Online portfolio selection research has so far focused mainly on minimizing regret defined in terms of wealth growth. Practical financial decision making, however, is deeply concerned with both wealth and risk. We consider online learning of portfolios of stocks whose prices are governed by arbitrary (unknown) stationary and ergodic processes, where the goal is to maximize wealth while keeping the conditional value at risk (CVaR) below a desired threshold. We characterize the asymptomatically optimal risk-adjusted performance and present an investment strategy whose portfolios are guaranteed to achieve the asymptotic optimal solution while fulfilling the desired risk constraint. We also numerically demonstrate and validate the viability of our method on standard datasets.
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