Measuring the behavioral component of financial fluctuaction. An analysis based on the S&P 500 - Caporin M., Corazzini L., Costola M. - November 8, 2013

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 of Padova (Italy)L. Corazzini, University of Padova (Italy)M. Costola, Ca' Foscari University of Venice (Italy)

ASSET 2013– Bilbao (ES). November 8, 2013.

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

Measuring the behavioral componentof financial fluctuations:

An analysis based on the S&P500.

Massimiliano Caporin Luca Corazzini Michele Costola

Ca’ Foscari University of Venice

Asset 2013, Bilbao.


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 byKahneman and Tversky (1979)

Each agent acts according to her own utility function (nointeractions between the agents)

The preferences of the two agents are expressed in terms ofperformance measures respectively related to the maximization oftheir utility functions (optimizing agents)

Given the two types of utility function, a different behavior of thetwo agents is expected solely on the losses (e.g high volatility in thefinancial markets)

The market is composed by these two types of investors


The Framework/2

Market

Rational investor Behavioral investor

Method

Blend in a Bayesian manner the two components through aweighting factor which monitors the relevance of behavioralexpectations

CRITERION: Estimate the optimal weighting factor which ismaximized from the past cumulative return of a k-asset portfolio.

The weighing factor is time varying (rolling evaluation)


Behavioral Finance

Different type of agents are distinguished in base of the expectations theyhave 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 beexplained using models where agents are not fully rational.

⇒ mistaken beliefs, they fail to update their beliefs correctly (badBayesians).(Overconfidence, Optimism, Representativeness, Convervatism,Anchoring...)

⇒ different preferences (e.g. loss aversion).

We consider at this purpose an agent with loss aversion.


Loss Aversion at the level of the individual stocks

Barberis et al. (2001) shows that an economy where investors are lossaverse over the fluctuations of individual stocks reflects more empiricalphenomena w.r.t an economy where investors are loss averse over thefluctuations of their stock portfolio.

Firm level returns have a high mean, are excessively volatile, and arepredictable 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 pastperformance returns.


CARA vs S-shaped Behavioral investor

CARA utility(rational investor)

S-shaped Behavioral utility(behavioral investor)

Generalized Sharpe-ratioa Z-ratiob

MaximumPrinciple

Optimal allocation between a riskyand a risk free asset in one periodhorizon.The solution is an increasing func-tion of a quantity that can be inter-preted as a performance measurec .

Optimal allocation between a riskyand a risk free asset in one periodhorizon.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).cPedersen and Satchel (2002).


The risk-adverse agent

The optimal decision rule for a rational investor is based on E(U) whereher 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’swealth.


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,maxa

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.


The Sharpe ratio

We can approximate the expected utility using the Taylor’s series.Intuitively,

E [U(w)] = −1 + aλE (x − rf )− λ2

2a2E (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 )− λ2E (x − rf )2a = 0 (6)

and the quantity that maximizes the expected utility function isproportional to the Sharpe Ratio,

a∗ =1

λ

µ− rfσ2

=1

λ

SR

σ. (7)


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 bebiased both in the measurement and in the ranking among theassets, Gatfaoui (2009).

Zakamouline et al. (2009a) derived a Generalized Sharpe Ratio (GSR)

E [U∗(w)] = −e− 12GSR

2

. (8)

The GSR is estimated using from expected utility (evaluated using akernel function to recover the returns density)

1

2GSR2 = −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 xd→ X ∼ N(µ, σ2).


The S-shaped Utility Function

A behavioural investor is a decision maker that discriminates anoutcome 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 thereference point and risk seeker below.

Kahneman and Tversky (1979) introduced the S-shaped utilityfunction starting from the evaluation of choices made by individualsover alternative lotteries


The Generalized a behavioral utility function

Zakamouline(2011) has introduced a generalized behavioural utilityfunction (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 theexistence of the first and second derivatives on the investor’s wealth.


