Mean-Variance Trade-Off in Portfolio Optimization

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Portfolio optimization with sign constraints

We return to the portfolio optimization problem. This time, we assume that shorting is not allowed. In other words, we can only invest in any given asset, or decide not to invest; but we can't borrow and hold any negative position. The problem we have introduced now takes the form

$min_x : x^T Sigma x ~:~ hat{r}^Tx = mu, ;; x ge 0,$

where

is our target for the nominal return, and Sigma

is a positive semi-definite matrix that contains information about the uncertainty on the assets’ returns (see here). The above is a QP.

CVX implementation

cvx_begin
    variable x(n,1);
    minimize( x'*S*x );
    subject to
       rhat'*x == mu;
       x >= 0;
cvx_end

Other constraints

Sector bounds

For example, we can put upper bounds on some elements of

, to limit the amount of investment in any single asset. We can also require that the total sum invested in a given sector (corresponding to say, the first

assets) does not exceed a fraction alpha

of the total sum invested. This can be modeled by

$min_x : x^T Sigma x ~:~ hat{r}^Tx = mu, ;; x ge 0, ;; sum_{i=1}^k x_ i le alpha left( sum_{i=1}^n x_i right).$

Diversification

We can also impose some diversification constraints. For example, we may impose that no group of

(with

) assets contain more than 80%

of the total invested. We write this as

$s_k(x) := sum_{i=1}^k x_{[i]} le 0.8 sum_{i=1}^n x_i,$

where $x_{[i]}$ is the

-th largest component of

, so that

is the sum of the

largest elements in

. The function s_k(x)

is convex, since it is the point-wise maximum of linear functions (hence it is polyhedral):

$s_k(x) = max_{(i_1,ldots,i_k) in {1,ldots,n}^k} : x_{i_1} + ldots + x_{i_k}.$

CVX implementation

cvx_begin
    variable x(n,1);
    minimize( x'*S*x );
    subject to
       rhat'*x == mu;
       x >= 0;
       sum_largest(x,10) <= 0.8*sum(x);
cvx_end

The above expression for s_k

simply states that the function s_k

is the pointwise maximum of the $C_{n,k}$ (

choose

) linear functions obtained by choosing any subset of

indices in {1,ldots,n}

. For example, with k=2

and

Since

is polyhedral, we can definitely represent the constraint s_k(x) le 0.8 sum_i x_i

as a finite set of affine inequality constraints, that is, a polyhedron. However the list of constraints is huge: for n=100

, we have to list more than $10^{13}$ constraints!

$s_k(x) = min_t : kt + sum_{i=1}^n max(0, x_i-t).$

In this form, the diversification constraint can be expressed as: there exist a scalar

and a

-vector

such that

$kt + sum_{i=1}^n u_i le 0.8 sum_{i=1}^n x_i, ;; u ge 0, ;; u ge x- t mathbf{1}.$

In the above, mathbf{1}

is the vector of ones in $mathbf{R}^n$ . Here, t,u

are additional variables that allow a QP representation with a moderate number of variables and constraints:

$min_{u,t,x} : x^TSigma x ~:~ begin{array}[t]{l} hat{r}^Tx = mu, ;; x ge 0, kt + displaystylesum_{i=1}^n u_i le 0.8 displaystylesum_{i=1}^n x_i, u ge 0, ;; u ge x- t mathbf{1}. end{array}$

The lesson here is that a polyhedron in a

-dimensional space, with an exponential number of facets, can be represented as a polyhedron in a higher dimensional space (in our case, with dimension 2n+1

), with a moderate number of facets (precisely 2n+1

facets). By adding just a few dimensions to the problem we are able to deal (implicitly) with a very high number of constraints.