Without 2024-2025 returns in their views, the paper's λ=0 portfolios earned net Sharpe of 0.019 for Markowitz and -0.021 for the LSTM, averaged across 100 seeds, over those two years. Even that comparison uses a universe screened for survival through December 2025. All remaining rows draw on knowledge of the returns in 2024 and 2025. The authors disclose this themselves.
What Kumar, Dey, Iyer and Juneja built
The starting point is Black-Litterman and the dependence of a max-Sharpe portfolio on its mean forecast. The paper expresses an estimated tangency portfolio's realized Sharpe as the best attainable Sharpe multiplied by a cosine. The angle is between the estimated and true optimal weight directions. The authors call a forecast on the straight line from the current estimate to the true mean a "directionally correct" view: (1-λ) times the historical mean plus λ times the truth. Under a shared covariance and with short sales permitted, Theorem 2 establishes that Sharpe cannot decline as λ increases. The curve flattens at λ=1.
The test uses daily NSE returns for the 20 largest firms by market cap at 31 December 2023. Eligible stocks must have prices on all 4,958 trading dates from January 2006 through December 2025. The authors estimate the historical mean over 2006-2023. Their "true" mean comes from realized returns over 2024-2025, the same window used for evaluation. They construct views on the grid λ = 0, 0.1,..., 1. An exponential tilt then reweights the historical daily return vectors, minimizing KL divergence subject to matching the view's mean.
Both allocators use this tilted sample. Markowitz estimates its mean and covariance, then solves for a long-only, fully invested max-Sharpe portfolio and maintains constant target weights. The end-to-end alternative is a one-layer, 16-unit LSTM with softmax output. It trains directly on a Sharpe ratio calculated with the same tilt weights, using 100 seeds per λ. Trading costs are 20 bps one-way on value traded. Markowitz earns net Sharpe of 0.019 at λ=0, rising to 1.774 at λ=0.5 and 1.936 at λ=1. Max drawdown improves from -25.0% to -13.8%. Averaged across seeds, the LSTM comes close, with 1.691 at λ=0.5 and 1.912 at λ=1.
How much foresight is "a small amount"?
The paper argues that "even a small amount of directionally correct information can materially improve investment decisions," and the results discussion places much of the improvement at small λ. Its abstract limits the claim to "a simple Markowitz setting". The authors ground it in Theorem 2 rather than relying solely on the long-only table.
Yet the table gives a different impression. Markowitz nets 0.204 at λ=0.1, roughly a tenth of the 1.936 ceiling, and 0.638 at 0.2. Most of the improvement comes between 0.2 and 0.5, from 0.638 to 1.774. Beyond 0.6, where Sharpe reaches 1.877, the curve is flat. The lower end is convex. Figure 1b, the paper's theoretical curve for these same 20 stocks, shows that shape too; the authors attribute its inflection to A+3B<0. For these stocks, then, the concavity needed for small λ to pay fails even in the simple setting named in the abstract.
More consequential is the interpretation of λ. Setting λ to 0.1 reduces the correction's size while preserving its exact direction. Each grid view aims straight from the 2006-2023 mean toward the realized 2024-2025 mean. At λ=1, both portfolios concentrate in BHARTIARTL, MARUTI and SUNPHARMA. The direction is exactly right at every grid point; no view in the experiment points the wrong way. As an interpolation weight, λ says nothing about the forecasting skill required to achieve, for example, the 1.027 at λ=0.3.
Markowitz leads the 16-unit LSTM after costs at all 11 view strengths
The authors acknowledge the ranking: "After costs, the LSTM performs slightly worse than Markowitz, particularly at intermediate λ, consistent with its greater turnover." The abstract nevertheless describes Markowitz as performing "as well as" the end-to-end model. The results discussion calls the curves "strikingly similar". At every grid point, the table supports the concession, both gross and net.
The net difference is 0.024 at λ=1, with 1.936 versus 1.912. It is largest at λ=0.4, where the comparison is 1.426 versus 1.235. At λ=0.3, the LSTM earns 0.885 net against 1.027 for Markowitz. It lags before costs too: 0.959 against 1.059. Turnover thus accounts for only about 0.042 of the 0.142 net difference at λ=0.3, weakening the authors' explanation. At λ=0, the LSTM's net Sharpe is -0.021 against 0.019. Daily LSTM turnover reaches 0.0200 at λ=0.2, compared with 0.0082 for Markowitz. By λ=1, the comparison narrows to 0.0074 versus 0.0066, as the LSTM allocation averaged across seeds collapses toward the constant Markowitz weights.
The experiment therefore shows a small LSTM rediscovering the static long-only max-Sharpe portfolio when trained on the tilted sample. The finding is about the tilt. Only at λ=0 does the network run without an oracle, and both approaches earn about zero there (-0.021 vs 0.019). The paper consequently cannot establish that either finds a forward signal unaided.
Even that narrower finding has two weaknesses. The reported LSTM results average 100 initializations. Averaging Sharpe across seeds preserves the mean, but averaging 100 seeds' weights makes convergence look smoother. Figure 3 therefore reveals little about the allocation of any individual network. The architecture was also "chosen after exploratory experiments"; we did not find a validation window for those experiments described separately from 2024-2025.
Theorem 2 uses unconstrained weights; the backtest stays long-only
Theorem 2 gives forecast, view and truth the same Σ and assumes unconstrained tangency weights. In the experiment, weights are long-only, and the tilt changes Σ_λ as well as the mean. The authors address this with Theorem 3. It bounds relative Sharpe loss with a term quadratic in the mean and covariance errors. The true mean must have norm at least m, and Σ minus mI must be positive definite. Both the mean error and the covariance error, measured in operator norm, must remain below m/2. The result bounds distance from the optimum; it establishes no monotonicity and does not address long-only portfolios.
The observed long-only Sharpe curve rises monotonically from 0.019 to 1.936, without a theorem in the paper guaranteeing that rise. The proof of Theorem 2 also closes by claiming concavity throughout [0,1] for any non-collinear view. This contradicts both the stated A+3B≥0 condition and Figure 1b.
Twenty large caps and an oracle
Costs make little difference here. Charging 20 bps at λ=0 reduces Markowitz Sharpe from 0.050 gross to 0.019 net. At λ=1, the reduction is from 1.957 to 1.936. The authors describe this assumption as conservative relative to India's 10 bps securities transaction tax per side. A constant-weight, long-only portfolio of 20 NSE large caps trades infrequently, although the printed list contains 19 names and a blank. Markowitz daily turnover ranges from 0.0066 to 0.0082 across λ. The requirement for prices through December 2025 introduces survivorship look-ahead.
An out-of-sample version would need a forecaster that remains close to the realized-mean direction across this cross-section for two years. The paper supplies no such forecaster and acknowledges the gap: "a natural next step is producing such views in practice, since our experiments constructed them using future information unavailable at decision time."
We did not backtest it. Our platform lacks the NSE universe and its market caps, and our history begins around 2010. Substituting US equities would make it a separate experiment. A view generator is the missing piece, so any test we ran would evaluate that forecaster rather than these oracle results.
The paper provides a clean sensitivity study of a perfectly aimed mean forecast.
The two rows without foresight earn 0.019 and -0.021.