A Custom Portfolio Optimizer, Up to 40% More Risk-Return Efficient Than Standard Heuristics

A spatial Branch-and-Bound built at ETH IFOR with swissQuant, delivering certified-optimal portfolios on a non-convex problem standard MIP solvers can only attempt heuristically

Modern asset managers want portfolios where every position carries a fair share of risk, not just a fair share of capital. The constraint that enforces this is mathematically nasty, and most off-the-shelf solvers either give up or return numbers no one can defend in front of a risk committee.

We worked with swissQuant and ETH Zurich's Institute for Operations Research to build a Branch-and-Bound algorithm that solves the problem to certified global optimality. The output isn't just a portfolio, it's a portfolio with a proof.

+40%

more risk-return efficient than the heuristic baseline on 33-asset MSCI World portfolios

2.5×

lower portfolio variance than the heuristic in the pure risk-minimization regime

Certified

global optimum on a non-convex QCQP standard MIP solvers cannot solve to optimality

Production-ready

documented Python implementation for swissQuant's valuation pipeline

The challenge

swissQuant's institutional clients required portfolios where each asset's marginal contribution to total volatility stays inside a tight tolerance. The math behind that constraint is non-convex, and off-the-shelf quadratic and conic solvers either return infeasible solutions or hit time limits as the asset universe grows.

Heuristics produce a point but no bound on how far that point sits from optimal. That's awkward when the result has to land on a portfolio manager's desk and be defended.

Our approach

We reformulated the problem as a Quadratically Constrained Quadratic Program and built a spatial Branch-and-Bound algorithm with custom convex relaxations on every node. Tight bound-tightening, smart branching rules, and a benchmark of six algorithmic variants on real MSCI World data identified the configuration that consistently wins.

Alongside the exact engine, we shipped two industry-friendly heuristics that warm-start the search or serve as a fast fallback when a certified optimum isn't worth the wait.

The outcome

Deliverable: a documented Python implementation of the spatial Branch-and-Bound on McCormick relaxations, the two warm-start heuristics, and the reproducible MSCI World test harnesses, ready to plug into swissQuant's internal valuation pipeline.

On real five-year MSCI World data (top-100 instruments, four-currency basket, daily log-returns Jan 2015 – Jan 2020), the algorithm closes a 40% objective gap left open by the iterative heuristic on 33-asset portfolios. In the most non-convex regime (n = 11, τ = 1) the heuristic returns a positive objective where the true optimum is negative, a 160% relative gap; our algorithm finds the certified optimum.

Technical deep dive

The model behind the result.

The model

We measure portfolio risk by volatility R(x) = √(xᵀΣx). For a long-only portfolio x, the marginal risk contribution of asset i follows from the Euler decomposition of volatility:

MRC_{i} (x) = x_{i} \cdot \frac{( Σ x ) _{i}}{x ^{⊤} Σ x}, i = 1 \sum n MRC_{i} (x) = R (x) .

Marginal risk contributions sum exactly to the portfolio's total volatility.

rMRC_{i} (x) = \frac{MRC _{i} ( x )}{R ( x )} = \frac{x _{i} ( Σ x ) _{i}}{x ^{⊤} Σ x} \leq c_{i} \forall i \in [n] .

Relative marginal risk: asset i may contribute at most a fraction cᵢ of the total risk.

x \geq 0 min s.t. τ x^{⊤} Σ x - (1 - τ) μ^{⊤} x x_{i} (Σ x)_{i} \leq c_{i} (x^{⊤} Σ x) \forall i \in [n] 1^{⊤} x = 1.

Problem (P): mean-variance with rMRC constraints. τ trades off risk against expected return μ.

y_{i} = (Σ x)_{i}, z_{i} = x_{i} y_{i}, z_{i} \leq c_{i} j = 1 \sum n z_{j} .

Reformulation: auxiliary variables y, z lift all the non-convexity into clean bilinear equalities.

z_{i} \geq z_{i} \geq z_{i} \leq z_{i} \leq x_{i}^{lb} y_{i} + x_{i} y_{i}^{lb} - x_{i}^{lb} y_{i}^{lb}, x_{i}^{ub} y_{i} + x_{i} y_{i}^{ub} - x_{i}^{ub} y_{i}^{ub}, x_{i}^{ub} y_{i} + x_{i} y_{i}^{lb} - x_{i}^{ub} y_{i}^{lb}, x_{i} y_{i}^{ub} + x_{i}^{lb} y_{i} - x_{i}^{lb} y_{i}^{ub} .

The McCormick envelope: four linear inequalities per asset that bound the bilinear term z_i = x_i y_i given box bounds on (x_i, y_i).

Each Branch-and-Bound node solves a convex relaxation built from these McCormick inequalities. Branching subdivides the box on x, the bounds on y are recomputed analytically, and the search terminates when the gap between the best feasible incumbent and the lowest open lower bound is below the tolerance.

Benchmark

Runtime of the best B&B configuration on MSCI World benchmarks.

Universe size	c-factor	τ ≤ 0.6	τ = 0.7	τ = 0.8	τ = 0.9	τ = 1.0
n = 11	1.5/n	22.3 s	11.0 s	12.7 s	14.1 s	17.3 s
n = 22	1.5/n	198.9 s	217.0 s	107.3 s	68.2 s	64.7 s
n = 33	1.5/n	674.3 s	1658.4 s	2219.0 s	34.4 s	57.3 s
n = 44	2/n	1223.6 s	1883.5 s	1232.2 s	761.5 s	598.2 s
n = 54	2/n	-	-	-	1238.8 s	621.1 s

Times in seconds, single-threaded relaxation calls, one-hour cap. Test bed: random subsets of the MSCI World top 100, daily log-returns Jan 2015 – Jan 2020, four-currency basket. Em-dashes indicate the configuration did not close the optimality gap within the budget.

From the record

Figure 2.1 from the report: the McCormick envelope for the bilinear term z = xy. The black curve is the original non-convex feasible region; the green and pink polytopes are the linear over- and under-estimators that the relaxation solves over.

Algorithm 1: the spatial Branch-and-Bound loop. At each node we solve the McCormick-relaxed program, prune by best-bound, and branch on the asset with the largest bilinear violation.

Figure 4.3 from the report: relative gap in objective between the iterative heuristic and the certified Branch-and-Bound optimum, as a function of risk-aversion τ on n = 33 assets. The gap peaks above 40% in the mid-τ regime where the problem is most non-convex.

Techniques

Spatial Branch-and-Bound
McCormick envelope relaxations
Bilinear reformulation of QCQP
Adaptive bound-tightening on relaxation duals
Risk-parity & risk-budgeting modeling
Mean-variance with non-convex risk constraints

Stack

Python
Gurobi
NumPy
MSCI World data pipeline

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