Download Reconsidering Behaviour in Finance

Behavioural Finance
Lecture 03
Finance Markets Behaviour
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Rational Choice in Finance Markets
1. Choose between
Your Choices?
A. $1000 with certainty; OR
Gamble
A
B
Number
B. 90% odds of $2000 & 10%
odds of -$1000
1
2. Choose between
A. $0 with certainty; OR
2
B. 50% odds of $150 and 50%
odds of -$100
3
3. Choose between
A. -$100 with certainty; OR
Total for
each option
B. 50% odds of $50; 50% odds
of -$200
• Take your time to work out which one to choose…
• Write down why you made your choices
Rational Choice in Finance Markets
• Tally of class
choices:
• Most people
choose:
Common Choices
Class Choices
“Rational” Choice
Gamble
A
B
Gamb
le
A
1
1
A
1
B
2
2
A
2
B
3
3
B
3
B
Total
Total
1
Total
Gamble
A
B
2
• So most people are irrational???
B
• Economic theory
says rational is…
0
3
Rational Choice in Finance Markets
• Why? A“rational person” maximises expected return:
– Sum of probabilities times returns:
Choices and Expected Return
Gamble
A
B
ER
1
1x1,000=
+$1,000
0.9x$2,000+ 0.1x-$1,000=
+$1,700
B>A
2
1x$0=
$0
.5x$150 + .5x-$100=
+$25
B>A
3
1x-$100=
-$100
.5x$50+.5x-$200=
-$75
B>A
• Why do most people not make these choices?
– Write down your guess as to why…
Rational Choice in Finance Markets
•
Straw Poll: Reasons for not “being rational”…
N
o.
Reason…
Campbelltown Class
Parramatta Day
Parramatta Evening
1
2
3
4
• Now an experiment: exactly the same choices EXCEPT
– Whichever option you choose is repeated 100 times
Rational Choice in Finance Markets
Class Choices
1. Choose between 100 repeats of Gamble
A
A. $1000 with certainty; OR
B. 90% odds of $2000 & 10%
1
odds of -$1000
2. Choose between 100 repeats of
2
A. $0 with certainty; OR
B. 50% odds of $150 and 50%
3
odds of -$100
3. Choose between 100 repeats of Total
A. -$100 with certainty; OR
B. 50% odds of $50 and 50%
odds of -$200
• Take your time to work out which one to choose…
• Write down why you made your choices
B
Rational Choice in Finance Markets
• Tally of class choices • This time •
economic
Class Choices
theory gets
Gamble
A
B
it right:
• You’re
1
(almost!)
all
rational…
2
Economic theory
says rational is…
“Rational” Choice
Gamble
Number
A
B
1
B
2
B
3
3
B
Total
Total
0
3
Rational Choice in Finance Markets
•
Subjective vs objective probability
– In a single gamble, you get one outcome OR the other
– Subjective expectations are still the given odds:
• E.g. One-off game with Gamble 1
A. $1000 with certainty; OR
B. 90% chance of $2000 & 10% chance of -$1000
– Actual outcome of A is “expected return” of $1,000
– Actual outcome of B is either $2,000 or -$1,000:
• You don’t get the “expected return” of $1,700
– Repeated gamble, you do get the expected return
• 100 repeats of gamble 1
A. 100x1000=$1,000 certain outcome per play; OR
B. 90x2000+10x-1000=$1,700 average per play
Rational Choice in Finance Markets
• Why the difference?
– One-off gamble fundamentally uncertain
• Probability of $2,000 in Gamble 3B is 90%...
• But outcome of single throw will be +$2K or -$1K
• Probability can’t tell you which one will occur
– Subjectively, odds are 90% you’ll get $2,000
– Objectively, you can’t tell which one will occur
• Outcome not “probable” but “uncertain”
• Problem of uncertainty key to failings of finance theory
• First, background to development of finance theory:
– Flawed use of “theory of games” developed by
physicist John von Neumann & economist Oscar
Morgenstern…
Rational Choice in Finance Markets
• von Neumann & Morgenstern developed “Expected
Return/Expected Utility” approach in Theory of Games
and Economic Behavior
– Intended to use “theory of games” to make economic
concepts like utility measureable
• Model then “borrowed” by economists to develop CAPM
• von Neumann & Morgenstern were
– Aware of distinction between objective & subjective
probability
– Insisted on using objective probability:
Rational Choice in Finance Markets
• “Probability has often been visualized as a subjective
concept more or less in the nature of estimation.
