The Arbitrage Pricing Model

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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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posted:5/4/2012
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