Solution in MATLAB Environment
Contents
Table 1. Comparison of Solutions of Problems 1-3
Case study Portfolio Optimization with Nonlinear Transaction Costs (see Formal Problem Statement) in MATLAB Environment is solved with mpsg_solver PSG function and includes three optimization problems (see Formal Problem Statement).
| • | Problem 1 considers linear transaction costs and includes the budget constraint with Polynomial Absolute function and power coefficients. |
| • | Problem 2 considers linear transaction costs, similar to Problem 1. However, it is based on linear version of the budget constraint (with linear functions). |
| • | Problem 3 includes the nonlinear budget constraint with Polynomial Absolute function and power coefficients. |
MATLAB code for Portfolio Optimization with Nonlinear Transaction Costs is in file CS_Nonlinear_Transaction_Costs_CVaR_POLYNOMABS.m.
Data are saved in files:
Solution of Problem in MATLAB
|
Formal Problem Statement |
Data |
|
|
Optimization Problem 1 (CS.1-CS.4) |
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Optimization Problem 2 (CS.5-CS.8) |
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Optimization Problem 3 (CS.9-CS.12) |
Let us describe the main operations. To run case study you need to do the following main steps:
In file CS_Nonlinear_Transaction_Costs_CVaR_POLYNOMABS_Toolbox.m:
problem_name = 'problem_1_Trans_Costs_Toolbox.mat';
Load the problem and data:
load(['.\' problem_name]);
tbpsg_export_to_workspace(toolboxstruc_arr);
Define problem statement:
problem_statement = sprintf('%s\n',...
'maximize',...
'Objective: objective_expected_gain_1, linearize = 1',...
' avg_g(matrix_loss_1)',...
'Constraint: <= 0, linearize = 1',...
' polynom_abs(matrix_polynom_abs_1)',...
' linear(matrix_initial_prices)',...
'Constraint: constraint_on_CVaR_1, <= 1000, linearize = 1',...
' cvar_risk(0.90000000, matrix_loss_1)',...
'Box: >= point_lowerbounds, <= point_upperbounds',...
'');
%Uncomment the following line to open the problem in Toolbox Window:
%tbpsg_toolbox(problem_statement,toolboxstruc_arr);
Optimize problem:
[solution_str1, outargstruc_arr1] = tbpsg_run(problem_statement,toolboxstruc_arr); |
Extract optimal point for the problem "problem_1_Trans_Costs":
point_variables = tbpsg_optimal_point_vars(solution_str1, outargstruc_arr1);
X_opt1 = tbpsg_optimal_point_data(solution_str1, outargstruc_arr1)';
fprintf('solution: %s\n', solution_str1);
problem_name = 'problem_2_Trans_Costs_Toolbox.mat';
Load the problem and data:
load(['.\' problem_name]);
tbpsg_export_to_workspace(toolboxstruc_arr);
Define problem statement:
problem_statement = sprintf('%s\n',...
'maximize',...
'Objective: objective_expected_gain_2, linearize = 1',...
' avg_g(matrix_loss_2)',...
'Constraint: <= 1000, linearize = 1',...
' cvar_risk_2(0.90000000, matrix_loss_2)',...
'Constraint: <= 0',...
' linear(matrix_budget)',...
'Box: >= point_lowerbounds, <= point_upperbounds',...
'');
%Uncomment the following line to open the problem in Toolbox Window:
%tbpsg_toolbox(problem_statement,toolboxstruc_arr);
Optimize problem:
[solution_str2, outargstruc_arr2] = tbpsg_run(problem_statement,toolboxstruc_arr); |
Extract optimal point for the problem "problem_2_Trans_Costs":
point_variables = tbpsg_optimal_point_vars(solution_str2, outargstruc_arr2);
X_opt2 = tbpsg_optimal_point_data(solution_str2, outargstruc_arr2)';
X_opt2_long = X_opt2(1:2:end);
X_opt2_short = X_opt2(2:2:end);
fprintf('solution: %s\n', solution_str2);
problem_name = 'problem_3_Trans_Costs_Toolbox.mat';
Load the problem and data:
load(['.\' problem_name]);
tbpsg_export_to_workspace(toolboxstruc_arr);
Define problem statement:
problem_statement = sprintf('%s\n',...
