To be continued Investment problem max E[W) · Subject to W=W(1+R) =∑0R
To be continued • Investment problem • Subject to { } [ ( )] max E U W1 ϖ i ∑ ∑ = = = = = + n i i n i P i i P R R W W R 1 1 1 0 1 ( 1 ) ω ω
Expected return and variance · Portfolio return ∑ OR=OR Expected return ER=∑OER=E(R) varlance var(Rp)=olo O
Expected return and variance • Portfolio return • Expected return • variance R R R n i P i i ' 1 = ∑ ω = ϖ = ' ( ) 1 ER ER E R n i P = ∑ ωi i = ϖ = ( ) var( ) ' ij R P σ ϖ ϖ Σ = = Σ
Portfolio optimization without riskless asset ∑m a Problem a subject to Use method of lagrange L=可Σ+(2-0)+1(1-01) · Minimize l First order conditions 0→20-y-l=0 0→ =0→m1-1=0
Portfolio optimization without riskless asset • Problem A • Use method of lagrange • Minimize L • First order conditions: { } '1 1 ' ' 2 1 min = = Σ r v ϖ ω µ µ ϖ ϖ ω p subject to i ' ( ' ) (1 '1) 2 1 r v L = ϖ Σ ϖ + γ µ p − ϖ µ + λ − ϖ = 0 ⇒ Σ − − 1 = 0 ∂ ∂ r r ϖ γµ λ ϖ L = 0 ⇒ ' − = 0 ∂ ∂ p L ϖ µ µ γ v = 0 ⇒ '1 − 1 = 0 ∂ ∂ v ϖ λ L
Minimum-Variance portfolio m*=1+少2p n1Σ+y72H-p2=0 1∑1+y eline A=121,B=121,C=pΣ Solution C-H,B u, A-B D D D=AC-B2≥0
Minimum-Variance portfolio 1 ' 0 * 1 1 1 1 1 Σ + Σ − = = Σ + Σ − − − − µ µ µ p λ µ γ ϖ λ γ µ r r r r r 1 1 1' 1 0 1 1 Σ + Σ − = − − λ γ µ r r r Solution 0 , 2 = − ≥ − = − = D AC B D A B D C p B µ p γ µ λ Define µ µ v v v v v v = Σ = Σ = Σ − − 1 1, 1 1, ' 1 1 A B C