• Since we propose to use it in constructing an individual,
numerical estimation of utility [more on this shortly…],
– the above view of probability would not serve our
purpose.
• The simplest procedure is, therefore, to insist upon the
alternative, perfectly well founded interpretation of
probability as
– frequency in long runs.” (von Neumann & Morgenstern
1944: 19 [Emphases added])
• This directive (and much else!) ignored by economists who
developed “Modern Finance Theory”…
Rational Choice in Finance Markets
• von Neumann & Morgenstern (abbreviated to vN&M)
– Critical of economic theory in general
– In particular rejected “indifference curves”:
• “the treatment by indifference curves implies
either too much or too little:
– if the preferences of the individual are not at all
comparable
• then the indifference curves do not exist.
– If the individual’s preferences are all
comparable
• then we can even obtain a (uniquely defined)
numerical utility
• which renders the indifference curves
superfluous.” (19-20)”
Rational Choice in Finance Markets
• Developed numerical measure of utility instead
– Argued that non-numerical economic concept of utility
was “immature”
– Gave examples of pre-numerical concepts in physics
– Measurement there resulted from good theory:
• “The precise measurement of the quantity and
quality of heat (energy and temperature) were the
outcome and not the antecedents of the
mathematical theory” (p. 3)
– i.e., develop a good theory, and what is currently
“ordinal” (ranking of preferences via
indifference curves) can become “cardinal”
(quantitative measure for utility)
Rational Choice in Finance Markets
• Their idea:
– Use gambles and objective probability to provide
numeric scale for utility
• “Numerical utility” model has same “curse of
dimensionality” problem as indifference curves
• But unlike economists, vN&M were aware of problem:
– “one may doubt whether a person can always decide
which of two alternatives—with the utilities u, v—he
prefers.
– But, whatever the merits of this doubt are, this
possibility—i.e. the completeness of the system of
(individual) preferences—must be assumed even for
the purposes of the ‘indifference curve method’.” (pp.
28-29)
Rational Choice in Finance Markets
• Model mirrored “Revealed Preference” numerically:
– Allowed precise statement of extend to which
consumer preferred one “shopping trolley” to another
Samuelson’s “Revealed Preference”
vN&M’s “Numerical Utility”
“Completeness”: Given any 2 bundles
of commodities A & B , consumer
can decide whether prefers A to B
(A≻B), B to A (B≻A), or is
indifferent between them (B≈A)
Ditto but now: if A≻B then there
is some numerical conversion of
utility v() such that v(A) greater
than v(B); Can put a numeric value
on utility of A and B
“Transitivity”: If (A≻B) and (B≻C)
then (A≻C)
Ditto but now: if A≻B then there
is some probability 0<a<1 such
that av(A) =v(B)
“Non-satiation”: More is preferred
to less
Probable (but not necessary)
consequences of numerically
measured utility
“Convexity”:Marginal utility positive
but falling as consumption of any
good rises
Rational Choice in Finance Markets
• Basic Idea:
– Arbitrarily assign value
• 0 to zero bananas
• 1 to 1 banana
– (just like setting freezing point of fresh
water=32ºF & freezing point of salt water=0ºF
in Fahrenheit scale)
– Then offer consumer repeated gamble of
• 1 banana for certain; OR
• a% odds of 2 bananas vs (1-a)% odds of no bananas
• If consumer accepts gamble when (say) a=0.7, then
utility of 1 banana = 0.7 times utility of 2 bananas:
Rational Choice in Finance Markets
• U(1 banana) = 1 “util”
• Accept repeated gamble between 1 for certain and 60% chance of 2
versus 40% chance of none:
– 1 banana gives you 60% the utility of 2 bananas
– 1 = 0.6 x U(2)
– U(2) = 1/0.6=1.67
– Marginal Utility of 2nd banana = 0.67
• Repeat same exercise
– Repeated gamble between
• 2 for certain and gamble between (0 and 3),
• 3 for certain and gamble between (0 and 4 bananas), etc.…
– Use odds at which gamble accepted to calculate numerical utility
for bananas:
Rational Choice in Finance Markets
• Adding up the bananas…
Number of Numeric Marginal Odds at which
bananas for “utils”
Utility
gamble for one
certain
more accepted
Calculation
0
0
0
N/A
N/A
1
1
1
60% odds of 2
bananas vs 40% of
no banana
1/0.6=1.67
2
1.67
0.67
78% odds of 3
1.67/0.78=2.141
3
2.141
0.471
92% odds of 4
2.141/0.92
4
2.327
0.186
97% odds of 5
2.327/0.97
5
2.399
0.072
99% odds of 6
2.399/0.99
Rational Choice in Finance Markets
• A numeric measure of utility & marginal utility:
Cardinal Utility from Odds of Gamble
3.00
Utility
2.50
Change in Utility
Utility
2.00
1.50
1.00
0.50
0.00
0
1
2
3
4
5
Number of bananas
• So “cardinal expected utility” meant to be a replacement
for “ordinal utility” indifference curve analysis…
Rational Choice in Finance Markets
• Instead, economists
– Ignored numeric utility proposal
– Combined new vN&M ideas:
• Expected Return
• Expected Utility etc.
