Decision-making under risk
and uncertainty
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OUTLINE
 The
Standard Economic Model
 Prospect Theory
 Reference Points
 Loss Aversion
 Diminishing Marginal Sensitivity
 Weighted Probability
 Ambiguity Aversion
 The Endowment Effect
 Emotion
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THE STANDARD ECONOMIC MODEL
 The
standard economic model under risk is
expected utility theory (EUT).
Decision-making under risk can be considered as a
process of choosing between prospects. In general
terms a prospect can be described mathematically as
(x1, p1 ;….. ; xn,pn ). EUT states that consumers will
behave in such a way that they will maximize the
preference function
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PROSPECT THEORY

Prospect theory was originally developed in the KT
paper of 1979 and then extended in 1992. PT
models choice as a two-phase process: the first
phase involves editing, and the second involves
evaluation.
1. Editing. This phase consists of a preliminary
analysis of the offered prospects that aims to yield a
simpler representation of these prospects.
2. Evaluation. The decision-maker evaluates each of
the edited prospects, and is assumed to choose the
prospect with the highest value.
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PROSPECT THEORY

The first scale v assigns to each outcome x a
number v(x), which reflects the subjective value of
that outcome. This scale entails an explanation of
reference points, loss-aversion, and diminishing
marginal sensitivity .

The second scale π associates with each probability
p a decision weight π(p), which reflects the impact
of p on the overall value of the prospect. This scale
entails an explanation of decision weighting or
weighted probability function.
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PROSPECT THEORY

The mathematical exposition of the KT (1979)
model
Consider the simple prospects of the form (x,p;y,q)
1. Regular prospects
V(x, p; y, q)= π (p)v(x) + π(q)v(y)
2. Strictly positive or strictly negative prospects
In the editing phase, such prospects are segregated
into two components: the riskless component and the
risky component
V(x, p; y, q)= v(y)+ π(p)[v(x)-v(y)]
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PROSPECT THEORY - REFERENCE
POINTS

In PT outcomes are defined relative to a reference
point. Thus the scale v measures the value of
deviations from that reference point, that is gains and
losses.

It is often assumed in analysis that the relevant point
is the current status of wealth or welfare, but this
need not be the case.
The relevant reference point may be the expected
status rather than the current status.
•
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PROSPECT THEORY - REFERENCE
POINTS
Reference points are also strongly influenced by the
status of others.
•
The reference point may not correspond to the current
level of wealth when a person has not yet adapted to the
current status.
For example, imagine a person has already lost
2,000 and is now facing a choice between a sure gain of
$1,000 and an even chance to win 2,000 or nothing.
How will he code the problem?
•
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PROSPECT THEORY - LOSS
AVERSION
In the words of KT(1979) the aggravation that one
experiences in losing a sum of money appears to be
greater than the pleasure associated with gaining the
same amount
v(x) < -v(-x) where x > 0

Some empirical evidence
• Asymmetric price elasticities of demand for consumer
goods

•
Disposition effect (Shefrin and Statman 1985)
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PROSPECT THEORY - SHAPE OF THE
UTILITY FUNCTION
In PT KT proposed a utility function that featured
diminishing marginal sensitivities in the domains of
both gains and losses.

This type of function generally implies risk aversion
in the domain of gains and risk-seeking in the domain
of losses.
v’’(x) < 0 for x > 0 and v’’(x) > 0 for x < 0

Reflection effect. The phenomenon that the preference
between negative prospects is the mirror image of the
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preference between positive prospects.

PROSPECT THEORY - SHAPE OF THE
UTILITY FUNCTION

Empirical evidence
choose between (4000, 0.8) and (3000)
choose between (-4000, 0.8) and (-3000)
The important differences between PT function and
the Markowitz function
Markowitz proposed a utility function that has convex
and concave regions in both the gain and the loss
domains. Jullien and Salanie (1997) found that the
utility function for small amount of money was convex.11
Besides people may be risk averse for very large losses.

PROSPECT THEORY - DECISION
WEIGHTING

As with some of the previous elements of PT,
decision weighting also features in other theories
prior to the original KT paper.
There are two reasons why decision weights may be
different form objective probabilities.
1. Estimation of probabilities (objective probabilities
are unknown)
Two examples of situations where people are often
bad at estimating probabilities: rare events and
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conditional probabilities.

PROSPECT THEORY - DECISION
WEIGHTING
2. Weighting of probabilities (objective probabilities are
known)
π(p) measures the impact of events on the desirability
of prospects and not the perceived likelihood of these
events.

There are a number of important characteristics of the
weighting function that were observed by KT.
1. π is an increasing function of p, with π(0)=0 and
π(1)=1.
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PROSPECT THEORY - DECISION
WEIGHTING
2. Subadditivity. This characteristic relates to situations
where p is small

the prospect (6000, 0.001) versus (3000, 0.002)
the prospect (-6000, 0.001) versus (-3000, 0.002)

In general terms the subadditivity principle can be
expressed as follows:
π(rp) > r π(p) for 0<r<1
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PROSPECT THEORY - DECISION
WEIGHTING
3. Subcertainty. Allais (1953) noted that people tend to
overweight outcomes that are considered certain,
relative to outcomes that are merely probable.

the prospect (2400) versus (2500, 0.33; 2400, 0.66)
the prospect (2500, 0.33) versus (2400, 0.34)

In general terms the subcertainty principle can be
expressed as
π(p) + π(1-p) <1
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PROSPECT THEORY-DECISION
WEIGHTING

One main implication is that preferences are
generally less sensitive to variations in probability
than EUT would suggest.
4. Subproportionality. The decision weighting functions
violate the axiom of EUT.

