Portfolio Allocation with Asset Bounds and Return Objectives

Summary

The document considers allocating a portfolio across five asset categories while imposing lower or upper bounds on selected weights. It asks whether maximizing expected return under such constraints can be formulated as linear programming. The replies distinguish that objective from risk-aware portfolio optimization: with expected returns as the sole objective and linear allocation constraints, the problem is linear. When short selling is prohibited, the unconstrained-by-risk solution tends to fill the permitted weight in the highest-return assets first, subject to the stated bounds.

The responses point to portfolio optimization software, including an R package designed to express allocation constraints, and mention quadratic optimization software for other formulations. They also recommend foundational study in active equity portfolio management. The discussion provides no data, worked allocation, or comparison of solvers. Maximizing expected return alone ignores risk and estimation uncertainty, so it does not establish that the resulting allocation is suitable in practice; a risk-aware objective would require a different formulation.

Key ideas

• Expected return is a linear objective in portfolio weights when asset returns are treated as fixed inputs.
• Upper and lower allocation bounds can be incorporated as linear constraints.
• Without short selling or a risk penalty, the optimizer tends to prioritize assets with the highest expected returns.
• Quadratic optimization is relevant when the objective or constraints introduce quadratic terms.
• The document gives software and reading suggestions but no worked example or empirical comparison.

Tags

# Portfolio optimisation by asset allocation


# Portfolio optimisation by asset allocation












I would like to optimize a portfolio allocation (maximizing the exposure or the expected return), but with asset constraints. (some parts of my portfolio cannot exceed a certain minimum or maximum). For example, 60% max on equity and 5% min on cash, say for liquidity. Let's say there are five assets, cash, US and UK equity, US fixed income, and US property.

How can I achieve that? Is there a way to turn the problem into a linear programming problem? or to approximate the results?

Any links or ideas are welcome.

## Answer by vonjd (score 6, accepted)

https://quant.stackexchange.com/a/30967

There are very powerful software solutions out there, so you should not reinvent the wheel.

One notable R package is PortfolioAnalytics.

You can find a very good introduction here, where your concrete constraints requirement is addressed in section 3.3, p. 6:

Benett, R.: Introduction to PortfolioAnalytics (2015)

## Answer by Sebapi (score 2)

https://quant.stackexchange.com/a/42871

You are formulating the problem of portfolio allocation as that of maximizing expected portfolio return without regard to risk.

This a linear programming problem.

If you are prevented from going short, the solution will be to max out your allocation of the best performing asset and then second best, etc...

Otherwise, you can use quadratic optimization software `cvxopt`.

## Answer by Atul Agarawal (score 1)

https://quant.stackexchange.com/a/30976

I agreed with the usage of PortfolioAnalytics package of R programming, but before that I must say read, read, read and try to understand the concept from book written on Active Equity Portfolio Management by Grinold and Kahn. Combination of both will gives you tremendous results of your objective. All the Best.

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