– With existing economic theory
• Indifference curves, etc.
– To develop “Modern Finance Theory” in 1950s & 60s…
• Vision of investors maximising non-numerical utility
given budget constraints…
“Modern” Finance?
• “Modern Finance Theory” has many components:
– Sharpe’s “Capital Asset Pricing Model” (CAPM)
– Modigliani-Miller’s “Dividend Irrelevance Theorem”
– Markowitz’s risk-averse portfolio optimisation model
– Arbitrage Pricing Theory (APT)
• “Modern” as compared to pre-1950 theories that
emphasised behaviour of investors as explanation of
stock prices, value investing, etc.
– Initially appeared to explain what “old finance” could
not
– But 50 years on, not so crash hot…
• Foundation is Sharpe’s CAPM:
• Objective: To “predict the behaviour of capital
markets”
The Capital Assets Pricing Model
• Method: Extend theories of investment under certainty
– to investment under conditions of risk
– Based on neoclassical utility theory:
• investor maximises utility subject to constraints where
utility is:
– Positive function of expected return ER
– Negative function of risk (standard deviation) sR
– Constraints are available spectrum of investment
opportunities as perceived by individual investor:
• Expectation of return on stock over time including
expected volatility of return & correlation with other
assets
– E.g. “I expect IBM to give a 6% return, with a
standard deviation of 3% & a minus 67% correlation
with CSR”…
The Capital Assets Pricing Model
Z inferior
to C
(lower ER)
and B
(higher
sR)
Investment
opportunities
“Efficient”
opportunities
on the edge
Indifference
curves
Increasing utility:
Higher expected
returns & lower risk
Optimal
combination
for this
investor
Border (AFBDCX) is Investment Opportunity Curve (IOC)
The Capital Assets Pricing Model
• IOC reflects correlation of separate investments.
Consider 3 investments A, B, C:
– A contains investment A only
Variance due to A
• Expected return is ERa,
• Risk is sRa
Perceived
correlation of A
– B contains investment B only
with B
• Expected return is ERb,
(varies between -1
• Risk is sRb
& +1)
– C some combination of a of A & (1-a) of B
• ERc=aERa + (1-a)ERb
s Rc  a .s Ra  (1  a ) s Rb  2. rab .a . 1  a .s Ra .s Rb
2
2
2
2
The Capital Assets Pricing Model
• If rab=1, C lies on straight line between A & B:
This is 1
s Rc  a 2 .s Ra 2  (1  a ) 2 s Rb 2  2. rab .a . 1  a .s Ra .s Rb
The Capital Assets Pricing Model
• If rab=1, C lies on straight line between A & B:
s Rc  a 2 .s Ra 2  (1  a ) 2 s Rb 2  2. rab .a . 1  a .s Ra .s Rb
 a 2 .s Ra 2  (1  a ) 2 s Rb 2  2.a . 1  a .s Ra .s Rb
This can be factored
The Capital Assets Pricing Model
• If rab=1, C lies on straight line between A & B:
s Rc  a .s Ra  (1  a ) s Rb  2. rab .a . 1  a .s Ra .s Rb
2
2
2
2
 a 2 .s Ra 2  (1  a ) 2 s Rb 2  2.a . 1  a .s Ra .s Rb

a .s
 (1  a ).s Rb   a .s Ra  (1  a )s Rb
2
Ra
Identical to straight line relation for expected return:
The Capital Assets Pricing Model
sR
W h en rab= 1
B
In v e sm
t en t
O ppo r tun ity C
C u rv e
A
ER
a
a
The Capital Assets Pricing Model
• If rab=0, C lies on curved path between A & B:
This is zero
s Rc  a 2 .s Ra 2  (1  a ) 2 s Rb 2  2. rab .a . 1  a .s Ra .s Rb
Hence this is zero
The Capital Assets Pricing Model
• If rab=0, C lies on curved path between A & B:
s Rc  a .s Ra  (1  a ) s Rb  2. rab .a . 1  a .s Ra .s Rb
2
2
2
2
 a 2 .s Ra 2  (1  a ) 2 s Rb 2
