Fin 501: Asset Pricing
                                                   Fin 501: Asset Pricing
                         Projections
                         Projections
      • States s=1,…,S  with s >0 
      • Probability inner product
      •  -norm                               (measure of length)
      13:30 Lecture 07    Mean-Variance Analysis and CAPM 
      13:30 Lecture 07    Mean-Variance Analysis and CAPM 
                           (Derivation with Projections)
                           (Derivation with Projections)
                                                                                         Fin 501: Asset Pricing
                                                                                         Fin 501: Asset Pricing
                                                                  )
                                                             shrink
                         y                                     axes                  y
                                x                                                     x
           x and y are -orthogonal iff [x,y]  = 0, I.e. E[xy]=0
                                                                
           13:30 Lecture 07                  Mean-Variance Analysis and CAPM 
           13:30 Lecture 07                  Mean-Variance Analysis and CAPM 
                                                (Derivation with Projections)
                                                (Derivation with Projections)
                                                                  Fin 501: Asset Pricing
                                                                  Fin 501: Asset Pricing
                           …Projections…
                            …Projections…
     •  Z space of all linear combinations of vectors z , …,z
                                                               1      n
     •                           S
        Given a vector y 2 R  solve
     •  [smallest distance between vector y and Z space]
        13:30 Lecture 07          Mean-Variance Analysis and CAPM 
        13:30 Lecture 07          Mean-Variance Analysis and CAPM 
                                    (Derivation with Projections)
                                    (Derivation with Projections)
                                                                                         Fin 501: Asset Pricing
                                                                                         Fin 501: Asset Pricing
                                        …Projections
                                         …Projections
                                                           y
                                                                 
                                                                Z
                                                              y
                     j
            E[ z]=0 for each j=1,…,n  (from FOC)
            ? z   
              Z
            y  is the (orthogonal) projection on Z
                     Z            Z
            y = y  + ’ , y  2 Z,  ? z
           13:30 Lecture 07                  Mean-Variance Analysis and CAPM 
           13:30 Lecture 07                  Mean-Variance Analysis and CAPM 
                                                (Derivation with Projections)
                                                (Derivation with Projections)
                                                   Fin 501: Asset Pricing
                                                   Fin 501: Asset Pricing
     Expected Value and Co-Variance…
     Expected Value and Co-Variance…
         squeeze axis by 
                                         (1,1)
                    x              [x,y]=E[xy]=Cov[x,y] + E[x]E[y]
                                            2             2  
                                   [x,x]=E[x ]=Var[x]+E[x]
                                           2 ½
                                   ||x||= E[x ]  
      13:30 Lecture 07     Mean-Variance Analysis and CAPM 
      13:30 Lecture 07     Mean-Variance Analysis and CAPM 
                            (Derivation with Projections)
                            (Derivation with Projections)