behavioral finance uncertain choices february 18, 2014 behavioral finance economics 437
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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 $
200 25 % chance of $ 800 or 75 % chance of zero
Behavioral Finance Uncertain Choices
Lotteries
A lottery has two things: A set of (dollar) outcomes: X1, X2, X3,…..XN
A set of probabilities: p1, p2, p3,…..pN
X1 with p1
X2 with p2
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” p1X1 + p2X2 + p3X3 +……..pNXN
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 ExaminationThursday, 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