Behavioral Finance Economics 437 Behavioral Finance Expected Utility

Behavioral Finance Economics 437 Behavioral Finance Expected Utility Feb 19 2019

Choices When Alternatives are Uncertain n Lotteries n Choices Among Lotteries n Maximize Expected Value n Maximize Expected Utility n Allais Paradox Behavioral Finance Expected Utility Feb 19 2019

What happens with uncertainty n Suppose you know all the relevant probabilities n Which do you prefer? n 50 % chance of $ 100 or 50 % chance of $ 200 n 25 % chance of $ 800 or 75 % chance of zero Behavioral Finance Expected Utility Feb 19 2019

Expected Value Calculates the Average Value: n 50 % chance of $ 100 or 50 % chance of $ 200 Expected Value = ½ times $100 plus ½ times $200, which equals $ 150 n 25 % chance of $ 800 or 75 % chance of zero n Expected Value = ¼ times $ 800, which equals $ 200 n n These two have the same “expected value. ” Are you indifferent between them? Behavioral Finance Expected Utility Feb 19 2019

How to decide which to choose? n Would you simply pick the highest “expected value, ” regardless of how low the probability of success might be, e. g. n 1/10 th chance of $ 2, 000 or 9/10 th chance of zero has an “expected value” of $ 200. n Would you pick this over the two previous choices, both of which have an “expected value” of $ 200? n If you are still indifferent between the three choices, then you probably order uncertain choices by their expected value. Behavioral Finance Expected Utility Feb 19 2019

Bernoulli Paradox n Suppose you have a chance to play the following game: n n n You flip a coin. If head results you receive $ 2 n Expected value is $ 1 dollar Suppose you get to continue flipping until your first head flip and that you receive 2 N dollars if that first heads occurs on the Nth flip. Exp Value of the entire game is: n $ 1 plus ¼($4) plus 1/8($ 8) plus ………. n Infinity, in other words This suggest you would pay an arbitrarily large amount of money to play this flipping game Would you? Behavioral Finance Expected Utility Feb 19 2019

So, how do you resolve the Bernouilli Paradox? Behavioral Finance Expected Utility Feb 19 2019

This lead folks to reconsider using “expected value” to order uncertain prospects n Maybe those high payoffs with low probabilities are not so valuable n This lead to the concept of a lottery and how to order different lotteries Behavioral Finance Expected Utility Feb 19 2019

Lotteries n A lottery has two things: n A set of (dollar) outcomes: X 1, X 2, X 3, …. . XN n A set of probabilities: p 1, p 2, p 3, …. . p. N n n Behavioral Finance 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) Expected Utility Feb 19 2019

For any lottery n We can define “expected value” n p 1 X 1 + p 2 X 2 + p 3 X 3 +……. . p. NXN n But “Bernoulli paradox” is a big, big weakness of using expected value to order lotteries n So, how do we order lotteries? Behavioral Finance Expected Utility Feb 19 2019

“Reasonableness” n Four “reasonable” axioms: n Completeness: for every A and B either A ≥ B or B ≥ A (≥ means “at least as good as” n Transitivity: for every A, B, C with A ≥ B and n 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 n n n 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 Expected Utility Feb 19 2019

Conclusion n If those four axioms are satisfied, there is a utility function that will order “lotteries” n Known as “Expected Utility” Behavioral Finance Expected Utility Feb 19 2019

For any two lotteries, calculate Expected Utility II n p U(X) + (1 – p) U(Y) n q U(S) + (1 – q) U(T) n U(X) is the utility of X when X is known for certain; similar with U(Y), U(S), U(T) Behavioral Finance Expected Utility Feb 19 2019

Expected Utility Simplified n Image that you have a utility function on all certain prospects n If only money is considered, then: Utility Money Behavioral Finance Expected Utility Feb 19 2019

Assume that Utility Function n Has positive marginal utility n Diminishing marginal utility (which means “risk aversion”) Behavioral Finance Expected Utility Feb 19 2019

So, begin with a utility function that values certain dollars n Then consider a lottery n Calculate Average Utilities Lotteries involving $ 1 and $ 2 $1 Behavioral Finance $2 Expected Utility Feb 19 2019

If th Behavioral Finance Expected Utility Feb 19 2019

Now, try this: n Choice of lotteries n Lottery C n n n Or, Lottery D: n n n 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 Expected Utility Feb 19 2019

Back to A and B n Choice of lotteries n n n Lottery A: sure $ 1 million Or, Lottery B: n 89 % chance of $ 1 million n 1 % chance of zero n 10 % chance of $ 5 million If you prefer B to A, then n n Behavioral Finance . 89 (U ($ 1 M)) +. 10 (U($ 5 M)) > U($ 1 M) Or. 10 *U($ 5 M) >. 11*U($ 1 M) Expected Utility Feb 19 2019

And for C and D n Choice of lotteries n n Lottery C n 89 % chance of zero n 11 % chance of $ 1 million Or, Lottery D: n 90 % chance of zero n 10 % chance of $ 5 million If you prefer C to D: n Then. 10*U($ 5 M) <. 11*U($ 1 M) n Behavioral Finance Expected Utility Feb 19 2019

Allais Paradox n Choice of lotteries n Lottery A: sure $ 1 million n Or, Lottery B: n n 89 % chance of $ 1 million 1 % chance of zero 10 % chance of $ 5 million Which would you prefer? A or B Behavioral Finance Expected Utility Feb 19 2019

Now, try this: n Choice of lotteries n Lottery C n n n Or, Lottery D: n n n 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 Expected Utility Feb 19 2019

So, if you prefer n B to A and C to D n It must be the case that: n . 10 *U($ 5 M) >. 11*U($ 1 M) n And n . 10*U($ 5 M) <. 11*U($ 1 M) Behavioral Finance Expected Utility Feb 19 2019

The End Behavioral Finance Expected Utility Feb 19 2019