Behaviouralizing Finance
CARISMA
February 2010
Hersh Shefrin
Mario L. Belotti Professor of Finance
Santa Clara University
Outline
• Paradigm shift.
• Strengths and weaknesses of
behavioural approach.
• Combining rigour of neoclassical
finance and the realistic psychologicallybased assumptions of behavioural
finance.
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Quantitative Finance
• Behaviouralizing
─Beliefs & preferences
─Portfolio selection theory
─Asset pricing theory
─Corporate finance
─Approach to financial market regulation
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Weaknesses in
Behavioural Approach
• Preferences.
─ Prospect theory, SP/A, regret.
─ Disposition effect.
•
•
•
•
•
Cross section.
Long-run dynamics.
Contingent claims (SDF: 0 or 2?)
Sentiment.
Representative investor.
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Conference Participants
Examples
• Continuous time model of portfolio selection with
behavioural preferences.
─ He and Zhou (2009), Zhou, De Georgi
• Prospect theory and equilibrium
─ De Giorgi, Hens, and Rieger (2009).
• Prospect theory and disposition effect
─ Hens and Vlcek (2005), Barberis and Xiong (2009), Kaustia
(2009).
• Long term survival.
─ Blume and Easley in Hens and Schenk-Hoppé (2008).
• Term structure of interest rates.
─ Xiong and Yan (2009).
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Beliefs
• Change of measure techniques.
─Excessive optimism.
─Overconfidence.
─Ambiguity aversion.
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Example:
Change of Measure is Log-linear
• Typical for a variance preserving, right
shift in mean for a normally distributed
variable.
• Shape of log-change of measure
function?
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99.84%
99.50%
99.16%
98.82%
8
-0.4
-0.6
-0.8
Consumption Growth Rate g (Gross)
106.19%
105.82%
105.46%
105.10%
104.74%
104.38%
104.03%
103.67%
103.32%
102.96%
102.61%
102.26%
101.91%
101.56%
101.22%
100.87%
100.53%
100.18%
-0.2
98.48%
98.15%
97.81%
97.48%
97.14%
96.81%
96.48%
96.15%
95.82%
Excessive Optimism
Sentiment Function
0.8
0.6
0.4
0.2
0
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99.84%
99.50%
99.16%
9
-0.4
-0.6
-0.8
Consumption Growth Rate g (Gross)
106.19%
105.82%
105.46%
105.10%
104.74%
104.38%
104.03%
103.67%
103.32%
102.96%
102.61%
102.26%
101.91%
101.56%
101.22%
100.87%
100.53%
100.18%
-0.2
98.82%
98.48%
98.15%
97.81%
97.48%
97.14%
96.81%
96.48%
96.15%
95.82%
Excessive Pessimism
Sentiment Function
0.8
0.6
0.4
0.2
0
Overconfidence
Sentiment Function
0.5
-0.5
-1
-1.5
-2
-2.5
Consumption Growth Rate g (Gross)
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106%
104%
103%
101%
99%
97%
96%
0
Preferences
• Psychological concepts
─Psychophysics in prospect theory.
─Emotions in SP/A theory.
• Inverse S-shaped weighting function,
rank dependent utility.
─Regret.
─Self-control.
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Prospect Theory Weighting Function
Based on Hölder Average
Ingersoll Critique
Prospect Theory Weighting Function
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
Decumulative Probability
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0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Inverse S in SP/A
Rank Dependent Utility
Functional Decomposition of Decumulative Weighting Function in SP/A Theory
1.2
1.0
h2(D)
0.8
h(D)
0.6
0.4
h1(D)
0.2
0.0
0
0.1
0.2
0.3
0.4
0.5
D
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0.6
0.7
0.8
0.9
1
Prospect Theory
• Tversky-Kahneman
(1992)
Prospect Theory Value Function
6
4
2
─Value function
• piecewise power
function
─Weighting function
• ratio of power
function to Hölder
average
─Editing / Framing
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9.
