This optimization should be used when a client's portfolio has drifted away from the model portfolio assigned to the client. The idea is to find the minimal amount of trades required to bring the portfolio within a prescribed tracking error constraint from the model.
Mathematically it is defined as:
$$\begin{array}{lll}\relax{min} \ \sum_{i} \ \ \overline{1}_{|t_{i}|} \\ \\ \relax{s.t.} && \left[ \ \relax{w}_{model} - (\relax{w}_{client} + \relax{t}) \ \right]^{T} \times \ \Sigma \ \times \left[ \ \relax{w}_{model} - (\relax{w}_{client} + \relax{t}) \ \right]\le \overline{\text{TE}}^{2}\\ \\&&\sum_{i} \relax{t}_{i} = 0 \\ \\&&\relax{t}_{i} \ge \relax{w}_{client} \\ \\ &&\relax{\small{other LCs}} \end{array}$$
Where %%\overline{1}_{|t_{i}|}%% is an indicator vector with ones when a security is traded and zeros elsewhere. By minimizing this indicator we are minimizing the amount of securities traded. This is a cardinality objective that requires a mixed-integer solver.
The constraints are:
- Tracking error constraint:
$$\left[ \ \relax{w}_{model} - (\relax{w}_{client} + \relax{t}) \ \right]^{T} \times \ \Sigma \ \times \left[ \ \relax{w}_{model} - (\relax{w}_{client} + \relax{t}) \ \right]\le \overline{\text{TE}}^{2}$$
where %%\Sigma%% is the covariance between unique positions from model and client combined. TE is the tracking error threshold.
- Self financing trades:
$$\sum_{i} \relax{t}_{i} = 0$$
Purchases have to be done with proceeds from sales.
- Long only:
$$\relax{t}_{i} \ge -\relax{w}_{client} , \forall i$$
Maximum sale for a position is the weight of the position in the portfolio.
- Other linear constraints:
To be supplied by user as per constraint definition