Behaviouralizing Finance CARISMA February 2010 Hersh Shefrin Mario L. Belotti Professor of Finance Santa Clara University

Behaviouralizing Finance

CARISMA

February 2010

Hersh Shefrin

Mario L. Belotti Professor of Finance

Santa Clara University

2Copyright, Hersh Shefrin 2010

Outline

• Paradigm shift.

• Strengths and weaknesses of behavioural approach.

• Combining rigour of neoclassical finance and the realistic psychologically-based assumptions of behavioural finance.


Quantitative Finance

• Behaviouralizing ─Beliefs & preferences─Portfolio selection theory─Asset pricing theory─Corporate finance─Approach to financial market regulation


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.


Conference ParticipantsExamples

• 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).


Beliefs

• Change of measure techniques.─Excessive optimism.─Overconfidence.─Ambiguity aversion.


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?


Excessive Optimism Sentiment Function

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

95.8

2%

96.1

5%

96.4

8%

96.8

1%

97.1

4%

97.4

8%

97.8

1%

98.1

5%

98.4

8%

98.8

2%

99.1

6%

99.5

0%

99.8

4%

100.

18%

100.

53%

100.

87%

101.

22%

101.

56%

101.

91%

102.

26%

102.

61%

102.

96%

103.

32%

103.

67%

104.

03%

104.

38%

104.

74%

105.

10%

105.

46%

105.

82%

106.

19%

Consumption Growth Rate g (Gross)


Excessive PessimismSentiment Function

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

95.8

2%

96.1

5%

96.4

8%

96.8

1%

97.1

4%

97.4

8%

97.8

1%

98.1

5%

98.4

8%

98.8

2%

99.1

6%

99.5

0%

99.8

4%

100.

18%

100.

53%

100.

87%

101.

22%

101.

56%

101.

91%

102.

26%

102.

61%

102.

96%

103.

32%

103.

67%

104.

03%

104.

38%

104.

74%

105.

10%

105.

46%

105.

82%

106.

19%



OverconfidenceSentiment Function

-2.5

-2

-1.5

-1

-0.5

0

0.5

96

%

97

%

99

%

10

1%

10

3%

10

4%

10

6%



Preferences

• Psychological concepts─Psychophysics in prospect theory.─Emotions in SP/A theory.

• Inverse S-shaped weighting function, rank dependent utility.

─Regret.─Self-control.


Prospect Theory Weighting Function

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

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 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

Decumulative Probability

Prospect Theory Weighting FunctionBased on Hölder Average

Ingersoll Critique


Functional Decomposition of Decumulative Weighting Function in SP/A Theory

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

D

h1(D)

h2(D)

h(D)

Inverse S in SP/A Rank Dependent Utility


Prospect Theory

• Tversky-Kahneman (1992)─Value function

• piecewise power function

─Weighting function • ratio of power

function to Hölder average

─Editing / Framing

Prospect Theory Value Function

-10

-8

-6

-4

-2

0

2

4

6

-10

-9.2

5-8

.5-7

.75 -7

-6.2

5-5

.5-4

.75 -4

-3.2

5-2

.5-1

.75 -1

-0.2

5 0.5

1.25 2

2.75 3.

54.

25 55.

75 6.5

7.25 8

8.75 9.

5

Gain/loss

Prospect Theory Weighting Function

0

0.2

0.4

0.6

0.8

1

1.2

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


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.


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 }


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.


Testing CPT vs. SP/AExperimental 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.


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.


Baseline: Aggressive Investor With Unbiased Beliefs

cj/c0 vs. g

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0.79

0.81

0.83

0.85

0.87

0.89

0.91

0.93

0.95

0.97

0.99

1.01

1.03

1.05

1.07

1.09

1.11

1.13

1.15

1.17

1.19

1.21

g

cj/

c0 cj/c0

g


How Would You Characteize an Investor Whose Return Profile Has

This Shape?cj/c0 vs. g

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.79

0.81

0.83

0.85

0.87

0.89

0.91

0.93

0.95

0.97

0.99

1.01

1.03

1.05

1.07

1.09

1.11

1.13

1.15

1.17

1.19

1.21

g

cj/

c0 cj/c0

g


Two Choices

• Aggressive underconfidence?

