Chapter 10 Capital Asset Pricing 10.1 An Optimization Problem Consider an agent who has initial wealth X 0 and wants to invest in the stock and money markets so as to maximize IE log X n : Remark 10.1 Regardless of the portfolio used by the agent, f k X k g 1 k=0 is a martingale under IP, so IE n X n = X 0 BC Here, (BC) stands for “Budget Constraint”. Remark 10.2 If  is any random variable satisfying (BC), i.e., IE n  = X 0 ; then there is a portfolio which starts with initial wealth X 0 and produces X n =  at time n .Tosee this, just regard  as a simple European derivative security paying off at time n .Then X 0 is its value at time 0, and starting from this value, there is a hedging portfolio which produces X n =  . Remarks 10.1 and 10.2 show that the optimal X n for the capital asset pricing problem can be obtained by solving the following Constrained Optimization Problem: Find a random variable  which solves: Maximize IE log  Subject to IE n  = X 0 : Equivalently, we wish to Maximize X !2 log  !  IP !  119 120 Subject to X ! 2  n !  !IP !  , X 0 =0: There are 2 n sequences ! in  .Callthem ! 1 ;! 2 ;::: ;! 2 n . Adopt the notation x 1 =  ! 1 ;x 2 =! 2 ; ::: ; x 2 n = ! 2 n : We can thus restate the problem as: Maximize 2 n X k=1 log x k IP ! k  Subject to 2 n X k=1  n ! k x k IP ! k  , X o =0: In order to solve this problem we use: Theorem 1.30 (Lagrange Multiplier) If x  1 ;::: ;x  m  solve the problem Maxmize f x 1 ;::: ;x m  Subject to g x 1 ;::: ;x m =0; then there is a number  such that @ @x k f x  1 ;::: ;x  m = @ @x k gx  1 ;::: ;x  m ; k =1;::: ;m; (1.1) and g x  1 ;::: ;x  m =0: (1.2) For our problem, (1.1) and (1.2) become 1 x  k IP ! k = n ! k IP ! k ;k=1;::: ;2 n ; 1:1 0  2 n X k=1  n ! k x  k IP ! k =X 0 : 1:2 0  Equation (1.1’) implies x  k = 1  n ! k  : Plugging this into (1.2’) we get 1  2 n X k=1 IP ! k =X 0 = 1  =X 0 : CHAPTER 10. Capital Asset Pricing 121 Therefore, x  k = X 0  n ! k  ;k=1;::: ;2 n : Thus we have shown that if   solves the problem Maximize IE log  Subject to IE  n  =X 0 ; (1.3) then   = X 0  n : (1.4) Theorem 1.31 If   is given by (1.4), then   solves the problem (1.3). Proof: Fix Z0 and define f x = log x , xZ: We maximize f over x0 : f 0 x= 1 x ,Z =0  x= 1 Z ; f 00 x=, 1 x 2 0; 8x2 IR: The function f is maximized at x  = 1 Z , i.e., log x , xZ  f x   = log 1 Z , 1; 8x0; 8Z0: (1.5) Let  be any random variable satisfying IE  n  =X 0 and let   = X 0  n : From (1.5) we have log  ,    n X 0   log  X 0  n  , 1: Taking expectations, we have IE log  , 1 X 0 IE  n    IE log   , 1; and so IE log   IE log   : 122 In summary, capital asset pricing works as follows: Consider an agent who has initial wealth X 0 and wants to invest in the stock and money market so as to maximize IE log X n : The optimal X n is X n = X 0  n , i.e.,  n X n = X 0 : Since f k X k g n k=0 is a martingale under IP, we have  k X k = IE  n X n jF k =X 0 ;k=0;::: ;n; so X k = X 0  k ; and the optimal portfolio is given by  k ! 1 ;::: ;! k = X 0  k+1 ! 1 ;::: ;! k ;H  , X 0  k+1 ! 1 ;::: ;! k ;T  S k+1 ! 1 ;::: ;! k ;H, S k+1 ! 1 ;::: ;! k ;T : . Chapter 10 Capital Asset Pricing 10.1 An Optimization Problem Consider an agent who has initial. produces X n =  . Remarks 10.1 and 10.2 show that the optimal X n for the capital asset pricing problem can be obtained by solving the following Constrained