Bansal's strongest rebalancing claim fails a check against the figures beside it. Monthly calendar rebalancing earns a reported mean Sortino of 9.62, yet the accompanying 5.9% mean and 0.91% downside deviation divide to 6.48. The author describes the simulated downside portfolio openly and acknowledges its limits. Even so, the row carrying the paper's best result cannot support that result with its own numbers.
Building the portfolio
The long-only portfolio minimizes downside semivariance, the mean squared shortfall below a 0% threshold. It must meet an expected-return floor and remain fully invested. SLSQP selects the weights, using 10 random restarts.
The simulation uses no return series. The paper assumes drifts, vols and correlations, describing them as historical estimates without identifying a sample period. Each asset follows a correlated geometric Brownian motion. There are 20,000 one-year paths and a 4% risk-free rate. The 6.76% return floor equals 4% plus half the gap to the simulated equity mean of 9.53%. That floor remains unchanged through the first four stages. According to the paper, every reported Sortino uses the threshold τ = 0.
The construction has five stages. It starts with four assets: equity, bonds, gold, and a VIX-like instrument whose drift is -5%, vol is 80% and correlation to equity is -0.70. The resulting Sortino is 2.04; the SPY-like asset scores 1.20.
Stage 2 introduces two puts written on the entire portfolio: an American put with a 92% strike and an arithmetic-average Asian put struck at 96%. Their combined premium is 0.81%. With both in place, the Sortino rises to 2.26 and maximum simulated loss shrinks from -23.7% to -8.8%.
In Stage 3, ten large-cap proxies replace the single equity holding, bringing the Sortino to 5.85. Stage 4 examines a covered call and three rebalancing rules. Stage 5 then constructs a 21-asset portfolio targeting 15%.
Bansal acknowledges the limits of this setup. GBM, the paper says, "understates tail risk relative to the true downside distribution." Every option is also priced at frictionless fair value by Monte Carlo.
We could not run this on our data. The puts reference the custom portfolio itself, with one referencing its arithmetic average; listed ETF options pay on a different underlying. Nor is there a tradable twin for the VIX-like holding. VX futures roll along a term structure absent from the GBM asset. Either substitution would therefore test a different hedge.
Why does the Sortino jump from 2.04 with four assets to 5.85 with thirteen?
Both portfolios have a 6.8% mean. Replacing the equity sleeve with ten names nevertheless cuts downside deviation from 3.31% to 1.16%. Starting from SPY's 1.20, this step accounts for 4.65 of the 8.42 points gained on the way to 9.62. It supplies more than half the total improvement.
Bansal makes the same attribution. His conclusion identifies the expansion from 4 to 13 assets as the study's largest Sortino increase. Section 4.3.2 assigns the gain to lower downside covariance under the assumed correlations. The paper thus concedes the role of the correlation inputs, for which it supplies only a 0.55 to 0.75 range of market loadings.
The ten names have drifts of 9 to 12% and vols of 17 to 25%. A three-factor model supplies their correlations. Whatever variance the market, rates and volatility factors leave unexplained is idiosyncratic, allowing the optimizer to average it away. Those inputs produce the fall from 3.31% to 1.16% with the mean unchanged at 6.8%, as the paper itself explains. Even naive 1/N across 13 assets earns 2.63, beating the optimized four-asset portfolio.
An implementer still needs a loading for each name. We found only ranges. We also found no statement that scoring uses paths different from those used by the optimizer. If the paths are shared, the same sample both fits and grades 13 weights, making part of the 5.85 an in-sample fit.
The drift convention also needs to be settled. Equation 2 specifies standard GBM with drift µ, whose expected simple return is e^µ - 1. For µ = 9%, that return is 9.42%; for 12%, it is 12.75%. The paper ascribes its 9.53% and roughly 12.75% to sampling variation. At 18% vol across 20,000 paths, the mean's standard error is about 0.14%. The observed 9.53% is therefore close to 9.42% and well away from 9%. This convention carries through to the 6.76% floor.
The calendar row cannot carry 9.62
Every 21 days, calendar rebalancing returns the portfolio to the optimizer's weights. Across 1,000 trials of 500 paths, its reported mean Sortino is 9.62. It beats buy-and-hold, at 5.81, in every trial.
