This document summarizes a study that measures the behavioral component of financial market fluctuations using a model with two types of investors - rational investors who maximize expected utility, and behavioral investors who have S-shaped utility functions. The model blends the asset selections of these two investor types using a Bayesian approach, with the rational investor preferences as the prior and behavioral investor preferences as the conditional. An empirical analysis is conducted using the S&P 500 to estimate the optimal weighting parameter between the two investor types that maximizes past cumulative returns.

1. Measuring the behavioral
component of financial
fluctuaction. An analysis
based on the S&P 500.
SYstemic Risk TOmography:
Signals, Measurements, Transmission Channels, and
Policy Interventions
M.Caporin, University ofPadova (Italy)
L. Corazzini, University of Padova (Italy)
M. Costola,Ca'Foscari University ofVenice (Italy)
ASSET 2013– Bilbao(ES).November8, 2013.

2. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
Measuring the behavioral component
of financial fluctuations:
An analysis based on the S&P500.
Massimiliano Caporin Luca Corazzini Michele Costola
Ca’ Foscari University of Venice
Asset 2013, Bilbao.

3. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
The Framework
We consider two agents in the market:
√
an agent with classical risk averse utility from EUT
√
an agent equipped with a S-shaped-like utility function introduced by
Kahneman and Tversky (1979)
Each agent acts according to her own utility function (no
interactions between the agents)
The preferences of the two agents are expressed in terms of
performance measures respectively related to the maximization of
their utility functions (optimizing agents)
Given the two types of utility function, a different behavior of the
two agents is expected solely on the losses (e.g high volatility in the
financial markets)
The market is composed by these two types of investors

4. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
The Framework/2
Market
Rational investor Behavioral investor
Method
Blend in a Bayesian manner the two components through a
weighting factor which monitors the relevance of behavioral
expectations
CRITERION: Estimate the optimal weighting factor which is
maximized from the past cumulative return of a k-asset portfolio.
The weighing factor is time varying (rolling evaluation)

5. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
Behavioral Finance
Different type of agents are distinguished in base of the expectations they
have about the future asset prices, Hommes (2006).
Traditional Paradigm: agents are rational.
⇒ Bayes’ law (they update their belief correctly),
⇒ they are consistent with Savage’s notion of subjective expected
utility.
i
u(xi )P(xi ).
Behavioral finance argues some financial phenomena can be
explained using models where agents are not fully rational.
⇒ mistaken beliefs, they fail to update their beliefs correctly (bad
Bayesians).
(Overconfidence, Optimism, Representativeness, Convervatism,
Anchoring...)
⇒ different preferences (e.g. loss aversion).
We consider at this purpose an agent with loss aversion.

6. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
Loss Aversion at the level of the individual stocks
Barberis et al. (2001) shows that an economy where investors are loss
averse over the fluctuations of individual stocks reflects more empirical
phenomena w.r.t an economy where investors are loss averse over the
fluctuations of their stock portfolio.
Firm level returns have a high mean, are excessively volatile, and are
predictable in the time series using lagged variables,
the premium to value stocks and to stocks with poor prior returns,
cross-sectional facts: the discount rate is function of the past
performance returns.

7. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
CARA vs S-shaped Behavioral investor
CARA utility
(rational investor)
S-shaped Behavioral utility
(behavioral investor)
Generalized Sharpe-ratioa
Z-ratiob
Maximum
Principle
Optimal allocation between a risky
and a risk free asset in one period
horizon.
The solution is an increasing func-
tion of a quantity that can be inter-
preted as a performance measurec
.
Optimal allocation between a risky
and a risk free asset in one period
horizon.
The solution is an increasing func-
tion of a quantity that can be inter-
preted as a performance measurec
.
aZakamouline and Koekebakker (2009a).
bZakamouline(2011).
c Pedersen and Satchel (2002).

8. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
The risk-adverse agent
The optimal decision rule for a rational investor is based on E(U) where
her risk-aversion is given by the concavity property of her wealth function.
The CARA utility function has been widely used in the financial literature,
U(W ) = −e−λW
(1)
where λ represents the coefficient of risk aversion and W the investor’s
wealth.

9. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
The Sharpe ratio
The Maximum Principle
Consider the wealth w of the investor at the begin of a period t0.
In particular,
a, wealth allocated in the risky asset x,
w − a, wealth allocated in the riskless asset.
At the end of the period t1 the investor’s wealth will be,
˜w = a × (1 + x) + (w − a) × (1 + rf ) = a × (x − rf ) + w × (1 + rf ) (2)
The investor’s objective,
max
a
E[U( ˜w)]. (3)
Therefore it will be,
E[U∗
( ˜w)] = E[−e−λ[a(x−rf )+w(1+rf )]
] = E[−e−λ[a(x−rf )]
× e−λw(1+rf )
q
]
(4)
where a∗
is independent from the initial wealth of the investor.

10. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
The Sharpe ratio
We can approximate the expected utility using the Taylor’s series.
Intuitively,
E[U( ˜w)] = −1 + aλE(x − rf ) −
λ2
2
a2
E(x − rf )2
+ O( ˜w3
) (5)
If we approximate up to the first two derivatives, the FOC is given by:
∂E[U( ˜w)]
∂a
= λE(x − rf ) − λ2
E(x − rf )2
a = 0 (6)
and the quantity that maximizes the expected utility function is
proportional to the Sharpe Ratio,
a∗
=
1
λ
µ − rf
σ2
=
1
λ
SR
σ
. (7)

11. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
Generalized Sharpe Ratio
The Sharpe Ratio
the quantity that maximizes the expected utility function,
when there is a departure from Gaussianity, the ratio begins to be
biased both in the measurement and in the ranking among the
assets, Gatfaoui (2009).
Zakamouline et al. (2009a) derived a Generalized Sharpe Ratio (GSR)
E[U∗
( ˜w)] = −e− 1
2 GSR2
. (8)
The GSR is estimated using from expected utility (evaluated using a
kernel function to recover the returns density)
1
2
GSR2
= −log(−E[U∗
( ˜w)]). (9)
The GSR takes into account the whole distribution of the risky asset x
and not two moments only, and GSR → SR, when x
d
→ X ∼ N(µ, σ2
).

12. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
The S-shaped Utility Function
A behavioural investor is a decision maker that discriminates an
outcome above and below a reference point (gains and losses).
This value function is concave in the gains and convex in the losses,
the decision maker is risk adverse in the outcome above the
reference point and risk seeker below.
Kahneman and Tversky (1979) introduced the S-shaped utility
function starting from the evaluation of choices made by individuals
over alternative lotteries

13. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
The Generalized a behavioral utility function
Zakamouline(2011) has introduced a generalized behavioural utility
function (piecewise linear plus power utility)
U(W ) =
(W − W0) − (γ+/α)(W − W0)α
, if W ≥ W0,
−λ(W0 − W ) + (γ−/β)(W0 − W )β
), if W < W0,
where
γ+ and γ− are real numbers,
λ > 0, α > 0 and β > 0 are parameters.
This utility function is continuous and increasing in wealth with the
existence of the first and second derivatives on the investor’s wealth.

14. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
The Z-Ratio
The author derives the performance measure which maximizes the utility
function,
Zγ−,γ+,λ,β =
E(x) − r − (1−(W − W0)λ − 1)LPM1(x, r)
β
γ+UPMβ(x, r) + λγ−LPMβ(x, r)
where x is the returns series of the asset and r is set to the risk–free rate.
LPM and UPM are respectively the lower and upper partial moments,
LPMn(x, r) =
r
−∞
(r − x)n
dFx (x),
UPMn(x, r) =
∞
r
(x − r)n
dFx (x),
where n is the order of the partial moment of x at the threshold r and
Fx (·) is the distribution function of x.

15. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
The utility function
A possible alternative is the utility function by Kahneman and Tversky
(1979),
(W − W0)α
if W ≥ W0,
−λ(W0 − W )β
if W < W0.
the investor with this utility exhibits loss aversion in the sense of
Kahneman and Tversky (1979) when λ > 1, α = β
−U(W0 − ∆W ) > U(W0 + ∆W ), ∀∆W > 0,
! As stressed by Zakamouline (2011) the existence of the z–ratio
requires β > α which implies no loss aversion in the utility.
We might consider loss aversion in a local sense defined by
K¨obberling and Wakker (2005) around the reference point.
λ =
U (W0−)
U (W0+)
,
when λ > 1, the investor exhibits loss aversion.

16. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
Agents choices
Agents’ preferences are obtained from the maximization of their
utility functions;
What they do:
They rank the M assets at the univariate level and then allocate
their wealth on a subset of them, the K best performing assets
Agents are not making choices directly at the portfolio level, but
then they allocate their wealth on a portfolio
Agents are not estimating the portfolio weights (i.e MV) but allocate
their wealth over the K assets with equal weights
This might be a simplification, but
Impact of different values of K is evaluated (K is not estimated)
The EW strategy outperforms in-sample out-of-sample optimized
portfolios, DeMiguel et al. (2009) and Duchin and Levy (2009).

17. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
Agents choices and the market
Our purpose is to focus on the entire market, as influenced by the
choices made by the two types of agents
We take thus the market point of view, or the view of an imaginary
optimizing agent willing to take investment decisions accounting for
both rational and behavioral points of view (with the same
allocation scheme adopted by rational/behavioral)
The question is: how much should I have trusted behavioral/rational
views to allocate my portfolio in the optimal way?
We thus introduce a framework where a parameter allows us to
monitor the relevance of behavioral choices (it will not be a relative
weight of behavioral agents nor the relative weight of behavioral
rankings over rational rankings)
At the market level we act starting from a rational point of view, in
such a way the impact of behavioral choices might be null in a
limiting case

18. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
The Bayesian Approach
The purpose of the model is to blend the selection of the assets coming
from a rational investor among an investment universe by conditioning it
according a behavioural component. The evaluation is performed in
terms of utility at the single asset level
Aggregated Measure
(Posterior)
Rational investor
(Prior)
Behavioral investor
(Conditional)
Generalized Sharpe-ratio Z-ratio

19. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
The Prior and the Conditional
Agents’ preferences
Agents are making asset ranks on the basis of the expected value of their
performance measure; under the assumption that this performance
measure for asset i follows
PMi ∼ iid(µi , δ2
i ). (10)
agents are making ranks on the basis of ˆµi .
Rational agents evaluate µ on sample data and under the estimated
GSR, they consider the following prior density
P (µi ) ∼ N GSRi , σ2
i (11)
Alternatively µi = GSRi + εi with V [εi ] = σ2
i

20. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
The Prior and the Conditional
Behavioral agents, the conditional distribution, evaluate providing a
different point of view on µ, that is
P (µi ) ∼ N ˆZi , ω2
i (12)
or, µi = ˆZi + ηi with V [ηi ] = ω2
i , and ranks based on ˆZi

21. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
The Prior and the Conditional
We further assume that εi and ηi are independent and that the
variance of εi is multiplied by a factor τ representing the degree of
confidence on the prior density
i
ηi
∼ N 0,
τσ2
i 0
0 ω2
i
(13)
In empirical evaluations, the parameters σ2
i and ω2
i are replaced by
their estimates, at the single asset level, obtained by bootstrap
methods
We have now all the elements to derive the posterior density for the
expected optimal performance measure at the market level; two
possible approaches: Bayesian approach or Theil Mixed Estimation;
both lead to the same result.

22. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
The Posterior
To aggregated expectations of behavioral and rational agents we
follow the Bayes theorem and determine the posterior density whose
mean µP
i and variance Mp
i are given as
µp
i = (τσ2
i )−1
+ (ω2
i )−1 −1
(τσ2
i )−1
GSR + (ω2
i )−1 ˆZi
Mp
i = (τσ2
i )−1
+ (ω2
i )−1 −1
.
(14)
In this framework µp
i represents the aggregated performance
measure coming from a mixture of the two type of agents; this
measure can be used at the market level to take investment
decisions, based on asset ranks (on µp
i )
However, asset ranks, to be computed, requires the evaluation of
one element: τ.

23. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
The estimation of τ∗
We determine τ by maximizing the following criterion function,
max
τ
f (τ) =
1
m
t
l=t−m+1

 1
K
j∈At (τ)
rj,l

 (15)
where:
At (τ) is the set of the K assets with highest score of the aggregate
measure µp
i ; rp,l = 1
K j∈At (τ) rj,l is the time l return of the equally
weighted portfolio over the assets included in At (τ),
The criterion function is the average return of the portfolio over last
m observations
Risk is not introduced in the criterion function to have an
investment strategy similar to that of the rational/behavioral agents
where risk is evaluated at the single asset level and enters only in the
performance measures and thus in the ranks (which are
risk-adjusted)

24. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
The interpretation of τ∗
The purpose is to estimate τ∗
for the aggregated measure µp
i
according the criterion function which optimizes the cumulative
return of K assets.
That is, we want to check for which value of τ∗
we would have
obtained the optimal cumulative return of a portfolio with K assets
for a given period.
In fact, a higher value of τ∗
would imply that the investor should
have correct her action towards a behavioral direction.
Conversely, a low value of τ∗
would imply that the investor should
have remained on her prior rational ranks.
Therefore, with the criterion function we are detecting the relative
importance of the behavioral choices over the rational ones.
τ∗
is in some sense related to impact of behavioral choices on the
market fluctuations in a given moment.

25. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
The dataset and the Model Settings
We consider the constituents at a given time t of the S&P500 from
January 1962 to February 2012 at a monthly frequency (source
CRSP/COMPUSTAT)
The model has been applied on rolling windows of 60 monthly
returns (shorter evaluation windows provide more noisy and volatile
evaluations of the performance measures)
Thus, we selected from the constituents of the S&P500 at a given t
solely the assets with at least 60 observations.
The variance of the measures is obtained using a block bootstrap
procedure (dimension of blocks 4).
We initially select K = 100 to reflect the dimension of a stock index
as S&P100 (which is equally weighted).

26. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
The filtered τ∗
To smooth the optimized τ∗
we filtered it with a local level model
τ∗
t = µt + t, t ∼ NID(0, σ2
)
µt+1 = µt + ξt, ξt ∼ NID(0, σ2
ξ)
(16)
µt is the unobserved level for t = 1, . . . , n,
t is the observation disturbance and ξt is the level disturbance a
time t.
For the investor equipped with an utility function similar to Kahneman
and Tversky (1979), we estimated ˆt ∼ NID(0, .4547) and
ˆξt ∼ NID(0, .001678).

27. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
The filtered τ∗
Figure: The bands represent the Economic Recessions according NBER.

28. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
The filtered τ∗
/2
Figure: The bands represent the Financial Crisis in the US based on
Kindleberfer and Aliber (2005).

29. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
τ∗
and investor sentiment
Our estimate of the confidence on rational priors can be linked to
the literature of market sentiment indices
Market sentiment indices monitor the general view on the market,
i.e. bullish/bearish
Moreover, sentiment indices can have a behavioral
component/interpretation
Linking the τ∗
to sentiment indices allows verifying how much
behavioral views are related to the evolution of market sentiment,
with an expected positive relation
Several indices of market sentiment: VIX (fear index), volume,
liquidity, and other (Baker and Wurgler, 2007)

30. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
Market sentiments and Economic indicators
Our factor τ,
⇒ can be interpreted as a quantity associated to agent’s overall
behaviour in period of market stress.
⇒ We can relate its evolution to other indicators that monitor the level
of financial stress,
FSIs capture the key features of market stress: i.e increases
uncertainty about fundamental value of asset, increased uncertainty
about behavior of other investors, increased information asymmetry,
Sentiment indicator (Baker and Wurgler, 2007),
NBER recessions (dummy),
Liquidity indicator (Paster and Stambaugh, 2001),
Industrial Production.

31. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
Regression Analysis

32. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
Regression Analysis

33. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
Regression Analysis

34. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
Regression Analysis

35. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
Regression Analysis

36. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
Regression Analysis

37. Intro Literature Review The rational agent The Behavioral agent The Model Empirical Analysis τ∗ and investor sentiment Conclusion
Conclusion
We estimated a time varying weighting factor on the S&P500
according on the maximization on the cumulative returns.
We found some interesting similarities between the local
maxima/minima of the estimated factor and economic recessions
(NBER) and financial crisis.
We found a relationship with the financial sentiment and other
economic indicators when analysing the relation between our
estimated parameter and sentiment indices
We can consider our indicator as a possible endogenous market
sentiment index.
Robustness checks: analyses on the role of K and with an
“opposite” utility function.
Use τ as a pricing factor at single-asset level or for macro-sector.

38. This project has received funding from the European Union’s
Seventh Framework Programme for research, technological
development and demonstration under grant agreement n° 320270
www.syrtoproject.eu