IX. Explaining Relative Prices
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Explaining Relative Prices
1.
CAPM – Capital Asset Pricing Model
2.
Non Standard Forms of the CAPM
3.
APT – Arbitrage Pricing Theory
2
Assumptions behind the CAPM
1.
2.
3.
4.
5.
No transaction costs
Assets are infinitely divisible
No personal taxes
Price Takers
Investors look only at expected return
and variances of their portfolio
6. Unlimited short sales
7. Unlimited lending and borrowing at the
riskless Rate
8. Homogenous Expectations about time
horizon
9. Homogenous expectations of expected
return, variance, and covariance
10. All assets are marketable
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Sharpe – Lintner – Mossine (CAPM)
Two Approaches to deriving
1. Economic intuition
2. Rigorous analysis
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5
6
A more rigorous proof
RP  RF
Max 
P
Lintner Equation


2 N

Rk  RF   X k k   X i ik 


i 1
ik


Rk  RF

 E  X  R


k
k
 Rk  Rk  Rk  
N
X
i 1
ik
i

 R  R  R  R 


k
k
i
i
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Rk  RF   E  Rk  Rk    X i  Ri  Ri  

N
i 1
Homogenous Expectations
Rk  RF   kM
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Rk  RF   kM
Must hold for all securities and
portfolios
RM  RF  

2
M
RM  R F
2
M
 kM
Rk  RF  2 RM  RF 
M

Rk  RF   k RM  RF

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Non standard forms of the CAPM
If the CAPM does a good job of explaining return
why bother
1.
May do a better job
2.
Even if the CAPM explains return; macro
behavior might not explain micro behavior –
e.g., everybody does not hold market
portfolio
3.
If we don’t include influences in the model,
e.g., taxes we can’t study the impact of their
influences on the model
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Modification of assumptions
1. Short sales
2. Riskless lending and
borrowing
3. Personal taxes
4. Non marketable assets
5. Heterogeneous expectations
6. Non price taking behavior
7. Multi period analysis
8. Consumption CAPM
Rolls critique
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Short Sales
Since under the standard CAPM nobody
short sells in equilibrium
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RF
Is a rate – such that if we could lend or
borrow at it we would hold the market
portfolio
Lintner Equation

Rk  RF   k RM  RF

But for zero beta
RZ  RF
so
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RZ  RM
by convexity of efficient frontier
Prof that Rz  RG return on global minimum
variance portfolio
2
C
2
d C
dX Z
XZ 


2 2
X Z Z
 1  X Z 
2
2 X Z Z
2
2
 2 M
2
M
0
2
 2 X Z M
0
2
M
2
2
 M  Z
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XZ
is greater than 0 and smaller than 1
Global minimum variance involves positive
investment in market and zero beta portfolios
and therefore, expected return must be in
between.
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Non Marketable Assets
CAPM
i 
covRi Rm 
2
m
Ri  RF 
Rm  RF
2
m
covRi Rm 
With nonmarketable assets (H)


Rm  RF
H
Ri  RF 
 covRi Rm  
covRi RH 
m


2  H 


 m   covRm RH 
 m 
Price of risk
*
amount of risk
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Test of Equilibrium Models

Ri  RF   i RM  RF

Expectations (1)
Test with realizations – expectations are an
average and on the whole correct
Market Model
Rit   i   i Rmt  eit
Ri   i   i Rm


Ri  Ri   i Rmt  Rm  eit
Substitution
Rit  RF   i Rmt  RF   eit
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Rit  RZ 1   i    i RMt  eit
Rit   i  RF 1   i    i RMt  eit
i   RZ  RF  1 i 
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Fama and Mac Beth
2
Rit   0t   1t  t   2t  i
1.
E  3t   0
2.
E  2t   0
3.
  3t Sei  it
E  1t   0
Auto correlation of
,
0t
, and 
1t
,
0
 0  RF  1  RM  RF
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Rolls Critique
Mathematically can show for any efficient portfolio

Rk  RZP   kP RP  RZP

“Unfortunately it has never been subject to an
unambiguous empirical test. There is considerable
doubt…that it will be.”
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POST TAX CAPM
R R  
i
F
i
 R
M
 R   T d  R
F
M
F
  T  d  R 
i
F
Ri  RF   0   1i   2  di  RF 
Ri  RF  0.0063  0.04211  0.236(di  RF )
t
Ti 
t gi
(2.631) (1.86)
t di  t gi
1  t gi
(8.62)
 .236
1
 t di
2
t di  .382
t gi  .19
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