Generalized risk parity portfolio optimization: an ADMM approach


The risk parity solution to the asset allocation problem yields portfolios where the risk contribution from each asset is made equal. We consider a generalized approach to this problem. First, we set an objective that seeks to maximize the portfolio expected return while minimizing portfolio risk. Second, we relax the risk parity condition and instead bound the risk dispersion of the constituents within a predefined limit. This allows an investor to prescribe a desired risk dispersion range, yielding a portfolio with an optimal risk–return profile that is still well-diversified from a risk-based standpoint. We add robustness to our framework by introducing an ellipsoidal uncertainty structure around our estimated asset expected returns to mitigate estimation error. Our proposed framework does not impose any restrictions on short selling. A limitation of risk parity is that allowing of short sales leads to a non-convex problem. However, we propose an approach that relaxes our generalized risk parity model into a convex semidefinite program. We proceed to tighten this relaxation sequentially through the alternating direction method of multipliers. This procedure iterates between the convex optimization problem and the non-convex problem with a rank constraint. In addition, we can exploit this structure to solve the non-convex problem analytically and efficiently during every iteration. Numerical results suggest that this algorithm converges to a higher quality optimal solution when compared to the competing non-convex problem, and can also yield a higher ex post risk-adjusted rate of return.

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    Short selling pertains to taking negative positions on our investments. ‘Long-only’ refers to portfolios where we can only take non-negative positions.

  2. 2.

    An estimated covariance matrix of financial assets has the useful property of being symmetric and positive semidefinite, provided sufficient data is used for estimation.


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Funding was provided by Natural Sciences and Engineering Research Council of Canada (Grant Number 455963).

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Correspondence to Roy H. Kwon.

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Costa, G., Kwon, R.H. Generalized risk parity portfolio optimization: an ADMM approach. J Glob Optim (2020).

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  • Non-convex optimization
  • Robust optimization
  • ADMM
  • Risk parity
  • Asset allocation