1 view bias towards ambiguity, expectile capm and the anomalies wei hu, zhenlong zheng

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View Bias towards Ambiguity, Expectile

CAPM and the Anomalies

Wei Hu, ZhenLong Zheng

Campbell (2000), Asset Pricing at the Millennium

Theorists develop models with testable predictions; empirical researchers document “puzzles” –stylised facts that fail to fit established theories –and this stimulates the development of new theories. Such a process is part of the normal development of any science.

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Motivation

Beta coefficient mean reverse Expected utility maximization axiomRisk preference & confidenceRisk & uncertainty Equity premium puzzle

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Main work

New concept of risk-reward measurement (Non-perfect information, view tendency)Revised expected utility maximization

axiomRedo Merton problemVLS econometrics method (GMM+VLS)Empirical Analysis

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A general framework of risk-reward measurement

-Concept

Mean & Variance (OLS) Median & Absolute deviation (LAD) Quantile & Weighted absolute deviation (Quantile regression)

Expetile & Variancile (???)

E(Reward)

D(Risk)

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A general framework of risk-reward measurement

-Definition

dxxfqxXMedian X

q)(||minarg)(

dxxfqxXE X

q)()(minarg)( 2

qX qX XX

qdxxfqxdxxfqxXQuantile )(||)(||)1(minarg)(

2 2( ) arg min (1 ) ( ) ( ) ( ) ( )X XX q X qqE X x q f x dx x q f x dx

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A general framework of risk-reward measurement-More detail

Quantile Expectile

)(1 XFq

0)(||)(||)1(

dq

dxxfqxdxxfqxdqX qX XX

0)()()1( qX qX XX dxxfdxxf

0)](1[)()1( qFqF XX

)(qFX

dxxfqx

dxxfqxdxxfqx

dxxfqxdxxfqx

XqXqX

qX qX XX

qX qX XX

)(]11)1[()(

)()()()1()(

)()()()()1(

2

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22

dxxfxdxxfq XqXqXXqXqX )(]11)1[()(]11)1[(

dxxf

dxxfxq

XqXqX

XqXqX

)(]11)1[(

)(]11)1[(

0)(]11)1[()( 2

dq

dxxfqxd XqXqX

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A general framework of risk-reward measurement

-Remark

( ) ( ) ( )X XE X q xf x dx

dxxf XqXqX

qXqXX

)(]11)1[(

11)1()(

2 2( ) arg min (1 ) ( ) ( ) ( ) ( )X XX q X qqE X x q f x dx x q f x dx

1 2 1 2

( ) ( )( ) (1 ) ( ( )) ( ) ( ( )) ( )X XX E X X E X

VAR X x E X f x dx x E X f x dx

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A general framework of risk-reward measurement

-Explanation

Perfect information vs Non-perfect information / E+D vs maxmin Question: same minimum? Answer: quantile Question: information fully used? Answer: No + inconvenience Expectile

Table 2.2-1 Perfect Information Based Decision Making

X state s1 s2 s3 s4 Sum Y state s1 s2 s3 s4 Sumpayoff 1 3 50 100 payoff -1000 3 50 100probability 0.05 0.15 0.7 0.1 1 probability 0.00001 0.09999 0.8 0.1 1E(X) 0.05 0.45 35 10 45.5 E(Y) -0.01 0.29997 40 10 50.28997D(X) 99.0125 270.9375 14.175 297.025 681.15 D(X) 11.03109 223.6118 0.067266 247.1087 481.8188

Table 2.2-2 Non- perfect Information Based Decision MakingX state s1 s2 s3 s4 Sum Y state s1 s2 s3 s4 Sumpayoff 1 3 50 100 payoff -1000 3 50 100

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A general framework of risk-reward measurement

-Intuition

Figure2.2-1 Probability Adjustment under Non-perfect Information (Pessimistic Investor)

( ) 50%

( ) ( ) 50%

( ) 50%

E x

E x E x

E x

[(1 )1 1 ] ( )( )

