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Empirical Financial Economics
Asset pricing and Mean Variance Efficiency
Eigenvalues and Eigenvectors
Eigenvalues and eigenvectors satisfy
Eigenvectors diagonalize covariance matrix
1,,
0,i i i i j
i j
i j
,
0,i
i j
i jor D
i j
1
1 ,G D then G G I GG
Normal Distribution results
Basic result used in univariate tests:
2
22 2
2
( , ) ( ,1)
(1, )
rr N z N
z Noncentral
Multivariate Normal results
Direct extension to multivariate case:
2 1 2 1
1
( , ) ( , )
' ' ( , )m
ii
r MVN G r z MVN G I
z z z r GG r r r Noncentral m
Mean variance facts
1 2
1 11 2 1 2
1 11 2 1 2
1 22 21 11 2 1 2
1. . ( ) (1 )
21
. . . :
,1
xMin x x
s t x E L x x E x x
x
F O C x x
E x a b cE b a bE
ac b ac bx b c
The geometry of mean variance
a
b
a
b
E
2 1a
1 1
2
1/1/
0
bx b
22
2
2a bE cE
ac b
Note: returns are in excess of the risk free rate
fr
Tests of Mean Variance Efficiency
Mean variance efficiency implies CAPM
For Normal with mean and covariance matrix ,
is distributed as noncentral Chi Square with
degrees of freedom and noncentrality
11/
1/
1/m
x bx
x x b x x Ex x
x b
z 1z z
m 1
MacBeth T2 test
Regress excess return on market excess return
Define orthogonal return Market efficiency implies ,
estimate .
; ,f m fy w y r r w r r
z
0Ez ̂
22
212
2
1
1 1 ˆˆVar 1 1ˆ
( )
T
tt
mTw
tt
ww
T TT w w
MacBeth T2 test (continued)
The T2 test statistic is distributed as noncentral Chi Square with m degrees of freedom and noncentrality parameter
The quadratic form is interpreted as the Sharpe ratio of the optimal orthogonal portfolio
This is interpreted as a test of Mean Variance Efficiency
Gibbons Ross and Shanken adjust for unknown
12 1ˆ ˆ ˆ1 mT
12 1ˆ1 mT
1
Gibbons, M, S. Ross and J. Shanken, 1989 A test of the efficiency of a given portfolio
Econometrica 57, 1121-1152
The geometry of mean variance
E
Note: returns are in excess of the risk free rate
fr
2 1
2 1
Multiple period consumption-investment problem
Multiperiod problem:
First order conditions:
Stochastic discount factor interpretation:
0
Max ( )jt t j
j
E U c
,( ) (1 ) ( )jt t i t j t jU c E r U c
, , ,
( )1 (1 ) ,
( )t jj
t i t j t j t jt
U cE r m m
U c
Stochastic discount factor and the asset pricing model
If there is a risk free asset:
which yields the basic pricing relationship
, , , , ,,
11 (1 ) (1 )
(1 )t f t j t j r t j t t j t t jf t j
E r m r E m E mr
1 (1 )
(1 ( )
(1 ) [ ] ( )
(1 ) ( )f f
E r m
E r m
E m E r m
r r E r m
Stochastic discount factor and mean variance efficiency
Consider the regression model
The coefficients are proportional to the negative of minimum variance portfolio weights, so
( ) ( )m E m r
1
(1 ) ( )( )
1( )
1
f f
ff
r r E r r
rr
2
2
(1 ) ( ) (1 ) ( )
(1 )(1 )
f f f MV
MV ff MV i
f MV
r r E r m b r E r r
rb r b
r
MVm a br
The geometry of mean variance
a
b
a
b
E
2 1a
1 1
2
1/1/
0
bx b
22
2
2a bE cE
ac b
Note: returns are in excess of the risk free rate
fr
Hansen Jagannathan Bounds
Risk aversion times standard deviation of consumption is given by:
“Equity premium puzzle”: Sharpe ratio of market implies a risk aversion coefficient of about 50
Consider
2(1 ) MV fm
f MV MVm MV
rr b
[(1 ) ] 1m
MV f fm m
m m MV
m a br m E r m
r r r r
Non negative discount factors
Negative discount rates possible when market returns are high
Consider a positive discount rate constraint:
1, 2 0
2MV MV MV MVr m
,
( )
(1 ) ( )
(1 ) ( ) (1 ) ( )( )
(1 )
M
f f
f f M
f M c c
m a b r c
r r E r m
r a E r b r E r r c
b r LPM
Stochastic discount factor and the asset pricing model
If there is a risk free asset:
which yields the basic pricing relationship
, , , , ,,
11 (1 ) (1 )
(1 )t f t j t j r t j t t j t t jf t j
E r m r E m E mr
1 (1 )
(1 ( )
(1 ) [ ] ( )
(1 ) ( )f f
E r m
E r m
E m E r m
r r E r m
Where does m come from?
