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Summary Behavioural Finance (FEB53108)
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Comprehensive summary of the minor Behavioural Finance (EUR)
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Week 1: Principles of rationality: expected utility theoremExpected utility
It is the product of probabilities times the utility of each outcome: ∑"!#$ 𝑝! 𝑢(𝑥! ). It can also
have a time dimension if the payoffs are received at different points in time:
∑"!#$ ∑&%#$ 𝛿% 𝑝!% 𝑢(𝑥!% ). The final wealth states matter, rather than just the outcomes of a
gamble. So, we add the initial wealth 𝑤 to any of the outcomes of the gamble when we
evaluate it
Expected utility theory (EUT)
A theory about how to make optimal decisions under risk. It is a normative theory, so it
describes how things should be, but not how things really are
Principles of economic rationality
Economic principles of rational choice are usually of three varieties:
- Explicit consistency principles (failures of axioms): behavior is not inconsistent with
normative preference axioms
- Implicit consistency principles (failures of invariance): things that shouldn’t matter don’t
matter
- Dominance principles: if one option is better than another (in an objective sense), people
choose the better one
Consistency vs. dominance principles
Consistency principles are examples of ‘formal rationality’. They are principles about the
consistency of people’s preferences and beliefs. Dominance principles are examples of
‘substantive rationality’, subject to values and an appeal to ethical norms. They are principles
about the content of people’s preferences and beliefs
Anomaly
An anomaly is a robust and systematic empirical finding that contradicts an economic
assumption or theory. An anomaly has to be:
- Robust, so the finding has been replicated by different researchers with different (but
related) examples
- Systematic, so the finding is not random but rather reveals behavior deviating from
rational choice theory in a particular direction
- An empirical finding, it is grounded in data collected in accordance with established
scientific standards
Axioms of rationality: transitivity
Given three alternatives to choose from, A, B and C, if A is preferred to B and B is preferred to
C, then A is preferred to C
Anomaly: intransitive indifference
The anomaly for transitivity is intransitive indifference, which means that if you are indifferent
for small changes, this would lead to being indifferent for a large change. However, transitivity
is not often violated
Axioms of rationality: stochastic dominance
Lottery A stochastically dominates lottery B if:
- For any outcome, x, lottery A gives at least as high probability as lottery B of receiving at
least x, and
- For some x, lottery A gives a higher probability of receiving at least x
If lottery A stochastically dominates lottery B, then lottery A is preferred to lottery B
Anomaly: ratio bias and stochastic dominance
The ratio bias is the tendency for people to judge a low probability event as more likely when
presented as a large-numbered ratio
,Anomaly: narrow framing and stochastic dominance
When a decision is split up, people make decisions that are not maximizing their utility when
these decisions are combined
Axioms of rationality: independence
Given three lotteries, A, B and C, and a constant 𝛼, where 0 < 𝛼 < 1, A is preferred to B if
and only if 𝛼𝐴 + (1 − 𝛼)𝐶 is preferred to 𝛼𝐵 + (1 − 𝛼)𝐶
Anomaly: the Allais paradox
A paradox that shows that in one situation people are risk averse and choose the save option,
and in another situation trade some risk for extra reward. This leads to a contradiction
Anomaly: the common ratio effect
The common ratio effect shows that people tend to change their decision when the
probabilities of winning two different lotteries are both scaled down by the same percentage
Axioms of rationality: probabilistic sophistication
A person’s beliefs about uncertain events can be represented by a unique subjective
probability distribution
Anomaly: Ellsberg’s paradox and ambiguity aversion
When people do not know the distribution of a probability, they tend to choose for the option
where they do know the distribution. This leads to a contradiction. Ambiguity aversion is an
aversion to choose alternatives that you have little information about
Axioms of rationality: description invariance
Given two ‘informationally equivalent’ descriptions of choice alternatives, A and B, if A is
