Chapter 12. 1: Modern portfolio Theory Fan longzhen
Chapter 12.1: Modern Portfolio Theory Fan Longzhen
Outline Mean-variance analySIS, Mean-variance analysis and utility maximization Does high moment matter
Outline • Mean-variance analysis; • Mean-variance analysis and utility maximization; • Does high moment matter?
Eliciting preference · Through experiment Consider gaining 50000 vs losing 10000 U(500001,U(-10000=0 Let gl be a 50/50 gamble 50.0000.5 10,000.5 Finding certainty equivalent Xl ofG1: XI=?, then U(XIFEUGII fine G2 and G3 similarly: G=50,0000.5 X,0.5 De x1 05 G3 This yield five points 10.0000.5 U(-10000 U(x2=0.75 U(xl)0.5 U(x3)=0.25 U(500001
Eliciting preference • Through experiment: • Consider gaining 50000 vs losing 10000 • U(50000)=1, U(-10000)=0 • Let G1 be a 50/50 gamble: • Finding certainty equivalent X1 of G1: X1=?, then • U(x1)=E[U(G1)]; • Define G2 and G3 similarly: • This yield five points: • U(-10000)=0 • U(x2)=0.75 • U(x1)=0.5 • U(x3)=0.25 • U(50000)=1 ⎩ ⎨ ⎧ − = 10 ,000 0.5 50 ,000 0.5 G1 ⎩ ⎨ ⎧ = 0.5 50,000 0.5 1 2 X G ⎩ ⎨ ⎧ − = 10,000 0.5 1 0.5 3 X G
Maximize expected utility and mean-variance analysis What about mean-variance preference? Investors like mean, dislike variance: V(u, 0=au-bo Consistent with expected utility? Consider second order taylor expansion W=0(+r) U(mn)=U(m(l+)=U()+()y+1/2(m2+ (a2)=EUun)=U(w)+U7(mh)E()+=U"(m)(E()2+va()+ a-=(42+a) V>0<1/b,V<0
Maximize expected utility and mean-variance analysis • What about mean-variance preference? • Investors like mean, dislike variance: • Consistent with expected utility? • Consider second order Taylor expansion 2 2 V ( µ,σ ) = a µ − b σ ( ) ( ( 1 )) ( ) '( ) 1 / 2 ''( ) ... ( 1 ); 2 0 0 0 0 0 = + = + + + = + U w U w r U w U w r U w r w w r ( ) 2 ''( )(( ( )) var( )) ... 2 1 ( , ) ( ) ( ) '( ) ( ) 2 2 2 0 0 0 2 µ µ σ µ σ ∝ − + = = + + + + b V EU w U w U w E r U w E r r > 0 < 1 / , < 0 µ µ Vσ V if b
Version 1 of the investment problem Two dates: 0 and l (today and tomorrow ) Current wealth Wo and future wealth W1 Preference U(Wi) No consumption, no income, no dynamics n assets R, R,R, with expected returns ER==(412,n) 5…:n variance 2d1 var( r) O 2
Version 1 of the investment problem • Two dates: 0 and 1 (today and tomorrow); • Current wealth W 0 and future wealth W 1; • Preference U(W1); • No consumption, no income, no dynamics; • n assets with expected returns • • variance { } R R R n , ,... 1 2 ( , ,..., )' ER = µ = µ1 µ2 µn v r ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = 2 1 2 2 2 21 2 12 1 2 1 ... ... ... ... var( ) n n n n n R σ σ σ σ σ σ σ σ σ r
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
Properties of mvps Characterization of mvp set2=*∑团 =m*(1+) 1+y 1+ 02=+p 2A2-2B4n+C parabola In the mean-standard deviation space, the curve is a hyperbola
Properties of MVPs • Characterization of MVP set: • In the mean-standard deviation space, the curve is a hyperbola p p λ γµ λϖ γϖ µ ϖ λ γ µ σ ϖ ϖ = + = + = Σ Σ + Σ = Σ − − r r r r *' 1 *' *' ( 1 *' 1 1 2 ） p p σ = λ +γµ 2 D A p B p C p − + = µ µ σ 2 2 2 parabola