The trade looks familiar. Utilities stay long most of the time. From June 2008 to June 2011, the short side is mainly materials, with a smaller financial-sector position. From 2020 through 2023, it is almost entirely financials. Yet the number of holdings depends on the eigenvector's scaling, which the paper settles only through arithmetic. The lower bound on the IPR uses weights of 1/p. The norm behind that choice is never named, and a standard eigen-solver returns a different one.
We could not trade the paper's market. Point-in-time S&P 500 membership is unavailable to us, so our runs use a point-in-time large-cap US universe formed from year-by-year capitalization, with daily prices from 2010 forward. The constituent-level figures below do not carry across. Starting in 2010 also cuts into the June 2008 to June 2011 materials episode.
The setup
The sample contains daily log returns for S&P 500 constituents from 6 January 2005 to 21 November 2024, spanning 5029 trading days and sourced from Yahoo Finance. Dropping firms with 5% or more missing trading days leaves p = 484 names. Membership is fixed at end-2024 and then carried backwards. Correlations use 504-day windows advanced in steps of 60 days, producing 76 windows.
For each window, the authors decompose the correlation matrix and treat every eigenvector as a portfolio. The Inverse Participation Ratio sums the fourth powers of its weights. Borrowed from scattering theory, quantum mechanics and particle physics, it measures how localized a wave function is. One dominant name raises it; equal distribution across all names lowers it. The authors then define N_eff = I^(-1/3), interpreting the result as the number of stocks that carry the portfolio, and ask whether those names still track the full portfolio.
The selection rule appears in two forms. The methodology ranks components of the expected-return vector in decreasing order and retains the first N_eff. A footnote instead selects weights by absolute value while keeping their original signs. Everything below follows the footnote because its reading supports the reported long and short sector results.
The paper focuses on the second-largest eigenvalue because of its counter behaviour relative to market spillover in the authors' earlier work, Molero González et al. 2025. Since the first eigenvalue dominates, the market is removed beforehand. Within each window, every stock is regressed by OLS on an equal-weighted 1/p market proxy using the Sharpe (1964) single-index model. The authors then decompose the residual correlation matrix. Its leading eigenvector v_1 serves as a proxy for the second raw eigenvector u_2.
The match is generally close. Pearson correlation exceeds 0.95 through most of the sample and runs between 0.90 and 0.95 from August 2010 to September 2012. It falls below 0.85 only from January to September 2007 and from September 2022 to June 2023. During the 2007 crisis and again in Covid, however, their eigenvalues move in opposite directions: λ_2 from the raw matrix declines as λ'_1 from the residual matrix rises.
That opposite-signed divergence is the paper's strongest result.
Which cube root?
An IPR normally maps to the participation number 1/I. This paper chooses I^(-1/3). Its argument starts from equal weights: assign every stock 1/p, obtain I = 1/p³, and the cube root recovers p = 484. That calculation requires an eigenvector rescaled so that its components sum to one.
A standard eigen-solver instead supplies unit L2 norm. Equal components are then 1/√484, making I = 1/484, and the cube root returns 7.85. The vector has stayed the same while the answers differ by about sixty times. The paper explicitly describes a portfolio equally weighted across all assets and derives the 1/p³ bound using weights of 1/p. It also says eigenvectors will appear in their normalised version, without specifying the norm applied to the empirical eigenvectors. We did not find that specification elsewhere.
For v_1, a third interpretation arises because the vector contains both signs. Scaling by the sum of raw components becomes unusable near dollar neutrality, where the divisor approaches zero. Scaling by the sum of absolute values is the sensible alternative, though the paper's 1/p³ lower bound then fails.
This constant determines the finding. N_eff sets the boundary between effective stocks and everything else. Under unit L2, the utilities trade may contain roughly eight names. Under sum-to-one scaling, it may extend across as many as 484. We raised a similar issue in a note on crypto hierarchical risk parity, where allocation rather than clustering drove the outcome (review).
What the 0.86 fit captures
For every window, the validation regresses the full eigenportfolio's expected return on that of the effective-only subset. The resulting R² is 0.79 for u_1, 0.82 for u_2 and 0.86 for v_1. Both selection and measurement occur within the same 504-day window. The regression therefore shows that removing small weights leaves expected return broadly unchanged, much as the IPR itself is meant to indicate.
The authors say this plainly. They do not formally assess how the information might enter actual portfolio construction or risk management, and out-of-sample performance appears in the future-work paragraph. Fair enough. Still, the conclusion says the method allows "improving transaction costs efficiency," a claim that needs a number. The effective set is estimated again every 60 days. Materials occupy the short leg through 2008, while financials occupy it through 2020. The paper prices neither borrow on those shorts nor turnover from a set changing four times a year.
Reimplementation gets messy
The eigen-decomposition leaves the sign of v_1 undetermined, and we did not find a stated sign convention. Because u_1 is all-positive, its sign fixes itself. The signed v_1 retains each stock's original sign, which means accepting whatever orientation the solver supplies. Reproducing the long/short split requires anchoring the sign independently, perhaps against overlap with the previous window or the utilities block. Figure 6 displays persistent long-utilities composition across windows, so the authors' pipeline evidently dealt with the issue. The text does not explain how.
Successive windows share 444 of 504 days. Consequently, 76 windows are nowhere near 76 independent observations. The angular-stability tests use this overlap, although the authors apply Benjamini-Hochberg FDR control, which is the right instinct. Another problem appears in the printed hyperspherical formula: each component is divided by the full radius r, while the accompanying text refers to the norm of the remaining components. The resulting angle sequences differ, and the atan2 argument contains a typo.
The constituent universe is fixed to end-2024 membership, with sectors assigned using Q1 2025 GICS. The authors acknowledge survivorship bias and cite Plerou et al. (2002) for the argument that dominant collective modes are insensitive to the chosen firm subset. They do not test that proposition here. Its direction matters for this result. A panel made from end-2024 survivors excludes the 2008 casualties, and financials is one of the two sectors favoured by the short leg.
The mechanism offers the transferable claim: whether the leading residual eigenvector localizes on defensives, and whether stress rotates the effective set.
The appendix carries the result I would keep. Third and fourth eigenvectors show no lasting sector dominance. Long and short positions coexist, while their composition fragments over the sample. The paper's literature review observes that researchers have often treated the eigenvector associated with the second-largest eigenvalue as a carrier of sector structure. In this sample, the third and fourth do not carry it.
One test would change my view of the rest. Choose the effective set on window j, hold it through window j+1, charge a per-share cost plus a borrow rate on the short leg, and report the Sharpe.