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 $
200
25 % 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: X1, X2, X3,…..XN
A set of probabilities: p1, p2, p3,…..pN




Behavioral Finance
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)
Preferences Part I
Feb 16
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
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

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



B ≥ C then A ≥ 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



Or, Lottery D:



89 % chance of zero
11 % chance of $ 1 million
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


Behavioral Finance
.89 (U ($ 1M)) + .10 (U($ 5M)) > U($ 1 M)
Or
.10 *U($ 5M) > .11*U($ 1 M)
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
Behavioral Finance
Preferences Part I
Feb 16