Index Tracking with Cardinality Constraints: A Stochastic Neural Networks Approach

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.

A metaheuristic-based framework for index tracking with practical constraints

“…A second approach was introduced to address this problem, namely, the joint approach. It integrates the two-step procedure into a single process, and it is optimized through various evolutionary heuristic or stochastic neural networks [3,21,37,46,63,65]. The last approach is to reformulate the sparse ITP into an alternative approximate function that is optimized through mixed-integer programming [4,10,43,45].…”

Nonasymptotic Estimates for Stochastic Gradient Langevin Dynamics Under Local Conditions in Nonconvex Optimization

“…As for the example from index tracking optimization, the mean squared tracking error is considered as the objective function (see, e.g. [20], [21]). Reparametrization is performed to remove the constraints on the parameter θ, which results in a nonconvex objective function.…”

“…min{ ċ,a/4}n nĉ(e min{ ċ,a/4} C2,1E[V 2 (θ 0 )] + 12E[V 4 (θ 0 )]) + exp(− ċ) ( C2,2 + 12c 3 (λmax + a −1 ) + 9v 4 (M 4 ) + 15) ≤ √ λ(e − min{ ċ,a/4}n/2 C2,3 E[V 4 (θ 0 )] + C2,4 ) = √ λ(e − ċn/2 C2,3 E[V 4 (θ 0 )] + C2,4 ),where the last inequality holds by applying the inequality e −αn (n + 1) ≤ 1 + α −1 , for α > 0 with α = min{ ċ, a/4}/2, and the last equality holds by noticing min{ ċ, a/4} = ċ with ċ given in(23). The explicit expressions for the constants C2,3 , C2,4 are given below: exp(− ċ) ( C2,2 + 12c 3 (λmax + a −1 ) + 9v 4 (M 4 ) + 15)(C13)with C2,1 , C2,2 given in (C12), ĉ, ċ given in Lemma 4.11, c 3 given in(20), and M 4 given in Lemma 4.Proof of Corollary 4.9 One notices that W 2 ≤ 2w 1,2 , then, by using similar arguments as in the proof of Lemma 4.8, one obtainsW 2 (L( ζλ,n t ), L(Z λ t )) ċ(n − k)/2)W 2 (L( θλ kT ), L( ζλ,k−1 kT )) + λ 1/4 √ 2ĉ n k=1 exp(− ċ(n − k)/2) 1 + E[V 4 ( θλ kT )] 1/E[V 4 ( ζλ,k−1 kT )]…”

“…≤ λ 1/4 (e − min{ ċ,a/4}n/4 C *2,3 E 1/2 [V 4 (θ 0 )] + C * 2,4 ) = λ 1/4 (e − ċn/4 C * 2,3 E 1/2 [V 4 (θ 0 )] + C * , C2,2given in (C12), ĉ, ċ given in Lemma 4.11, c 3 given in(20), and M 4 given in Lemma 4.4. This completes the proof.Proof of Lemma 4.10 By using Lemma 4.7 and 4.8, one obtainsW 1 (L( θλ − ċn/2 C2,3 E[V 4 (θ 0 )] + C2,4 )Proof of Lemma 4.11 To obtain the contraction constant ċ, we apply the arguments in the proof of[15, Theorem 2.2] to SDE(17).…”

Partial Index Tracking: A Meta-Learning Approach

“…It is the most intuitive index tracking approach and provides perfect tracking performance in a frictionless market. However, it leads to high transaction costs due to a large number of index constituents, frequent rebalancing (e.g., daily rebalance), churn in index members, and illiquid assets (Zheng et al, 2020a;Strub & Baumann, 2018).…”