Long-Short Risk Budget Optimization with Signal-Directed Weights

Summary

The document presents a long-short risk-budgeting formulation attributed to a UBS paper on trend-following and risk parity. It maximizes a signal-weighted sum of logarithms of absolute portfolio weights, limits portfolio volatility, requires each position’s sign to match its signal, and constrains total absolute exposure to one. The central question is how to solve this optimization problem given its stated nonconvexity.

No solution method, comparison of algorithms, implementation, or empirical results are provided. The formulation is useful as a starting point for studying portfolio construction, but the note leaves practical questions open, including how to handle zero signals or weights and how estimates of expected returns and covariance affect the result. Readers should consult the cited paper or further analysis before using the objective in a trading system.

Key ideas

• The objective weights log absolute positions by the magnitudes of their signals.
• The sign constraints align long and short positions with the corresponding signal directions.
• A portfolio volatility ceiling and a unit gross-exposure constraint bound the allocation.
• The document identifies nonconvexity as the main computational challenge but does not propose a solver.

Tags

# How to perform a long short risk budget optimization?


# How to perform a long short risk budget optimization?












In their paper titled "Trend-Following meets Risk-Parity" UBS proposed an optimization algorithm for performing risk budgeting for long short portfolios. The formulation was as follows:

$$ Maximize \sum |\mu_i| \log(|w_i|) $$ Subject to:

- $\sqrt(w^T \Sigma w) \leq \sigma_{max}$

- $w_i \gt 0, if \mu_i \gt 0$

- $w_i \lt 0, if \mu_i \lt 0$

- $\sum |w_i| =1$

The optimization problem is not convex. What is the best way to solve it? Any code suggestions?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.