ap5 asset pricing
TRANSCRIPT
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Asset Pricing
Zheng Zhenlong
Chapter 5.
Mean-variance frontier
and beta representations
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Main contents
Expected return-Beta representation
Mean-variance frontier: Intuition and Lagrangian
characterization
An orthogonal characterization of mean-variance frontier
Spanning the mean-variance frontier
A compilation of properties of
Mean-variance frontiers for m: H-J bounds
***,, xRR e
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5.1 Expected Return-BetaRepresentation
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representation
....)( bibaiaiRE
... , 1, 2,...i a b it i ia t ib t t R t T
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Remark(1)
In (1), the intercept is the same for all assets.
In (2), the intercept is different for different asset.
In fact, (2) is the first step to estimate (1).
One way to estimate the free parameters is to runa cross sectional regression based on estimation of beta
is the pricing errors
( ) ....i
ia a ib b iE R
,
i
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Remark(2)
The point of beta model is to explain the variation in
average returns across assets.
The betas are explanatory variables,which vary asset
by asset. The alpha and lamda are the intercept and slope in the
cross sectional estimation.
Beta is called as risk exposure amount, lamda is the
risk price.
Betas cannot be asset specific or firm specific.
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Some common special cases
If there is risk free rate,
If there is no risk-free rate, then alpha is called (expected)zero-beta
rate.
If using excess returns as factors,
(3)
Remark: the beta in (3) is different from (1) and (2).
If the factors are excess returns, since each factor has beta of one on
itself and zero on all the other factors. Then,
( ) ..., 1, 2,...ei ia a ib bE R i N
( ) ( ) ( ) ..., 1, 2,...ei a bia ib
E R E f E f i N
fR
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5.2 Mean-Variance Frontier: Intuition
and Lagrangian Characterization
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Mean-variance frontier
Definition: mean-variance frontier of a given set of assets is
the boundary of the set of means and variances of returns on
all portfolios of the given assets.
Characterization: for a given mean return, the variance is
minimum.
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With or without risk free rate
tangencyrisk asset frontier
original assets
( )R
)(RE
fR
mean-variance
frontier
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exist?
Theorem: So long as the variance-covariance matrix of
returns is non singular, there is mean-variance frontier.
Intuition Proof:
If there are two assets which are totally correlated and havedifferent mean return, this is the violation of law of one
price. The law of one price implies the existence of mean
variance frontier as well as a discount factor.
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Zheng ZhenlongMathematical method: Lagrangian
approach
Problem:
Lagrangian function:
])')([(
),(
11',',.,'min }{
ERERE
REE
wuEwtswww
1)'(''wuEwwwL
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Mathematical method: Lagrangian
approach(2)
First order condition:
If the covariance matrix is non singular, the inverse
matrix exists, and
01Ew
w
L
11'1'1
1'',1)1('1'1
,)1(''
),1(
11
11
1
1
1
u
E
EEEEw
uEEwE
Ew
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Mathematical method: Lagrangian
approach(3)
In the end, we can get
1'1,1','
,2
)var(
,)(1)(
,,
111
2
2
2
1
22
CEBEEA
BAC
ABuCuR
BAC
BuABCuEw
BAC
BuA
BAC
BCu
p
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Remark
By minimizing var(Rp) over u,giving
min var 1 1/ , 1/(1' 1)u B C w
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5.3 An orthogonal characterization ofmean variance frontier
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Introduction
Method: geometric methods.
Characterization: rather than write portfolios as combination
of basis assets, and pose and solve the minimization problem,
we describe the return by a three-way orthogonal
decomposition, the mean variance frontier then pops outeasily without any algebra.
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Theorem: two-fund theorem for MVF
* *mv eR R R
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Proof: Geometric method
* *i e iR R R n
0
R=space of return (p=1)
Re =space of excess return (p=0)
R*R*+wiRe*
Re*
E=0 E=1 E=2
Rf=R*+RfRe*
NOTE:123111
1
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Proof: Algebraic approach
Directly from definition, we can get
*
* *
2 2 * * 2
( ) ( ) 0
( ) ( ) ( )
( ) ( ) ( )
0
i e i
i i e
i i e i
i
E n E R n
E R E R w E R
R R w R n
n
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Zheng ZhenlongDecomposition in mean-
variance space
*R
)()()()( 22*22*2 ie nEREwRERE
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5.4 Spanning the mean variance
frontier
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frontier
With any two portfolios on the frontier. we can span the
mean-variance frontier.
