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Behavioral Finance

Chapter 1 Introduction &

Expected Utility Theory

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Neoclassical Economics

Fundamental assumptions 1. People have rational preferences across

possible outcomes or states of nature. 2. People maximize utility and firms

maximize profits. 3. People make independent decisions based

on all relevant information.

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Rational preferences

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Exercise Q1: When eating out, Rory prefers spaghetti over a

hamburger. Last night she had a choice of spaghetti and macaroni and cheese and decided on the spaghetti again. The night before, Rory had a choice between spaghetti, pizza, and a hamburger and this time she had pizza. Then, today she chose macaroni and cheese over a hamburger. Does her selection today indicate that Rory’s choices are consistent with economic rationality? Why or why not?

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Exercise Answer

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Utility maximization

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Utility maximization

Utility over money: u(w2) > u(w1) if w2 > w1 To arrive at her optimal choice, an individual

considers all possible bundles of goods that satisfy her budget constraint (based on wealth or income), and then chooses the bundle that maximized her utility As an example of a single good, utility ftns are

often defined in relation to wealth

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Utility function (u(w) = ln(w)) over wealth

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Added wealth at low income levels increases utility more than added wealth at high income levels

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Risk vs. Uncertainty

Says that individuals should act when confronted with decision-making under uncertainty in a certain way. – Normative: describes how people should

rationally behave – cf. positive: how people actually behave

Theory is really set up to deal with risk, not uncertainty: – What is the difference b/w risk and uncertainty?

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Risk vs. Uncertainty

Expected utility theory is set up to deal with risk: – Risk is when you know what the outcomes could

be, and can assign probabilities – Uncertainty is when you can’t assign

probabilities; or you can’t even come up with a list of possible outcomes Risk is measurable using probability, but uncertainty is not!

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Wealth outcomes Say there are a given number of states of the world:

– A. rain or sun – B. cold or warm leading to 4 states: e.g., rain and cold

And individuals can assign probabilities to each of these states: – Probability of rain+cold is .1, etc.

Say income (or wealth) level can be assigned to each state of world. Think of an ice cream vendor: – Rain+cold: $100/day – Sun+warm: $500/day

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Prospects Prospect: a series of wealth or income levels

and associated probabilities E.g.: $500 w/ probability .8 & $2,000 w/

probability .2 – P1(.8, 500, 2,000)

When 2nd option is zero, let’s write: – P2(.8, 500)

Expected utility theory comes from a series of assumptions (axioms) on these prospects: – Probability-weighted expected value of the different

possible utility levels

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Expected utility

Say one has to choose between two prospects. Based on assumptions such as ordering and

transitivity (and others), it can be shown that when such choices over risky prospects are to be made, people should act as if they are maximizing expected utility:

U(P) = pr A * u(wA) + (1-pr A) * u(wB) Can generalize to more than two outcomes: U(P) = pr A * u(wA) + pr B * u(wB) + pr C * u(wC)

+…

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1- 14 Transitivity and completeness of preferences over prospects

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Expected utility example

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Properties of utility functions

Certain properties of utility functions can be demonstrated: – Upward-sloping – Unique up to a positive linear transformation:

allows one to set u(lowest outcome)=0 and u(highest outcome)=1, which can be useful for proving certain things

Other properties such as differentiability (implying continuity) are often assumed.

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Risk Attitude

Most people avoid risk in most circumstances People are, however, willing to assume risk if

they are compensated for it. – E.g. choosing b/w two stocks w/ the same

expected return invest in the one w/ the lower risk

– If you take on a riskier investment, you will demand a higher return: trade-off b/w risk and return

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Risk aversion assumption

This comes from frequent observation that most people most of the time are not willing to accept a fair gamble: Would you be willing to bet me $100 that

you can predict a coin flip? – Most would say no. – And if one of you says yes, I will say no, since

I am risk averse. Risk aversion implies concavity of utility.

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Example A person is given the choice between two

scenarios, one with a guaranteed payoff and one without. – In the guaranteed scenario, the person receives $50. – In the uncertain scenario, a coin is flipped to

decide whether the person receives $100 or nothing.

– The expected payoff for both scenarios is $50, meaning that an individual who was insensitive to risk would not care whether they took the guaranteed payment or the gamble.

However, individuals may have different risk attitudes. 19

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Example (continued) risk-averse (or risk-avoiding) - if he or she

would accept a certain payment (certainty equivalent) of less than $50 (for example, $40), rather than taking the gamble and possibly receiving nothing. risk-neutral - if he or she is indifferent between

the bet and a certain $50 payment. risk-loving (or risk-seeking) - if the

guaranteed payment must be more than $50 (for example, $60) to induce him or her to take the guaranteed option, rather than taking the gamble and possibly winning $100.

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Example (continued)

The average payoff of the gamble, known as its expected value, is $50. The dollar amount that the individual would

accept instead of the bet is called the certainty equivalent The difference between the expected value and

the certainty equivalent: risk premium. i) for risk-averse individuals, it becomes positive,

ii) for risk-neutral persons it is zero, and iii) for risk-loving individuals their risk premium becomes negative. 21

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Expected utility of a prospect

E.g. Consider prospect P7: – P7(.4, 50,000, 1,000,000)

Use expected utility formula:: – U(P7) = 0.40u(50,000) + 0.60u(1,000,000) – Using logarithmic utility function:

U(P7) = 0.40(1.6094) + 0.60(4.6052) = 3.4069 Use expected value of wealth(w), or prospect

P7 – E(w)=0.4(50,000)+.6(1,000,000)=620,000= E(P7)

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Expected utility of a prospect

Consider prospect P7 (continued) – P7(.4, 50,000, 1,000,000)

Graph also shows utility of exp. value of the prospect: u(E(w)) = ln(62) = 4.1271

Utility of the expected wealth is greater than the expected utility of the prospect

u(E(w)) = u(62) = 4.1271 > U(P7)=3.4069 A Risk-averse person has a concave utility

ftn. u(E(P)) > U(P)

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Expected utility on graph

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Certainty equivalents A risk-averse is willing to sacrifice for

certainty. Certainty equivalent: that wealth level that

leads decision-maker to be indifferent between a particular prospect and a certain wealth level. We need to solve for w below: U(P7) = 3.4069 = u(w) w = 30.17 A risk seeker has a convex utility ftn. A risk neutral person has a linear utility ftn.

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A number of violations of expected utility have been discovered. Most famous is Allais paradox. Alternative theories have been developed

which seek to account for these violations. Best-known is prospect theory of Daniel

Kahneman and Amos Tversky.

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Exercise Q2: Consider a person with the following utility

function over wealth: u(w) = ew, where e is the exponential function (approximately equal to 2.7183) and w = wealth in hundreds of thousands of dollars. Suppose that this person has a 40% chance of wealth of $50,000 and a 60% chance of wealth of $1,000,000 as summarized by P(0.40, $50,000, $1,000,000).

a. What is the expected value of wealth? b. Construct a graph of this utility function. c. Is this person risk averse, risk neutral, or a risk seeker? d. What is this person’s certainty equivalent for the

prospect?

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Exercise Answer

A2: a. What is the expected value of wealth? E(w) = .4 * .5 + .6 * 10 = 6.2 U(P) = .4e0.50 + .6e10 = 13,216.54

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Exercise Answer A2: b. Construct a graph of this utility function.

The function is convex.

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Exercise Answer

A2: c. Is this person risk averse, risk neutral, or a risk seeker?

Risk seeker because graph is convex. d. What is this person’s certainty equivalent for

the prospect? ew = 13,216.54 gives w = 9.4892244 or

$948,922.44