GA, How to modify the requied return to requied risk in a Genetic Algorithm

As you may see, this code have a requied return as a constraint, but I'm triying to modify it, to give a requied risk instead of a requied return.

If someone can help me I would be amazing, I've been trying to modify the code but I haven't had many results.

Thanks.

format long g
filename = 'https://www.mathworks.com/matlabcentral/answers/uploaded_files/835520/Returns.xlsx';

data = readmatrix(filename);

nAssets = size(data, 2);

%Returns and covariance
mu = mean(data);
mu.'

sigma = cov(data);

%formulate the problem/optimization
r_target = 0.004; %r_target is the required return
f = zeros(nAssets, 1);    %there is no constant
A = [-mu; -eye(nAssets)] %besides the returns we forbid short selling
b = [-r_target; zeros(nAssets, 1)] % required return and weights greater/eqauls 0
Aeq = ones(1, nAssets)   %All weights should sum up...
beq = 1                  %... to one (1)

%solve the optimization
fcn = @(w)MaxReturn(w,mu);

[w, fval, flag, output] = ga(fcn, nAssets, A, b, Aeq, beq)

if isempty(w)
   warning('could not find any solution')
else
    %print the solution
    fprintf(2, 'Risk: %.3f%%\n', sqrt(w*sigma*w')*100);
    fprintf(2, 'Ret: %.3f%%\n', w*mu'*100);
end

function f = MaxReturn(w,mu)
  f = w * mu';
end

Answer 1

[Aman]

Hi Santiago,

As per my understanding, you want to now get the required risk instead of the required returns.

In order to do so, you need to make the below two changes:

1. Instead of maximizing the return, you now need to minimize the portfolio risk.
2. Instead of putting a cap on the minimum return, you now need to cap the maximum allowable risk.

With the above-mentioned changes, the updated code will look like below:

format long g
filename = 'https://www.mathworks.com/matlabcentral/answers/uploaded_files/835520/Returns.xlsx';

data = readmatrix(filename);

nAssets = size(data, 2);

% Returns and covariance
mu = mean(data);
mu.';

sigma = cov(data);

% formulate the problem/optimization
risk_target = 0.004; % risk_target is the maximum allowable risk
f = zeros(nAssets, 1);    % there is no constant in the objective function for risk minimization

% Objective function is now to minimize risk, so we do not use mu in constraints
A = -eye(nAssets); % Forbid short selling
b = zeros(nAssets, 1); % Weights greater/equal 0

Aeq = ones(1, nAssets); % All weights should sum up...
beq = 1;                % ... to one (1)

% solve the optimization
fcn = @(w)PortfolioRisk(w, sigma);

options = optimoptions('fmincon', 'Display', 'iter', 'Algorithm', 'sqp');
[w, risk, flag, output] = fmincon(fcn, ones(nAssets, 1)/nAssets, A, b, Aeq, beq, [], [], [], options);

if isempty(w)
   warning('could not find any solution')
else
    % print the solution
    fprintf(2, 'Risk: %.3f%%\n', sqrt(w'*sigma*w)*100);
    fprintf(2, 'Ret: %.3f%%\n', w'*mu'*100);
end

function risk = PortfolioRisk(w, sigma)
  risk = sqrt(w' * sigma * w); % Minimize this function
end

I hope it works for you!