文档介绍：Chapter : Modern Portfolio
Theory
Fan Longzhen
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(50000)=1, U(-10000)=0
• Let G1 be a 50/50 gamble: ⎧ 50 ,000
G1 = ⎨
⎩− 10 ,000
• Finding certainty equivalent X1 of G1: X1=?, then
• U(x1)=E[U(G1)];
⎧50,000 ⎧X1
• Define G2 and G3 similarly: G2 = ⎨ G3 = ⎨
X −10,000
• This yield five points: ⎩ 1 ⎩
• U(-10000)=0
• U(x2)=
• U(x1)=
• U(x3)=
• U(50000)=1
Maximize expected utility and mean-variance analysis
µ
• What about mean-variance preference? σ
µ
• Investors like mean, dislike variance: V ( , 2 ) = a − bσ 2
• Consistent with expected utility?
• Consider second order Taylor expansion
w = w0 (1+ r);
µ 2
U (σw) = U (w0 (1+ r)) = U (w0 ) +U '(w0 )r +1/ 2U ''(w0 )r +...
1
V( , 2 ) = EU(w) =U(w ) +U'(w )E(r) + U''(w )((E(r))2 + var(r))+...
0 0 2 0
µ µ
b µ
∝−( 2 +σ 2 )
2
V > 0 if µ < 1/ b, Vσ< 0
Version 1 of the investment problem
• Two dates: 0 and 1 (today and tomorrow);
• Current wealth W0 and future wealth W1;
• Preference U(W1);
• No consumption, no e, no dynamics;
• n assets {} R , R ,... R µ with expected returns
1 2 n µ µ
• v σr
ER = = (σ 1, 2 ,..., µn )'
•variance σσ
⎛ 2 ... ⎞
⎜ 1 σ 12 1 n ⎟
2 σ
r ⎜ 21 2 ... 2 n ⎟
var( R ) = ⎜σ⎟
⎜... σ⎟
⎜ 2 ⎟
⎝ n 1 n 2 ... σ n ⎠
To be continued
• Investment problem
max E[U (W1)]
• Subject to {}ϖ i
W1 = W0 (1+ RP )
n
RP = ∑ω i Ri
i=1
n
∑ωi =1
i=1
Expected return and variance
• Portfolio return
n ω
RP = ∑ i Ri =ϖ' R
i=1
• Expected return
n ω
ERP = ∑ i ERi =ϖ' E(R)
i=1
•variance
var(RP ) =ϖ'Σϖ
Σ= (σ ij )
Portfolio optimization without riskless asset
1
'Σ
minϖ 2
{}ωi ϖ
• Problem A subject to
ω v
ϖµ' = p
ϖµ
γ r
• Use method of lagrangeµ ϖ'1 =1
ϖµ
1 λ v