Portfolio Selection

Contents

Problem description

A consultant in finance has to choose for one of his wealthy female clients a certain number of shares in which to invest. She wishes to invest $ 100,000 in 6 different shares. The consultant estimates for her the return on investment that she may expect for a period of six months. The following table gives for each share its country of origin, the category (T: technology, N: non-technology) and the expected return on investment (ROI). The client specifies certain constraints. She wishes to invest at least $ 5,000 and at most $ 40,000 into any share. She further wishes to invest half of her capital in European shares and at most 30% in technology. How should the capital be divided among the shares to obtain the highest expected return on investment?

List of shares

+--+-------+--------+------------+
|Nr|Origin |Category|Expected ROI|
+--+-------+--------+------------+
| 1|Japan  |   T    |   5.3%     |
| 2|UK     |   T    |   6.2%     |
| 3|France |   T    |   5.1%     |
| 4|USA    |   N    |   4.9%     |
| 5|Germany|   N    |   6.5%     |
| 6|France |   N    |   3.4%     |
+--+-------+--------+------------+

Variables

budget                     Budget
mininvest                  Minimal investment
maxinvest                  Maximal investment
catinvest1min              Minimal investment in category One (N)
idx1cat                    Index of category One
catinvest2max              Maximal investment in category Two (T)
idx2cat                    Index of category Two
returns                    Expected ROI

Reference

Applications of optimization... Gueret, Prins, Seveaux

% Marcus Edvall, Tomlab Optimization Inc, E-mail: tomlab@tomopt.com
% Copyright (c) 2005-2009 by Tomlab Optimization Inc., $Release: 7.2.0$
% Written Oct 7, 2005.   Last modified Apr 8, 2009.

Problem setup

budget        = 100000;
mininvest     = 5000;
maxinvest     = 40000;
catinvest1min = 0.5;
idx1cat       = [0 1 1 0 1 1]';

catinvest2max = 0.3;
idx2cat       = [1 1 1 0 0 0]';

returns       = [5.3 6.2 5.1 4.9 6.5 3.4]';

n   = length(returns);
buy = tom('buy',n,1);

% No variables are binary.
bnds = {mininvest <= buy <= maxinvest};

% Budget constraints
con1 = {sum(buy.*idx2cat) <= catinvest2max*budget};
con2 = {sum(buy.*idx1cat) >= catinvest1min*budget};
con3 = {sum(buy) == budget};

% Objective
objective = -sum(returns.*buy/100);

constraints = {bnds, con1, con2, con3};
options = struct;
options.solver = 'cplex';
options.name   = 'Portfolio Selection';
Prob = sym2prob('lp',objective,constraints,[],options);
Prob.MIP.SC = 1:Prob.N; % All variables are semi-continuous

PriLev = 1;
Result = tomRun('cplex', Prob, PriLev);
sol = getSolution(Result);

if PriLev > 0
    invest = sol.buy;
    for i = 1:length(invest),
        if invest(i) ~= 0,
            disp(['invest $ ' num2str(invest(i)) ' in share ' num2str(i)])
        end
    end
end

% MODIFICATION LOG
%
% 051201 med   Created.
% 060116 per   Added documentation.
% 060125 per   Moved disp to end
% 090308 med   Converted to tomSym

===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
Problem: ---  1: Portfolio Selection            f_k   -5930.000000000000000000
                                       sum(|constr|)      3.000000000000000000
                              f(x_k) + sum(|constr|)  -5927.000000000000000000
                                              f(x_0)      0.000000000000000000
 ==>  Number of variables violating lower bound   3.  Number of variables violating upper bound   0

Solver: CPLEX.  EXIT=0.  INFORM=101.
CPLEX Dual Simplex LP solver
Optimal integer solution found

FuncEv    3 
invest $ 30000 in share 2
invest $ 30000 in share 4
invest $ 40000 in share 5