Constrained Risk Budgeting Portfolios: Theory, Algorithms, Applications & Puzzles

36 Pages Posted: 25 Feb 2019 Last revised: 1 Mar 2019

See all articles by Jean-Charles Richard

Jean-Charles Richard

Eisler Capital

Thierry Roncalli

Amundi Asset Management; University of Evry

Date Written: February 8, 2019

Abstract

This article develops the theory of risk budgeting portfolios, when we would like to impose weight constraints. It appears that the mathematical problem is more complex than the traditional risk budgeting problem. The formulation of the optimization program is particularly critical in order to determine the right risk budgeting portfolio. We also show that numerical solutions can be found using methods that are used in large-scale machine learning problems. Indeed, we develop an algorithm that mixes the method of cyclical coordinate descent (CCD), alternating direction method of multipliers (ADMM), proximal operators and Dykstra's algorithm. This theoretical body is then applied to some investment problems. In particular, we show how to dynamically control the turnover of a risk parity portfolio and how to build smart beta portfolios based on the ERC approach by improving the liquidity of the portfolio or reducing the small cap bias. Finally, we highlight the importance of the homogeneity property of risk measures and discuss the related scaling puzzle.

Keywords: Risk budgeting, large-scale optimization, Lagrange function, cyclical coordinate descent (CCD), alternating direction method of multipliers (ADMM), proximal operator, Dykstra's algorithm, turnover, liquidity, risk parity, smart beta portfolio

Suggested Citation

Richard, Jean-Charles and Roncalli, Thierry, Constrained Risk Budgeting Portfolios: Theory, Algorithms, Applications & Puzzles (February 8, 2019). Available at SSRN: https://ssrn.com/abstract=3331184 or http://dx.doi.org/10.2139/ssrn.3331184

Jean-Charles Richard

Eisler Capital ( email )

16 St. James's Street
London, SW1A1ER

Thierry Roncalli (Contact Author)

Amundi Asset Management ( email )

90 Boulevard Pasteur
Paris, 75015
France

University of Evry ( email )

Boulevard Francois Mitterrand
F-91025 Evry Cedex
France

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