The Number of Stocks Before (n) and After Portfolio Optimization (k): The Heuristic k ≈ sqrt(n)
Bibliographic record
Abstract
This study focuses on the relationship between the number of securities (n) pre-selected for mean-variance portfolio optimization and the number of optimal securities (k). We propose a heuristic k ≈ square root (n) based on empirical research optimizing different sized (n) portfolios. That is, a sample selection of 20-30 securities should yield a portfolio of about five optimal securities, and an initial sample of 500 securities, should result in an optimal portfolio of about 22. We focus on the tangent portfolio that maximizes the return-to-risk ratio. The heuristic finds its support, rationale, and logic in the numerical properties and statistical nature of the optimization. More specifically, the heuristic seems to originate in the dynamic convergence patterns observable in many statistical processes, especially in standard deviations. It is also supported by available results in the literature. Our “square root” heuristic functions as part of the wider family of approximation around the power law, where some variables (authors, securities, people) receive a disproportionate share of a given collection of items – see, for example, Pareto’s principle, Zipf’s law, Lotka’s law, Price’s square root law, Simon’s law, etc. The heuristic provides assistance not only in anticipating the number of optimal securities chosen by the mean-variance optimizer but also in suggesting selectivity in the effort of pre-selecting securities prior to the optimization and in sharpening portfolio-based approaches to investing in general. In sum, the heuristic k ≈ sqrt(n) seems helpful at all levels of portfolio management.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.018 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".