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