# The Arbitrage Pricing Model by ert554898

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```									The Arbitrage Pricing Model

Lecture XXVI
A Single Factor Model
away from the specific form of the
 Abstracting
CAPM model, we posit a single factor model
written as
K
zi  ai   bik f k   i
k 1

 In this model, the random return on an investment zi is
a linear function of some random factor fi and an
idiosyncratic term i.
                                      
E   i   E f k  E   i j   E  i f k  E f k f l  0
E   i2   si2  Si2

 
E f k2  1
 Abstracting   away from the idiosyncratic risk

zi  ai  bi fi
 Ifthe bis of two assets are the same, then the ais must
be the same for an arbitrage free model.
 Suppose we are interested in forming a portfolio of two
assets with different bis, bi  bj , bi  0, bj  0
z  w  ai  bi f   1  w   a j  b j f 
 wai  wbi f  a j  wa j  b j f  wb j f
  w  ai  a j   a j    w  bi  b j   b j  f
                                               

 Computing   the mean and variance of this portfolio
yields
E  z   w  ai  a j   a j


V  z   E  w  ai  a j   a j    w  bi  b j   b j  f       
2

                                               

                  
 w  ai  a j   a j
2

 E w  a  a   a   E  w  a  a   a   w  b  b   b  f  
2
i       j        j              i     j       j              i   j   j

E   w  b  b   b  f   w  a  a   a 
2
2
         i       
j        j               i       j       j

  w  bi  b j   b j 
2

                      
 Holding   the variance of the portfolio equal to zero, we
find

 w  bi  b j   b j   0
2

                      
w  bi  b j   b j  0
bj
w 
*

b j  bi
 bj                              b j                   
z            ai  a j   a j            b  bj   bj  f
 b  b  i

 b j  bi 
                                 j i 
                         

b j  ai  a j 
                            a j  R the riskfree rate
b j  bi
a  a   R  a
i        j                   j

R  ai
b  b  b
j      i               j            bi
R  zi  ai  a j 

bj      b j  bi 
Rz 
a  a  b   i           j

b  b 
i
j           i
j

R
a  a  b  z
i       j

b  b  j       i
j       i

zi    R
a  a  b
i            j
 1        
a  a 
i   j

b  b 
j           i
j
b  b 
j   i

zi  R  1bi

zi  0  1bi
Multifactor Models:
 Suppose  that asset returns are generated by a two
factor linear model:

zi  ai  bi f1  ci f 2
A   portfolio of these assets then yields

w z  wa  wb f  wc f
i
i i
i
i i
i
i i 1
i
i i 2
 Again    to minimize systematic risk

wb  wc
i
i i
i
i i   0

 If   the portfolio is riskless, then it yields zero profit
R   wi ai
i

wa  R  0
i
i i
 Given      w 1  w R  R
i
i
i
i

 a1  R a2  R a3  R  w1   0 
                         
  wi  ai  R   0   b1       b2     b3  w2    0 
i                      c               c3  w3   0 
 1        c2             
 The   matrix

 a1  R a2  R a3  R 
                      
   b1     b2     b3 
 c               c3 
 1        c2          
must be singular, or the first row must be a linear
combination of the last two rows
 a1  R a2  R a3  R 
                      
 b1       b2     b3   R1  1 R2  2 R3  0
 c               c3 
 1        c2          

zi  R  ai  R  1bi  2ci i
or
zi  0  1bi  2 ci i

```
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