The Capital Assets Pricing Model
• If rab=0, C lies on curved path between A & B:
s Rc  a 2 .s Ra 2  (1  a ) 2 s Rb 2  2. rab .a . 1  a .s Ra .s Rb
 a 2 .s Ra 2  (1  a ) 2 s Rb 2

a .s
Ra
 (1  a )s Rb 
2
Straight line relation
Hence lower risk for diversified portfolio
(if assets not perfectly correlated)
The Capital Assets Pricing Model
sR
W h en rab< 1
In v e sm
t en t
O ppo r tun ity C
C u rv e
B
Fall in sR due to
diversification
when investments
are not perfectly
correlated
A
ER
a
a
The Capital Assets Pricing Model
• Sharpe assumes riskless asset P with ERP=pure interest
rate, sRP=0.
– Assumes limitless borrowing/lending at riskless
interest rate = return on asset P
• Investor can form portfolio of P with any other
combination of assets
– Investor can therefore move to anywhere along PfZ
line by borrowing/lending…
The Capital Assets Pricing Model
•
•
Efficiency: maximise expected return & minimise risk given constraints
One asset combination will initially dominate all others:
Only asset combination
which can efficiently
be combined with
riskless asset P in
a portfolio
The Capital Assets Pricing Model
• To this point, Sharpe has theory of a single investor
• However…
– “Riskless” lending rate will differ for each investor
– Expectations of future returns will differ too…
Fred Nurk
Jane Nguyen
Bill Gates
sR
Z
CSR
IBM
BHP
P
sR
IBM Z
BHP
CSR
ER
P
ER
sR
Z
IBM
BHP
CSR
P
ER
• How to go from theory of individual investor to theory of
market?...
– You guessed it…
– Sharpe assumes all investors are (almost!) identical...
The Capital Assets Pricing Model
• “In order to derive conditions for equilibrium in the capital market
we invoke two assumptions.
• First, we assume a common pure rate of interest, with all investors
able to borrow or lend funds on equal terms.
• Second, we assume homogeneity of investor expectations:
– investors are assumed to agree on the prospects of various
investments—the expected values, standard deviations and
correlation coefficients described in Part II.
• Needless to say, these are highly restrictive and undoubtedly
unrealistic assumptions.
• However, since the proper test of a theory is not the realism of its
assumptions but the acceptability of its implications,
• and since these assumptions imply equilibrium conditions which form
a major part of classical financial doctrine,
• it is far from clear that this formulation should be rejected—
especially in view of the dearth of alternative models leading to
similar results.” (Sharpe 1964, pp. 433-434)
The Capital Assets Pricing Model
• Defended by appeal to Friedman’s “Instrumentalism”:
– “the proper test of a theory is not the realism of its
assumptions but the acceptability of its implications”
• Bad version of a bad methodology (discussed in
History of Economic Thought lecture—click for link)
• Another example of “proof by contradiction”
– IF have to assume identical investors to get a Capital
Assets Market Line
– THEN there can’t be a Capital Assets Market Line
• Could have been heuristic step to more general model
– See methodology lecture…
• But instead…
The Capital Assets Pricing Model
• CAPM based on absurd counter-factual assumptions that
all investors:
– Agree with each other about every stock; AND
– Have limitless ability to borrow at risk-free rate; AND
– Their expectations about the future are correct!
