Summary Behavioral finance HW bach2

LES 1: EXPECTED VALUE & EXPECTED UTILITY THEORY

 EXPECTED VALUE
o Probability
= a number between 0 and 1 that indicates a likelihood that a particular outcome will occur,
0 means the event is impossible, 1 means it is certain (bv: flipping a coin is 0,5)
 the probability of all possible events sum to 1
 binary prospects with two prospects (x , y) and probability (p):
 ∼ = indifference
 ≻ = strict preference

o Expected value
= the value of each possible outcome times the probability of that outcome
= 𝐸𝑉(𝑥, 𝑝; 𝑦) = 𝑝𝑥 + (1 – 𝑝)𝑦
= 𝐸𝑉(𝑥𝑖, 𝑝𝑖) = ∑ p i x i
Bv: investment x considered by an investor: buying some shares cheaply, then resell them at higher
price. Assume one share costs 30p and there are three possible outcomes, each p≈0.33:
a) Shares close at flotation cost : 30p ➔ investor gets nothing (0 p payoff)
b) Shares close at 60 p ➔ investor gets 30p payoff
c) Shares close at 90 ➔ investor gets 60p payoff
=> EV: 0.33*0 + 0.33*30 + 0.33*60 ≈ 30p (Expected value is in term of payoffs not closing prices!)
=> This is a “fair bet” : the cost is equal to the expected value



o St Petersburg paradox
= A coin is tossed. If it comes up heads, you are paid €2. Then the coin is tossed again. If it comes up
heads again, you are paid €4= 22; and so on. When the coin comes up tails the game is over.
 overall people want max to pay €25 to play this gamble

 expected value is infinite:

 but still some people don’t want to play… -> expected
utility



o Expected utility (a solution to the St. Petersburg paradox by Daniel Bernoulli)
= the satisfaction or pleasure a person derives from consuming a good, service, level of wealth
(ex: the first euro’s winning means more than the last ones// poor people get more satisfaction from
winning money)
 Decreasing marginal utility: utility increases as consumption increases but at a diminishing rate =

, EXPECTED UTILITY = 𝐸𝑈(𝑥𝑖, 𝑝𝑖)= ∑ p iU ( x i )
 note difference with expected value = 𝐸𝑉(𝑥𝑖, 𝑝𝑖) = ∑ p i x i
 where the assumption is that U′(x) > 0 and U′′(x) < 0 = utility increases in outcomes (money),
but at a decreasing rate
 u(x) = ln(x) would solve the St. Petersburg paradox

o 4 axioms:
1) Completeness: a decision maker has defined preferences can always decide:
= either x ≽ y or x ≼ y ∀ x,y
2) Transitivity: a decision maker’s preferences are consistent:
= if x ≽ y and y ≽ z then x ≽ z
3) Continuity: there exists a probability p where y is equally good as px + (1 − p)z
= if x ≽ y ≽ z then ∃ p such that y ∼ px + (1 − p)z
4) Independence: An indifference between two prospects holds also if both prospects are mixed with a
common third prospect or outcome Z: x ≽ y implies (x,p;Z) ≽ (y,p;Z) ∀ p,Z meaning that if you prefer x
to y, adding or mixing another prospect should not change your preference

o Attitudes towards risk
 expected utility of a prospect will be smaller than the utility of the expectation

= concavity of the utility function and risk aversion
imply each-other

 utility is given by a linear combination of the
utilities of the outcomes

 C = certainty equivalent
 RP = EV – ce = risk premium


 Certainty equivalent
= to find the sure amount of money that makes a decision maker indifferent between playing the
prospect and obtaining that amount (that’s the amount that makes you indifferent from playing or
taking the money)
=
= U(CE) = W(p) * UX + (1-W(p)) * UY --------> met W(p) = (CE-Y)/(X-Y)

 if ce < 𝐸𝑉(𝑥𝑖 , 𝑝𝑖) = Σ𝑖 𝑝𝑖 𝑥𝑖 : the agent is risk averse, RP = EV - ce > 0
 if ce = 𝐸𝑉(𝑥𝑖 , 𝑝𝑖) = Σ𝑖 𝑝𝑖 𝑥𝑖 : the agent is risk neutral
 if ce > 𝐸𝑉(𝑥𝑖 , 𝑝𝑖) = Σ𝑖 𝑝𝑖 𝑥𝑖 : the agent is risk seeking (loving), RP = EV - ce < 0




