A 0.161 Sharpe advantage across nine stocks and 260 trading days is the study's sole cell where a deep reinforcement learning agent beats the strongest classical benchmark. The wider sweep in Cluster 0 looks better for DRL at first: benchmark mean Sharpe is 1.26, and all four agents average above it. Table A.4 changes the picture. Against HRP, A2C records -0.408, PPO -0.339 and SAC -0.371, each significant under Jobson-Korkie at p<0.05. Three of the four agents lose outright to HRP in the cluster meant to support the DRL case. Sahu and Bhandari acknowledge this in the conclusion, describing the rankings as directionally consistent with their hypothesis and "statistically inconclusive." The paper's more interesting contribution is the conditioning variable.
The mechanism has two stages. The first measures how far each stock's return process departs from a martingale. The authors take 45 Nifty 50 constituents with uninterrupted daily data from 2010 to 2026, then fit a fuzzy autoregression to daily log returns using a rolling 126-day window and BIC-selected lag. Every AR coefficient is a triangular fuzzy number, an interval with a peak rather than a point estimate. The Fuzzy Market Inefficiency Measure (FMIM) applies the usual ratio of summed absolute coefficients to one plus that sum to those fuzzy coefficients. A high FMIM means past returns contain more information about future returns.
Twelve descriptors compress each stock's FMIM path: mean, median, standard deviation, p10, p90, IQR, skewness, kurtosis, the annualised OLS trend slope and its R2. The remaining two are the COVID-window peak and the COVID-to-full-sample mean ratio. After standardisation, a three-component Gaussian mixture clusters them over 50 restarts. It labels 9 firms Most Efficient, with average FMIM 0.0710; 27 Moderate, at 0.0844; and 9 Least Efficient, at 0.0944.
We could not trade the Nifty 50 names ourselves. Our run uses a US large-cap universe, adapts the idea and does not test the paper's claim.
The second stage is a controlled horse race within each bucket. Four Stable-Baselines3 agents, PPO, A2C, SAC and DDPG, trade through a Gymnasium portfolio environment. Every asset has nine features: 1-, 5- and 20-day returns, 20-day volatility, RSI, MACD, log P/E, log P/B and a market-wide FinBERT sentiment score. The agents use a 10-day lookback. Softmax weights have a 0.40 cap per name, while turnover incurs 0.1% one-way on the L1 weight change.
Three rewards are tested: log cumulative return, rolling annualised Sharpe, and Sharpe minus half the absolute 5% CVaR. Six classical benchmarks run beside them: equal weight, buy and hold, MVO max-Sharpe, MVO minimum variance, HRP, and Black-Litterman with sentiment views. Both MVO variants use Ledoit-Wolf shrinkage, a 252-day window and 21-day rebalancing. Five rebalancing benchmarks pay the same 0.1% one-way cost. Buy and hold pays none. Training ends at end-2022, validation spans 2023 to 2024, and testing begins in 2025 and covers 260 trading days.
Where should the money come from? The authors answer directly: "We therefore do not interpret DRL outperformance as evidence of alpha generation or return prediction." Any DDPG advantage is attributed to daily reweighting as volatility and covariance shift. The tradeable proposition is therefore a meta-rule that selects an allocator according to the universe's informational regime.
One narrow win
Within the nine-stock efficient cluster, Sharpe-reward DDPG produces 34.72% annualised, 76.84% cumulative, Sharpe 1.7523, volatility 13.98% and maximum drawdown -15.07%. HRP follows with Sharpe 1.591 and 30.80% annualised. The full row is less clean. Cluster 0 gives DRL the Sharpe, Sortino and return wins, while HRP takes Calmar, 2.3562 to 2.3037, and drawdown, -13.07% to -15.07%.