• Consequence of identical accurate expectations and
identical access to limitless borrowing “assumptions”:
– spectrum of available investments/IOC identical for
all investors
– P same for all investors
– PfZ line same for all investors
– Investors only differ by preferences for risk:
• distribute along line by borrowing/lending according
to own risk preferences:
The Capital Assets Pricing Model
Thrill seeker...
Risk neutral…
Highly
risk-averse
The Capital Assets Pricing Model
• Next, the (perfect) market mechanism
– Price of assets in f will rise
– Price of assets not in f will fall
– Price changes shift expected returns
– Causes new pattern of efficient investments aligned
with PfZ line:
The Capital Assets Pricing Model
Capital market line
Range of efficient asset
combinations after
market price adjustments:
more than just one
efficient portfolio
The Capital Assets Pricing Model
• Theory so far applies to combinations of assets
• Individual assets normally lie above capital market line
(no diversification)
• Can’t relate between ERi & si
• Can relate ERi to “systematic risk”:
• Investment i can be part of efficient combination g:
– Can invest (additional) a in i and (1-a) in g
• a=1 means invest solely in i;
• a=0 means some investment in i (since part of
portfolio g);
• Some a<0 means no investment in i;
• Only a=0 is “efficient”
The Capital Assets Pricing Model
Single investment
i which is part of
portfolio g
Efficient
combination g
Additional investment
in i is zero (a=0) here
The Capital Assets Pricing Model
• Slope of IOC and igg’ curve at tangency can be used to
derive relation for expected return of single asset
E Ri  P  rig
s Ri
s Rg

 E Rg  P

• This allows correlation of variation in ERi to
variation in ERg (undiversifiable, or systematic, or
“trade cycle” risk)
• Remaining variation is due to risk inherent in i:
The Capital Assets Pricing Model
Risk peculiar
to asset i
Higher return for assets more strongly
affected by trade cycle (systematic risk)
The Capital Assets Pricing Model
• Efficient portfolio enables investor to minimise asset
specific risk
• Systematic risk (risk inherent in efficient portfolio) can’t
be diversified against
• Hence market prices adjust to degree of responsiveness
of investments to trade cycle:
– “Assets which are unaffected by changes in economic
activity will return the pure interest rate; those which
move with economic activity will promise appropriately
higher expected rates of return.” [OREF II]
The Capital Assets Pricing Model
• Crux/basis of model: markets efficiently value
investments on basis of expected returns/risk tradeoff
• Modigliani-Miller extend model to argue valuation of
firms independent of debt structure (see OREF II)
• Combination: the “efficient markets hypothesis”
• Focus on portfolio allocation across investments at a
point in time, rather than trend of value over time
• Argues investors focus on “fundamentals”:
– Expected return; Risk; Correlation
• So long as assumptions are defensible…
– common pure rate of interest
– homogeneity of investor expectations
• Sharpe later admits to qualms…
The CAPM: Reservations
• “People often hold passionately to beliefs that are
far from universal.
• The seller of a share of IBM stock may be
convinced that it is worth considerably less than
the sales price.
• The buyer may be convinced that it is worth
considerably more.” (Sharpe 1970)
• However, if we try to be more realistic:
– “The consequence of accommodating such
aspects of reality are likely to be disastrous in
terms of the usefulness of the resulting
theory...
The CAPM: Reservations
• “The capital market line no longer exists.
• Instead, there is a capital market curve–linear over
some ranges, perhaps, but becoming flatter as [risk]
increases over other ranges.
• Moreover, there is no single optimal combination of
risky securities; the preferred combination depends
upon the investors’ preferences...
• The demise of the capital market line is followed
immediately by that of the security market line.
• The theory is in a shambles.” (Sharpe 1970 emphasis
added)
The CAPM: Evidence
• Sharpe’s qualms ignored & CAPM took over economic
theory of finance
• Initial evidence seemed to favour CAPM
– Essential ideas:
• Price of shares accurately reflects future earnings
– With some error/volatility
• Shares with higher returns more strongly
correlated to economic cycle
– Higher return necessarily paired with higher
volatility
• Investors simply chose risk/return trade-off that
suited their preferences
– Initial research found expected (positive) relation
between return and degree of volatility
– But were these results a fluke?
