Asset Pricing
Zheng Zhenlong
1
CHAPTER 6
Relation between Discount
Factors,Betas,and Mean-Variance Frontiers
01:51
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Main contents
• we will draw the connection between discount factors,meanvariance frontiers, and beta representations,then we will show
how they transform between each other,because these three
representations are equivalent.
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Transformation between the three
representations
• p  E(mx)   .
m, x , R , orR  Re
p  E (mx)
p  E (m x) 
x  proj(m | X )
R
R  x E( x2 )
 p  E (mx)
m  a  bRmv
p  E(mx)
R , R  Re
Transformation between the three
representations(2)
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•   p  E (mx) . If we have an expected return-beta model
 bf
with factors f , m
then
linear in the factors psatisfies
 E(mx)
.
• If a return is on the mean-variance fron-tier,then there is an
expected return-beta model with that return as reference
variable.
Transformation between the three
representations(2)
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6
6.1 From Discount Factors to Beta
Representations
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Beta representation using m
1  E(mRi )  E(m) E( Ri )  cov(m, Ri )
i
1
cov(
m
,
R
)
 E ( Ri ) 

E (m)
E (m)
  1 E(m)
Multiply and divide by var(m),define
,we get:
cov(m, R i )
var(m)
E(R )    (
)(
)     i ,m m
var(m)
E (m)
i
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•
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Theorem
•
R  x E( x2 )
E ( R i )     i , x  x 

E ( R i )     i , R E ( R )  

1  E(mRi )  E( x Ri )  E( x ) E( Ri )  cov(x , Ri )

i

i

1
cov(
x
,
R
)
1
cov(
x
,
R
)
var(
x
)
E ( Ri ) 



E ( x )
E ( x )
E ( x )
var(x ) E ( x )
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Proof
E ( R i )     i , x  x 
•
R  x E( x2 )
*2

i
*2

i

E
(
R
)
cov(
R
,
R
)
E
(
R
)
cov(
R
,
R
)
var(
R
)
E ( Ri ) 



E ( R )
E ( R )
E ( R )
var( R ) E ( R )
var(R )
E(R )   
E ( R )


E ( R i )     R i , R E ( R )  

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状态2回报
Rf
1
R*
x*
P=1(收益率）
pc
状态1回报
Re*
P=0(超额收益率）
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状态2回报
Rf
1
R*
x*
P=1(收益率）
pc
状态1回报
Re*
P=0(超额收益率）
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Special case
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14
6.2 From Mean-Variance Frontier to a
Discount Factor and beta Representation
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Theorem
•
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Proof
•
m  a  bR  a  b( R  Re  n)
1  E(mR )  aE( R )  bE( R2 )
0  E(mRe )  aE( Re )  bE( Re2 )  (a  b) E( Re )

1
a
,b  

2
 E(R )  E(R )
 E ( R )  E ( R2 )
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Proof(2)
•
  ( R  R e  n)
m
E ( R )  E ( R2 )
xi  yi R   i Re  ni
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Proof(3)
•
(  R  R e  n)( y i R   i R e  ni )
E (m x )  E (
)

2
E ( R )  E ( R )
i
 y i E ( R )  y i E ( R2 )  E (nni )
E (mx ) 
 E ( R )  E ( R2 )
i
E (nni )
y 
 E ( R )  E ( R2 )
i
n
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Note
  E( R2）E( R )  1 E( x )
• If the denominator is zero, i.e., if
,this construction cannot work.
• If there is a risk-free rate, we are ruling out the case
R mv  R   R f R e  R f
• If there is no risk-free rate, we must rule out the case
Rmv  R  ( E( R*2 ) / E( R* )) Re
(the “constant- mimicking portfolio return”).
• 证毕。
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21
6.3Factor Models and Discount
Factors
01:51
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•
m  1  b f  E( f )
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Theorem
•
m  1  b  f  E( f ) ,0  E(mRe )
E ( Re )   i 
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Proof
• From (6.7),
0  E(mRe )  E( Re )  b cov(f , Re )
• Here we get (6.8)
E ( R e )  b cov(f , R e )
 b var( f ) var( f ) 1 cov(f , R e )   
• where
   var( f )b
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Theorem
•
m  a  bf ,1  E (m Ri )
E( Ri )    i
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Proof
•
1
cov(m, R) 1 E ( Rf )b
E ( R) 

 
E (m)
E (m)
a
a
i  E( ff )1 E( fRi )
1 E ( Rf )b 1 E ( Rf ) E ( ff ) 1 E ( ff )b
E ( R)  
 
a
a
a
a
1
E ( ff )b
  
a
a
 
1
1
1
 ,    cov( ff )b   E (mf )
E (m) a
a
E( Ri )    i
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Proof(2)
•
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•
   p 

E( f ) 
f  E ( f )     p( f ) 




