Hsieh's invariance theorem carries the paper. His dominance result over fixed-fraction feedback is proved directly, while the attached SPY/TLT simulation offers thin empirical evidence. It resamples returns i.i.d. from within the fitted support and provides no tabulated values. The pathwise drawdown guarantee therefore covers the return set supplied by the modeller.
The construction begins with drawdown's path dependence. Hsieh augments the state with current account value and its running peak. Safe wealth remains at or above (1 minus d_max) times that peak. His central theorem gives the necessary and sufficient condition for an action to preserve the safe set under every return in an assumed compact support. The position's worst supported one-stage loss, including costs, can be no larger than the cushion between current wealth and the floor. In formal terms, the support function of the negated position plus the cost term must not exceed M(k)V(k). The modulator M(k) equals (d_max minus current drawdown) divided by (1 minus current drawdown).
Positive homogeneity then gives an exact factorization. Every safe action equals the cushion times a direction chosen from a fixed normalized set that is independent of wealth.
The safe geometry is scale-free.
With stagewise-independent returns, Hsieh reduces the two-dimensional wealth-and-peak problem to a scalar Bellman recursion in z = V/V_max over [1 minus d_max, 1]. A measurable maximizing selector exists, which establishes an optimal state-feedback policy. He also identifies the constant-gain linear policies satisfying the same limit: the gain's worst-case one-stage loss must obey h(K) at most 1 minus (1 minus d_max)^(1/N). The cushion policy weakly dominates that class. For horizons of at least two stages, dominance is strict whenever the linear gain has positive expected one-stage net return. This stronger result uses the paper's Section 5 assumption that the conditional mean is constant, E[X(k) | F_k] = mu almost surely.
How the controller takes risk
The control u(k) is a vector of dollar positions across m risky assets. Wealth follows V(k+1) = V(k) + u(k)'X(k) - eps'|u(k)|, with X(k) denoting the vector of stage-k asset returns and eps the per-period proportional cost rate. The portfolio holds long SPY and TLT exposure sized to the cushion, while costs apply to the absolute position.
Its benchmark is the fixed-fraction rule u(k) = K V(k), where exposures remain constant fractions of current wealth. This is the familiar Markowitz-style, Kelly-style object a trader already runs. Hsieh's cushion rule sizes exposure from V(k) minus (1 minus d_max) times the running peak. After a loss, that base is much smaller than current wealth.
The data comprises 2,910 daily joint return vectors for SPY and TLT, stated as January 2015 through July 2026, then resampled i.i.d. across stages. Figure 1 reports three horizons. At every evaluated positive d_max, the modulated policy has higher expected terminal return, though the paper gives no tabulated values.
Any construction of our own needs a caveat up front. Nobody has a valid known support for future returns. A backtest can only evaluate a conservative approximation of Hsieh's condition, using a rolling or stress-calibrated return set in place of the unknown support. The Bellman result also relies on stagewise independence and an estimated return distribution. Either assumption can fail in live trading. A flat proportional cost remains a placeholder because daily OHLCV provides no spreads or realized impact for calibration. Anything we run is a construction from Hsieh's idea, not a test of his guarantee.
The empirical section stays small: Two ETFs, SPY and TLT. Its support contains the 2,910 observed daily joint return vectors, all assigned equal probability, from the stated January 2015 through July 2026 window, which we could not verify. The linear benchmark uses the empirical mean vector from those same days. The horizons are 21, 63 and 252 trading days. Costs are 0.5 basis points per asset per stage. Drawdown limits range from 0 to 0.60 in steps of 0.05, with 300,000 independently resampled paths for each configuration.
Figure 1 places the modulated policy above the linear-programming-optimized linear benchmark at every positive evaluated drawdown limit. Yet the plotted horizontal axis ends at 50%, while the stated grid extends to 0.60. We did not find tabulated values anywhere in the text. From the three panels of Figure 1, the vertical axes reach roughly 5% (N = 21), 15% (N = 63) and 60% (N = 252).
