09. Alternative Ways of Setting Up the Problem

Alternative Ways of Setting up the Optimization Problem

Earlier, we introduced you to one way to set up the optimization problem, and we showed you one possible objective function to use. However, there is not one single “right” way to do this; there are many alternative ways you could set the problem up.

The first formulation we saw was:

objective: minimize -\mathbf{\alpha}^T\mathbf{x} ,
with a constraint on risk: -\mathbf{x}^T(\mathbf{B}^T\mathbf{FB} + \mathbf{S})\mathbf{x} < c .

As you know, the risk constraint would be one of many constraints, such as the long-only constraint, market neutral constraint, leverage constraint, factor, and individual weight constraints—we’ve discussed these previously. These would potentially apply to all the problem formulations we’ll discuss here.

We also saw that a regularization term might be added to the objective to enforce greater diversification:

objective: minimize -\mathbf{\alpha}^T\mathbf{x} + \lambda \left |\mathbf{x} \right|_2

If the above formulations don’t produce satisfactory results, a common alternative is the following:

objective: minimize \left | \mathbf{x} - \mathbf{x}^* \right |

where \mathbf{x}^* represents a set of target weights: the weights on assets in an ideal portfolio. The goal here would be to get the weights as close as possible to the set of target weights while respecting a set of constraints. How do we generate these target weights? As an example, these target weights might be values thought to be proportional to future returns for each asset, in other words, an alpha vector.

Thus, we’d calculate

\mathbf{x}^* = \frac{\mathbf{\alpha} - \bar{\alpha}}{\sum_i \left | \alpha_i \right |}

where \bar{\alpha} is the mean of the values of \alpha .

All of these formulations seek to optimize an objective function that is based on the alpha model, constrained (in part) by a constraint on risk as defined by the risk model. However, you might also structure the problem differently.

One option is to maximize predicted return and minimize risk by including terms representing each quantity in the objective function, with the tradeoff between goals governed by a parameter. That objective function might look like this:

objective: minimize -\mathbf{\alpha}^T\mathbf{x} + \lambda [\mathbf{x}^T(\mathbf{B}^T\mathbf{FB} + \mathbf{S})\mathbf{x}]

In this case, you avoid the question of what limit to place on risk. However, this is not necessarily better as you need to specify a lambda which makes sense. However this is a significant benefit in this kind of penalty formulation: you avoid the problem of an infeasible optimization due to possibly conflicting constraints.

Another possibility is to include the alpha and risk models in the objective as follows:

objective: minimize (\mathbf{x} - \mathbf{x}^ )^T(\mathbf{B}^T\mathbf{FB} + \mathbf{S})(\mathbf{x} - \mathbf{x}^ )

This objective is similar to the objective where we minimize the distance of the portfolio weights from the target portfolio weights, \left | \mathbf{x} - \mathbf{x}^* \right | , except that in this case, the objective is to minimize not only the distance between the portfolio weights and the target portfolio weights, but the risk introduced by this distance. In this case, we find the weights such that the difference between the risk of the ideal portfolio and the achievable portfolio is minimized—the whole term is basically the tracking error. This objective function is convex; if the weights match the weights on the target portfolio, the value of the function is 0. The idea is the same as those of the objectives above, but this objective is more theoretically elegant. In fact this term does not need to be in the objective, it could also be implemented as constraint.