Then r(w)cr(x).

11
Expected Utility IV(10)
12
Expected Utility IV(11)

• Theorem 2 If vx c, where c is a constant,
  then r(v)r(x). Therefore, the magnitude of the
  risk aversion coefficient is unaffected by the
  use of incremental rather absolute returns.
• Example Suppose that a study of U.S. farmers
  gives a risk aversion coefficient of r.0001/
  (U.S.) Application to the Australian farmers
  whose dollar is worth .667 of the U.S. dollar is
  r.0000667/ Australian