The CAPM: Evidence
• Volatile but superficially exponential trend
– As it should be if economy growing smoothly
12000
DJIA 1920-2001
10000
8000
6000
4000
2000
0
11/25/1921
-2000
8/4/1935
4/12/1949
12/20/1962
8/28/1976
5/7/1990
1/14/2004
The CAPM: Evidence
• Sharpe’s CAPM paper published 1964
• Initial CAPM empirical research on period 1950-1960’s
– Period of “financial tranquility” by Minsky’s theory
• Low debt to equity ratios, low levels of speculation
– But rising as memory of Depression recedes…
• Steady growth, high employment, low inflation…
• Dow Jones advance steadily from 1949-1965
– July 19 1949 DJIA cracks 175
– Feb 9 1966 DJIA sits on verge of 1000 (995.15)
• 467% increase over 17 years
– Continued for 2 years after Sharpe’s paper
• Then period of near stagnant stock prices
The CAPM: Evidence
• Dow Jones “treads water” from 1965-1982
– Jan 27 1965: Dow Jones cracks 900 for 1st time
– Jan 27 1972: DJIA still below 900! (close 899.83)
• Seven years for zero appreciation in nominal terms
• Falling stock values in real terms
– Nov. 17 1972: DJIA cracks 1000 for 1st time
• Then “all hell breaks loose”
– Index peaks at 1052 in Jan. ‘73
– falls 45% in 23 months to low of 578 in Dec. ’74
– Another 7 years of stagnation
– And then “liftoff”…
The CAPM: Evidence
• Fit shows average exponential growth 1915-1999:
• index well above or below except for 1955-1973
Log of Dow Jones Industrial Average 1915-1999 plus last ten years...
4.5
4
3.5
y = 6E-05x + 1.4228
R2 = 0.9031
Crash of ’73: 45%
fall in 23 months…
Sharpe’s paper published
Jan 11 ’73: Peaks at 1052
Log Closing Value
3
Dec 12 1974: bottoms at 578
2.5
Bubble takes off in ‘82…
2
CAPM fit doesn’t look so hot any more…
1.5
1
Steady above trend growth 19491966: Minsky’s “financial tranquility”
CAPM fit to this data looks pretty good!
0.5
0
7/5/1914
4/13/1926
1/20/1938
10/29/1949
8/7/1961
5/16/1973
Date
2/22/1985
12/1/1996
9/9/2008
6/18/2020
Anomalies mount…
• For CAPM to describe reality:
– At the individual level
• All investors have to maximise expected utility
– Exhibit risk-return tradeoff
– At the systemic level
• Stock market has to follow “random walk with drift”
• Only determinant of stock’s price can be market
(efficient) return, riskless return, and stock’s beta
• Experiments like earlier ones challenge individual rule
– Most individuals breach risk/return tradeoff rule…
– Reaction of economists & psychologists to breaches
gave rise to “Behavioural Economics & Finance”
• But even here misunderstanding of what vN&M
tried to do distorted development of alternative
Anomalies mount…
•
Behavioural “anomalies”—people not maximising expected
return—initially explained by “preference for risk”
1. “Choose between
A. $1000 with certainty; OR
B. 90% odds of $2000 & 10% odds of -$1000”
– “Rational” person would choose B (expected return
$1700) over A
– Vast majority choose A over B
– Explanation: majority is “risk averse”
– Actively dislikes risk, chooses A to minimise it
– Problem: “risk preference reversal”…
Anomalies mount…
• Problem 1: Choose between two
alternatives:
– A: do nothing
Your Choices?
– B a gamble with:
Choice
• 50% chance of winning $150; Problem
Number
A
B
• 50% chance of losing $100.
• Problem 2: Choose between two
1
alternatives:
2
– A: Lose $100 with certainty
Total for
– B: a gamble with:
each option
• 50% chance of winning $50;
• 50% chance of losing $200
• Record your choices…
Anomalies mount…
• Did they look like this?:• Or this? • Or this?
