Behavioral Finance Preferences Part I Feb 16 Behavioral Finance Economics 437

Behavioral Finance Preferences Part I Feb 16 Choices When Alternatives are Uncertain Lotteries Choices Among Lotteries Maximize Expected Value Maximize Expected Utility Allais Paradox

Behavioral Finance Preferences Part I Feb 16 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 Preferences Part I Feb 16 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 Preferences Part I Feb 16 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 Preferences Part I Feb 16 “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 Preferences Part I Feb 16 Conclusion If those four axioms are satisfied, there is a utility function that will order “lotteries” Known as “Expected Utility”

Behavioral Finance Preferences Part I Feb 16 For any two lotteries, calculate Expected Utility 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 Preferences Part I Feb 16 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 Preferences Part I Feb 16 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 Preferences Part I Feb 16 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 Preferences Part I Feb 16 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 Preferences Part I Feb 16 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 Preferences Part I Feb 16 Oops Choose Between: A: A sure gain of $ 240 B: 25 % gain of $ 1,000 and 75 % chance to gain nothing Choose Between: C: A sure loss of $ 750 D: 75 % chance to lose $ 1,000 and 25 % chance to lose nothing If you chose A & D: 25 % gain of $ 240 and 75 % chance to lose $ 760 {B & C} dominates {A & D} 25 % gain of $ 250 and 75 % chance to lose $ 760

Behavioral Finance Preferences Part I Feb 16 The End