Measuring the behavioral component of financial fluctuations: an analysis based on the S&P 500.

Massimiliano Caporin

University of Padova

Department of Economics and Management “Marco Fanno”

Luca Corazzini

University of Padova

Department of Economics and Management “Marco Fanno”

Michele Costolaa

University of Padova and Ca’ Foscari University of Venice

Department of Economics and Management “Marco Fanno”

Abstract

We estimate a Bayesian, mixture model of financial investments with

two categories of agents: one rational (with a CRRA utility function) and

one behavioral (with an S-shaped, loss averse value function). Agents take

investment decisions by ranking the alternative assets according to their

performance measures. We estimate the evolution of the relative weight

of the behavioral component over time by using monthly data on the con-

stituents of the S&P 500 index from January 1962 to April 2012. Our results

confirm the existence of a significant behavioral component, which is more

likely to emerge during recessions. We find a strong correlation between the

estimated relative weight series and the VIX index. Thus, our estimates

substantially explain financial expectations.

Robustness CheckWe consider an utility function which behaves in an opposite way of the

S-shaped utility function (with no loss-aversion). It is concave in the domain of

losses (risk adverse) and convex in the gains (risk seeking). We replicate our

analysis considering the performance measure underlying this inverse-S-shaped

utility function, the ratio proposed by Tibiletti and Farinelli (2003).

The filtered τ ∗ (dotted) and the VIX (solid)

aThe research leading to these results has received funding from the European Union, Seventh Framework Programme FP7/2007-2013 under grant agreement SYRTO-SSH-2012-320270.

The Framework� We consider two agents in the market:√

the classical risk averse agent from EUT,√

an agent equipped with a S-shaped utility function introduced by Kahne-

man and Tversky (1979).

� The preferences of the two agents are expressed in terms of performance

measures respectively related to the maximization of their utility functions

(optimizing agents).

� Each agent acts according to her utility function (no interactions between

the 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 represented by these two types of investors.

Method

� Blend in a Bayesian manner the two components through a weighting factor.

� 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).

Classical and Behavioral financial economicsDifferent 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 SEU.∑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, Anchor-

ing...)

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

� We consider at this purpose an agent with loss aversion.

The risk adverse investorThe 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.

CRRA utility function provides a performance measure consistent with a market

in equilibrium, Zakamouline et al.(2009).

U(W ) =

1ρW

1−ρ, if ρ > 0, , ρ 6= 1

lnW if ρ = 1(1)

where ρ measures the degree of relative risk aversion.

� Mehra and Prescott (1985) shows that in order to be consistent with the

observed equity premium, ρ must be pretty high (around 30).

� when ρ is quite high, the relative preference for the moments of the distribu-

tion are similar to those of CARA utility function, Zakamouline et al.(2009).

� for computational convenience, we use CARA instead of CRRA.

Zakamouline et al. (2009a) have proposed following the conjecture of Hodges (1998) a

Generalized Sharpe Ratio (GSR) as

E[U∗(w̃)] = −e−12GSR

2. (2)

From the maximization of the expected utility,

E[U(w̃)] = E[− e−λ(x−rf)] = max

a

∫ ∞−∞−e−λa(x−rf)f̂h(x)dx (3)

where f̂h(x) is the estimated kernel density function.

Consequently, the GSR is obtained by the numerical optimization,

GSR =√−2 log(−E[U∗(w̃)]). (4)

We consider this ratio as the performance measure for the rational investor,

� it take into account all distribution moments of the risky asset x,

�GSR→ SR, when xd→ X ∼ N(µ, σ2).

The behavioral investor with S-shaped utilityZakamouline(2011) has generalized a behavioural utility function with a piecewise linear

plus power utility function,

U(W ) =

1+(W −W0)− (γ+/α)(W −W0)α, if W ≥ W0,

−λ(1−(W0 −W ) + (γ−/β)(W0 −W )β), if W < W0,

The authors derives the performance measure which maximizes the utility function,

Zγ−,γ+,λ,β,1 =E(x)− r − (1 λ− 1)LPM1(x, r)

β

√γ+UPMβ(x, r) + λγ−LPMβ(x, r)

The estimated factor τ ∗

� The purpose is to estimate the optimal weighting factor τ∗ for the aggregated measure

according a criterion function which optimizes the cumulative return of k = 100 assets.

� In practice, we want to check for which value of τ∗ we would have obtain 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 remain on her prior.

� Therefore, with our criterion function we are detecting which component between the

CRRA investor and the behavioral would have performed better in the market.

� In some sense, this τ∗ can describe part of the market behavior in a given moment.

The τ ∗ and VIX relationship

V IXt = c + β1τ∗t + β2h

1/2t + ηt (5)

where h1/2t is the volatility from an APARCH(1, 1, 1) model for the S&P500 returns

(control variable). xt = µ + h1/2t εt,

hδt = ω + α (|εt−1| + γεt−1)δ + βhδt−1.(6)

Estimation Resultsyear 62-70 71-80 81-90 91-00 01-12 All

c 1.00 1.39 1.20 1.011 1.039 1.14s.e 0.07 0.15 0.12 0.03 0.08 0.03

Skewness -0.07 0.05 0.58 0.00 0.03 1.08Kurtosis 2.10 1.84 1.76 2.00 1.80 3.27

Min 0.89 1.12 1.06 0.96 0.91 0.89Max 1.12 1.65 1.43 1.06 1.16 1.65

TOBIT regression for the filtered τ ∗ using the S-shaped utility function.

Estimated Robust s.e tStat pValue R2p

(Intercept) -0.30 0.059 -5.13 0.00

τ∗t 0.31 0.06 5.00 0.00 0.09

h1/2t 1.25 0.10 13.06 0.00 0.41

R2 0.59

R̄2 0.59 F-test 194.15 0.00The dependent variable is the VIX and the regressors are the filtered τ ∗ and

the volatility h1/2t from an APARCH(1, 1, 1) model for the S&P500 rets.

Estimate SE tStat pValue

(Intercept) 0.00 0.00 4.01 0.00

rτ 0.90 0.02 51.79 0.00

R2 0.83

R̄2 0.83 F-test 2682.66 0.00Regression where the dependent variable is the S&P500 equally weighted

return and the explicative variable is return from the selection of the

aggregated measure according τ ∗ for each period.

The filtered τ ∗

The bands represent the Economic Recessions according NBER.

The filtered τ ∗ (dotted) and the VIX (solid)

Different Agents in the Market

CRRA utility

(rational investor)

S-shaped Behavioral utility

(behavioral investor)

Generalized Sharpe-ratio

(Prior)

Z-ratio

(Conditional)

The Market

(Posterior)