Our review of the paperwe backtested it

A drawdown floor turns the Kelly fraction into a state variable

Adapting the absorbing-boundary sizing map to daily US ETFs, and where it saturates

2026-08-04 · 4 min read · US ETFs and liquid US equities using daily price data

Read the paper on arxiv

Our backtest of this idea

Our automated quick test, not the paper's

Boundary-Aware Kelly First-Passage Position Sizing for Liquid US ETFs

Backtest period 2020-01-01 to 2025-10-08 · hypothetical, net of modelled costs

Why these figures are not the paper's (3)

The paper reports no results of its own

This is a theoretical paper — derivations and proofs, with no measurement on market data. The backtest below is a strategy we built from its idea, not a test of anything the authors claimed.

Run on a different market than the paper

The paper does not study a real market instrument; it analyzes a stylized binary multiplicative return process. A practical backtest would adapt the mechanism to daily US ETF or equity returns by estimating an up/down return distribution and applying the same absorbing-boundary position-sizing logic. The substitution is acceptable because the core mechanism is path-dependent wealth evolution relative to a continuation floor, not a market-specific feature such as futures carry, order-book dynamics, or term structure.

The paper's own figures describe its universe and do not carry over to ours.

Our own audit found this run does not follow the paper faithfully (4)

The exposure grid max 0.3 and Kelly reference f_K=0.10 are tagged from_paper but calibrated for the paper's unit-scale return magnitudes a=b=1; applied to daily ETF returns where a_hat,b_hat≈0.01, the symmetric Kelly fraction is f_K=(2p_hat-1)/a_hat≈(2*0.52-1)/0.01=4.0, roughly 40x the grid ceiling of 0.3, so the constrained optimizer saturates at 0.3 (then the 0.1 position cap) for every positive-edge ETF and the boundary-compression content the strategy claims to trade is deleted by a scale-specific constant.
Time-base decoupling (category 8): a_hat and b_hat are per-day return magnitudes (R_t=close_t/close_{t-1}-1, 252-day lookback), but the lattice horizon T=50 is carried verbatim from the paper's abstract period count while the rebalance is weekly, so first-passage risk is propagated over 50 daily-scale steps (~10 weeks) yet the position is only held ~5 days before recomputation — a factor of ~10 between the horizon that governs the absorption term and the actual holding period.
Replaced defining construct: the paper's d=log(W0/L), L, S describe a single multiplicative wealth process, but the spec computes each ETF's f_star on its own single-asset binary lattice against the shared PORTFOLIO trailing-wealth floor L_t (0.8*peak); with each ETF capped at 0.1 across 14 names, no single-ETF lattice trajectory equals the actual portfolio wealth path, so the boundary/absorption term entering each f_star does not describe the portfolio-level absorption event it is claimed to model.
The inactive-boundary condition d>-T*log(1-f*a) instantiated at the spec's own scale (a_hat≈0.01, f≈0.3, T=50) gives threshold d≈0.15, while the trailing floor at 0.8 of peak puts d_t=log(W0/L_t) near log(1/0.8)=0.22 whenever wealth is at its peak; combined with grid saturation this means the boundary mechanism is dynamically irrelevant except in a narrow drawdown band, so the paper's headline compression rarely arises under the chosen parameters.

These are our findings about our own implementation, not criticisms of the paper. Read the figures below as a description of what we ran.

Jan 2020Total 16.4%Oct 2025

Sharpe: 0.68
Total Return: 16.4%
Max Drawdown: -8.3%
CAGR: 2.7%
Volatility: 4.2%
Trades: 2,950

A finite continuation floor makes the growth-optimal exposure depend on where you are, not just on your edge. Any risk manager running against a real liquidation threshold (margin, a mandate drawdown, fund closure) already half-believes it: you should not size to the same fraction near the floor as you do far above it.

The paper makes no empirical claim. There is no traded instrument, no sample, no Sharpe of theirs to sit beside anything. It is an exact numerical study of one object: a finite-horizon binary multiplicative process with a lower absorbing boundary L, a residual value S assigned on breach, and a single fixed exposure f chosen before the path runs. Every figure I quote from our own work is ours alone, from an adaptation to a market the paper never touches, and the paper's numbers describe its lattice and nothing else.

