On US equities, the 126-day window beat both adaptive blends, even though Liu and Hsieh correctly price a cost those blends can incur. Their equity tables show Fixed Share trailing plain Hedge in both samples. The paper's accounting result deserves attention; its tracking algorithm has a harder case to make.
How the windows become a portfolio
How much history should a rolling mean-variance portfolio use? Liu and Hsieh let the allocator choose among windows instead of settling on one. Each candidate window j produces a long-only Markowitz portfolio from the last j daily returns. A 10 bp L1 penalty discourages moves away from yesterday's holdings. The paper calls these portfolios its "experts".
A second layer assigns the experts weights q(t). It updates them after each day using exponential weighting on each expert's loss: negative PnL plus that expert's turnover cost. Hedge makes that update and stops. Fixed Share also moves a fraction α of the weight toward uniform, keeping every window in contention as it tracks an expert that may change over time.
That ability to change is the intended edge. The synthetic test gives returns a changing memory: 32 simulated stocks over 2,000 business days, driven by a 63-day moving average until day 1,000 and a 5-day average afterward. The real-data tests use eight windows spanning 5 to 126 days. They cover 30 DJIA stocks from 2021/03/04 to 2024/02/25 and 481 S&P 500 stocks from 2020/01/02 to 2026/08/01. Both include the BIL T-bill ETF as cash and charge 10 bps.
A cost the expert losses miss
Weight changes can make the traded book turn over while its component portfolios stay still. Move weight between two stationary window portfolios and the allocator has made a trade. Theorem 3.1 adds that cost to the usual expert-loss regret: its bound includes the sum of c(t)‖q(t) - q(t-1)‖₁. A two-expert example, with each expert holding a single asset, reaches regret of exactly 2c(2). The coefficient of 1 on the extra term therefore cannot be improved. Anyone trading a meta-allocator over sub-strategies should account for it.
Lemma 3.3 limits daily weight movement to η times that day's loss range plus 2α(1 - 1/n). The latter allowance comes from mixing. In Corollary 3.4, it can add up to 2α(1 - 1/n) per unit of cost rate each day, even without a market shift. Actual cost can be lower; it vanishes when the weights are already uniform.
What does the guarantee buy?
The comparison portfolio follows the best expert sequence with at most K switches. As the paper states, each segment pays only its selected expert's own turnover. The benchmark can therefore jump from the 5-day book to the 126-day book without paying for that switch. Its "no tracking regret" claim uses that comparison.
The tuned guarantee also relies on information unavailable in live trading. Proposition 3.5 chooses optimal η and α using c(t) across the whole horizon and a switching budget set beforehand. For the equity runs, the authors use K(T) = ⌊T / (log T (log log T)^(1+ε))⌋ because it grows more slowly than T. The static Hedge bound is 2√(A_T log n); A_T adds the squared daily loss ranges and cost terms across the horizon. Sublinear regret means the average gap to the benchmark eventually shrinks. The static bound limits how far the blend can fall behind the best single window. It offers no promise that the blend will beat that window, and neither equity sample shows it doing so.
The equity numbers
On the DJIA, Hedge returned 41.6977% with a -12.6326% maximum drawdown (MDD). Fixed Share returned 38.1074% with -12.9282%. UP earned 48.9870% (MDD -20.3914%) and EG 48.5459% (MDD -20.3837%). The authors disclose that these baselines retain settings "originally derived under frictionless assumptions", leaving UP and EG untuned at 10 bps. The main text acknowledges their higher returns and presents the result as a return and drawdown trade-off. The appendix says the framework "progressively accumulates a higher account value" on the DJIA without identifying the comparison. Against UP (48.9870%) and EG (48.5459%), its account value is lower. On the S&P 500, Hedge's 578.6997% exceeds UP's 205.2483% and EG's 205.7900%, alongside a -56.0818% drawdown versus UP's -38.3269%.
The strongest return comparison comes from the windows already inside the allocator. On the DJIA, the 126-day expert earned 157.8256% with a -22.2768% drawdown; the 5-day expert earned 109.8576%. On the S&P 500, Hedge returned 578.6997% (MDD -56.0818%) and Fixed Share 471.0522% (MDD -57.8097%). The 126-day expert returned 1881.7486% (MDD -60.4582%), and the 41-day expert returned 1173.9300%. In both samples, the 126-day expert had a deeper drawdown than Hedge.
Fixed Share trailed Hedge by 3.59 points on the DJIA and 107.65 on the S&P 500.
The reported weights help explain the blends' lower returns. The authors write that "the weight distribution remains largely uniform, with the proportion assigned to j = 126 being marginally higher than that of the others." Such a blend spreads exposure across windows, and the volatility figures reflect that. DJIA daily std was 0.0115 for Hedge and 0.0117 for Fixed Share, versus 0.0140 to 0.0174 for individual windows. On the S&P 500, the blends recorded 0.0364 and 0.0372, versus 0.0440 to 0.0476. We did not find a fixed equal-weight blend among the baselines. That comparison would show more clearly what the learning step contributed.
The Sharpe column raises another question. DJIA Hedge has a reported Sharpe of -2.4214, although its daily mean is 0.0533% and its std is 0.0115. Those figures imply a raw annualized ratio near 0.74. Getting a negative value would require a daily deduction above 0.0533%, roughly 13% a year. We did not find a Sharpe definition in the paper.
Stock selection across the full period
Both real-data tests combine US single stocks with BIL. We did not run our own test: a fair one requires daily histories for hundreds of single stocks.
For the S&P universe, the authors retain 481 names after they "remove all the changed stocks" over 2020 to 2026. That full-period selection introduces survivorship into the backtest. The published list includes GEV and VLTO, neither of which existed as a listed stock at the 2020/01/02 start. With 481 stocks, a 5-day covariance has rank at most 4. The 5-day expert's risk penalty is consequently flat in almost every direction, leaving the long-only constraint and cost term to do the work.
The paper describes its empirical studies as an illustration of why window size and transaction costs need modelling, and disclaims forecasts for specific instruments. Theorem 3.1 and its sharp two-expert example make the case for charging weight turnover. The equity runs do not isolate the benefit of doing so. A real-market sample in which Fixed Share beats Hedge after paying its own mixing turnover would change our view of the tracking algorithm.