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To achieve robustness of risk across different assets, risk parity investingrules, a particular state of risk contributions, have grown in popularity overthe previous few decades. To generalize the concept of risk contribution fromthe simple covariance matrix case to the continuous-time case in which theterminal variance of wealth is used as the risk measure, we characterize riskcontributions and marginal risk contributions on various assets as predictableprocesses using the Gateaux differential and Doleans measure. Meanwhile, therisk contributions we extend here have the aggregation property, namely thattotal risk can be represented as the aggregation of those among differentassets and $(t,\omega)$. Subsequently, as an inverse target -- allocating risk,the risk budgeting problem of how to obtain policies whose risk contributionscoincide with pre-given risk budgets in the continuous-time case is alsoexplored in this paper. These policies are solutions to stochastic convexoptimizations parametrized by the pre-given risk budgets. Moreover,single-period risk budgeting policies are explained as the projection of riskbudgeting policies in continuous-time cases. On the application side,volatility-managed portfolios in [Moreira and Muir,2017] can be obtained byrisk budgeting optimization; similarly to previous findings, continuous-timemean-variance allocation in [Zhou and Li, 2000] appears to be concentrated interms of risk contribution.

Continuous-Time Risk Contribution of the Terminal Variance and its Related Risk Budgeting Problem