Classical allocators lead elsewhere. In the 27-stock moderate cluster, MVO minimum variance has the best annualised return at 26.37%, Sharpe at 1.5361, Sortino at 2.75 and Calmar at 2.1607. A2C supplies the strongest DRL Sharpe, 1.3576. A minimum-variance objective producing the cluster's highest raw return deserves another look. HRP tops the least efficient cluster with Sharpe 1.2561 and a -12.01% drawdown. Its best DRL rival reaches 0.8954, below the benchmark mean of 0.90. For the twelve algorithm-cluster combinations, the paper reports that "only one achieves DRL superiority on Sharpe ratio, a success rate of 8.3%."
DRL does claim a genuine tail result. In the least efficient cluster, SAC reduces maximum drawdown by 12.8% and A2C by 7.9%. SAC gives up 0.51 Sharpe to HRP in exchange. The classical extremes in the same cluster are severe: MVO max-Sharpe earns 41.88% annualised with 48.23% volatility and a -36.83% drawdown, while Black-Litterman reaches 57.44% volatility for -41.26%. Allocator choice matters most when prices are noisiest, and this spread shows how much.
Inference weakens the headline. Jobson-Korkie rejects Sharpe equality for most pairs, yet every 95% bootstrap interval spans roughly [-2.0, +2.0] and crosses zero. The +0.161 DDPG-versus-HRP gap is marked insignificant. The authors disclose that cost constraints limited significance testing to the Sharpe-reward specification. Those tested Sharpes differ from the best-performing configurations in the main results table. They argue that the 260-day window creates the wide intervals and that "the overall ranking of the strategies doesn't change." Readers find that caveat in the conclusion and limitations section. The abstract omits it and states that "deep learning models perform best in highly efficient markets where signals are weak but consistent."
The sensitivity baseline changes the picture
The ranking defence carries much of the argument, which makes the hyperparameter check central. The authors write, "Importantly, the main conclusion of the study remains unchanged across all tested settings", followed by "DDPG continues to achieve the best risk adjusted performance in the most efficient cluster". Table A.3 gives a different level comparison. Across three learning-rate settings, DDPG's efficient-cluster Sharpes are 0.7256, 0.7465 and 1.0084. Every result falls below HRP's 1.591. The 0.7465 baseline is also the weakest baseline among the four agents in that cluster, behind PPO at 1.0782, A2C at 1.1116 and SAC at 1.0888.
SAC and DDPG received fewer training steps in this exercise, according to the paper, so these figures should be interpreted as relative rankings rather than converged levels. The caveat is fair. As a level comparison, however, Table A.3 does not display the configuration supporting the headline. We encountered a similar shape in our earlier article on Wasserstein allocation, where the headline Sharpe was one point selected from a small grid. This result comes from one seed, and we could find no seed-variance results.
Can FMIM separate tradeable regimes?
Mean FMIM for the clusters is 0.0710, 0.0844 and 0.0944. The entire spread is 0.023 on a measure bounded at 1, while the paper's own scale classifies anything below 0.5 as moderate inefficiency. GMM silhouette is 0.1952. Agreement with K-Means reaches 60%. After PCA reduction to seven components covering 95.1% of variance, only 51.1% of firms retain their full-sample labels.
Collinearity is substantial. The paper reports that "six level and distributional shape features such as: mean, median, standard deviation, p90, skewness, and kurtosis of FMIM are exhibited VIF values exceeding 10", and mean FMIM reaches 147.6. In effect, the twelve features repeatedly view one collinear efficiency dimension. The authors discuss this openly, arguing that GMM uses covariance structure and that diffuse boundaries support soft membership.
The proposed live rule exposes the weak separation. FMIM is recomputed over a 252-day window and reviewed every 21 days. Hard thresholds then route universes below 0.075 to DDPG, from 0.075 to 0.095 to HRP or MVO minimum variance, and above 0.095 to HRP. These cutoffs lie between in-sample cluster means separated by 0.013 and 0.010. The reported within-cluster FMIM standard deviations are 0.0653, 0.0688 and 0.0785, leaving threshold gaps of roughly a fifth of a within-cluster standard deviation.