 Someone risk averse would reject a fair gamble
 Someone risk seeking would accept a fair gamble

, LES 2: EXPECTED UTILITY THEORY PARADOXES

 INSURANCE AND GAMBLING
 clarification that EUT could explain coexistence of gambling and insurance
 a utility function that has ‘double inflection point’ = concave, convex, concave again
 concave part: below some target wealth w0, investors
display a diminishing marginal utility in wealth.
 convex part: above some target income w0, investors
display an increasing marginal utility in wealth; every
additional € makes them happier
 at wealth level w0 is person expected to both gamble, and
to take out insurance

=> The main argument: convex part may propel individuals from one social class to another
while concave part keeps the social class unchanged.
=> The marginal utilities of moving the social ladder is increasing,
while the marginal utility of increase in wealth in the same social class in decreasing

 EUT AND FINAL WEALTH
 Markowitz found that preferences changed systematically depending on the amount at stake
 for losses: agents prefer paying a small sure amount to avoid a larger loss, but this choice
pattern tends to invert as outcomes are scaled up in absolute terms
 this effect is INDEPENDENT of the initial wealth of the respondents
=> conclusion: utility shouldn’t be defined over wealth/income, but over changes in wealth

 w(0) = current wealth

A: Risk seeking for small stakes
B: Risk averse for large stakes
C: Risk averse for small losses
D: Risk seeking for large losses

,  FRAMING EFFECT
= when equivalent descriptions of the same situation leads to different choices.
 experiment: choose an option from each problem:
Problem 1) You are given a cash gift of EUR 200. And are asked to choose one of 2 options:
A: obtaining EUR 50 for sure
B: 25% chance to win an additional EUR 200 and a 75% chance of winning nothing
Problem 2) You are given a cash gift of EUR 400. And are asked to choose one of 2 options:
C: Losing EUR 150 for sure.
D: 75% probability of losing EUR 200 and a 25% probability of losing nothing

=> most people choose A and D
=> this choice pattern violates expected utility theory because EUT assumes people have
consistent choices regardless of the frame

 ALLIAS PARADOX
Imagine you have the choice between two gambles to win millions of money:
.

 In prospect A, €1m can be obtained for sure (100%),
choosing the prospect and losing (1%) could trigger regret
 In prospect B, probability difference is small and
outcome difference is large: tempting to choose the larger
outcome with smallest probability difference
 This implies that a 1% chance is not equal to a 1%
chance, depending on where it falls
= THE INTUITION
 FORMAL PROOF:
prospect B: 0.11 U(€1m) + 0.89 x U(€0m) < 0.10 x U(€5m) + 0.90U(€0m)
0.11 U(€1m) < 0.10 x U(€5m) + 0.90U(€0m) - 0.89 x U(€0m)
1x U(€1m)- 0.89 x U(€1m) < 0.10 x U(€5m) + 0.90 x U(€0m) - 0.89 x U(€0m)
1x U(€1m) < 0.10 x U(€5m) + 0.89 x U(€1m) + 0.01x U(€0m)
Prospect A: 1x U(€1m) > 0.10 x U(€5m) + 0.89x U(€1m) + 0.01x U(€0m)
 100% kans op 1 miljoen is toch NIET kleiner dan de andere kans
= The results are contradictory and violate Expected Utility Theory

 ELLSBERG PARADOX (decision under ambiguity)
Ambiguity aversion: difference between risk and uncertainty
= probabilities are observable under risk and are unobservable under uncertainty
 Example: There are two urns, the first 50 black and 50 red balls, the second 100 red or black balls in
unknown proportion. Choose a colour on which to bet, and an urn. Then extract one ball to win €100.
If a ball of the declared colour appears you get €100 and else 0. Which do you pick?
 Most people prefer to bet on the urn with the known proportion of colours; they are usually
willing to leave some money on the table for this preference
 This contradicts (subjective) expected utility theory: you should pick the likelier (subjective
believe) ball if you think the urn is unfair
 punknown(Red) +punknown(Black) < pknown(Red) +pknown(Black) = 1 => IMPOSSIBLE
 This contradicts the principle that probabilities must sum to 1, i.e. the subjective probabilities