Research

The math behind a sparse, fully-invested, long-only index tracker

We solve a high-dimensional convex problem with non-negativity and simplex constraints via a custom ADMM splitting. λ controls the bias-variance trade-off between tracking accuracy and portfolio sparsity.

The optimisation problem

\min_{w \in \mathbb{R}^p}\;\tfrac{1}{2}\|Xw - y\|_2^2 + \lambda\|w\|_1 \quad \text{s.t.}\quad w \ge 0,\; \mathbf{1}^\top w = 1

$X$ is an $n \times p$ matrix of constituent returns, $y$ the index return, $w$ the desired sparse, simplex-constrained portfolio weights.

L1 vs L2 — why corners give sparsity

The L1 ball's vertices align with coordinate axes — so the constraint hyperplane $y=Xw$ almost surely touches it at a vertex (sparse). The L2 ball is rotationally symmetric and has no such vertices.

ADMM updates (the three steps)

w^{k+1} = (X^\top X + \rho I)^{-1}\bigl(X^\top y + \rho(z^k - u^k)\bigr)

z^{k+1} = \mathrm{prox}_{(\lambda/\rho)\|\cdot\|_1, +}\!\bigl(w^{k+1} + u^k\bigr)

u^{k+1} = u^k + w^{k+1} - z^{k+1}

We Cholesky-factorise $X^\top X + \rho I$ once per ρ-update, so each iteration costs $O(p^2)$ instead of $O(p^3)$ .

Pick your λ — interactive

Slide to see the bias–variance trade-off live: tighter regularisation (right) → fewer stocks → higher tracking error.

λ / λ_max

0.100

20 stocksOOS TE 3.80%R² 0.961

Demo data — when running against a live API, switching to /api/proxy/api/v1/lambda-path repaints the chart.

Convergence (real residuals)

Primal and dual residuals from a real synthetic 100×300 ADMM solve. Decay is sub-linear at the start, geometric near the optimum (Boyd §3.4).