1. Risk Averse
Problem
Number
Choice
A
1
X
2
Total
2. Risk Seeking
B
Problem
Number
A
1
X
1
X
X
2
X
2
2
Total
2
Total
B
Problem
Number
Choice
3. Risk Reversal
A
Choice
B
X
1
1
• Most people looked like 3:
– “Irrational” re risk too:
• Risk avoiding in one case
• Risk seeking in the other…
• Result didn’t make sense in either neoclassical (“risk
averse vs risk seeking”) or vN&M (numerical utility)
terms…
Anomalies mount…
• If people normally choose A over B in Problem 1 then:
– U($0) > U(0.5x$150+0.5x-$100)
• Using vN&M axioms we can rewrite this as:
– U($0) > 0.5xU($150)+0.5xU(-$100)
– “Utility of zero exceeds 0.5 times utility of
$150 plus 0.5 times utility of -$100”
• If people normally choose B over A in Problem 2 then:
– U(-$100) < U(0.5x$50+0.5x-$200)
• Using vN&M axioms we can rewrite this as:
– U(-$100) < 0.5xU($50)+0.5xU(-$200)
– “Utility of zero is less than 0.5 times utility of
$50 plus 0.5 times utility of -$200”
• Inconsistent in vN&M terms because axioms are linear in
money: adding fixed sum shouldn’t alter outcome:
Anomalies mount…
• If U(-$100) < U(0.5x$50+0.5x-$200), then add $100:
• Then U($0) < U(0.5x$150+0.5x-$100)
– U($0) < 0.5xU($150)+0.5xU(-$100)
• So if someone chooses A over B in Problem 1, vN&M say:
– U($0) > 0.5xU($150)+0.5xU(-$100)
• And if they choose B over A in Problem 2, vN&M say:
– U($0) < 0.5xU($150)+0.5xU(-$100)
• These are inconsistent:
• Preference reversal even in vN&M terms!
• May look like “cheating” to add $100;
– But same result turns up in single experiment…
Anomalies mount…
Your Choices?
• Problem 3. Choose between:
Problem
Choice
– A: Lose $45 with certainty
A
B
– B: 50% chance of -$100 and 50%
3
chance of $0
4
• Problem 4. Choose between:
Total
– A: 10% chance of -$45 and 90%
chance of $0
Commonest Choice
– B: 5% chance of -$100 and 95%
Problem
Choice
chance of $0
A
B
• A is “rational choice” in both cases:
3
Expected return
Problem
Choice
A
B
3
-$45
0.5x-$100=-$50
4
0.1x-$45=-$4.5
0.05x-$100=-$5
X
4
X
Total
1
1
• A/B choice pair
gives expected
utility reversal…
Anomalies mount…
•
•
•
•
•
•
Choosing 3B implies that:
U(-$45) < U(0.5x-$100+0.5x$0); or
1.0xU(-$45) < 0.5xU(-$100) + 0.5xU($0)
Choosing 4A implies that:
U(0.1x-$45 + 0.9x$0) > U(0.05x-$100 + 0.95x$0); or
0.1xU(-$45) + 0.9xU($0) > 0.05xU(-$100) + 0.95xU($0)
– Subtract 0.9xU($0) from both sides to yield:
• 0.1xU(-$45) > 0.05xU(-$100) + 0.05xU($0)
– Multiply both sides by 10 to yield:
• 1.0xU(-$45) > 0.5xU(-$100) + 0.5xU($0)
• Since most people choose 3B and 4A, this implies
• 1.0xU(-$45) < 0.5xU(-$100) + 0.5xU($0) AND
• 1.0xU(-$45) > 0.5xU(-$100) + 0.5xU($0): contradiction
Anomalies mount…
• Or is it?
– “Contradiction” disappears if
examples applied as vN&M
insisted they should be…
• Problem 5. Choose between 100
repeats of either:
– A: Lose $45 with certainty
OR
– B: 50% chance of -$100 and
50% chance of $0
• Problem 6. Choose between 100
repeats of either:
– A: 10% chance of -$45 and
90% chance of $0 OR
– B: 5% chance of -$100 and
95% chance of $0
Your Choices?
Problem
Choice
A
B
5
6
Total
Expected Return over 100
plays
Problem
Choice
A
B
5
-$4500
-$5000
6
-$450
-$500
From risk to uncertainty
• vN&M framework intended to derive numeric alternative
to indifference curves
– Suffers same core problem (impossibility of forming
complete set of preferences);
– But valid with repeated choices to derive model of
utility
• NOT devised to handle “one-off” choices where even
given probability data, each single outcome is
fundamentally uncertain
• A model of behaviour in finance must consider
uncertainty
– Next week…