# 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