But the same table gives it a 5.9% mean and 0.91% downside deviation, which divide to 6.48. All three outcome cells, 5.9%, 0.91% and -12.2% max loss, duplicate the covered-call row.
Averaging trial-level ratios can give a different answer from dividing pooled moments. Denominators estimated from 500-path trials are noisy, so a gap is possible. The adjacent rows provide a measure of its size here. In Table 8, buy-and-hold has 6.8% over 1.16%, yielding 5.86, versus a trial mean of 5.81. The difference is 0.05. For the regime row, a 6.6% mean divided by 0.78% downside deviation gives 8.46, compared with the reported 8.58 (8.49 in the summary table).
Only calendar rebalancing has a discrepancy of 3.14 points. The paper supplies no per-trial distribution, and its only pooled figures for the rebalanced portfolio are those yielding 6.48.
Until that distribution appears, 9.62 remains unsupported.
The summary table creates further confusion. Regime rebalancing appears twice, at 8.49 and 8.58, while calendar rebalancing has no row. Momentum is listed at 4.21, versus 4.29 in the text. The code-availability statement reads "Not applicable."
The explanation for the ranking is sensible. Under stationary GBM, the regime trigger activates 6.7 times a year. The paper attributes these triggers to sampling noise rather than genuine regime shifts. Bansal explains the level of performance through periodic rebalancing: it prevents the portfolio from wandering away from the semivariance-minimizing weights w*. Drift control is real. It cannot account for a calendar row that misses its own arithmetic by 3.14 points when the base and regime rows differ by 0.05 and 0.12.
Early exercise leaves the floor unresolved
The floor K - p·e^{rT} follows path by path from the terminal put payoff. It holds for a European put.
The American leg raises a different question. A benchmark test gives 4.23 from the Longstaff-Schwartz pricer, against 4.03 for the Black-Scholes European put. Within the overlay, the 92% put costs 0.45% of the portfolio. Early exercise occurs on 6.9% of paths, averaging day 188 of 252. We found no rule specifying the portfolio's holdings after exercise. The proceeds could be reinvested, or the put could be rolled.
An implementer must choose between two readings. The simulation may charge the American premium while recording a European payoff at maturity. Alternatively, it may honour early exercise, leaving the -8.5% floor soft on the 6.9% of paths that exit before day 252. Our arithmetic favours the first reading. The American-only max loss of -8.5% equals the strike less the grown premium: 92 minus 0.45 × 1.04, about 91.5. The combined -8.8% likewise matches 92 minus 0.81 × 1.04. The text also describes the 92% strike as "10% OTM".
It is 8% out of the money.
The target pays for the protection
Stage 5 brings in eight growth assets with drifts of 13 to 22%. The portfolio targeting 15% records 3.70% semiDD, the downside-deviation measure, a Sortino of 4.06 and a -28.8% max loss. HiYield accounts for 36.6% and REIT for 30.6%. Another 9.5% remains in the vol instrument.
Each overlay reduces the Sortino. The American put alone takes it to 3.59, with max loss of -11.5%; the Asian put alone gives 3.86 and -24.6%. Buying both puts leaves a 12.1% mean, a Sortino of 3.24 and max loss of -12.6%.
The 112% covered call earns 3.39% of income. Adding it takes the mean to 7.4%, the Sortino to 3.23 and max loss to -9.1%. In effect, the call exchanges 4.7 points of mean for 3.5 points of worst-case improvement. Downside efficiency barely changes.
Bansal fairly describes the full overlay as a tradeoff. He says the covered call entails "giving up the 15% return objective". Two other passages fit poorly with that account. The prose says that without the call the "Sortino stays lower (3.24)", although 3.24 is the higher value. The overlay table also claims the call "raises mean return."
The paper's own frontier offers a cleaner comparison. Unhedged, its 7% portfolio has a Sortino of 5.76 and max loss of -13.3%. The overlaid 15% portfolio finishes with a 7.4% mean, Sortino 3.23 and -9.1%. For an investor ending up near 7% anyway, the exchange is 2.5 Sortino points surrendered for 4.2 points less worst-case loss.
We would choose the 7% frontier portfolio over the overlaid book. Re-estimating the frontier on paths held out from the optimizer could change our view.