(3000) to (4000, 0.8) and (3000, 0.25) to (4000, 0.2)

In general terms the subproportionality principle can
be expressed as
π(pq)/ π(p) < π(pqr)/ π(pr) 0< p, q, r <1
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PROSPECT THEORY - DECISION
WEIGHTING
For a fixed ratio of probabilities, the ratio of the
corresponding decision weights is closer to unity when
the probabilities are low than when they are high (0.25
is judged more similar to 0.2 than 1 is to 0.8).

KT (1992) proposed an inverted S-shaped weighting
functions for both gains and losses. It predicts riskseeking for gains of low probability and risk-aversion
for gains of high probability, with this pattern being
reversed for losses.

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AMBIGUITY AVERSION

In reality, the probability of many events is not
defined precisely and must usually be estimated
subjectively when making a decision.
When it comes to analyzing cases with undefined
probability distribution, the equivalent of the classical
utility hypothesis of Von Neumann and
Morgenstern(1944) is the theory of subjective utility
proposed by Savage (1954).

Ambiguity aversion: empirical psychological studies
suggest that most people try to avoid lotteries with
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undefined probability distribution.
AMBIGUITY AVERSION

People often take actions that run counter to the
theory of subjective utility. The most often cited
example of such irrationality is the Ellsberg’s paradox.

Ellsberg (1961). Subjects are presented with two urns.
Urn 1 contains 100 balls, some of which are red,
some blue. The ratio of blue and red balls is not
known. Urn 2 contains the total of 100 balls—50 red
and 50 blue. Respondents are asked to choose
(1) a1 versus a2 (2) b1 versus b2
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AMBIGUITY AVERSION

In both choices most people prefer to draw from the
urn with defined probability distribution choosing
lottery a2 and b2 . Such behavior is irrational.

Heath and Tversky (1991) suggest that the degree of
ambiguity aversion may depend on how competent a
decision maker feels in the field where he is supposed
to estimate probability. The higher qualifications he
thinks he has, the less concerned and more ready to
accept the ambiguous situation he will be.
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THE ENDOWMENT EFFECT

Status Quo bias: Samuelson and Zeckhauser (1988)
documented that preferences may be heavily
dependent on the status quo present when making a
decision. We are incredibly often reluctant to take
steps that would change the current situation. We are
biased toward maintaining the status quo even though
our preferences would be completely different if we
were to make the same choices without any
information on the current state of affairs.
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THE ENDOWMENT EFFECT

Related to the status quo effect is the so-called
endowment effect . It consists in people attaching
more value to things they currently have than to
identical objects that are not in their possession.

Kahneman et al. (1990) demonstrated that for the
owner of a thing the discrepancy between the sales
price (he would be willing to sell) and the purchase
prices (he would be prepared to pay) occurred not
because the suggested purchase price was deflated
but because owners inflated the sales price.
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THE ENDOWMENT EFFECT

Loewenstein and Kahneman (1991) argue that the
endowment effect does not result from assessing a
given thing as particularly attractive, but first and
foremost from the discomfort related to parting with
something we already received.
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EMOTION

Mood and Weather
Psychologists argue that the decisions we make may
be heavily dependent on emotional states. In general,
people in good mood are more optimistic in their
judgments and more willing to take risks.

The role of emotions in risk perception and decision
making was systematically expanded on by
Loewenstein et al. (2001). Dowling and Lucey (2005)
also provide an overview of literature on the role of
emotions and feelings in decision making.
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EMOTION

It is widely documented that mood depends on
weather conditions. More daylight means less
depression, less skepticism and more optimism,
happiness, and well-being.

Observations made by psychologists inspired studies
into weather’s influence on stock market returns.
•
Saunders (1993) proved that there is a statistical
relationship between the level of cloud cover over
New York City and changes of the Dow Jones
Industrial index as well NYSE/AMEX indexes.
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EMOTION
•
With total cloud cover, returns for the indexes were
usually below average. On sunny days (cloud cover
up to 20%), indexes usually increased more than on
average.
•
Hirshleifer and Shumway (2003) carried out more
comprehensive studies to analyze the influence of
weather on index changes on 26 stock markets. The
negative correlation between the level of could cover
and rates of return was basically confirmed on 22
stock markets, although with relatively low statistical
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significance.
EMOTION
•
One general remark about testing weather impact on
stock returns relates to the implicit assumption that
traders are geographically concentrated in the area
where the exchange is located and the weather is
observed.

Regret: Regret is a psychological reaction to making
a choice whose outcomes proved disadvantageous
(Bell, 1982). The feeling of regret will be especially
strong when it turns out that an alternative, previously
rejected in favor of the wrong decision, would have
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brought desired results.
EMOTION
We usually are less regretful of lost profits we have
not earned because we did not decide to act than of
the same losses suffered as a result of a wrong
decision.
 The feeling of regret will also be much greater if the
wrong decision is taken as an exception from rules or
habits normally adhered to.
 Regret will also be exacerbated if the wrong decision
was made individually. When this happens, one
cannot put the blame on anyone else or identify
external factors that could be made responsible for28
the failure.

EMOTION
Disappointment. An emotion that is close to regret
but usually has less of an impact on the decisionmaking process is disappointment. Disappointment is
experienced when the results of a choice fall short of
decision maker’s expectations.

Greed and Fear
Representatives of behavioral finance argue that
investors experience two strong, contradictory feelings
when making decisions in the capital market. Greed is
related to the prospect of enrichment and it is the main
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causative factor for accepting risky investments.

EMOTION
Fear is a negative feeling triggered by the possibility of
suffering a loss and it works in an opposite direction,
discouraging risky behavior. It is a sort of emergency
brake preventing investors from taking excessive risk
when pursuing profits.
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