What the lattice pins down

The specification is tight where it matters. Wealth evolves as W(t+1)=W(t)(1+f R), with R equal to +b (prob p) or -a (prob 1-p). The benchmark is L=100, S=10, so a residual ratio rho=0.1, T=50, p=0.55, a=b=1. In the symmetric case the no-boundary Kelly fraction is f_K = 2p-1 = 0.10, and that is the reference line for everything else. The objective, expected absorbed log terminal wealth, is evaluated by exact recombining-lattice propagation rather than Monte Carlo, scanning 601 exposures on [0,0.3] across 80 distances on [0.05,3.0], with a tie tolerance of 1e-12 that keeps the smallest exposure. An implementer can rebuild the exposure map from that alone. Nothing is hand-waved in the propagation.

The result is clean. Far from the boundary the optimal exposure converges to 0.10 and the implied apparent CRRA coefficient approaches gamma=1, the log-growth benchmark. Near a costly boundary the optimizer picks exposure well below 0.10, which a model that omits the floor reads as sharply higher risk aversion. Same preferences, same returns, only the distance to the threshold changed. The identification caution is the real contribution: estimated risk aversion is partly a measurement of boundary proximity, not taste.

The haircut, with one exception

Two features complicate the picture, and the paper is honest about both. When rho approaches one, the residual truncates downside while favorable paths keep their multiplicative upside, and a narrow near-boundary region can show optimal exposure above f_K = 0.10. A limited-liability-like reversal falling out of terminal payoff geometry, concentrated at small d. And the horizon is not a scaling knob: absorption events nest, {tau<=T1} is contained in {tau<=T2}, so P(tau<=T) is nondecreasing in T and longer horizons widen the compressed region. The authors decline to prove global monotonicity of f* in d, T or rho, and say plainly that the finite lattice produces shallow competing maxima and local irregularities. The restraint reads as well judged.

Where does the implementer choose?

The paper leaves three things to judgment, all named in its own limitations section. The exposure is fixed ex ante, not a dynamic feedback policy; the authors note a control formulation could change the magnitude. Returns are strictly binary and i.i.d. And the mapping to apparent CRRA is benchmark specific to the symmetric binary case, so the numerical level of inferred curvature does not transfer. None of these is hidden. They dictate every adaptation decision downstream.

Why the compression collapsed in our build

We swapped the binary process for daily returns on 14 liquid US ETFs (SPY, QQQ, IWM, the sector XL-series), estimating up/down probability p and mean gain/loss b, a from a rolling 252-day window, then running the same T=50 absorbing lattice against a trailing floor L=0.8 times peak wealth with rho=0.1. Long-only, weekly rebalance, 10% per-name cap, 2020-01-01 to 2025-10-08. Commissions at $0.004 per share, zero modelled slippage. The headline figures sit above this text, and they are ours over that 2020 to 2025 window, not the paper's, because the paper reports none.

Read them as equity beta, not as evidence of the mechanism. The honest problem with what we built: daily ETF gain/loss coefficients run around 0.01, so the symmetric no-boundary Kelly fraction comes out near 4.0. Our 0.30 grid ceiling and then the 10% cap force every positive-edge ETF to saturate at the cap. The compression curve from the paper's Figures 3 and 5, the whole point, never touches the traded weights. On top of that we run a 50-step daily first-passage objective but rebalance after about five trading days, so the horizon geometry does not map to the holding period. What we shipped is a capped long-only sector book with a hard cash-out below an 80% trailing floor. The boundary logic survives only as that cash-out switch.

All of that describes our one automated pass, and the fix is not subtle: match the exposure scale before the floor can bite, using a per-period edge and volatility that put the unconstrained fraction back inside the grid, and align the lattice horizon to the rebalance frequency. Done that way, the idea is buildable as a drawdown-aware sizing overlay on top of an existing sizer, dialing exposure down as measured wealth nears a real continuation threshold. It will not manufacture edge where there is none. Its job is to keep a genuine edge from sizing itself into the floor.