Choosing Between Combined and Separate Portfolio Optimization Objectives

Summary

The document asks how to decide whether a portfolio consideration belongs inside an existing objective function or should be optimized as a separate objective. It starts with a conventional formulation that seeks higher expected return while controlling covariance-based risk, subject to a fully invested weight constraint. It then introduces an existing portfolio with positions that may have short- or long-term gains or losses, and a tax-cost function for changing each holding.

Two formulations are compared: subtracting tax cost from the return objective, or minimizing aggregate tax cost as an additional objective alongside return and risk. The question also points to portfolios balancing yield and growth or multiple risk measures. It does not provide a decision rule, solution method, or empirical comparison. In practice, the choice depends on how priorities and trade-offs are represented, including whether tax cost can be expressed on a compatible scale or should instead be controlled separately. The example frames the modeling problem but leaves those choices open.

Key ideas

• The example balances expected return against covariance-based portfolio risk.
• Tax consequences of changing existing positions can be represented with a cost function.
• Tax cost can be included in the return objective or optimized as a separate objective.
• Separate objectives make trade-offs explicit, while a combined objective requires a way to weigh unlike quantities.
• The document raises the modeling choice but does not supply a general decision rule.

Tags

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


# 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)

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.