Stein’s lemmaIf the vector ft+1 and rt+1 are jointly
Normal
Taylor series expansionLinear term: CAPM, higher order
terms? Put option payoff
11 1
( )( )
( )t
t tt
u cm g f
u c
1 1 1 1 1 1 1
1 1
[( ) ( )] [ ( )] [( ) ]
. . ( [ ( )] )t t t t t t t
t f ft t t
E r g f E g f E r f
r i e the APT assumes E g f exists
21 1 1( ) ...t t tg f a bf cf
1( ) ( )t Mg f a b r c
Multivariate Asset Pricing
Consider
Unconditional means are given by
Model for observations is
m m m mr b f e
r Bf e
fr B
fr r B Bf e
Principal Factors
Single factor caseDefine factor in terms of
returnsWhat factor maximizes
explained variance?
Satisfied by with criterion equal to
r f e ( )f w r
2 2
1
. . . : ( ) ( ) 0
m
i fw
i
w wMax
w w
F O C w w w w w w
jw k j
Principal Factors
Multiple factor caseCovariance matrix Define and the
first columnsThen This is the “principal factor”
solutionFactor analysis seeks to
diagonalize
Satisfied by with criterion equal to
r Bf e
efB B D *B D
*
kB k* * * *
k kB B B B
Importance of the largest eigenvalue
The Economy
1 1
( ) 11, ,
( )
it it i t ki kt it
i
i b
r b f b f
E bi m
Cov b D
What does it mean to randomly select security i?
Restrictive?
Harding, M., 2008 Explaining the single factor bias of arbitrage pricing models in finitesamples Economics Letters 99, 85-88.
k Equally important factors
Each factor is priced and contributes equally (on average) to variance:
Eigenvalues are given by
2 2
22 2
1 2
22 2
2 2
21
1, ,
( 1) 1(1 )
( 1) 1(1 )
fj f
b
k b
k m
j k
Rm km
k R
Rm
k R
Important result
The larger the number of equally important factors, the more certain would a casual empirical investigator be there was only one factor!
1
22
1k b
dkdm
ddm
Numerical example
2
2
2
1
2
1
4
: .1235
.0045
.01:
0.00063456
0.00000158
0.0
b
k
k m
k
Brown and Weinstein R
d
dmd
dmd
dm
What are the factors?
Where W is the Helmert rotation:
*1
*2
*
1 1 2 1 2 3 1 ( 1)
1 1 2 1 2 3 1 ( 1)
1 0 2 2 3 1 ( 1)
1 0 0 ( 1) ( 1)
s
k
B BWD
k k k
k k k
W k k k
k k k k
b
b
The average is one and
the remaining average to zero
Implications for pricing
Regress returns on factor loadings
Suppose k factors are priced:
Only one factor will appear to be priced!
1 11 1 1
111 2 1 2
1 2
* *1 2
ˆ ˆ( )
ˆ( ) , 0
( ) ( ) ( ) 2
( ) 2 ( ) 0
k
k
k
B B B r Var B B
Var B B I where
If t t t
Then t k and t
Application of Principal Components
Yield curve factors: level, slope and curvature
1 1 11 1
2 2 2 * *2 3
3 3 33 2
4 4 4
* *,
t t t
t t tt t
t t tt t t t
t t tt t
t t t
t t
y Bf e
y e ef f
y e ef f B f e
y e ef f
y e e
B B f f where
1 0 0
0 0 1 .
0 1 0
Note I
A more interesting example
Yield curve factors: level, slope and curvature
1 1 11 1
2 2 2* * *2 2
3 3 3*3 3
4 4 4
* *
?
?
?
?
,
t t t
t t tt t
t t tt t t t
t t tt t
t t t
t t
y Bf e
y e ef f
y e ef f B f e
y e ef f
y e e
B B f f wher
1 0 0
1 10 .
2 21 1
02 2
e Note I
Application of Principal Components
Procedure:
1. Estimate B* using principal components
2. Choose an orthogonal rotation to minimize a function
that penalizes departures from
*( )
. .
Min h B
s t I
B
(.)h
Conclusion
Mean variance efficiency and asset pricing
Important role of Sharpe ratioImplicit assumption of
Multivariate NormalityLimitations of data driven
approach