Validation · #360760 · Cross-asset

Gold/Bitcoin lazy-factor

Optimizing Investment Strategies with Lazy Factor and Probability Weighting, 2023

Shuo Han, Yinan Chen & Jiacheng Liu · arXiv q-fin

The rule

Each day, forecast tomorrow's gold and Bitcoin prices with ARIMA, pick the max-Sharpe cash/gold/Bitcoin mix from 10,000 Monte-Carlo portfolios, then anchor it: final weights are 70% plain equal-weight plus 30% of that ideal — the 'lazy factor'. Rebalance daily, $1,000 start, Sep 2016 → Sep 2021.

Fails $136,216→$4,374 claimed → measured final wealth, from $1,000

136×

Claimed multiple$1,000 → $136,216 at a '2.5% max retracement'

4.4×

Honest replication$4,374 — with a −47% max drawdown

10×

Ceiling with tomorrow's pricesdeliberate look-ahead under their own rule still lands 13× short

AssetGold + Bitcoin + cash

StrategyARIMA → MC mean-variance, lazy blend

Period2016–2021, daily

CostsIn the objective; rates never stated

Start$1,000

Assets tradedXAU BTCLBMA gold PM fix (trades only on London fix days) and Bitcoin (7 days a week) — the paper's own two sources, from their public mirrors. Bitcoin alone did 74.6× over the window.

The exact rules

Each day (after a one-month warm-up)	→	Fit ARIMA(2,1,2) on the trailing ≤100 gold and Bitcoin prices; forecast tomorrow's price
Expected return = ln(forecast ÷ today)	→	Draw 10,000 random cash/gold/BTC weight mixes; keep the max Sharpe net of trading costs (covariance from the trailing two months)
The lazy factor	→	Final weights = 0.7 × (⅓, ⅓, ⅓) + 0.3 × ideal — then trade to them, paying commissions on every move
Gold market closed (weekends, UK holidays)	→	Gold position frozen; the cash/Bitcoin sleeve re-splits half-and-half plus the tilt

The backtest, re-run

Inside the model

Position mix: avg 33.7% cash / 33.3% gold / 33.0% BTC — the lazy anchor dominates; max BTC weight ever reached 37.4%
Win rate: 53.9% of days · 55.7% of months
Skew / kurtosis: −0.3 / 10.3 daily — crypto tails, unmitigated
Best / worst day: +8.6% / −13.4%
Annual returns: 2017 +142% · 2018 −35% · 2019 +28% · 2020 +71% · 2021 +16% — one bull year carries everything
Turnover / costs: 1.2%/day one-way; $838 of commissions on a $4,374 outcome
Internal consistency: the paper's own three headline numbers disagree: 136.2× ≡ 167%/yr, the text says 27.04%, and '135.2161%' is the multiple misprinted as a percent

The validation ladder

C0	Replicate	Honest replication of the stated rule: $4,374 (4.4×, 34%/yr) — 3.2% of the claimed $136,216. The claim exceeds the no-leverage bound before any simulation: Bitcoin did 74.6× and the stated blend caps Bitcoin at 53%. Even feeding tomorrow's exact prices into the optimizer, the stated rule reaches only $10,191; the claim is unreachable at any forecast quality.
C1	Honesty	Under identical accounting, a plain ⅓/⅓/⅓ daily-rebalanced mix ends at $5,228 — the 'optimized' strategy loses to its own do-nothing anchor by 20%. Bitcoin buy-and-hold ends at $73,942. The paper benchmarks against neither.
C2	Deflate	Active returns vs equal-weight across the whole δ/Monte-Carlo search surface: best t-statistic −1.91 — every tilt level subtracts value. Deflated Sharpe 0.00. There is no edge to deflate.
C3	Crisis	Inside the paper's own sample: −44%/yr through the 2018 crypto winter (−47% drawdown), −48%/yr annualized through the COVID crash, −68%/yr in the May-2021 flush. The claimed 2.5% max retracement coexists with none of this.
C5	Frictions	The strategy trails plain equal-weight at every commission level, including zero. Costs aren't the story — the ARIMA tilt is pure noise, and higher costs paradoxically shrink the damage by suppressing trading.
C6	Decay	True post-publication frontier (Sep 2021 → Jul 2026): 8.1%/yr with a −39% max drawdown through the 2022 crypto crash — vs 8.09%/yr for plain equal-weight. The active edge going forward is +2bp of noise; the 2.5%-MDD claim was demolished within 14 months of publication.
C7	Originality	Daily returns regress on Bitcoin + gold with R² = 0.994: β(BTC) 0.33, β(gold) 0.33, alpha −8.7%/yr (t = −7.5). The strategy is the ⅓/⅓/⅓ benchmark minus fees and forecast noise.

Reproduces claimArithmetically possibleStatistically realOriginal vs betaCrisis-robustBeats its own benchmark

How we rebuilt it

Data: LBMA gold PM fix from the LBMA itself + Blockchain.com Bitcoin market price (the series behind 'NASDAQ's Bitcoin data') — the paper's own sources, public mirrors.
Method: ARIMA(2,1,2) next-price forecasts → 10,000-draw Monte-Carlo tangency with costs in the objective → 0.7 equal-weight / 0.3 ideal lazy blend; unit-based wealth accounting per the paper's Eq 6.
Universe: Two assets plus cash; gold trades only on London fix days (the paper's non-trading-day rule replicated).
Regime stated: None — a single 2016–2021 window in which Bitcoin rose 75×; every crypto crash inside it contradicts the 2.5% retracement claim.

Deviations from the paper

Commission rates never stated (in a paper titled '…with Transaction Costs') → the MCM-2022-C competition rates the setup matches: 1% gold, 2% Bitcoin; swept 0–3% and the conclusion is cost-invariant.
ARIMA order undisclosed beyond one printed gold fit → ARIMA(2,1,2) both assets, re-fit daily on trailing ≤100 observations, strictly past-only; the look-ahead bracket covers every possible forecast quality.
Covariance lookback 'two months' → 60 calendar days; Monte-Carlo sampling law undisclosed → uniform on the weight simplex, fixed seed. MC path noise moves outcomes by percents, not the 31× gap.
Risk-free for our reported Sharpe: 13-week T-bill; the paper reports no Sharpe at all.

ProvenanceBeta

Engine: v1
Blocks: 5 new (LBMA gold + Blockchain.com BTC adapters, ARIMA next-price signal, MC-tangency lazy sizing, three-asset daily backtest), 5 reused (metrics, G1/G2/G3 gates, regime)
Data: LBMA PM fix + Blockchain.com market price · 2016–2021 paper window + 2021–2026 frontier · run_id 7
Source: arXiv 2306.07928 ↗ · LBMA precious-metal prices ↗ · Blockchain.com charts API ↗
Tests: 5 known-value tests (closed-form accounting, EW collapse, cost arithmetic, no-look-ahead, tangency direction) + multi-seed noise-forecast negative control
Reproduce: view code ↗