View Bias towards Ambiguity,
Expectile CAPM
and the Anomalies
Wei Hu,
ZhenLong Zheng
1
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
Motivation





Beta coefficient mean reverse
Expected utility maximization axiom
Risk preference & confidence
Risk & uncertainty
Equity premium puzzle
3
Main work
New concept of risk-reward measurement
(Non-perfect information, view tendency)
 Revised expected utility maximization axiom
 Redo Merton problem
 VLS econometrics method (GMM+VLS)
 Empirical Analysis

4
A general framework of risk-reward
measurement
-Concept
E
(Reward)
D
(Risk)

Mean & Variance
(OLS)
Median & Absolute deviation
(LAD)
Quantile & Weighted absolute deviation (Quantile regression)

Expetile & Variancile


(???)
5
A general framework of risk-reward
measurement
-Definition
Median( X )  arg min  | x  q | f X ( x)dx


q
Quantile ( X )  arg min(1   )  | x  q | f X ( x)dx    | x  q | f X ( x)dx


X q
X q
q
E ( X )  arg min  ( x  q ) 2 f X ( x) dx




q
E ( X )  arg min (1   )  ( x  q) 2 f X ( x)dx    ( x  q) 2 f X ( x)dx 


X q
X q
q
6
A general framework of risk-reward measurement
-More detail


Quantile
d (1   ) | x  q | f X ( x)dx    | x  q | f X ( x)dx


X q
X q
0
dq
(1   )
X q
f X ( x)dx   
X q
f X ( x)dx  0
(1   ) FX (q)   [1  FX (q)]  0
  FX (q)
q  F ( )
1
X
Expectile
(1   ) 
X q

X q
( x  q) 2 f X ( x)dx   
X q
( x  q) 2 (1   ) f X ( x)dx  
X q
( x  q) 2 f X ( x)dx
( x  q) 2 f X ( x)dx
  ( x  q) 2 [(1   )1 X  q   1 X  q ] f X ( x)dx
d   ( x  q) 2 [(1   )1X q  1X q ] f X ( x)dx

 0
dq
q [(1   )1X q  1X q ] f X ( x)dx   x[(1   )1X q  1X q ] f X ( x)dx
q
 x[(1   )1
 [(1   )1
X q
X q
 1X q ] f X ( x)dx
 1X q ] f X ( x)dx
7
A general framework of risk-reward
measurement
-Remark

E ( X )  q 
 X ( ) 

X
( ) xf X ( x ) dx
(1   )1X q   1X q
 [(1   )1
X q
  1X q ] f X ( x)dx
E ( X )  arg min (1   )  ( x  q) 2 f X ( x)dx    ( x  q) 2 f X ( x)dx 


X q
X q
q
VAR ( X )  (1   )
X  E ( X )
( x  E1 ( X ))2 f X ( x)dx   
X  E ( X )
( x  E1 ( X ))2 f X ( x)dx
8
A general framework of risk-reward
measurement
-Explanation
Table 2.2-1
X state
payoff
probability
E(X)
D(X)
Perfect Information Based Decision Making
s1
s2
1
3
0.05
0.15
0.05
0.45
99.0125 270.9375
Table 2.2-2
X state
payoff





s3
s4
50
0.7
35
14.175
Sum
100
0.1
10
297.025
1
45.5
681.15
Y state
payoff
probability
E(Y)
D(X)
s1
s2
s3
s4
Sum
-1000
3
50
100
0.00001 0.09999
0.8
0.1
1
-0.01 0.29997
40
10 50.28997
11.03109 223.6118 0.067266 247.1087 481.8188
Non- perfect Information Based Decision Making
s1
1
s2
3
s3
50
s4
100
Sum
Y state
payoff
s1
-1000
s2
3
s3
50
s4
100
Sum
Perfect information vs Non-perfect information / E+D vs maxmin
Question: same minimum?
Answer: quantile
Question: information fully used?
Answer: No + inconvenience  Expectile
9
A general framework of risk-reward
measurement
-Intuition
Figure2.2-1 Probability Adjustment under Non-perfect Information (Pessimistic Investor)
E ( X )  q 
 x[(1   )1
 [(1   )1
  1X q ] f X ( x)dx
X q
X q
 1X q ] f X ( x)dx
 E ( x)

E ( x )   E ( x )
 E ( x)

  50%
  50%
  50%
10
A general framework of risk-reward
measurement
-Comparison
11
A general framework of risk-reward
measurement
-Property

n
n
i 1
i 1
En ( X i )   E1 ( X i )
n
n
i 1
i 1
E ( X i )   E1 ( X i )
1
n
n
Info_ premium E ( X i )  E ( X i )
n
1
i 1
i 1
12
A general framework of risk-reward
measurement
-Property

13
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
14
Expectile CAPM
- Modelling
J [W (t ), t ]  max E
{C( ) , w( ) }
n
( , t )
T
{ U1[C ( ),  ]d  U 2 [W (T ), T ]}
t
J [W (T ),T ]  U 2 [W (T ),T ]
St: boundary condition:
n
budget equations:
W (t )   wi (t0 )
i 1
assumption2:
Pi (t )
[W (t0 )  C (t 0 )h], t  t0  h , h  0
Pi (t 0 )
dPi (t )
  i (t )dt   i (t ) dti ,
Pi (t )
V( nn)  [ il ],
 il   i l  il ,
i  1,2,, n
i, l  1,2,, n
15
Expectile CAPM Model
- Result

16
Expectile CAPM Model
-Model specification
Figure 3.2-1
 -adjusted risk-reward projection
i 
D
B
A1
B2
B1
systematic risk is the weighted
average of exposed risk and
potential risk
i
rf
 2~iM   iM
dt
 2 ~2
   , i  1,2,, n
2

  M   M
M  M   r f
dt
A
A2

C
O
17
New approach to explain equity premium
puzzle

Equity premium puzzle can be explained in a way that
people are pessimistic when there is no perfect
information in the postwar US
E ( R mv  R f )
9%  1%

a

(

ln
c
)
 0.5  1%
 ( R mv )
16%
P 
P 
Dt

 P   r f   C P  9% 1%  16% 252  0.5 1% 16%
Pt
dt

Dt  P
2

  r f   C P 2 1  CP
 sign( CP )  CP )
Pt
dt

9% 1% 16% 252  0.5 1% 16%( 0.962  0.2)
18
How to estimate VCAPM?


Why new econometrics model?
Model correct specification requires
E ( | X )  0

But
E( | X )  0  E ( | X )  0iff   50%
19
VLS

Comparison



VLS vs.OLS
VLS vs. WLS
VLS vs.Quantile regression
20
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
How to estimate VCAPM?

22
Empirical Results(1)

Assume risk aversion is constant 3,then from

Dt  P
2
P  
  r f   C P 2 1  CP
 sign( CP )  CP )
Pt
dt


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.
23
Is View tendency mean reversion?
0.53
Mean of View tendency=0.49735
View tendency
0.52
0.5
t
View endency(t)
0.51
0.49
0.48
0.47
0.46
0
100
200
300
400
500
period t
600
700
800
900
24
Empirical Results(2)
~  



1
iM
iM
 : 2 ~ 2

2
  M   M
2

From spectral analysis. 13 of 20 stocks share a
compatible periodicity with view tendency.
25
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
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