# How do I determine what is a separate objective in a multi-objective portfolio optimization?

Is there a general rule to determining when to separate objectives when developing a multi-objective portfolio optimization? For example, one might start with a standard portfolio optimization of maximizing expected return, while minimizing some risk metric. E.g:

$\text{maximize} \sum_{i=1}^N w_i*r_i$

$\text{minimize} \sum_{i=1}^N\sum_{j=1}^n w_iw_j\sigma_{ij}$

$\text{subject to} \sum_{i=1}^N w_i = 1$

Now, lets make things more complicated. Lets assume that the portfolio is an existing portfolio with long and short term gains/losses. We can define a tax cost function, $T_i(w_i)$ which returns the tax cost of a change in weight from holding i's current position to the new position dependent on long/short term gains/losses. One thought is to simply wrap this into the first objective above:

$\text{maximize} \sum_{i=1}^N w_i*r_i - T_i(w_i)$

Or, one could create a third objective:

$\text{minimize} \sum_{i=1}^N T_i(w_i)$

What should one keep in mind while making a decision between wrapping a new objective into an existing objective function, versus creating a new objective to solve on? (other applications might be when creating a portfolio that aims for both yield and growth, or minimizing on multiple risk metrics)

• it depends on how much control you would like to have over the optimization. If you wrap the new criteria into an existing function it might get difficult to really interpret it and to work with it. It is really less a finance question and more of a numerics question. – Probilitator Mar 14 '14 at 8:22
• As often with optimization trial and error might be the solution. Some criteria might turn out to be "well-wrappable" others might not. – Probilitator Mar 14 '14 at 8:24