The Markowitz edge here depends on a tightly constrained trading rule. Its forty-year simulation also rests on just seventeen observed years. Those restrictions deserve more attention than the headline Sharpe.

Inside the variance ceiling

Mešťan uses seven U.S.-listed ETFs: four equity funds (SPY, IJR, VGK, EEM) and three bond funds (LQD, TLT, BIL). The sample contains 887 weekly returns, covering January 2009 to December 2025. Each year, the rule estimates means and a covariance matrix using the trailing 156 weeks. It chooses the long-only portfolio with the highest estimated return, keeping estimated variance at or below equal weighting's and each ETF weight at or below 40%. The matrix receives a ridge term when needed. After that, weights drift for twelve months before the next estimation.

The return signal is simply the trailing three-year mean. Using the covariance matrix, the rule tilts toward past winners while staying within 1/N's risk budget. SPY frequently reaches the 40% cap, the paper reports, while some regional equity sleeves often receive zero weight. The practical trade is a slow tilt toward previous winners, mostly U.S. large caps, within a balanced risk budget.

The comparison set includes an eleven-point fixed grid running from 100/0 to 0/100, with equal weights inside each asset class. It also includes plain 1/N and inverse volatility with a 30% cap per ETF. There are two evaluations. A historical out-of-sample run covers 6 January 2012 to 31 December 2025 and charges 10bp per unit of one-way turnover. Then 1,000 moving-block bootstrap paths, built with 13-week blocks, are evaluated at 10, 20, 30 and 40 years.

Historically, Markowitz earns a 0.778 Sharpe and 7.87% certainty equivalent (CEQ, mean less a variance penalty with gamma of 3). Equal weighting records 0.554 and 5.53%; 60/40 records 0.558 and 5.67%. All-equity compounds faster: 10.08% versus Markowitz's 9.29%. Its drawdown is also deeper, 35.21% versus 26.89%. Across the bootstrap horizons, Markowitz leads on median Sharpe throughout, with values from 0.722 to 0.737.

Which simple portfolio deserves to be the hurdle?

With four equity funds and three bond funds, 1/N begins each year with four-sevenths, about 57%, in equity. Mešťan acknowledges the issue: fund count mechanically determines 1/N's equity allocation. Comparing every rule with the entire fixed grid is therefore the paper's best design decision.

The gains against the two paired benchmarks narrow as the benchmark takes more equity risk. The median paired CEQ advantage over 1/N is 0.64 to 0.69pp annually. Over 60/40, it is 0.49 to 0.55pp. Against 100/0, the median ranking reverses. All-equity produces median CEQ of 7.83%, 7.89% and 7.65% at 20, 30 and 40 years, exceeding Markowitz's 7.46 to 7.47%. At ten years, 90/10 and 100/0 tie at 7.745% and 7.742%.

I think the variance ceiling accounts for much of this. Median volatility for Markowitz runs from 10.72 to 10.76%; for 100/0, it runs from 17.97 to 18.06%. Gamma-3 utility rewards the larger equity allocation in this sample. The optimizer cannot take that route while remaining within 1/N's variance limit.

The paper acknowledges the ranking: high-equity fixed portfolios retain the highest median CEQ. The abstract confines its claim to risk-adjusted performance. The conclusion says optimization "can improve how risk is compensated, but it cannot determine how much risk the investor should bear." I agree with that framing. The question for the rest of this review is how much evidence supports the compensation claim.

An oddity remains. Markowitz's historical CEQ of 7.87% exceeds all-equity's 6.94%, yet the bootstrap puts all-equity ahead. Bootstrap 100/0 also compounds at 11.68% over ten years, compared with 10.08% in the backtest. The simulations draw on the full 2009 to 2025 record. That would include the 2009 to 2011 recovery, which the backtest never trades, and is consistent with the stronger equity draws.

A 0.64pp paired median gain, a 0.45pp gap between medians

Comparing the rows in Table 3 gives a ten-year CEQ gap of 7.27% minus 6.82%, or 0.45pp, between Markowitz and 1/N. The reported 0.64pp instead takes the median of the differences on each path. With common paths, that is the appropriate statistic. Readers still need to know that the figures differ. By 40 years they converge: 0.72 between medians and 0.69 paired.

The probability of improvement makes the more honest headline. Markowitz beats 1/N on CEQ in 67.3% of ten-year paths. One path in three favors the other side. The corresponding ten-year probability against 60/40 is 63.5%. Bootstrap drawdown differences are small: the median paired drawdown is 0.42 to 0.72pp worse than 1/N and 0.21 to 0.35pp better than 60/40. The historical drawdown gap is larger, at 26.89% versus 23.85% for 1/N.

Trading costs pose little threat to the finding. Markowitz's median annual turnover is 29.6 to 30.8%, compared with 4.1 to 4.5% for 1/N. Annual trading keeps the CAGR drag to about 3.2 to 3.4bp with 10bp costs, and 8.1 to 8.4bp with 25bp costs. The historical table does, however, report Markowitz turnover of 18.35. We did not find an explanation for the difference from the bootstrap's roughly 30%.

Forty simulated years still draw on seventeen observed ones

All bootstrap paths reuse the same 2009 to 2025 history. Within that setup, long-run average returns on each path converge toward the sample averages. A rule estimating means over 156 trailing weeks thus keeps pursuing a stationary target it repeatedly hits. The paper interprets the rise in win probability, from 67.3% at ten years to 82.6% at forty, as an "increasingly reliable risk-adjusted advantage as the horizon lengthens".

I read that rise as an expected consequence of the setup. Longer paths give the fixed sample means more room to overwhelm block noise. Any persistent difference in sample means could generate a win rate that increases with horizon, including a difference between two fixed allocations. The rising probability therefore says more about consistency with 2009 to 2025 than about the next four decades.

This also weakens the Holm-adjusted p below 0.001. A thousand paths resampled from a single history can make any persistent sample difference appear significant. The author acknowledges the record's length, describing it as "short relative to the simulated 40-year horizon". Other stated limitations include a single currency environment, the assumption that blocks inform future regimes, and use of a single 156-week window with gamma fixed at 3. These limitations leave unanswered how much 1,000 resampled paths can establish about significance.

We could not test this ourselves. We lack price history for LQD, TLT or BIL, and substituting other bond proxies would test a different rule.

I would change my view if the identical rule, using 156 weeks, a 40% cap and the 1/N variance ceiling, beat 60/40's CEQ outside 2009 to 2025 in a sample where SPY was not the fund pinned at the 40% cap.