# maximisation of the utility function , portfolio optimization

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Az.Sa on 6 Feb 2023
Edited: Torsten on 8 Feb 2023
Hi,
I am trying to miximize a utility function of an investor , I use CRRA utility , , , is known such that
, , , , γ is risk averse with constant value and ( R_1,...,R_4 ) is the matrix returns in excel file with 4 asstes and 249 observations and
So , everthing is known except from the weights ( x_1,...,x_4) , I need to find the optimal weights \$x_i\$ which maximize expected utility with constraint
I don't know how to solve this optimization problem , does this problem nonlinear optimization problem ?
Thank you
##### 2 CommentsShow 1 older commentHide 1 older comment
Az.Sa on 7 Feb 2023
I updated my question with more details , I hope it's clear now

Torsten on 7 Feb 2023
Calculate W1*Rt1, W2*Rt2,...,W4*Rt4. Let Wi*Rti be maximum. Then (assuming gamma < 1) x_i = 1, x_j = 0 for i~=j is optimal.
Torsten on 8 Feb 2023
Edited: Torsten on 8 Feb 2023
is there any different function I can apply it to compare the results ?
You could try "ga".
For completeness, you should add the bound constraints
lb = zeros(4,1);
ub = ones(4,1);
as lower and upper bound constraints for the x(i):
W = ...;
R = ...;
gamma = ...;
Aeq = [1 1 1 1];
beq = 1;
lb = zeros(4,1);
ub = ones(4,1);
u = @(x) 1/(1-gamma)*x.^(1-gamma);
obj = @(x)-sum(u(x.'*W*R));
x = fmincon(obj,0.25*ones(1,4),[],[],Aeq,beq,lb,ub)

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