Folio Lab/Documentation

Sparse Markowitz (L1)

Brodie–Daubechies–De Mol–Giannone–Loris sparse mean-variance — regularises the Markowitz objective with an L1 penalty to encourage concentrated, sparse portfolios that allocate to a small number of high-conviction names.

Overview

Markowitz portfolios spread weight across many assets — including small allocations that rarely justify their transaction cost. Brodie et al. (2009) show that adding an L1 penalty to the variance objective produces sparse portfolios with non-zero weights on only a small, data-driven subset of assets. The L1 term acts as a regulariser that also stabilises the solution against estimation error in .

Folio Lab's implementation enforces long-only weights, which means the L1 norm reduces to on the simplex. The optimiser still benefits from the penalty during the interior-point search, and additional constraints control concentration and gross leverage.

Mathematical Formulation

With historical mean , Ledoit–Wolf covariance , return tradeoff , and L1 weight :

subject to:

Default values are , , and a maximum leverage . Folio Lab solves this with cvxpy using CLARABEL with ECOS / SCS as fallback solvers, jittering by for numerical stability.

Advantages & Limitations

Advantages

  • Sparse output: Concentrated, transaction-friendly portfolios.
  • Regularised: L1 stabilises against noisy .
  • Convex: Reliable global optimum.
  • Tunable concentration: controls sparsity.

Limitations

  • Hyperparameter tuning: The trade-off / penalty pair is sensitive.
  • Long-only redundancy: Under and , the L1 term reduces to a constant.
  • Solver-dependent: Near-singular covariance can stall solvers.
  • Concentration risk: Sparse portfolios can underperform if the chosen set is wrong.

References

  • Brodie, J., Daubechies, I., De Mol, C., Giannone, D., & Loris, I. (2009). "Sparse and stable Markowitz portfolios." PNAS, 106(30), 12267-12272.
  • DeMiguel, V., Garlappi, L., Nogales, F. J., & Uppal, R. (2009). "A generalized approach to portfolio optimization: Improving performance by constraining portfolio norms." Management Science, 55(5), 798-812.
  • Tibshirani, R. (1996). "Regression shrinkage and selection via the lasso." JRSS-B, 58(1), 267-288.