Behavioral Finance Uncertain Choices February 18, 2014 Behavioral Finance Economics 437

Behavioral Finance Uncertain Choices Choices When Alternatives are Uncertain Lotteries Choices Among Lotteries Maximize Expected Value Maximize Expected Utility Allais Paradox

Behavioral Finance Uncertain Choices What happens with uncertainty Suppose you know all the relevant probabilities Which do you prefer? 50 % chance of $ 100 or 50 % chance of $ % chance of $ 800 or 75 % chance of zero

Behavioral Finance Uncertain Choices Lotteries A lottery has two things: A set of (dollar) outcomes: X 1, X 2, X 3,…..X N A set of probabilities: p 1, p 2, p 3,…..p N X 1 with p 1 X 2 with p 2 Etc. p’s are all positive and sum to one (that’s required for the p’s to be probabilities)

Behavioral Finance Uncertain Choices For any lottery We can define “expected value” p 1 X 1 + p 2 X 2 + p 3 X 3 +……..p N X N But “Bernoulli paradox” is a big, big weakness of using expected value to order lotteries So, how do we order lotteries?

Behavioral Finance Uncertain Choices “Reasonableness” Four “reasonable” axioms: Completeness: for every A and B either A ≥ B or B ≥ A (≥ means “at least as good as” Transitivity: for every A, B,C with A ≥ B and B ≥ C then A ≥ C Independence: let t be a number between 0 and 1; if A ≥ B, then for any C,: t A + (1- t) C ≥ t B + (1- t) C Continuity: for any A,B,C where A ≥ B ≥ C: there is some p between 0 and 1 such that: B ≥ p A + (1 – p) C

Behavioral Finance Uncertain Choices Conclusion If those four axioms are satisfied, there is a utility function that will order “lotteries” Known as “Expected Utility”

Behavioral Finance Uncertain Choices For any two lotteries, calculate Expected Utility II p U(X) + (1 – p) U(Y) q U(S) + (1 – q) U(T) U(X) is the utility of X when X is known for certain; similar with U(Y), U(S), U(T)

Behavioral Finance Uncertain Choices Allais Paradox Choice of lotteries Lottery A: sure $ 1 million Or, Lottery B: 89 % chance of $ 1 million 1 % chance of zero 10 % chance of $ 5 million Which would you prefer? A or B

Behavioral Finance Uncertain Choices Now, try this: Choice of lotteries Lottery C 89 % chance of zero 11 % chance of $ 1 million Or, Lottery D: 90 % chance of zero 10 % chance of $ 5 million Which would you prefer? C or D

Behavioral Finance Uncertain Choices Back to A and B Choice of lotteries Lottery A: sure $ 1 million Or, Lottery B: 89 % chance of $ 1 million 1 % chance of zero 10 % chance of $ 5 million If you prefer B to A, then.89 (U ($ 1M)) +.10 (U($ 5M)) > U($ 1 M) Or.10 *U($ 5M) >.11*U($ 1 M)

Behavioral Finance Uncertain Choices And for C and D Choice of lotteries Lottery C 89 % chance of zero 11 % chance of $ 1 million Or, Lottery D: 90 % chance of zero 10 % chance of $ 5 million If you prefer C to D: Then.10*U($ 5 M) <.11*U($ 1M)

Behavioral Finance Uncertain Choices So, if you prefer B to A and C to D It must be the case that:.10 *U($ 5M) >.11*U($ 1 M) And.10*U($ 5 M) <.11*U($ 1M)

Behavioral Finance Uncertain Choices First Mid Term Examination Thursday, Feb 20, 2014 Covers all reading listed on the syllabus Covers all lectures through Feb 11. No materials needed. Answers are written directly on the exam. No calculators, notes or anything else but something to write with, are permitted. There will be plenty of extra space available on the exam itself

Behavioral Finance Uncertain Choices The End