Big Data Challenges of High-Dimensional Continuous-Time Mean-Variance Portfolio Selection and a Remedy.

Investors interested in the global financial market must analyze financial securities internationally. Making an optimal global investment decision involves processing a huge amount of data for a high-dimensional portfolio. This article investigates the big data challenges of two mean-variance optimal portfolios: continuous-time precommitment and constant-rebalancing strategies. We show that both optimized portfolios implemented with the traditional sample estimates converge to the worst performing portfolio when the portfolio size becomes large. The crux of the problem is the estimation error accumulated from the huge dimension of stock data. We then propose a linear programming optimal (LPO) portfolio framework, which applies a constrained ℓ1 minimization to the theoretical optimal control to mitigate the risk associated with the dimensionality issue. The resulting portfolio becomes a sparse portfolio that selects stocks with a data-driven procedure and hence offers a stable mean-variance portfolio in practice. When the number of observations becomes large, the LPO portfolio converges to the oracle optimal portfolio, which is free of estimation error, even though the number of stocks grows faster than the number of observations. Our numerical and empirical studies demonstrate the superiority of the proposed approach.

Roy's safety-first portfolio principle in financial risk management of disastrous events.

Roy pioneers the concept and practice of risk management of disastrous events via his safety-first principle for portfolio selection. More specifically, his safety-first principle advocates an optimal portfolio strategy generated from minimizing the disaster probability, while subject to the budget constraint and the mean constraint that the expected final wealth is not less than a preselected disaster level. This article studies the dynamic safety-first principle in continuous time and its application in asset and liability management. We reveal that the distortion resulting from dropping the mean constraint, as a common practice to approximate the original Roy's setting, either leads to a trivial case or changes the problem nature completely to a target-reaching problem, which produces a highly leveraged trading strategy. Recognizing the ill-posed nature of the corresponding Lagrangian method when retaining the mean constraint, we invoke a wisdom observed from a limited funding-level regulation of pension funds and modify the original safety-first formulation accordingly by imposing an upper bound on the funding level. This model revision enables us to solve completely the safety-first asset-liability problem by a martingale approach and to derive an optimal policy that follows faithfully the spirit of the safety-first principle and demonstrates a prominent nature of fighting for the best and preventing disaster from happening.