preferred to B under one description, then A is preferred to B under the other. Three routes
to describing choice alternatives are via language, symbolic/mathematical notation and visual
presentation/imagery. Violations of description invariance are called framing effects
Axioms of rationality: context independence
Different forms of context independence:
- Choice set independence (independence of irrelevant alternatives)
- Default independence
- Position independence
- Independence of irrelevant information
Anomaly: the compromise effect
Compromise effect suggests that a product will have a higher chance to be chosen from a
product choice set when its attributes are not the extremes
Anomaly: the middle position effect
People tend to choose the option in the middle
Anomaly: attraction effect
The attraction effect refers to a phenomenon in which adding an irrelevant alternative into
an existing choice set increases the proportion of people choosing an alternative from the
original set
Anomaly: default effects
The default effect explains the tendency for an agent to generally accept the default option
in a strategic interaction
Axioms of rationality: consequentialism
Consequentialism says that only final outcomes (‘consequences’) matter. The implication is
that the utility of wealth level, w, today is the same regardless of whether w was higher
yesterday or lower yesterday
Anomaly: reference dependence and loss aversion
If someone gained €500 yesterday to bring his wealth to level w, he tends to feel different
than if he lost €500 yesterday
, Anomaly: Rabin’s paradox
Suppose a decision maker rejects the gamble (𝑤 − €100, 0.5; 𝑤 + €125, 0.5) at every wealth
level. Then if the decision maker is a risk-averse expected utility maximiser, the decision
maker will reject the following gamble: (𝑤 − €1000, 0.5; 𝑤 + €1 million, 0.5)
Axioms of rationality: risk aversion
When given a choice between any lottery L, and a certain amount of money €C such that €𝐶 =
𝐸(𝐿), a decision maker is risk-averse if the decision maker prefers C over L.
Anomaly: skewness preferences and the ‘fourfold pattern’
Skewness preference means that people avoid losses that are certain, and seek for getting
money that is certain. The fourfold pattern means that people tend to go for certainty when
it is either high probability gains or low probability losses, and tend to take risks when low
probability gains or high probability losses
Assumption Behavioral anomaly
Transitivity Non-transitive indifference
Stochastic dominance Non-transparent dominance, ratio bias
Independence axiom Allais paradox, common ratio effect
Probabilistic sophistication Ellsberg paradox
Description invariance Gain-loss framing effects, presentation effects
Context independence Compromise effect, attraction effect, default, middle position
Consequentialism Rabin’s paradox, loss aversion
Risk aversion Fourfold pattern
Week 1: Prospect theory
Formal presentation
Consider a binary lottery 𝑝, of the form (𝑥, 𝑝; 𝑦, 1 − 𝑝). The expected value of the lottery is
𝐸(𝑝) = 𝑝𝑥 + (1 − 𝑝)𝑦. The expected utility of the lottery is 𝑈(𝑝) = 𝑝𝑢(𝑥) + (1 − 𝑝)𝑢(𝑦).
The prospect theory value of the lottery is 𝑉(𝑝) = 𝑤(𝑝)𝑣(𝑥) + E1 − 𝑤(𝑝)F𝑣(𝑦) when either
𝑥 > 𝑦 ≥ 0 or 0 ≥ 𝑦 > 𝑥, or 𝑉(𝑝) = 𝑤(𝑝)𝑣(𝑥) + 𝑤(1 − 𝑝)𝑣(𝑦) when 𝑥 > 0 > 𝑦
Probability weighting function
𝑤(𝑝) is the probability weighting function and captures the idea that people tend to overreact
'!
to small probability events, but underreact to large probabilities. 𝑤(𝑝) = ('! )($*')! )"/!
where 𝛾 determines the curvature of the function
Value function
𝑣(𝑥) is the s-shaped value function and is asymmetrical because of loss aversion. 𝑣(𝑥) = 𝑥 ,
if 𝑥 ≥ 0 and 𝑣(𝑥) = −𝜆(−𝑥)- if 𝑥 < 0. Typically, 𝛼, 𝛽 ≤ 1 and 𝜆 > 1. 𝛼 represents risk
preferences for gains. 𝛽 represents risk preferences for losses. 𝜆 represents loss aversion. Two
widely used special cases are 𝛼 = 𝛽 = 1 and 𝛼 = 𝛽 < 1
Explained by prospect theory
The prospect theory can explain violations of independence, description invariance,
consequentialism and risk aversion. So, it can explain the Allais paradox, common ratio effect,
Ellsberg paradox, framing effects, Rabin’s paradox, loss aversion and the fourfold pattern
Emotion-based theory: regret theory
It is an alternative explanation for the Allais paradox. The basic idea is that people choose
options that avoid or reduce regret. Its limitations are that it violates transitivity and first order
stochastic dominance
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