1 view bias towards ambiguity, expectile capm and the anomalies wei hu, zhenlong zheng
TRANSCRIPT
1
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.
2
3
Motivation
Beta coefficient mean reverse Expected utility maximization axiomRisk preference & confidenceRisk & uncertainty Equity premium puzzle
4
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
5
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)
6
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
7
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
22
22
dxxfxdxxfq XqXqXXqXqX )(]11)1[()(]11)1[(
dxxf
dxxfxq
XqXqX
XqXqX
)(]11)1[(
)(]11)1[(
0)(]11)1[()( 2
dq
dxxfqxd XqXqX
8
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
9
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
10
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
11
A general framework of risk-reward measurement-Comparison
12
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
13
14
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
15
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
16
17
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
18
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
19
How to estimate VCAPM?
Why new econometrics model?Model correct specification requires
But
( | ) 0 ( | ) 0 50%E X E X iff
( | ) 0E X
20
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.
21
22
How to estimate VCAPM?
23
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
24
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
25
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.
26
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.
27
28
Thanks!welcome to visit:
http://efinance.org.cn