The Z-Ratio

The author derives the performance measure which maximizes the utilityfunction,

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)ndFx(x),

UPMn(x , r) =

∫ ∞r

(x − r)ndFx(x),

where n is the order of the partial moment of x at the threshold r andFx(·) is the distribution function of x .


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 ofKahneman 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–ratiorequires β > α which implies no loss aversion in the utility.

We might consider loss aversion in a local sense defined byKobberling and Wakker (2005) around the reference point.

λ =U ′(W0−)

U ′(W0+),

when λ > 1, the investor exhibits loss aversion.


Agents choices

Agents’ preferences are obtained from the maximization of theirutility functions;

What they do:

They rank the M assets at the univariate level and then allocatetheir wealth on a subset of them, the K best performing assetsAgents are not making choices directly at the portfolio level, butthen they allocate their wealth on a portfolioAgents are not estimating the portfolio weights (i.e MV) but allocatetheir 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 optimizedportfolios, DeMiguel et al. (2009) and Duchin and Levy (2009).


Agents choices and the market

Our purpose is to focus on the entire market, as influenced by thechoices made by the two types of agents

We take thus the market point of view, or the view of an imaginaryoptimizing agent willing to take investment decisions accounting forboth rational and behavioral points of view (with the sameallocation scheme adopted by rational/behavioral)

The question is: how much should I have trusted behavioral/rationalviews to allocate my portfolio in the optimal way?

We thus introduce a framework where a parameter allows us tomonitor the relevance of behavioral choices (it will not be a relativeweight of behavioral agents nor the relative weight of behavioralrankings over rational rankings)

At the market level we act starting from a rational point of view, insuch a way the impact of behavioral choices might be null in alimiting case


The Bayesian Approach

The purpose of the model is to blend the selection of the assets comingfrom a rational investor among an investment universe by conditioning itaccording a behavioural component. The evaluation is performed interms of utility at the single asset level

Aggregated Measure(Posterior)

Rational investor(Prior)

Behavioral investor(Conditional)

Generalized Sharpe-ratio Z-ratio


The Prior and the Conditional

Agents’ preferences

Agents are making asset ranks on the basis of the expected value of theirperformance measure; under the assumption that this performancemeasure for asset i follows

PMi ∼ iid(µi , δ2i ). (10)

agents are making ranks on the basis of µi .

Rational agents evaluate µ on sample data and under the estimatedGSR, they consider the following prior density

P (µi ) ∼ N(GSR i , σ

2i

)(11)

Alternatively µi = GSR i + εi with V [εi ] = σ2i



Behavioral agents, the conditional distribution, evaluate providing adifferent point of view on µ, that is

P (µi ) ∼ N(Zi , ω

2i

)(12)

or, µi = Zi + ηi with V [ηi ] = ω2i , and ranks based on Zi



We further assume that εi and ηi are independent and that thevariance of εi is multiplied by a factor τ representing the degree ofconfidence on the prior density(

εiηi

)∼ N

(0,

[τσ2

i 00 ω2

i

])(13)

In empirical evaluations, the parameters σ2i and ω2

i are replaced bytheir estimates, at the single asset level, obtained by bootstrapmethods

We have now all the elements to derive the posterior density for theexpected optimal performance measure at the market level; twopossible approaches: Bayesian approach or Theil Mixed Estimation;both lead to the same result.


The Posterior

To aggregated expectations of behavioral and rational agents wefollow the Bayes theorem and determine the posterior density whosemean µP

i and variance Mpi are given as

µpi =

[(τσ2

i )−1 + (ω2i )−1

]−1 [(τσ2

i )−1GSR + (ω2i )−1Zi

]Mp

i =[(τσ2

i )−1 + (ω2i )−1

]−1.

(14)

In this framework µpi represents the aggregated performance

measure coming from a mixture of the two type of agents; thismeasure can be used at the market level to take investmentdecisions, based on asset ranks (on µp

i )

However, asset ranks, to be computed, requires the evaluation ofone element: τ .