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•
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Factor-mimicking porfolios
•
p  E(m x)  E(bfx)  E(bproj( f | X ) x)  E(bf  x)
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•
E( Ri )       ' 
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•
fi

proj( f i | X )

p proj( f i | X )
E ( R ei )   i , f     i , f  E ( f  )
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33
6.4 Discount Factors and
Beta Models to
Mean-Variance Frontier
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•
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•
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36
6.5 Three Risk-free Rate
Analogues
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•
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•
E( R R )  E( R ) E( R )
E ( R R )
E ( R2 )
1
  E(R ) 




E(R )
E(R )
E ( x )

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•
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E (R )
 =E(R*2)/E(R*)
E ( R )
R
R


利用相似三角形
其长度为 E ( R *2 )
 (R )
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•

  E( R2 ) E( R )
E ( R 2 ) 
E ( R 2 ) E ( R  )
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•
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•
E ( R2 )

e
E(R ) 

E
(
R
)


E
(
R
)

E(R )

E ( R2 )  E ( R ) 2
var(R )



e
E(R )E(R )
E ( R ) E ( R e )

var(
R
)
e
R  R 
R
E ( R ) E ( R e )
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Minimum-Variance Return
• The risk-free rate obviously is the minimum -variance return
when it exists. When there is no risk-free rate, the minimumvariance return is
R min. var.

E
(
R
)

e
R 
R
1  E ( R e )
(6.15)
• Taking expectations,
E(R
min. var.
E ( R )
E ( R )
e
)  E(R ) 
E(R ) 
e
1  E(R )
1  E ( R e )

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•
R
min. var.

 R  E( R

min. var.
)R
e

min var(R  R e )  E ( R  R e ) 2  E 2 ( R  R e )

 E ( R2 )   2 E ( R e2 )  E 2 ( R )  2E ( R ) E ( R e )   2 E 2 ( R e )

E
(
R
)
e
e

e


0  E( R ) 1  E( R )  E( R ) E( R ),
1  E ( R e )


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Constant-Mimicking Portfolio Return
•
Rˆ 
proj(1 | X )
p proj(1 | X )
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•
ˆ  R  R e
R
R e
2
E
(
R
) e
 R 
R

E(R )
E ( R ) 
 proj(1 | X ) 
R
2
E(R )
E ( R )
p proj(1 | X ) 
E ( R2 )
proj(1 | X )  R
e
E ( R )

R
2
E(R )
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Risk-Free Rate
• Here we will show that if there exists a risk-free rate,then
all the zero-beta return, minimum-variance return,and
constant-mimicking portfolio return reduce to the risk-free
rate.
• These other rates are:
• Constant-mimicking:
2
E ( R ) e

ˆ
RR 
R

E(R )
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• Minimum-variance: R

min. var.
E ( R )
e
R 
R
1  E ( R e )

var(R )
R 
R e

e
E(R )E(R )

• Zero-beta: R
• And the risk-free rate:
R f  R   R f R e
(6.19)
• To establish that there are all the same when there is a riskfree rate, we need to show that:
R
f
E ( R2 )
E ( R )
var(R )




e
E(R )
1  E(R )
E ( R ) E ( R e )
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•
Rf
1
E ( R2 )



E( x )
E ( R )
R f  E( R )  R f E( Re )
E ( R 2 )

f
e

E
(
R
)

R
E
(
R
)

E(R )
E ( R2 )  E ( R ) 2
var(R )
R 


e
E(R )E(R )
E ( R ) E ( R e )
f
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51
6.6 Mean-Variance Special Cases
with No Risk-Free Rate
01:51
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• There exist special cases for the equivalence
theorems,that is,when the expected discount
factor,price of a unit payoff,or risk-free rate is zero
or infinity.
• If risk-free rate is traded or the market is
complete,then it won’t be a problem; however,in an
incomplete market in which no risk free rate is
0 make
E ( M )itsure

traded,we must pay attention to it and
that
The special case for a meanvariance frontier to a discount
factor
•


R  E( R2 ) E( R ) Re
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The special case for meanvariance frontier to a beta model
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• We can use any return on the mean-variance frontier as the
reference return for a single-beta representation,except the
minimum-variance return.
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56
Theorem:
•

E ( Ri )   R m v  i , Rm v E ( R mv )   R m v

E( Ri )  E( R )   i E( Re )
01:51
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•

cov(R i , R mv )  cov ( R   R e ), ( R    i R e )

 var(R  )  i var(R e )  (   i ) E ( R  ) E ( R e )

 var(R  )  E ( R  ) E ( R e )   i  var(R e )  E ( R  ) E ( R e )
E ( R  ) E ( R e )
E ( R  ) E ( R e )


var(R e )
E ( R e  2 )  E ( R e ) 2
E ( R )

1  E ( R e )

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