How to use quadprog for constrained maximization (portfolio opitization)

Hi,

I use a student version of Matlab for solving a portfolio optimization problems (optimization toolbox, I don't have a financial toolbox). My current problem is to maximize:

(x'*r-rf)/sqrt(x'*C*x)

where: x- vector of equities shares in a portfolio;

r - vector of expected returns;

rf- rish free rate;

C - covariance matrix

In short I try to maximize a Sharpe ratio and find a tangency portolio. To do that I use a fmincon:

[x,fval]=fmincon(@(x)-(x'*r-rf)/sqrt(x'*C*x),x0,[],[],Aeq,beq,lb,ub,[],options);

Unfortunately I think that a solution which I receive is not really a tangency portfolio. I think a solver fails to find a global minimum. That is why I would like to use a quadprog (x = quadprog(H,f,A,b,Aeq,beq,lb,ub,x0,options). I found it effective in finding a minimum variance portfolio (minimazing sqrt(x'*C*x)). The problem is I don't know if it is possible to use a quadprog having a function shown above. Can anyone help me to parameterize my problem using a quadprog?

Thank you very much in advance.

[Matt J]

I think a solver fails to find a global minimum.

What are you using as an initial guess, x0? Also, what are you doing to prevent x'*C*x from getting too close to zero, where the derivatives are infinity?

[Piotr]

At first I use vector of zeros for x0 and use interion point algorithm, than I set x0=x and use sqp algorithm with tolcon,tolfun,tolx set to 1e-20. I found this method fast and at least a solution seems to be "good". If I started with sqp or active-set from the beginning Matlab couldn't find a solution even after several hours.I will add that my cov matrixes are quite big, from 100x100 to 450x450.

[Piotr]

I will add one more thing. I set lb to be a vector of zeros so I don't allow a short sellign and I set ub to be a vector of 0.03 so I limit each equity to have maximum 3% share in a portfolio.

Answer 1

[Matt J]

To answer your question, I don't think your function can be rewritten as quadratic, however, if the goal is to minimize

    x'*C*x/(x'*r-rf)^2

it might be a useful approximation to minimize the difference between the numerator and the denominator

    min.  x'*C*x - (x'*r-rf)^2

This is a quadratic program, and can be solved with quadprog with

     H = C-r(:)*r(:).'
     f = -2*r*rf

If nothing else, this might give you a better initial guess x0 for fmincon than what you are using now. You could also try using the minimum variance portfolio as your initial guess.

In either case, when using fmincon, I would try rewriting your minimization objective as

    x'*C*x/(x'*r-rf)^2

maybe with appropriate additional constraints on x'*r-rf. Avoiding the sqrt() operations gets rid of differentiability issues.

[Piotr]

Thanks Matt. I will try all of your suggestions. I think I will start with rewriting my minimization object and with minimal variance portfolio as my x0. I will have to think some time about your first idea cause it's not obvious to me right know. But anyway, thanks for prompt reply.

[Matt J]

You could also rewrite the objective as

 min. -(x'*r-rf)^2/x'*C*x

These reformulations are only equivalent to your original problem if there is some constraint that x'*r-rf>=0. Otherwise, squaring or taking reciprocals of the objective has some dangers...

[Piotr]

It turned out that rewriting has not helped. The result is exactly the same as with old function. When I tried to use minimum variance portfolio as x0, Matlab gave me an error that function is undefined at initial pooint, fmincon can't continue. A result looks good (most of equities with 0 or very close to 0 share, rest with a share close to the upper bound). When I checked it graficaly and compare it with the minimum variance portfolio (I checked on graph which equities were choosen by both portfolios) I thought that it is impossible that sets of choosen equities are so different. But now when I see that the result is the same with two approaches I think that maybe an efficient set looks different than I expected and the result is actualy all right.

[Piotr]

I also checked your quadprog proposition. It gave the result and I wonder if it is not the best solution. But anyway, I used it as x0 for my fmincon optimization and one more time I received the same result as on the beginning. But different from quadprog result.

[Matt J]

It turned out that rewriting has not helped. The result is exactly the same as with old function. When I tried to use minimum variance portfolio as x0, Matlab gave me an error that function is undefined at initial pooint

The objective

     -(x'*r-rf)^2/x'*C*x

should be defined everywhere, as long as C is strictly positive definite. You should investigate why the value is unexpectedly undefined.

[Piotr]

From what I know every covariance matrix is positive semi-definite. I calculated my matrix in excel using monthly returns of my equities. Maybe it is the matter that before uploading it into Matlab I didn't round values so I have a huge matrix with values having even 15 decimal places and Matlab has problem to handle that.

During a weekend when I will have more time I will try to dig deeper into your quadprog idea. I'm quite new to Matlab and don't really understand your Matlab syntax (especialy the code for H).

Anyway, thank you very much for all your help. You already gave me a lot of ideas.

[Matt J]

I'm quite new to Matlab and don't really understand your Matlab syntax (especialy the code for H).

It may be helpful to study the effect of the (:) operation on different matrices, e.g.,

    >> x=[1 2 3; 5 6 7].'

    x =

         1     5
         2     6
         3     7

    >> x(:)

    ans =

         1
         2
         3
         5
         6
         7

    >> x(:).'

    ans =

         1     2     3     5     6     7