'maximize',...
'Objective: objective_expected_gain_1, linearize = 1',...
' avg_g(matrix_loss_1)',...
'Constraint: <= 0',...
' linear(matrix_initial_prices)',...
' polynom_abs(matrix_polynom_abs_2)',...
'Constraint: <= 1000, linearize = 1',...
' cvar_risk(0.90000000, matrix_loss_1)',...
'Box: >= point_lowerbounds, <= point_upperbounds',...
'');
%Uncomment the following line to open the problem in Toolbox Window:
%tbpsg_toolbox(problem_statement,toolboxstruc_arr);
Optimize problem:
[solution_str3, outargstruc_arr3] = tbpsg_run(problem_statement,toolboxstruc_arr); |
Extract optimal point for the problem "problem_3_Trans_Costs":
point_variables = tbpsg_optimal_point_vars(solution_str3, outargstruc_arr3);
X_opt3 = tbpsg_optimal_point_data(solution_str3, outargstruc_arr3)';
fprintf('solution: %s\n', solution_str3);
Display table comparing solutions for three problems:
fprintf('Compare Optimal Solutions for Three Problems');
fprintf('\n\n Component Problem_1 Problem_2 Problem_2 Problem_3\n');
fprintf(' long position short position\n');
for j=1:1:length(point_variables)
fprintf('%6s%16f%15f%14f%14f\n', point_variables{j},X_opt1(j),X_opt2_long(j), X_opt2_short(j),X_opt3(j));
end
Table 1. Comparison of Solutions of Problems 1-3
Component Problem_1 Problem_2 Problem_2 Problem_3
long position short position
aa 57.866184 57.866184 0.000000 28.339480
aep -39.603960 0.000000 39.603962 -39.603960
agc -31.434185 0.000000 31.434185 -15.512097
aig 27.027027 27.027027 0.000000 0.414658
ait -38.787879 0.000000 38.787880 -16.678772
alt -113.074200 0.000000 113.074200 -28.269952
arc -27.777778 0.000000 27.777779 -15.525287
avp 68.669528 68.669525 0.000000 6.759822
axp 26.890756 26.890757 0.000000 26.890756
ba 61.185468 61.185467 0.000000 -3.449192
bac -36.363636 0.000000 36.363636 -12.295229
bax -34.632035 0.000000 34.632034 -17.094627
bc 143.497760 143.497760 0.000000 23.195000
bcc 86.253369 86.253372 0.000000 13.426171
bdk 47.832586 47.832584 0.000000 -5.961267
bel -38.882139 0.000000 38.882137 -13.848223
bhi 96.969697 96.969696 0.000000 16.299233
bmy 41.612484 41.612484 0.000000 4.332810
bni -66.945607 0.000000 66.945610 -23.324559
bs -250.000000 0.000000 250.000000 -17.791169
c 32.045530 32.045528 0.000000 -4.868256
cbs 80.808081 80.808083 0.000000 21.398613
cen -71.032313 0.000000 71.032310 -17.018208
cgp -56.737589 0.000000 56.737587 -11.072844
cha 69.414317 69.414314 0.000000 20.873344
ci 30.888031 30.888031 0.000000 -3.481200
cl 55.603745 55.603745 0.000000 0.837950
col 99.688474 99.688477 0.000000 -2.454808
cpb -38.049941 0.000000 38.049942 -24.789605
csc 41.290323 41.290321 0.000000 -5.254600
csco 71.748879 71.748878 0.000000 26.882552
dal -41.343669 0.000000 41.343670 -7.657693
dd -36.488027 0.000000 36.488026 -7.103296
dis -79.207921 0.000000 79.207924 -9.141350
dow 23.038157 23.038157 0.000000 -3.582221