5
8
75
25
8.
7.
5
75
6.
5
5.
25
3.
5
4.
2
75
2.
25
5
0.
5
1.
-1
.2
-0
5
-2
.5
-1
.7
5
-4
.2
-3
5
.2
-5
.5
-4
.7
5
5
-7
.7
-6
5
.5
-7
.2
-8
-9
-1
0
0
-2
-4
-6
-8
-10
Gain/loss
Prospect Theory Weighting Function
1.2
1
0.8
0.6
0.4
0.2
0
0
0.05 0.09 0.14 0.18 0.23 0.27 0.32 0.36 0.41 0.45 0.5 0.54 0.59 0.63 0.68 0.72 0.77 0.81 0.86 0.9 0.95 0.99
Probability
14
SP-Function in SP/A
Rank Dependent Utility
n
SP =  (h(Di)-h(Di+1))u(xi)
i=1
• Utility function u is defined over gains and
losses.
• Lopes and Lopes-Oden model u as linear.
─ suggest mild concavity is more realistic
• Rank dependent utility: h is a weighting function
on decumulative probabilities.
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The A in SP/A
• The A in SP/A denotes aspiration.
• Aspiration pertains to a target value  to
which the decision maker aspires.
• The aspiration point might reflect status
quo, i.e., no gain or loss.
• In SP/A theory, aspiration-risk is measured
in terms of the probability
A=Prob{x }
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Objective Function
• In SP/A theory, the decision maker
maximizes an objective function
L(SP,A).
• L is strictly monotone increasing in both
arguments.
• Therefore, there are situations in which
a decision maker is willing to trade off
some SP in exchange for a higher
value of A.
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Testing CPT vs. SP/A
Experimental Evidence
• Lopes-Oden report that adding $50
induces a switch from the sure prospect
to the risky prospect.
• Consistent with SP/A theory if A is
germane, but not with CPT.
• Payne (2006) offers similar evidence
that A is critically important, although
his focus is OPT vs. CPT.
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Behaviouralizing Portfolios
• Full optimization using behavioural
beliefs and/or preferences.
• What is shape of return profile relative
to the state variable?
• In slides immediately following, dotted
graph corresponds to investor with
average risk aversion.
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Baseline: Aggressive Investor With
Unbiased Beliefs
cj/c0 vs. g
1.8
1.6
1.4
1.2
cj/c0
1
cj/c0
g
0.8
0.6
0.4
0.2
g
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1.
21
1.
19
1.
17
1.
15
1.
13
1.
11
1.
09
1.
07
1.
05
1.
03
1.
01
0.
99
0.
97
0.
95
0.
93
0.
91
0.
89
0.
87
0.
85
0.
83
0.
81
0.
79
0
How Would You Characteize an Investor
Whose Return Profile Has
This Shape?
cj/c0 vs. g
1.4
1.2
1
cj/c0
0.8
cj/c0
g
0.6
0.4
0.2
g
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1.
21
1.
19
1.
17
1.
15
1.
13
1.
11
1.
09
1.
07
1.
05
1.
03
1.
01
0.
99
0.
97
0.
95
0.
93
0.
91
0.
89
0.
87
0.
85
0.
83
0.
81
0.
79
0
Two Choices
• Aggressive underconfidence?
• Aggressive overconfidence?
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CPT With Probability Weights
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CPT With Rank Dependent Weights
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SP/A With Cautious Hope
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Associated Log-Change of Measure
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Caution!
Quasi-Optimization
• Prospect theory was not developed as
a full optimization model.
• It’s a heuristic-based model of choice,
where editing and framing are central.
• It’s a suboptimization model, where
choice heuristics commonly lead to
suboptimal if not dominated acts.
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Behaviouralizing
Asset Pricing Theory
• Stochastic discount factor (SDF) is a
state price per unit probability.
• SDF  M = /.