The classification itself is cross-sectional, a taxonomy of stocks. The live rule asks that taxonomy to operate as a time-series regime switch. We did not find a walk-forward backtest of the paper's four-step framework; the authors present it as a way to operationalise the findings.
Ex ante labels move
Primary cluster assignments use full-sample FMIM data, including the test period, as the authors state. Their pre-2023 clustering check agrees for 71.1% of firms and has silhouette 0.1625. Roughly 13 of 45 stocks would receive different labels under an ex ante process. A handful of changes can materially reshape the two nine-stock clusters carrying the result. We did not find a portfolio comparison rerun with those pre-2023 labels.
On survivorship, the authors say the fixed sample "may introduce some degree of survivorship bias". The underlying setup selects a 45-firm universe for uninterrupted data from 2010 to 2026. Agents rebalance daily and pay a flat 10bp one-way cost, with no market impact or slippage model.
Our experiment used another market
We could not reconstruct the paper's market. The 45 Nifty 50 series, Refinitiv fundamentals and India-specific news history are unavailable to us. Our news tables begin around 2020 rather than 2010, so our own code scored sentiment instead of reproducing the authors' FinBERT news pipeline. The fuzzy fitting objective and its magnitude also remain open in the method, which defers to Askari and Hajizadeh: "For further details on the estimation of the FMIM, refer Askari & Hajizadeh". Training seeds are unreported.
All remaining choices are ours. We did not implement DDPG, the rolling-Sharpe reward, the Sharpe-CVaR reward or daily rebalancing. The following results are a US-equity adaptation of the idea and do not verify the paper's figures.
Our universe was the point-in-time top 200 US large caps, held long only and fully invested, with weekly rebalancing from January 2020 to July 2024. We estimated FMIM through a 126-day rolling fuzzy/interval AR whose lags 1 to 10 were selected by BIC. A moving-block bootstrap with 200 replications produced the coefficient intervals. An expanding point-in-time three-component GMM relabelled regimes weekly. The allocator was a Ridge actor surrogate, retrained quarterly on cumulative log return. Costs were 10bp on the full L1 weight change plus $0.004 a share with a $1 minimum. We modelled no slippage and made 13,510 trades.
The run delivered 5.15% CAGR, Sharpe 0.78, realised volatility 7.19%, maximum drawdown -11.72% and beta 0.09 to SPY. For its efficient-cluster DDPG portfolio, the paper reports 34.72% annualised, Sharpe 1.7523 and volatility 13.98%, net of its own 0.1% one-way cost over 2025 to 2026. A direct comparison between the two sets of figures would be invalid. The country and universe differ. Decisions are weekly rather than daily, and our linear surrogate replaces a continuous-action agent. The sample periods do not overlap. Our result is an unconditional all-regime portfolio, while theirs selects the single best of ten strategies by three clusters by three rewards.
Our realised volatility is roughly half the paper's, and beta of 0.09 leaves the portfolio barely exposed to the market. Much of the return gap can arise from sizing and breadth, with softmax weights distributed across 200 names, before signal quality enters. Our period also includes the 2020 crash and 2022. The paper's 260-day test excludes both.
This was one automated pass. A weak outcome primarily reflects our implementation. The Ridge surrogate cannot create the continuous-action reweighting that the authors credit for their advantage. Weekly rebalancing also removes the daily response to volatility and covariance changes, the paper's stated mechanism. The available evidence does not let us fully explain the size of the gap.
The paper's direction remains an interesting hypothesis: adaptive agents fare better where prices are most efficient, while hierarchical risk parity fares better where they are least efficient. That result cuts against the habit of aiming machine learning at the market's noisiest corner. A convincing test would repeat the comparison across 500 to 1,000 test days. Labels would need to be formed ex ante and then used in the portfolio comparison itself, going beyond the current assignment check. The 0.075 and 0.095 thresholds would also need to be fixed before evaluation begins, since the paper currently reads them from the evaluation window's cluster means.
Our backtest stops at 2024-07-01, and everything after that date is deliberately left untouched so the same strategy can be checked out of sample later.