ek -26.315789 0.000000 26.315790 -15.276536
etr -64.257028 0.000000 64.257027 -16.333934
f -46.376812 0.000000 46.376812 -7.314124
fcx 164.948450 164.948460 0.000000 4.783043
fdx 88.765406 88.765404 0.000000 11.745142
flr -51.036683 0.000000 51.036682 -8.309632
gd -37.037037 0.000000 37.037037 -9.550204
ge 26.600166 26.600166 0.000000 4.112174
gm -36.363636 0.000000 36.363636 -5.628974
hal 70.640177 70.640175 0.000000 8.347402
het 159.203980 159.203980 0.000000 11.842007
hig -42.895442 0.000000 42.895443 -8.488540
hm -147.465440 0.000000 147.465440 -19.532824
hnz -35.714286 0.000000 35.714287 -15.196241
hon 31.434185 31.434185 0.000000 9.465887
hrs -65.708419 0.000000 65.708420 -11.534697
hwp 40.201005 40.201004 0.000000 10.703998
ibm 32.000000 32.000000 0.000000 9.794812
iff -60.037523 0.000000 60.037525 -5.884921
intc 47.690015 47.690014 0.000000 10.042008
ip -44.137931 0.000000 44.137932 -7.304685
jnj -0.000000 0.000000 0.000000 -4.378575
km -179.775280 0.000000 179.775280 -6.052459
ko -33.934252 0.000000 33.934254 -9.875256
ltd 87.671233 87.671234 0.000000 5.875608
lu 63.492063 63.492062 0.000000 10.726641
may -56.974051 0.000000 56.974052 -7.056461
mcd 68.891400 68.891403 0.000000 -2.211304
mer 43.478261 43.478260 0.000000 8.029168
mkg 97.859327 97.859329 0.000000 4.197196
mmm 23.719379 23.720615 0.000000 -3.963917
mob -0.000000 0.000000 0.000000 -4.080268
mrk -31.295868 0.000000 31.295868 -7.209837
msft 38.415366 38.415367 0.000000 6.871320
mtc -38.461538 0.000000 38.461540 -8.962128
nsc -68.522484 0.000000 68.522484 -7.960004
nsm 216.216220 216.216220 0.000000 14.219438
nt 128.514060 128.514050 0.000000 11.545924
one -47.690015 0.000000 47.690014 -5.450827
orcl 116.550120 116.550120 0.000000 9.564157
oxy -93.023256 0.000000 93.023254 -7.600155
pep -65.708419 0.000000 65.708420 -6.407939
pg -26.711185 0.000000 26.711185 -3.687962
pnu -40.201005 0.000000 40.201004 -6.313073
prd -36.249525 0.000000 36.246004 -3.130691
ral -67.796610 0.000000 67.796608 -7.544621
rok 58.823529 58.823528 0.000000 5.348785
rtnb -35.437431 0.000000 35.437431 -4.948689
s -47.761194 0.000000 47.761192 -6.988060
slb 41.184041 41.184040 0.000000 -0.566637
so -64.516129 0.000000 64.516129 -7.593499
t -51.689604 0.000000 51.689606 -3.813024
tan 87.312605 87.312607 0.000000 20.745351
tek 126.984130 126.984120 0.000000 7.859000
txn 79.899965 79.899963 0.000000 9.561655
ucm -51.696284 0.000000 51.696285 -6.449936
uis 94.674556 94.674553 0.000000 6.331597
usb -54.888508 0.000000 54.888508 -5.845273
utx 52.545156 52.545155 0.000000 4.464782
wfc -23.964565 0.000000 23.969716 -3.043909
wmb 71.910112 71.910110 0.000000 2.563881
wmt 72.153456 72.153458 0.000000 3.226680
wy 46.647230 46.647232 0.000000 -2.648977
xon -27.397260 0.000000 27.397261 -5.053101
xrx -46.920821 0.000000 46.920822 -4.092645
Analysis of this table shows that problem 1 and problem 2 are equivalent.