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DCC: Differentiable Cardinality Constraints for Partial Index Tracking
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Proposes a differentiable, floating-point-aware cardinality constraint (DCC_fpp) using a sigmoid approximation so that SLSQP can solve partial index replication directly, with a condition on the steepness hyperparameter a.
Problem
Partial index replication requires a cardinality constraint on portfolio weights, which makes index tracking non-convex, non-differentiable and NP-hard; existing methods are heuristics or black-box neural networks.
Approach
Approximate the discontinuous binary count of nonzero weights with a differentiable sigmoid centred at a weight cutoff epsilon, so the cardinality constraint can be enforced within a standard SLSQP solver, with a steepness constant a chosen to guarantee the true cardinality bound.
Key results
- On S&P 100, DCC_fpp achieves MAE 3.9155 at K=20, 3.5385 at K=25, and 2.3922 at K=30, versus SNN 5.8007, 4.0457, 4.9006.
- Forward and backward selection yield MAE of roughly 7.7 to 8.9, performing worst among methods.
- The paper reports the sigmoid steepness a must be at least 138,157 to satisfy all conditions under 64-bit floating-point precision.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use DCC_fpp with SLSQP instead of SNN | Partial index tracking · S&P 100, K=20 | MAE: 5.8007 -> 3.9155 | Helped |
| Use DCC_fpp with SLSQP instead of SNN | Partial index tracking · S&P 100, K=25 | MAE: 4.0457 -> 3.5385 | Helped |
| Use DCC_fpp with SLSQP instead of SNN | Partial index tracking · S&P 100, K=30 | MAE: 4.9006 -> 2.3922 | Helped |
Surprising findings
- Backward selection's MAE increases at K=30 (8.6562) relative to K=25, which is not monotone in K.
Limitations
- The sigmoid steepness a required for guaranteed cardinality is very large (about 138,157), which may cause numerical issues in optimization.
- The complexity argument relies on SLSQP's internal O(N^3) cost, and the claim of an exact solution is not established for a non-convex problem.
- Experiments are limited to three indices and a single solver (SLSQP) with a simple backtest.
Experimental setup
Datasets: S&P 100 daily returns (2018-2023, Yahoo Finance) S&P 500 daily returns (2018-2023, Yahoo Finance) KOSPI 100 daily returns (2018-2023, Yahoo Finance)
Benchmarks: S&P 100 index tracking MAE S&P 500 index tracking KOSPI 100 index tracking
Metrics: mean absolute error cumulative return volatility Sharpe ratio maximum drawdown runtime
Baselines: Forward Selection Backward Selection SNN
Builds on: SLSQP SNN partial replication
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