Implementing Markowitz Portfolio Optimization in Matlab

Hi people,

I'm new to Matlab and I'm trying to implement the following Markowitz portfolio optimization problem in Matlab:

As this is a quadratic optimisation problem I figured I should use the Optimization Toolbox's quadprog() function. I'm having trouble formulating the objective function however and I can't entirely picture how many rows and especially columns it should have. Any help would be much appreciated!

Thanks, Tom

Answer 1

[Varun Gunda]

As given in the example in the following link:

http://in.mathworks.com/help/optim/ug/quadprog.html

H would be a nxn matrix, f - nx1 matrix and A- 1xn matrix and b 1x1 matrix.

Answer 2

[Aurele Turnes]

As noted in the doc page for quadprog, the objective function that quadprog expects has this form:

 obj = (x' * H * x)/2 + f' * x

You need to build the matrices H and f to pass your objective to quadprog.

To see how to do this, expand the matrix multiplications above (using pseudo-math notation below):

 obj = sum_(i=1^n) sum_(j=1^n) H_(i,j)/2 * x_i * x_j + \sum_(i=1^n) f_i * x_i

Identifying the coefficient in front of each unknown term with your equation, you can easily see that:

 H_(i,i) = 2* \lambda \sigma_i^2, for i = 1..n
 H_(i,j) = 2* \lambda \sigma_(i,j) for i,j = 1..n, i \neq j
 f_i = -(1-\lambda) * \bar{r}_i