The fixed fraction exhausts its loss budget
Apply Hsieh's linear-gain condition, h(K) at most 1 minus (1 minus d_max)^(1/N). With a 20% limit and N = 252, the permitted worst-case one-stage loss is 1 minus 0.8^(1/252), about 8.85 basis points of wealth. At the same limit and N = 21, it rises to about 1.06% of wealth, twelve times larger.
The cushion policy avoids this horizon compression. At a fresh peak, drawdown is zero and the modulator equals d_max. Its one-step budget is therefore a worst-case loss of 20% of wealth, and each new peak resets that budget.
No data was needed for the dominance result. The paper proves that the linear class sits inside drawdown modulation. In the proof of Theorem 5.3, any feasible LTI gain K can be replicated for a given d_max by setting gamma(k) = K/M(k) whenever M(k) is positive. The comparison puts a restricted form of the rule against the full rule.
At the floor, exposure vanishes
Exposure scales with the cushion. A loss near the limit leaves the policy trading at a small fraction of its peak size, so recovery must proceed from that reduced exposure. Exactly at the boundary, the cushion is zero and zero is the only safe action, given the optimality assumption that no nonzero position earns a nonnegative payoff under every supported return. Hsieh exposes the mechanism through the boundary case. At d_max = 0, both policies set u(k) = 0, leaving wealth at V_0 throughout the horizon.
Expected terminal wealth is the sole objective. We did not find variance, Sharpe or turnover for either policy. The mean consequently hides the outcome distribution and the frequency with which the overlay shrinks toward zero exposure.
Can the support survive a crash?
The theory states its boundary clearly. It assumes a known nonempty compact support with probability one. Hsieh calls the safety analysis distribution-free because the prescribed drawdown limit holds along every supported return path and hence with probability one. That probability-one statement inherits the support assumption. The abstract supplies its own qualifier: "against all supported returns."
Hsieh also gives the right answer within those terms. Conditional on the support, his condition is necessary and sufficient, so no conservatism is lost inside the box. The unresolved number is the box itself.
For the simulation, the support consists of 2,910 realized daily return vectors. Its support function is the support function of their convex hull, and the same observations estimate the mean. A single day outside that set removes the cap. Live use means widening the box to admit events such as a 20% down day in SPY or an overnight gap. A wider box contracts the normalized action set and cuts the position allowed for any given cushion. The paper sweeps d_max over {0, 0.05,..., 0.60} and plots expected terminal return against it, thereby pricing a tighter limit. We found no sensitivity analysis for either policy under a wider support set X.
One figure carries the empirical case
The simulation samples joint return vectors independently across stages. This strips out serial dependence and time-varying SPY/TLT correlation. Every simulated return also lies inside the support by construction, leaving the experiment unable to breach the guarantee even in principle.
The curves come from different calculations. For the linear benchmark, expected return has the closed form (1 + K'mu minus eps'|K|)^N minus 1. The modulated policy is obtained from a numerically approximated recursion and evaluated across 300,000 paths. We did not find Monte Carlo standard errors or the grid resolution. The paper also provides no comparison with other discrete-time drawdown-constrained policies, although Hsieh discusses Klass and Nowicki on the Grossman-Zhou strategy losing its optimality in discrete time, along with Hernández-Bustos and Hernández-Hernández on long-run growth under a drawdown constraint.
Hsieh does report the finding that most weakens the case for his added machinery. For N = 21 and N = 63, he writes that the numerically computed optimal policy uses an approximately constant gamma across stages and states. A constant-gamma cushion rule therefore already produces the improvement shown in those two panels. He says so plainly.
Our construction cannot test the guarantee
We built a version of the cushion rule, with strict limits on what the exercise can establish. Historical prices can provide only a rolling or stress-calibrated box, so our backtest evaluates a conservative approximation. It cannot verify the pathwise guarantee, which requires a valid known support for future returns. The Bellman result assumes stagewise independence and an estimated return distribution, and both may be misspecified in live trading. The 0.5 basis point flat cost remains a placeholder because daily OHLCV contains no spreads or realized impact for calibration. The universe contains two highly liquid ETFs. We found no position limit beyond the safety constraint itself.
One result would change my mind: a widened support fitted outside the test window that still leaves the cushion policy ahead of an optimized fixed fraction, net of turnover. The theorem remains exact either way. The guarantee's price is the open number.