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Investment Investment with Linear with Linear
ProgrammingProgramming
Xuan HuangXuan Huang
Investment 101Investment 101
• Return – capital gain or loss.
• Risk – No free lunch!
• Securities – bond, stock, etc.
• Mutual fund Portfolio –risk diversification; don’t put all your eggs in one basket.
LP application in InvestmentLP application in Investment
• Portfolio Management– Mutual fund portfolio optimization– Bond portfolio improvement
• Asset and Liability Management (ALM)
Portfolio OptimizationPortfolio Optimization
• W.F. Sharpe, 1967.
• H. Konno and H. Yamazaki, 1991.
• M.R. Young, 1998.
Portfolio OptimizationPortfolio Optimization
• xj – decision variables, j= 1,2…n• Rj – rate of return. rjt , t= 1,2…T;
E(Rj ) = rj = Σtrjt /T
• Expected return: E(ΣjRjxj)=Σjrjxj
• Risk of Return: Var(ΣjRjxj)
Famous MarkowitzFamous Markowitz’’s Model s Model (1952)(1952)
( ) ( )1 1
01
01
minimize ,...
subject to
0 1,...,
nn j jj
nj jj
njj
j
f x x Var R x
r x M
x M
x j n
ρ
=
=
=
=
≥
=
≥ =
∑∑∑
Famous MarkowitzFamous Markowitz’’s Model s Model (1952)(1952)
( ) ( )1 1
01
01
minimize ,...
subject to
0 1,...,
nn j jj
nj jj
njj
j
f x x Var R x
r x M
x M
x j n
ρ
=
=
=
=
≥
=
≥ =
∑∑∑
Famous MarkowitzFamous Markowitz’’s Model s Model (1952)(1952)
( ) ( )1 1
01
01
minimize ,...
subject to
0 1,...,
nn j jj
nj jj
njj
j
f x x Var R x
r x M
x M
x j n
ρ
=
=
=
=
≥
=
≥ =
∑∑∑
Famous MarkowitzFamous Markowitz’’s Model s Model (1952)(1952)
( ) ( )1 1
01
01
minimize ,...
subject to
0 1,...,
nn j jj
nj jj
njj
j
f x x Var R x
r x M
x M
x j n
ρ
=
=
=
=
≥
=
≥ =
∑∑∑
Sharpe remarked (1971)Sharpe remarked (1971)……-If the essence of the portfolio analysis problem could be adequately captured in a form for linear programming methods, the prospect for practical application would be greatly enhanced.
Konno and YamazakiKonno and Yamazaki’’s s treatmenttreatment
min f(x)=x2 min g(x)=|x|Monotonic transform
Konno and YamazakiKonno and Yamazaki’’s s treatmenttreatment
T
t=1 1minimize /
subject to ......
njt jj
a x T=∑ ∑
Konno and YamazakiKonno and Yamazaki’’s s treatmenttreatment
T
t=1 1minimize /
subject to ......
njt jj
a x T=∑ ∑
minimize | |subject to ......
x
Konno and YamazakiKonno and Yamazaki’’s s treatmenttreatment
T
t=1 1minimize /
subject to ......
njt jj
a x T=∑ ∑
minimize | |subject to ......
x
......
yy x y− ≤ ≤
Konno and YamazakiKonno and Yamazaki’’s s treatmenttreatment
T
t=1 1minimize /
subject to ......
njt jj
a x T=∑ ∑
minimize | |subject to ......
xminimize subject to ......
yy x y− ≤ ≤
Konno and YamazakiKonno and Yamazaki’’s s treatmenttreatment
T
t=1
1
1
01
01
minimize /
subject to t=1,2,...T
t=1,2,...T
0
t
nt jt jj
nt jt jj
nj jj
njj
j
y T
y a x
y a x
r x M
x M
x
ρ
=
=
=
=
≥
≥ −
≥
=
≥
∑∑∑
∑∑
1,...,
y 0 t 1,...,t
j n
T
=
≥ =
YoungYoung’’s treatments treatment
• At t, expected return of the portfolio is Σjrjtxj
• Maximize ( min Σjrjtxj )t
YoungYoung’’s treatments treatment
n
j=1tmaximize min
subject to ......jt jr x∑
YoungYoung’’s treatments treatment
n
j=1tmaximize min
subject to ......jt jr x∑
pM
YoungYoung’’s treatments treatment
n
j=1tmaximize min
subject to ......jt jr x∑
n
j=1
t
p
jt j p
M
r x M≥ ∀∑
YoungYoung’’s treatments treatment
n
j=1tmaximize min
subject to ......jt jr x∑
n
j=1
maximize
subject to t
......
p
jt j p
M
r x M≥ ∀∑
SharpeSharpe’’s Improvement of s Improvement of the modelthe model
• Modify the objective function to
Z=(1-λ)Return- λRisk
• Adding more constraints with industry consideration.
ReferenceReference• H. Konno and H. Yamazaki, 1991.
– “Mean-Absolute Deviation Portfolio Optimization Model and its Application to Tokyo Stock Market”.
• W.F. Sharpe, 1967.– “A linear Programming Algorithm for Mutual
Fund Portfolio Selection”. • M.R. Young, 1998.
– “A Minimax Portfolio Selection Rule with Linear Programming Solution”.
Thank you!Thank you!