Why Efficient Frontier Formulations Use Equality or Inequality Constraints

Summary

The document explains the role of equality and inequality constraints in alternative portfolio-selection formulations. It compares maximizing return subject to a variance limit with minimizing variance subject to a minimum return, and relates these formulations to utility maximization. Under the conditions behind an efficient frontier, these approaches can identify the same portfolio even though their constraints are written differently.

An inequality expresses a bound, such as variance being no greater than a chosen level or return being at least a target. In the paper discussed, equality is used in one formulation as part of an argument about equivalence across formulations; the answer suggests that equality can also be easier to handle analytically. The exchange provides intuition rather than a proof or a full account of assumptions, so equivalence should not be taken as universal across all portfolio constraints and objectives.

Key ideas

• Efficient portfolios can be characterized by maximizing return for a variance bound or minimizing variance for a return target.
• An inequality constraint sets a limit on risk or a minimum acceptable return.
• Equality constraints may simplify analytical treatment in a particular formulation.
• The claimed equivalence applies to the efficient-frontier argument and depends on its setup.

Tags

# Portfolio Selection formulation


# Portfolio Selection formulation












I was just wondering why in http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1601412 on page 22, the constraint (48) is a strict equality for the minimum variance formulation. Whereas in a different formulation (but equivalent as proven in the paper) (45) is an inequality.

Is there any intuition behind this ?

Best, Jcl

## Answer by John (score 2, accepted)

https://quant.stackexchange.com/a/17756

That part of the paper is showing why the efficient frontier is the same regardless of whether you are maximizing utility, maximizing returns given variance, or minimizing variance given returns.

Inequality constraints tend to be a bit more work to deal with analytically, so that might be a reason why they use the equality constraint on one of them.

Usually you get the same portfolio regardless of whether you use the inequality constraint or the equality constraint. I don't think this paper is really trying to evaluate equality vs. inequality or anything. They're focused more on the general point. The inequality constraint is just telling you that the variance must be less than some amount. The equivalent when minimizing variance given returns would be an inequality constraint such that the return of the portfolio is greater than or equal to some amount.

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