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