How Portfolio Norm Constraints Affect Regularization and Trading Costs
Summary
The document explains why portfolio optimizers may constrain or penalize the L1 or L2 norm of asset weights. It connects these constraints to regularization in regression: L1 and L2 penalties can limit estimation error and produce more stable weights, while also acting as generalized restrictions that may limit short selling. The discussion presents this motivation as an analogy and notes uncertainty about whether norm minimization is a universal principle.
The answers distinguish the norms’ portfolio implications. With weights summing to one, an L1 norm above one implies short positions; allowing larger values can permit more holdings and potentially raise monitoring and rebalancing costs. A larger L2 norm can signal concentrated exposure because large weights contribute disproportionately. Neither norm is inherently bad: the tradeoff depends on the optimizer, transaction-cost definition, and investor constraints, and the text offers conceptual explanations rather than empirical comparisons.
Key ideas
- L1 and L2 portfolio constraints are motivated by regression regularization and may reduce estimation error.
- With weights summing to one, an L1 norm above one implies that the portfolio contains short positions.
- A larger L1 norm can allow more positions and may increase monitoring and rebalancing costs.
- A larger L2 norm can indicate concentrated exposure because large weights contribute disproportionately.
- Norm constraints are tradeoffs, not universal goals; their value depends on portfolio objectives and costs.
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Full text
# Why is a smaller portfolio norm better?
# Why is a smaller portfolio norm better?
If the norm of the portfolio weight vector, $\frac{1}{p}\sum_{i=1}^n |w_i|^p$ for $p=1,2$, of portfolio A is 0.6, and the norm of portfolio B is 0.4, then portfolio B is considered more attractive because its portfolio norm is lower.
What is the intuition behind this? What's so great about having a small portfolio norm, and why should minimizing the mere value of a portfolio's norm be the main aim of improving asset allocation?
Are there any cases where a larger portfolio norm is instead better to have?
## Answer by Dhruv Mahajan (score 5)
https://quant.stackexchange.com/a/60358
Norm constraints are motivated by regularisation in regression analysis. L1 and L2 norm are similar to Ridge and Lasso Regression. The author who first introduced this method argued that it will reduce estimation error since you can think of portfolio optimisation (unconstrained) weights as OLS regression estimates, so the same logic of minimising the L1,L2 norm follows here too.
But I don’t know if minimising the norm is considered a norm in the academia (no pun intended) or if it was a more author-specific belief.
Another thing is that when you constraint the norm of a portfolio, authors show that you automatically satisfy some constraints like short-selling etc., so it’s a more generalised case of those.
You can find the paper here : http://facultyresearch.london.edu/docs/DeMiguelGarlappiNogalesUppal20070715.pdf
## Answer by Myoujin (score 2)
https://quant.stackexchange.com/a/72034
Short answer: High L1 or L2 norm weights are not per se bad, but can be associated with higher transaction costs and a higher concentration ratio.
Long answer: Generally, we require an efficient portfolio with K assets to follow the constraint $\sum_{i=1}^K w_i = 1$.
In order for the L1 norm $||w||_1$ to be greater than one, such a portfolio must have at least one short sell. This is due to the nature of the constraint mentioned beforehand. Allowing for higher values of the L1 norm thus allows for the potential inclusion of more assets, which then can cause higher transaction costs when the portfolio needs to be rebalanced. The more assets you control, the more monitoring cost will occur and the more likely it is that you have to do some changes in your portfolio weights when rebalancing takes place. However, this is heavily dependent on how your transaction costs are defined.
The L2 norm $||w||_2$ can be relevant for measuring the concentration of your portfolio. High levels of the L2 norm might coincide with high exposures to one certain asset due to the nature of the second order polynomial, e.g. a weight of 2 will have $2^2=4$ times the impact than a weight of 1 with $1^2=1$ in the L2 norm. In portfolio management, companies tend to avoid being invested in only a few companies with large quantities due to issues with risk exposure, compliance etc.
For further information, I suggest the paper Sparsity and stability for minimum-variance portfolios by Husmann, Shivarova and Steinert (2022) who also use a L1 constraint and measure the impact of it by the parameter $\delta$.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.