1. Markowitz Model For 3-stock portfolio, short selling allowed Eg. R A = 20% R B = 10% R C = 8%  2 A =100  2 B =25  2 C =16  AB = 15  AC = 20  BC = 4

2. Markowitz Model Form the Lagrangian

3. Markowitz Model: Solution For R* = 15% w A = 50.214% w B = 48.715% w C = 1.075% Standard Dev = 6.22% For R* = 20% w A = 91% w B = 56% w C = -47% (short sale) Standard Dev = 9.47%

4. Single Index Model R t = A + BR mt + e t E(R t ) = A + BE(R mt ) Cov(e i,e k )=0 E(e i )=0 Cov(e i,R m )=0  2 p =  P i [R i - E(R)] 2 by substitution  2 =  P i [A + BR mt +e i -A - BE(R m )] 2  2 =  P i {B[R mi -E(R m )] + e i } 2  2 =  P i {B2[R mi -E(R m )] 2 +e i 2 +2B[R mi -E(R m )] e i }  2 p = B 2 p  2 m +  2 e

5. Single Index Model (Cont’d)  2 p = B 2 p  2 m +  2 ep We also know that Bp= Cov(Rp, Rm)/  2 m Bp=  wjBj  2 ep =  w j 2  2 ej Hence, the portfolio variance is:  2 p = (  x j B j ) 2  2 m +  x j 2  2 ej

6. Single Index Model (Cont’d) For 3-stock portfolio, short selling allowed Eg. R A = 20% R B = 10% R C = 8%  2 A  =20  2 B  =15  2 C  =9  A = 1.5  B = 0.75  C = 0.50

7. Single Index Model (Cont’d) Form the Lagrangian

8. Single Index Model Solution When R* = 15% w A = 0.57 w B = 0.08 w C = 0.35 Standard Deviation =