• Price of any one-period security Z is
qZ = Z = E{MZ}
Et[Ri,t+1 Mt+1] = 1
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Graph of SDF
What’s This?
• x-axis is a state
variable like
aggregate
consumption
growth.
• y-axis is M.
• SDF is linear.
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How About This?
Logarithmic Case?
• x-axis is a state
variable like logaggregate
consumption
growth.
• y-axis is log-M.
• Relationship is
linear.
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Empirical SDF
• Aït-Sahalia and Lo (2000) study
economic VaR for risk management,
and estimate the SDF.
• Rosenberg and Engle (2002) also
estimate the SDF.
• Both use index option data in
conjunction with empirical return
distribution information.
• What does the empirical SDF look like?
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Aït-Sahalia – Lo’s SDF Estimate
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Rosenberg-Engle’s SDF Estimate
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Behavioral Aggregation
• Begin with neoclassical EU model with
CRRA preferences and complete
markets.
• In respect to judgments, markets
aggregate pdfs, not moments.
─Generalized Hölder average theorem.
• In respect to preferences, markets
aggregate coefficients of risk tolerance
(inverse of CRRA).
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Representative Investor Models
• Many asset pricing theorists, from both
neoclassical and behavioral camps,
assume a representative investor in
their models.
• Aggregation theorem suggests that the
representative investor assumption is
typically invalid.
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Typical Representative Investor:
Investor Population Heterogeneous
• Violate Bayes rule, even when all
investors are Bayesians.
• Is averse to ambiguity even when no
investor is averse to ambiguity.
• Exhibits stochastic risk aversion even
when all investors exhibit CRRA.
• Exhibits non-exponential discounting
even when all investors exhibit
exponential discounting.
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Formally Defining Sentiment 
General Model
Measured by the random variable
 = ln(PR(xt) / (xt)) + ln(R/ R,)
• R, is the R that results when all traders hold
objective beliefs
• Sentiment is not a scalar, but a stochastic
process < , >, involving a log-change of
measure.
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Neoclassical Case, Market Efficiency
=0
• The market is efficient when the
representative trader, aggregating the
beliefs of all traders, holds objective
beliefs.
─i.e., efficiency iff PR= 
• When all investors hold objective beliefs
 = (PR/) (R/ R,) = 1
and
 = ln() = 0
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Decomposition of SDF
m  ln(M)
m =  - R ln(g) + ln(R,)
Process <m, >
─Note: In CAPM with market
efficiency, M is linear in g with a
negative coefficient.
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Overconfident Bulls &
Underconfident Bears
ln SDF & Sentiment
60.00%
50.00%
40.00%
30.00%
ln(SDF)
20.00%
Sentiment
Function
10.00%
ln(g)
-20.00%
-30.00%
Gross Consumption Growth Rate g
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106.19%
105.82%
105.46%
105.10%
104.74%
104.38%
104.03%
103.67%
103.32%
102.96%
102.61%
102.26%
101.91%
101.56%
101.22%
100.87%
100.53%
100.18%
99.84%
99.50%
99.16%
98.82%
98.48%
98.15%
97.81%
97.48%
97.14%
96.81%
96.48%
96.15%
-10.00%
95.82%
0.00%
How Different is a Behavioural SDF
From a Traditional Neoclassical SDF?
Behavioral SDF vs Traditional SDF
1.2
1.15
Behavioral SDF
1.1
1.05
1
0.95
Traditional Neoclassical SDF
0.9
0.85
Aggregate Consumption Growth Rate g (Gross)
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106%
105%
104%
103%
103%
102%
101%
100%
99%
98%
97%
97%
96%
0.8
It’s Not Risk Aversion in the Aggregate
• Upward sloping portion of SDF is not a
reflection of risk-seeking preferences at
the aggregate level.
• Time varying sentiment  time varying
SDF.
• After 2000, shift to “black swan”
sentiment and by implication SDF.