• Aggressive overconfidence?


CPT With Probability Weights


CPT With Rank Dependent Weights


SP/A With Cautious Hope


Associated Log-Change of Measure


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.


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


Graph of SDFWhat’s This?

• x-axis is a state variable like aggregate consumption growth.

• y-axis is M.

• SDF is linear.


How About This?Logarithmic Case?

• x-axis is a state variable like log-aggregate consumption growth.

• y-axis is log-M.• Relationship is

linear.


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?


Aït-Sahalia – Lo’s SDF Estimate


Rosenberg-Engle’s SDF Estimate


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).


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.


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.


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.


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


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.


ln SDF & Sentiment

-30.00%

-20.00%

-10.00%

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

95.8

2%

96.1

5%

96.4

8%

96.8

1%

97.1

4%

97.4

8%

97.8

1%

98.1

5%

98.4

8%

98.8

2%

99.1

6%

99.5

0%

99.8

4%

100.

18%

100.

53%

100.

87%

101.

22%

101.

56%

101.

91%

102.

26%

102.

61%

102.

96%

103.

32%

103.

67%

104.

03%

104.

38%

104.

74%

105.

10%

105.

46%

105.

82%

106.

19%

Gross Consumption Growth Rate g

ln(g)

Sentiment Function

ln(SDF)

Overconfident Bulls & Underconfident Bears


Behavioral SDF vs Traditional SDF

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

96

%

97

%

97

%

98

%

99

%

10

0%

10

1%

10

2%

10

3%

10

3%

10

4%

10

5%

10

6%

Aggregate Consumption Growth Rate g (Gross)

Behavioral SDF

Traditional Neoclassical SDF

How Different is a Behavioural SDF From a Traditional Neoclassical SDF?


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.



Taleb “Black Swan” SentimentOverconfidence

Sentiment Function

-2.5

-2

-1.5

-1

-0.5

0

0.5

96

%

97

%

99

%

10

1%

10

3%

10

4%

10

6%



Barone Adesi-Engle-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.


SDF for 2002, 2003, Garch on Left, Gaussian on Right


Continuous Time Modeling

• 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.

ln SDF & Sentiment

-30.00%

-20.00%

-10.00%

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

95.8

2%

96.1

5%

96.4

8%

96.8

1%

97.1

4%

97.4

8%

97.8

1%

98.1

5%

98.4

8%

98.8

2%

99.1

6%

99.5

0%

99.8

4%

100.

18%

100.

53%

100.

87%

101.

22%

101.

56%

101.

91%

102.

26%

102.

61%

102.

96%

103.

32%

103.

67%

104.

03%

104.

38%

104.

74%

105.

10%

105.

46%

105.

82%

106.

19%

Gross Consumption Growth Rate g

ln(g)

Sentiment Function

ln(SDF)

• 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]


Gross Return to Mean-variance Portfolio:Behavioral Mean-Variance Return vs Efficient Mean-Variance Return

75%

80%

85%

90%

95%

100%

105%

110%

96

%

97

%

99

%

10

1%

10

3%

10

4%

10

6%


Me

an

-va

ria

nc

e R

etu

rn

Behavioral MV Portfolio Return

Neoclassical Efficient MV Portfolio Return

How Different are Returns to a Behavioural MV-Portfolio From Neoclassical Counterpart?


MV Function Quadratic2-factor Model, Mkt and Mkt2

Gross Return to Mean-variance Portfolio:Behavioral Mean-Variance Return vs Efficient Mean-Variance Return

0.95

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

95

.82

%

96

.64

%

97

.48

%

98

.31

%

99

.16

%

10

0.0

1%

10

0.8

7%

10

1.7

4%

10

2.6

1%

10

3.4

9%

10

4.3

8%

10

5.2

8%

10

6.1

9%


Me

an

-va

ria

nc

e R

etu

rn

Efficient MV Portfolio Return

Behavioral MV Portfolio Return

Return to a Combination of the Market Portfolio and Risk-free Security


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.


Summary of Key PointsBehaviouralizing 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.

Documents

Behaviouralizing Finance CARISMA February 2010 Hersh Shefrin Mario L. Belotti Professor of Finance Santa Clara University