At λ = 0.01, this optimizer gives a trader almost exactly 1/N after extra solver work. Bhat, Sheikh and Khanday largely acknowledge it. Their case rests on the λ-sensitivity analysis: it traces a systematic regularization path and gives dispersed capital an explicit optimization rationale. Yet the path is in-sample, and the rationale depends on a λ chosen with "no formal cross-validation". Anyone trying to reproduce the result should start where regularization changes the weights, then ask whether that region was tested out of sample.

The portfolio the authors built

A long-only, fully invested minimum-variance (MV) optimizer can concentrate in a few assets when those assets minimize estimated variance. The authors add Shannon entropy and minimize w'Σw + λΣ w_i ln w_i. The added term is negative entropy: it is convex on the simplex and reaches its minimum at equal weight. As λ rises, the solution moves from MV toward 1/N. The authors call the result ERMV, entropy-regularized minimum variance. It uses no return forecast. Any advantage must come from better-conditioned weights.

The universe contains 12 U.S.-listed ETFs: SPY, QQQ, IWM, EEM, VEA, GLD, SLV, TLT, IEF, HYG, LQD and VNQ. Yahoo adjusted closes from 1 January 2015 to 31 July 2026 yield 2,907 daily simple returns, and Σ is the plain sample covariance. The paper compares three static portfolios: equal weight (EW), MV and ERMV at λ = 0.01. A 21-point log grid from 1e-5 to 1 maps the λ path. Further checks use a rolling backtest with a 252-day estimation window and weights held 21 days, a circular block bootstrap of Sharpe differences, and risk-free rates of 0%, 2% and 4.5%.

In a separate section, the authors build a correlation-distance minimum spanning tree and rolling 126-day spectral entropy, the entropy of the correlation matrix's normalized eigenvalues. Neither network measure enters the weights. The reported result is limited: ERMV removes MV's concentration and matches EW's Sharpe.

Why does MV end up in two bond funds?

Full-sample long-only MV puts 61.87% in IEF and 38.13% in HYG. The other ten ETFs receive weights the paper calls effectively zero within floating-point precision. That follows from the objective. IEF has the universe's lowest daily standard deviation, 0.00413, or about 6.6% annualized. HYG's 0.005134 becomes about 8.1% annualized. Together they run at 5.35%, below either fund alone; their correlation must therefore be below about 0.8, the ratio of their volatilities. MV's annualized return was 2.42%, versus 7.94% for EW. Its max drawdown was -17.91%, versus -26.11%.

The paper holds the risk-free rate at 4.5% throughout 2015 to 2026. That constant rate flatters the comparison without changing the ordering: MV's Sharpe falls to -0.3887, and the EW-minus-MV gap nearly triples, from 0.2357 at 0% to 0.6866 at 4.5%. Even at 0%, MV's 0.4523 trails EW's 0.6880.

The bend in the λ path

Rebuild this calculation first.

EW's 11.53% annual volatility corresponds to a daily variance of roughly 5.3e-5. At λ = 0.01, the entropy term near uniform weights is about 0.01 × 2.48, or 0.025. Its curvature there is λN = 0.12, hundreds to thousands of times the daily variances in Σ. Covariance has little influence on the solution: weights range from 0.0828 to 0.0841, and N_eff (the exponential of entropy) reaches 11.99 out of 12. The authors describe this as strong diversification regularization rather than a subtle covariance-entropy compromise. They also call λ = 0.01 a benchmark with "no formal cross-validation" behind it.

Smaller λ values change the portfolio. At λ = 1e-5, volatility is 6.33%, the largest weight is 42.02%, and N_eff is 5.80. At 1e-4, those figures are 9.31%, 14.76% and 11.16. Between 1e-5 and 1e-4, N_eff rises from 5.80 to 11.16 while volatility rises from 6.33% to 9.31%, roughly half the distance from MV to EW. The earlier move, from MV's roughly two effective assets to 5.80, lies below 1e-5 and outside the paper's grid. Volatility continues from 9.31% at 1e-4 to 11.50% at 0.01. Above 0.1, the weights scarcely move: L2 distance from EW is 0.000166 at 0.1 and 0.000017 at 1.

The path's Sharpe figures use the full sample: -0.0979 at 1e-5 and 0.1883 at 1e-4, both with 4.5% rf. The rolling backtest uses only λ = 0.01. We did not find an out-of-sample result for an intermediate λ. That omission matters most where ERMV differs from EW, at or below 1e-4; at λ = 0.01 and above, it effectively tracks EW.

EW and ERMV look alike in the Sharpe test

With 4.5% rf, full-sample Sharpe is 0.2979 for EW and 0.2966 for ERMV. Their difference, -0.0013, has a 21-day-block bootstrap interval of [-0.0029, 0.0000]. The paper treats the interval as containing zero and notes that its upper endpoint is zero only to four decimals. A bootstrap mean of -0.0014 leans against ERMV. In the rolling out-of-sample run, EW returns 9.20% at a 0.4000 Sharpe; ERMV returns 9.15% at 0.3978, a -0.0022 gap.

These figures support the authors' conclusion that ERMV matches EW without beating it. The close result is unsurprising when portfolio weights differ by under a tenth of a percentage point. The authors note that even modest costs could erase the difference; the run charges none. Comparisons with MV say more about the allocation. ERMV minus MV is 0.6853, with a bootstrap interval of [0.2636, 1.1792]; EW minus MV is 0.6866, with [0.2634, 1.1809]. The authors acknowledge that these intervals mostly compare 1/N with two bond funds.

That bootstrap uses only the 4.5% rate. At 0%, the EW-minus-MV gap is 0.6880 - 0.4523 = 0.2357, below the 0.2634 lower bound of the 4.5%-rate interval. Some of the 0.6866 gap comes from the risk-free-rate assumption. Full-sample results also rely on one Σ estimated across the entire period, an in-sample choice. The rolling section addresses that issue, and its qualitative ordering holds.

Implementation questions

March 2020 in the network measures

Rolling spectral entropy averages 1.505, with a standard deviation of 0.118. It reaches 1.093 on 17 March 2020. Its 2020 average, about 1.38, recovers to about 1.63 in 2021. The measure behaves as expected during stress. The authors claim no predictive power for it, and the static weights never respond to it.

We could not run this ourselves. The MV corner consists of IEF and HYG, with TLT (long Treasuries) and LQD (investment-grade corporates) also in the universe. We hold no bond price data. Substituting other low-volatility assets would test a different allocation.

ERMV at λ near 1e-4 needs an out-of-sample comparison with EW after costs. A positive gap there would change our view of this paper.