Portfolio Optimisation using a mean/ mean absolute deviation model (linear program)

Hey everybody, I am new to Matlab. I have recently used quadprog for the standard mean-variance model to optimise a portfolio. I want to try several models and am currently trying to model the linear program -> mean / mean absolute deviation. Mathematically speaking :

[URL=http://www.pic-upload.de/view-20298258/Screen-Shot-2013-08-05-at-01.05.47.png.html][IMG]http://www10.pic-upload.de/thumb/05.08.13/3hz1okoiiv24.png[/IMG][/URL]

in case that didn't work :

http://www.pic-upload.de/view-20298258/Screen-Shot-2013-08-05-at-01.05.47.png.html

S = 872 observations I have for 9 stocks. delta is the target return.

% This script will calculate and plot the efficient frontier for a % mean - mean absolute deviation model in the Konno context.

%   Take sample 'Sample' and calculate the mad -> transpose PeriodMAD
%   and store variable in the workspace

PeriodMAD = mad(Sample);
PeriodMAD = PeriodMAD';

assignin('base','PeriodMAD',PeriodMAD);

%   get number of Assets ( i am using 9 stocks)

nAssets = numel(PeriodMAD);

%   Set the first targetreturn

targetreturn = 0.000001;

%   Not sure if that is right -> allow none of the MAD values to lower than
%   0

nonnegativityMAD = zeros(nAssets, 1);

%   Conformity with the linproq notation

f = PeriodMAD;

%   Set some restrictions -> as shown in the graphic I have provided. NOt
%   sure if I set them up correctly to be honest.
%   PeriodDeviations includes nothing more than Observation at time(i) -
%   mean of the asset (PeriodDeviations -> (9,1) vector)
%   PeriodReturns -> ln returns for the observations 782 -> vector (9,1)

Aeq = [ PeriodDeviations
        -PeriodDeviations
        -PeriodMAD
        -PeriodReturns   ];

Beq = [ PeriodMAD
        -PeriodMAD
        nonnegativityMAD
        -targetreturn];

%   Set lower and upper boundaries for the weights -> 0<= x <= 1    

lb = zeros(nAssets,1);
ub = ones(nAssets, 1);

%   Copied that line from the quadprog example given on the internet
%   changed interior-point-convex to interior-point

options = optimset('Algorithm','interior-point');
options = optimset(options,'Display','iter','TolFun',1e-10);

for i=1:100000;

      x = linprog ( f,[],[],Aeq,Beq,lb,ub,[], options );

      %   Not sure if that is correct. Store respective mean in Output(1,i)
      %   do the same for the portfolio MAD -> is that correct?
      %   This is a modified "Sharpe- Ratio" or Konno Ratio. the quotient of
      %   the former two variables 
      %   Finally store the weights x in Output as well -> 9 variables
      %   therefore Output(5,i) to Output(13,i)

      Output(1,i) = x' * PeriodReturns;
      Output(2,i) = x' * PeriodMAD;
      Output(3,i) = (Output(1,i)/Output(2,i));

      Output(5,i) = x(1,1);
      Output(6,i) = x(2,1);
      Output(7,i) = x(3,1);
      Output(8,i) = x(4,1);
      Output(9,i) = x(5,1);
      Output(10,i) = x(6,1);
      Output(11,i) = x(7,1);
      Output(12,i) = x(8,1);
      Output(13,i) = x(9,1);

      %   now increment targetreturn -> repeat the entire process as long as
      %   linprog finds a solutions (converges to a solution) otherwise
      %   stop/break

      targetreturn = targetreturn + 0.000001;

      if(exitflag == 1)

          continue

      else

          break

      end

end

%   Find highest KonnoRatio - Quotient of Returns/Risk stored as
%   maxKonnoRatio then find position in Output

maxKonnoRatio = max(Output(3,:));

[row col] = find(maxKonnoRatio == Output(3,:));

%   finally create and display an array that contains all the information
%   for the optimal allocation. 

OptimalAll = [  Output(1,col)
                Output(2,col)
                Output(3,col)
                0
                Output(5,col)
                Output(6,col)
                Output(7,col)
                Output(8,col)
                Output(9,col)   ]

%   and hopefully draw the efficient frontier

plot(Output(2,i),Output(1,i))

I am stuck. I have modified so much stuff already and I keep getting error messages. currently this is the most recent one :

EDU>> MAD Error using linprog (line 231) The number of rows in Aeq must be the same as the number of elements of beq.

Error in MAD (line 37) x = linprog ( f,[],[],Aeq,Beq,lb,ub,[], options );

EDU>> MAD Error: File: MAD.m Line: 108 Column: 32 Unbalanced or unexpected parenthesis or bracket.

EDU>>

if you could please help me. Am I at least close to a proper solution ? Btw. : I will annualise those daily data observations - as soon as I can solve the basic problem.

I would be very grateful for some advice and/or help. Thank you so much!

Daniel

Answer 1

[Shashank Prasanna]

Beq should be a column vector, but in your the above example it seems to be a row vector.

Beq = Beq(:); % Forces it to be a column vector

Take a look at the documentation of linprog for more information on how Aeq and Beq should be specified.

Equality constraints should be of the form:

Aeq*w = Beq, where

Aeq -> m x n

Beq -> m x 1

w -> n x 1

w is the weights you are trying to solve for.

Answer 2

[Alejandra Pena-Ordieres]

You can use the PortfolioMAD functionality to compute optimal portfolios using the mean-absolute devaition as the risk measure.

To get started with the PortfolioMAD object, see the examples in Creating the PortfolioMAD Object.