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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?

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  • $\begingroup$ Your formula yields $L_1$ norm for $p=1$ but not $L_2$ norm for $p=2$. $\endgroup$
    – Richard Hardy
    Jan 6, 2021 at 14:14
  • $\begingroup$ you're right I'll have to correct the exponent etc $\endgroup$
    – develarist
    Jan 6, 2021 at 16:32
  • $\begingroup$ More attractive to who? $\endgroup$
    – steveo'america
    Jan 6, 2021 at 17:40
  • $\begingroup$ an investor who wants a portfolio to have good out-of-sample performance, i.e. live up to its in-sample promise. For example, the minimum-variance portfolio's norm will always be smaller than the max-Sharpe ratio portfolio norm, which reflects the well-known fact that the latter traditionally has very poor out-of-sample performance $\endgroup$
    – develarist
    Jan 6, 2021 at 19:35

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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

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  • $\begingroup$ back to the example and numerical intuition, is it true that when confronted with two portfolio norms, an investor should always choose the portfolio with the smaller norm? and is there ever a situation when a larger portfolio norm is more sought after $\endgroup$
    – develarist
    Jan 6, 2021 at 5:00
  • $\begingroup$ I don’t think you can look at the norm in isolation. Like in lasso or ridge regression you optimise Sum of squares + Norm, the norm in that case would “shrink” the co-efficients towards zero ensuring non-relevant features do not get any weights. Similarly in portfolio optimisation adding norm to the whole portfolio optimisation problem would have a similar shrinking effect. $\endgroup$
    – Dhruv Mahajan
    Jan 6, 2021 at 6:15
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    $\begingroup$ I think norm can be a part of a portfolio optimisation problem, but you cannot judge a portfolio on the basis of norm alone, that doesn’t make any sense in my opinion. $\endgroup$
    – Dhruv Mahajan
    Jan 6, 2021 at 6:16
  • $\begingroup$ during optimization, sure, the objective function is sum of squares + norm, but after any optimization (shrunk or not shrunk), the norm of any resulting portfolio weight vector can still be computed (shrunk or not). This post-optimization norm is what I have been referring to. We have a vector and simply measure its norm. Having prior knowledge that the vector had been shrunk will tip us off that it will have a small norm before we measure it, fine, but even if it hadn't been shrunk, its norm can still be measured! what is the intuition of finding a norm being smaller than another norm? $\endgroup$
    – develarist
    Jan 6, 2021 at 6:45
  • $\begingroup$ even if judging a portfolio based on its calculated norm might not make sense, the fact is that we will always observe that the norm of the minimum-variance portfolio will always be smaller than the max-Sharpe ratio portfolio, for example, (and will perform better out-of-sample) yet neither of them were shrunk during optimization. This is what I am trying to understand. The calculated norms must therefore have some sort of meaning, just like how the variance of the min-variance portfolio always has lower return variance than the max-Sharpe portfolio $\endgroup$
    – develarist
    Jan 6, 2021 at 6:47
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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 $\delta$.

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