Mean-constrained portfolio optimization under lambda expected shortfall: A loss-level decomposition approach

Kunpeng Zhu; Yanjing Guo; Kunpeng Zhu; Yanjing Guo

doi:10.3934/jimo.2026202

Journal of Industrial and Management Optimization

2026, Volume 22, Issue 11: 6003-6033. doi: 10.3934/jimo.2026202

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Research article

Mean-constrained portfolio optimization under lambda expected shortfall: A loss-level decomposition approach

Kunpeng Zhu ,
Yanjing Guo ^,

School of Management Science and Engineering, Nanjing University of Information Science and Technology, 219 Ningliu Road, Nanjing, Jiangsu Province, China

Received: 07 August 2026 Revised: 22 October 2026 Accepted: 23 October 2026 Published: 27 October 2026
Primary 90C26; Secondary 90C25, 90C90, 91B30, 91G10

Conventional expected shortfall portfolio optimization requires an externally specified confidence level, which can make optimized portfolios sensitive to the chosen confidence level. We developed a mean-constrained lambda expected shortfall portfolio model that replaces the fixed confidence level with a loss-dependent confidence function, together with an $ \ell_2 $-regularized extension intended to stabilize portfolio weights. The resulting joint empirical formulations are generally nonlinear and nonconvex. We addressed this difficulty through a loss-level decomposition that reduces the original problems to convex fixed-confidence expected shortfall subproblems and a one-dimensional outer optimization over the loss level. For the basic model, the monotone-crossing structure yields a bisection procedure. We also developed an interval branch-and-bound procedure applicable to both models, including the regularized case in which the monotone-crossing structure is not guaranteed. Numerical experiments on four equity datasets showed that the proposed models endogenously select effective confidence levels and achieve competitive out-of-sample performance relative to fixed-confidence benchmarks, with the regularized extension further reducing portfolio turnover.
- lambda expected shortfall,
- conditional value-at-risk,
- portfolio optimization,
- loss-level decomposition,
- interval branch-and-bound
Citation: Kunpeng Zhu, Yanjing Guo. Mean-constrained portfolio optimization under lambda expected shortfall: A loss-level decomposition approach[J]. Journal of Industrial and Management Optimization, 2026, 22(11): 6003-6033. doi: 10.3934/jimo.2026202

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Abstract

Conventional expected shortfall portfolio optimization requires an externally specified confidence level, which can make optimized portfolios sensitive to the chosen confidence level. We developed a mean-constrained lambda expected shortfall portfolio model that replaces the fixed confidence level with a loss-dependent confidence function, together with an $ \ell_2 $-regularized extension intended to stabilize portfolio weights. The resulting joint empirical formulations are generally nonlinear and nonconvex. We addressed this difficulty through a loss-level decomposition that reduces the original problems to convex fixed-confidence expected shortfall subproblems and a one-dimensional outer optimization over the loss level. For the basic model, the monotone-crossing structure yields a bisection procedure. We also developed an interval branch-and-bound procedure applicable to both models, including the regularized case in which the monotone-crossing structure is not guaranteed. Numerical experiments on four equity datasets showed that the proposed models endogenously select effective confidence levels and achieve competitive out-of-sample performance relative to fixed-confidence benchmarks, with the regularized extension further reducing portfolio turnover.

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