# Mean Variance Portfolio Selection

## Contents

## Problem description

An investor wishes to invest a certain amount of money. He is evaluating four different securities (assets) for his investment. The securities are US Treasury Bills (‘T-bills’), a computer hardware company, a computer software company, and a high-risk investment in a theater production. He estimates the mean yield on each dollar he invests in each of the securities, and also adopts the Markowitz idea of getting estimates of the variance/covariance matrix of estimated returns on the securities. (For example, hardware and software company worths tend to move together, but are oppositely correlated with the success of theatrical production, as people go to the theater more when they have become bored with playing with their new computers and computer games.) The return on theatrical productions are highly variable, whereas the T-bill yield is certain. The estimated returns and the variance/covariance matrix are given in the table below.

Estimated returns and variance/covariance matrix

+----------------+--------+--------+--------+-------+ | |Hardware|Software|Show-biz|T-bills| +----------------+--------+--------+--------+-------+ |Estimated return| 8 | 9 | 12 | 7 | +----------------+--------+--------+--------+-------+ |Hardware | 4 | 3 | -1 | 0 | |Software | 3 | 6 | 1 | 0 | |Show-biz | -1 | 1 | 10 | 0 | |T-bills | 0 | 0 | 0 | 0 | +----------------+--------+--------+--------+-------+

Question 1: Which investment strategy should the investor adopt to minimize the variance subject to getting some specified minimum target yield?

Question 2: Which is the least variance investment strategy if the investor wants to choose at most two different securities (again subject to getting some specified minimum target yield)?

## Variables

estreturn Estimated return of securities covmat The variance/covariance matrix target Minimum target yield maxassets Number of securities

## 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

estreturn = [8 9 12 7]'; covmat = [ 4 3 -1 0;... 3 6 1 0;... -1 1 10 0;... 0 0 0 0]; target = 8.5; n = length(estreturn); buy = tom('buy',n,1); % Bounds bnds = {0 <= buy <= 1}; % Cost constraints con1 = {estreturn'*buy >= target}; con2 = {sum(buy) == 1}; % Objective objective = 0.5*buy'*covmat*buy; constraints = {bnds, con1, con2}; options = struct; options.solver = 'cplex'; options.name = 'Mean Variance Portfolio Selection'; sol1 = ezsolve(objective,constraints,[],options); buybin = tom('buybin',n,1,'int'); bnds2 = {0 <= buybin <= 1}; maxassets = 2; con3 = {sum(buybin) <= maxassets}; con4 = {buy - buybin <= 0}; constraints = {bnds, con1, con2, con3, con4}; sol2 = ezsolve(objective,constraints,[],options); PriLev = 1; if PriLev > 0 names = ['Hardware'; 'Software'; 'Show-biz'; 'T-bills ']; disp('Answer to Question 1:') for i = 1:length(sol1.buy), disp([' invest ' num2str(sol1.buy(i)*100) ... '% of the capital in ' names(i,:) ]) end disp('Answer to Question 2 (limited number of securities):') x = sol2.buy; x(find(x < 1e-6)) = 0; for i = 1:length(x), if x(i) ~= 0, disp([' invest ' num2str(x(i)*100) ... '% of the capital in ' names(i,:) ]) end end end % MODIFICATION LOG % % 051202 med Created. % 060117 per Added documentation. % 060125 per Moved disp to end % 090308 med Converted to tomSym

Problem type appears to be: qp ===== * * * =================================================================== * * * TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2010-02-05 ===================================================================================== Problem: 1: Mean Variance Portfolio Selection f_k 0.362379808516659960 f(x_0) 0.000000000000000000 Solver: CPLEX. EXIT=0. INFORM=1. CPLEX Barrier QP solver Optimal solution found FuncEv 12 GradEv 12 ConstrEv 12 Iter 12 CPU time: 0.015625 sec. Elapsed time: 0.015000 sec. Problem type appears to be: miqp ===== * * * =================================================================== * * * TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2010-02-05 ===================================================================================== Problem: --- 1: Mean Variance Portfolio Selection f_k 0.450000000000000120 sum(|constr|) 0.000000000000000009 f(x_k) + sum(|constr|) 0.450000000000000120 f(x_0) 0.000000000000000000 Solver: CPLEX. EXIT=0. INFORM=101. CPLEX Branch-and-Cut MIQP solver Optimal integer solution found FuncEv 15 GradEv 15 Iter 4 Answer to Question 1: invest 15.1442% of the capital in Hardware invest 4.3269% of the capital in Software invest 25.2404% of the capital in Show-biz invest 55.2885% of the capital in T-bills Answer to Question 2 (limited number of securities): invest 30% of the capital in Show-biz invest 70% of the capital in T-bills