[(1 )1 1 ] ( )

X q X q X

X q X q X

x f x dxE X q

f x dx

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A general framework of risk-reward measurement-Comparison

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A general framework of risk-reward measurement

-Property

n

ii

n

ii

n XEXE1

1

1

)()(

n

ii

n

ii XEXE

1

1

1

1 )()(

)()(_1

1

1

n

ii

n

ii

n XEXEpremiumInfo

A general framework of risk-reward measurement-Property

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Expectile CAPM Model-Assumption

Assumption1: time interval between each decision is infinitesimal

Assumption2: prices are diffusion processes

Assumption3: only consumption and portfolio process are controllable

Assumption4: No exogenous endowment

Assumption5: Homogenous investors

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Expectile CAPM- Modelling

St: boundary condition:

budget equations:

assumption2:

]}),([]),([{max]),([ 21),(},{ )()(

TTWUdCUEttWJT

t

nt

wC

]),([]),([ 2 TTWUTTWJ

0,,])()([)(

)()()( 000

010

hhtthtCtWtP

tPtwtW

i

in

ii

nidttdtttP

tdPiii

i

i ,,2,1,)()()(

)(

nliV illiililnn ,,2,1,,],[)(

Expectile CAPM Model - Result

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Expectile CAPM Model-Model specification

systematic risk is the weighted average of exposed risk and potential risk

A

B

A1 B1 C O A2 B2

Figure 3.2-1 -adjusted risk-reward projection

nir

dt

rdt

MM

iMiM

fMM

fii

,,2,1,~

~

222

2

D

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New approach to explain equity premium puzzle

)ln()(

)(ca

R

RREmv

fmv

PCfP

t

tP r

dtP

D

))(1 22CPCPCPPC

fP

t

tP signr

dtP

D

%15.0%16

%1%9

%16%15.0252%16%1%9

Equity premium puzzle can be explained in a way that people are pessimistic when there is no perfect information in the postwar US

)2.096.0%(16%15.0252%16%1%9 2

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How to estimate VCAPM?

Why new econometrics model?Model correct specification requires

But

( | ) 0 ( | ) 0 50%E X E X iff

( | ) 0E X

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VLSComparison VLS vs.OLS VLS vs. WLS VLS vs.Quantile regression

How to estimate VCAPM?

We establish the VLS methodology by listing all the assumptions, finding new estimators, and proving the asymptotic consistency and normality in large sample analysis. We develop the hypothesis testing by the case of conditional homoskedadticity and heteroskedasticty. We estimate and test the expectile based unconditional CAPM theory through the conditional GMM being restricted by a view bias based linear condition.

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How to estimate VCAPM?

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Empirical Results(1) Assume risk aversion is constant 3,then from

We get find theta is between o.47 to 0.53 with mean 0.497 using US. post war data. The periodicity is 60 months.

))(1 22CPCPCPPC

fP

t

tP signr

dtP

D

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0 100 200 300 400 500 600 700 800 9000.46

0.47

0.48

0.49

0.5

0.51

0.52

0.53

period t

Vie

w tendency(t

)Is View tendency mean reversion?

Mean of View tendency=0.49735

View tendency

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Empirical Results(2)

From spectral analysis. 13 of 20 stocks share a compatible periodicity with view tendency.

1222

2

~

~:

MM

iMiM

Contributions

We define the expectile and variacile We revise the expectation utility

maximization axiom into an expectile utility maximization axiom.

We redo Merton Problem under the expectile framework, and extend the CAPM theory.

we develope a new econometrics methodology, the view bias adjusted least square (VLS) to test the extended CAPM theory.

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Contributions

we demonstrate the advantage of the expectile based asset pricing theory through empirical application.

Our approach solves the two categories of anomalies within one integrated and extended asset pricing theoretical framework.

The advantage of our approach is that not only does the expectile take the merits of quantile, but also the expectile based asset pricing framework takes the merits of expectation framework.

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