The estimation of τ ∗

We determine τ by maximizing the following criterion function,

maxτ

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 aggregatemeasure µp

i ; rp,l = 1K

∑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 lastm observations

Risk is not introduced in the criterion function to have aninvestment strategy similar to that of the rational/behavioral agentswhere risk is evaluated at the single asset level and enters only in theperformance measures and thus in the ranks (which arerisk-adjusted)


The interpretation of τ ∗

The purpose is to estimate τ∗ for the aggregated measure µpi

according the criterion function which optimizes the cumulativereturn of K assets.

That is, we want to check for which value of τ∗ we would haveobtained the optimal cumulative return of a portfolio with K assetsfor a given period.

In fact, a higher value of τ∗ would imply that the investor shouldhave correct her action towards a behavioral direction.Conversely, a low value of τ∗ would imply that the investor shouldhave remained on her prior rational ranks.

Therefore, with the criterion function we are detecting the relativeimportance of the behavioral choices over the rational ones.

τ∗ is in some sense related to impact of behavioral choices on themarket fluctuations in a given moment.


The dataset and the Model Settings

We consider the constituents at a given time t of the S&P500 fromJanuary 1962 to February 2012 at a monthly frequency (sourceCRSP/COMPUSTAT)

The model has been applied on rolling windows of 60 monthlyreturns (shorter evaluation windows provide more noisy and volatileevaluations of the performance measures)

Thus, we selected from the constituents of the S&P500 at a given tsolely the assets with at least 60 observations.

The variance of the measures is obtained using a block bootstrapprocedure (dimension of blocks 4).

We initially select K = 100 to reflect the dimension of a stock indexas S&P100 (which is equally weighted).


The filtered τ ∗

To smooth the optimized τ∗ we filtered it with a local level model{τ∗t = µt + εt , εt ∼ NID(0, σ2

ε )

µt+1 = µt + ξt , ξt ∼ NID(0, σ2ξ)

(16)

µt is the unobserved level for t = 1, . . . , n,

εt is the observation disturbance and ξt is the level disturbance atime t.

For the investor equipped with an utility function similar to Kahnemanand Tversky (1979), we estimated εt ∼ NID(0, .4547) andξt ∼ NID(0, .001678).


The filtered τ ∗

Figure: The bands represent the Economic Recessions according NBER.


The filtered τ ∗/2

Figure: The bands represent the Financial Crisis in the US based onKindleberfer and Aliber (2005).


τ ∗ and investor sentiment

Our estimate of the confidence on rational priors can be linked tothe 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 behavioralcomponent/interpretation

Linking the τ∗ to sentiment indices allows verifying how muchbehavioral 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)


Market sentiments and Economic indicators

Our factor τ ,⇒ can be interpreted as a quantity associated to agent’s overallbehaviour in period of market stress.⇒ We can relate its evolution to other indicators that monitor the levelof financial stress,

FSIs capture the key features of market stress: i.e increasesuncertainty about fundamental value of asset, increased uncertaintyabout behavior of other investors, increased information asymmetry,

Sentiment indicator (Baker and Wurgler, 2007),

NBER recessions (dummy),

Liquidity indicator (Paster and Stambaugh, 2001),

Industrial Production.


Regression Analysis


Regression Analysis


Regression Analysis


Regression Analysis


Regression Analysis


Regression Analysis


Conclusion

We estimated a time varying weighting factor on the S&P500according on the maximization on the cumulative returns.

We found some interesting similarities between the localmaxima/minima of the estimated factor and economic recessions(NBER) and financial crisis.

We found a relationship with the financial sentiment and othereconomic indicators when analysing the relation between ourestimated parameter and sentiment indices

We can consider our indicator as a possible endogenous marketsentiment 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.

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

Economy & Finance

Measuring the behavioral component of financial fluctuaction. An analysis based on the S&P 500 - Caporin M., Corazzini L., Costola M. - November 8, 2013