Theory and Multi-Asset Evidence
Proof and backtest code accompanying the research paper submitted to Applied Mathematical Finance (Taylor & Francis)
Among all leverage sequences with the same total leverage applied to a Dollar Cost Averaging (DCA) entry schedule, the non-decreasing (increasing) sequence is the unique optimal allocation — it minimises the weighted average entry price and maximises the reference-to-entry span.
Example: For sequences with average leverage 6×:
| Strategy | Sequence | Avg Entry | Span B | P/L @ 75% Recovery |
|---|---|---|---|---|
| Increasing | (2, 4, 6, 8, 10) | 86.7 | 13.3% | +9.57% |
| Uniform | (6, 6, 6, 6, 6) | 90.0 | 10.0% | +5.56% |
| Decreasing | (10, 8, 6, 4, 2) | 93.3 | 6.7% | +1.82% |
The proof uses the Rearrangement Inequality (Hardy, Littlewood & Pólya, 1952) and is parameter-independent.
Backtest on 9.5 million one-minute bars across 5 markets (2021–2026):
| Market | Increasing Sharpe | Uniform Sharpe | Decreasing Sharpe | Inc. Liquidations |
|---|---|---|---|---|
| XAUUSD (Gold) | 0.66 | 0.40 | -0.00 | 3 |
| BTCUSDT | -0.09 | -2.41 | -0.79 | 62 |
| ETHUSDT | -0.73 | -4.69 | -0.94 | 135 |
| SPY (S&P 500) | 1.60 | 1.52 | 0.59 | 2 |
| QQQ (NASDAQ 100) | 3.36 | 2.92 | 0.90 | 0 |
The increasing sequence ranks first in Sharpe ratio and fewest liquidations in every market. Approximate replication p-value: (1/3)⁵ ≈ 0.004.
├── dca_backtest.py # Primary XAUUSD backtest engine
├── dca_multi_asset.py # Multi-asset backtest (BTC, ETH, SPY, QQQ)
├── DCA_AMF_manuscript.tex # LaTeX source of the research paper
├── references.bib # BibTeX references
└── README.md
pip install pandas numpy matplotlib pyarrowpython dca_backtest.py- Reads
XAUUSD_1m.csv(1-minute candlestick data) - Outputs equity curves, annual P/L, P/L distributions as PNG
- Prints full performance table for all three leverage strategies
python dca_multi_asset.py- Reads XAUUSD, BTCUSDT, ETHUSDT CSVs and SPY/QQQ parquet files
- Outputs per-asset equity curves and cross-asset comparison charts
| Asset | Format | Required Columns |
|---|---|---|
| XAUUSD | CSV | datetime, open, high, low, close |
| BTCUSDT / ETHUSDT | CSV | Date, Open, High, Low, Close |
| SPY / QQQ | Parquet (directory) | timestamp, open, high, low, close |
| Parameter | Value | Description |
|---|---|---|
f |
3% | Initial entry drawdown from rolling high |
M |
15% | Maximum drawdown threshold (final entry) |
n |
4 | Number of DCA intervals (5 total entries) |
W |
7,200 bars | Lookback window for rolling reference high (≈5 days) |
| Cooldown | 480 bars | Minimum gap between new DCA sequences (8 hours) |
| Capital | $100/step | Margin per DCA entry ($500 total per trade) |
| Max hold | 20,160 bars | Maximum position hold time (14 days) |
INCREASING = [2, 4, 6, 8, 10] # Optimal — proven by Proposition 1
UNIFORM = [6, 6, 6, 6, 6] # Baseline
DECREASING = [10, 8, 6, 4, 2] # WorstAll sequences sum to 30 (average 6×) — comparisons reflect only the distribution of leverage, not the total amount.
Assign lower leverage to earlier (higher-priced) DCA entries and higher leverage to later (lower-priced) entries.
This is not a heuristic — it is the provably optimal allocation.
Ayush Pandey (2026). Optimal Leverage Sequencing in Leveraged Dollar Cost Averaging: Theory and Multi-Asset Evidence. Submitted to Applied Mathematical Finance, Taylor & Francis.
Contact: ayushpandey85986@gmail.com
This repository is provided for academic research purposes only. Nothing in this repository constitutes financial advice. Leveraged derivatives trading carries significant risk of loss, including total loss of deposited margin. Past backtest performance does not guarantee future results.