Consider
/
)1()(
,
,0,
*****
**
**
wy
yRRyRRwRwRR
RRR
RRR
e
e
e
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5.5 A compilation of properties of R*,
Re*, and x*
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Properties(1)
Proof:
,)(
1
)( 2*2*
xERE
)(/1)(
)()(
,)(
,)(
2*2*
**
2*
2*
**2*
2*
**
xExE
RxERE
xE
RxR
xE
xR
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Properties(2)
Proof:)( 2*
**
RE
R
x
)()(
,)(
2*
*2***
2*
**
RE
RxERx
xE
xR
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Properties(4)
If a risk-free rate is traded,
If not, this gives a zero-beta rate interpretation.
*2
* *
1 ( )
( ) ( )
f E RR
E x E R
A P i i
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Properties(5)
has the same first and second moment.
Proof:
Then
* * * *2( ) ( ) ( )e e e eE R E R R E R
*eR
))(1)(()()()var( **2*2** eeeee RERERERER
A t P i i
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Properties(6)
If there is risk free rate,
Proof:
** eff RRRR
*
* * *
*
* *
1 (1| ) (1| )
11 (1| ) 1
e
e
f
f
e
f f f
proj R proj R
R proj R RR
R R R R
R1R
R
A t P i i
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If there is no risk free rate
Then the 1 vector can not exist in payoff space since it is riskfree. Then we can only use
*
*
* *
* *
**
*2
(1 | ) ( (1 | ) | ) ( (1 | ) | )
(1 | ) (1 | )
(1 | ) (1 | )
(1 | ) )
( )(1 | )
( )
e
e
e
proj X proj proj X R proj proj X R
proj R proj R
R proj X proj R
proj X R
E Rproj X R
E R
E( x
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Properties(7)
Since
We can get
* 1
* * *
' ( ')
( ) ( )
x p E xx x
p x E x x
pxxEp
xxxEp
xp
xR
1
1
*
**
)'('
''
)(
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Zheng ZhenlongMean-variance frontier for m: H-J
bounds
The relationship between the Sharpe ratio of an excess returnand volatility of discount factor.
If there is risk free rate,
)(
|)(|
)(
)(
,1|)()()()(|||
,0)()()()()(
,
,
e
e
e
e
Rm
e
Rm
ee
R
RE
mE
m
RmREmE
RmREmEmRE
e
e
fRmE /1)(
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Jagannathan bounds
For any hypothetical risk free rate, the highestSharpe ratio is the tangency portfolio.
Note: there are two tangency portfolios, the higher
absolute Sharpe ratio portfolio is selected. If risk free rate is less than the minimum variance
mean return, the upper tangency line is selected,and the slope increases with the declination of risk
free rate, which is equivalent to the increase ofE(m).
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g
Zheng Zhenlong
Duality
A duality between discount factor volatility and Sharpe ratios.
{ } { }
( ) ( )min max
( ) ( )e
e
eall m that price x X all excess returns R in X
m E R
E m R
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A li i i f H J
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g
Zheng ZhenlongAn explicit expression for H-J
bounds
Proof:
))()(())'()(()( 12 xEmEpxEmEpm
2 2 2
2 1
1 1' 2 2
1' 2
1 2
( ) ( ) ( )
( ) 2 ( ) [[( ( ) ( )] ( ( ))]
[( ( ) ( )) ' ( ( ))( ( )) ' ( ( ) ( ))] ( ) ( )
( ( ) ( )) ' ( ( ) ( )) ( )
( ( ) ( )) ' ( ( ) ( )) ( )
( (
m E m E m
E m E m E p E m E x x E x
E p E m E x x E x x E x p E m E x E m
p E m E x p E m E x
p E m E x p E m E x
p E m
1) ( )) ' ( ( ) ( ))E x p E m E x
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Special case
If unit payoff is in payoff space,
The frontier and bound are just And
This is exactly like the case of state preference neutrality
for return mean-variance frontiers, in which the frontierreduces to the single point R*.
*
m x
* 1 (1| ) 0e proj X
)()( *22 xm
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Some development
H-J bounds with positivity. It solves
This imposes the no arbitrage condition.
Short sales constraint and bid-ask spread is developed by
Luttmer(1996).
A variety of bounds is studied by Cochrane and Hansen(1992).
2min ( ), . . ( ), 0,m s t p E mx m E m
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