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Taleb “Black Swan” Sentiment
Overconfidence
Sentiment Function
0.5
-0.5
-1
-1.5
-2
-2.5
Consumption Growth Rate g (Gross)
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106%
104%
103%
101%
99%
97%
96%
0
Barone AdesiEngle-Mancini (2008)
• Empirical SDF based on index options data for
1/2002 – 12/2004.
• Asymmetric volatility and negative skewness
of filtered historical innovations.
• In neoclassical approach, RN density is a
change of measure wrt , thereby “preserving”
objective volatility.
• In behavioral approach RN density is change
of measure wrt PR.
• In BEM, equality broken between physical and
risk neutral volatilities.
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SDF for 2002, 2003,
Garch on Left, Gaussian on Right
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Continuous Time Modeling
ln SDF & Sentiment
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50.00%
40.00%
30.00%
ln(SDF)
20.00%
Sentiment
Function
10.00%
106.19%
105.82%
105.46%
105.10%
104.74%
104.38%
104.03%
103.67%
103.32%
102.96%
102.61%
102.26%
101.91%
101.56%
101.22%
100.87%
100.53%
99.84%
100.18%
99.50%
99.16%
98.82%
98.48%
98.15%
97.81%
97.48%
97.14%
96.81%
96.48%
-10.00%
96.15%
0.00%
95.82%
• E(M) is the discount
rate exp(-r) associated
with a risk-free security.
• m=ln(M)
• Take point on realized
sample path, where M
is value of SDF at
current value of g.
• dM has drift –r with
fundamental
disturbance and
sentiment disturbance.
• r>0  expect to move
down the SDF graph.
60.00%
ln(g)
-20.00%
-30.00%
Gross Consumption Growth Rate g
• Fundamental
disturbance relates to
shock to dln(g).
• Sentiment disturbance
relates to shift in
sentiment.
• Marginal optimism
drives E(dm) >0.
Risk Premiums
Risk premium on security Z is the sum
of a fundamental component and a
sentiment component:
-cov[rZ g-]/E[g-] + (fundamental)
ie(1-hZ)/hZ +
(sentiment)
ie-i
(sentiment)
where
hZ = E[ g- rZ]/ E[g- rZ]
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How Different are Returns to a Behavioural
MV-Portfolio From Neoclassical Counterpart?
Gross Return to Mean-variance Portfolio:
Behavioral Mean-Variance Return vs Efficient Mean-Variance Return
110%
105%
Mean-variance Return
100%
Neoclassical Efficient MV Portfolio Return
95%
90%
Behavioral MV Portfolio Return
85%
80%
Consumption Growth Rate g (Gross)
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106%
104%
103%
101%
99%
97%
96%
75%
MV Function  Quadratic
2-factor Model, Mkt and Mkt2
Gross Return to Mean-variance Portfolio:
Behavioral Mean-Variance Return vs Efficient Mean-Variance Return
1.03
1.02
Efficient MV Portfolio Return
Mean-variance Return
1.01
1
0.99
Behavioral MV Portfolio Return
0.98
Return to a Combination of the Market Portfolio and
Risk-free Security
0.97
0.96
Consumption Growth Rate g (Gross)
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106.19%
105.28%
104.38%
103.49%
102.61%
101.74%
100.87%
100.01%
99.16%
98.31%
97.48%
96.64%
95.82%
0.95
When a Coskewness Model
Works Exactly
• The MV return function is quadratic in
g, risk is priced according to a 2-factor
model.
• The factors are g (the market portfolio
return) and g2, whose coefficient
corresponds to co-skewness.
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Summary of Key Points
Behaviouralizing Finance
• Paradigm shift.
• Strengths and weaknesses of
behavioural approach.
• Agenda for quantitative finance?
• Combine rigour of neoclassical finance
and the realistic psychologically-based
assumptions of behavioural finance.
Copyright, Hersh Shefrin 2010
52