[Paper Review] Index Tracking with Cardinality Constraints: A Stochastic Neural Networks Approach

Yu Zheng, Bowei Chen+2 more|arXiv (Cornell University)|Nov 12, 2019

Financial Markets and Investment Strategies31 references4 citations

TL;DR

This paper proposes a stochastic neural network approach with a novel reparametrisation method to solve index tracking with cardinality constraints, significantly reducing tracking error compared to established methods. The method achieves superior performance across multiple portfolio sizes using S&P 500 data over 12 years, outperforming forward/backward selection and market-cap weighting while maintaining comparable risk-adjusted metrics.

ABSTRACT

Partial (replication) index tracking is a popular passive investment strategy. It aims to replicate the performance of a given index by constructing a tracking portfolio which contains some constituents of the index. The tracking error optimisation is quadratic and NP-hard when taking the L0 constraint into account so it is usually solved by heuristic methods such as evolutionary algorithms. This paper introduces a simple, efficient and scalable connectionist model as an alternative. We propose a novel reparametrisation method and then solve the optimisation problem with stochastic neural networks. The proposed approach is examined with S&P 500 index data for more than 10 years and compared with widely used index tracking approaches such as forward and backward selection and the largest market capitalisation methods. The empirical results show our model achieves excellent performance. Compared with the benchmarked models, our model has the lowest tracking error, across a range of portfolio sizes. Meanwhile it offers comparable performance to the others on secondary criteria such as volatility, Sharpe ratio and maximum drawdown.

Motivation & Objective

• To address the NP-hard nature of index tracking with ℓ₀ norm constraints (portfolio cardinality) using a scalable, efficient alternative to heuristic methods.
• To develop a reparametrisation technique enabling gradient-based optimisation of sparse portfolio weights via stochastic neural networks.
• To evaluate the proposed method’s performance against widely used benchmarks across diverse portfolio sizes and risk metrics.
• To demonstrate the method’s consistency and scalability through large-scale backtesting over a 12-year horizon.

Proposed method

• A novel reparametrisation method transforms the ℓ₀-constrained optimisation problem into a differentiable form suitable for stochastic gradient descent.
• Stochastic neural networks are used to learn sparse portfolio weights by sampling from a parameterised distribution over stock weights.
• The optimisation objective minimises tracking error (quadratic loss) under constraints: non-negative weights, sum-to-one capital allocation, and ℓ₀ norm for cardinality control.
• The model is trained using stochastic gradient descent with reparametrised backpropagation to enable end-to-end learning of sparse, high-performance portfolios.
• Backtesting is performed using a sliding window approach on S&P 500 data from 2006 to 2018, with portfolio sizes K ∈ {30, 40, 50}.
• Performance is evaluated using tracking error, volatility, Sharpe ratio, and maximum drawdown, with comparisons to forward selection, backward selection, and market-cap weighting baselines.

Experimental results

Research questions

• RQ1 Can a stochastic neural network with a novel reparametrisation method effectively solve the NP-hard index tracking problem with ℓ₀ cardinality constraints?
• RQ2 How does the proposed method compare to traditional heuristics like forward and backward selection in terms of tracking error across varying portfolio sizes?
• RQ3 Does the stochastic neural network approach maintain consistent performance across multiple random seeds and long-term backtesting periods?
• RQ4 To what extent does the method preserve key risk metrics (volatility, Sharpe ratio, drawdown) relative to the benchmark index?

Key findings

• The proposed stochastic neural network method achieves the lowest tracking error across all tested portfolio sizes (K = 30, 40, 50) compared to forward selection, backward selection, and largest market capitalisation methods.
• The method consistently outperforms benchmarks, with tracking errors predominantly showing portfolio outperformance (positive error in PE plots), indicating robustness and accuracy.
• Performance is highly consistent across 100 runs with different random seeds, as shown by narrow confidence intervals in the stochasticity analysis.
• The model maintains comparable performance to baselines on secondary metrics: it slightly outperforms on Sharpe ratio and maximum drawdown, and slightly underperforms on volatility, but remains within acceptable bounds.
• The method demonstrates scalability and efficiency, avoiding the high computational cost of evolutionary algorithms while delivering superior tracking accuracy.
• Over a 12-year backtesting window (2006–2018), the model shows strong generalisation and stability, confirming its practical viability for real-world index tracking.