异尺度与多重分形下的投资组合分配
Portfolio Allocation under Heterogeneous Scales and Multifractality
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中文总结 AI 辅助
该研究针对金融信号的异尺度与多重分形特性，构建基于MFCCA的投资组合分配模型，可降低多资产的回撤、VaR等风险且不损失收益，适用于多尺度耦合的复杂系统资源分配。
中文摘要 AI 辅助
金融信号间的交叉相关性既非无尺度也非与振幅无关：它们随测量的时间尺度和主导平均值的波动幅度而变化。我们利用该结构构建投资组合分配模型，其中风险函数是多重分形交叉相关分析（MFCCA）的带符号波动函数，由尺度$s$和波动阶数$q$索引。与MFDCCA类准则在聚合前校正局部去趋势协方差不同，MFCCA保留其符号，因此同向和反向变动成分对风险的贡献符号相反；当$q=2$时，所得二次型与投资组合序列自身的去趋势波动函数一致，将均值-方差准则恢复为尺度依赖极限。利用双成分ARFIMA和马尔可夫切换多重分形过程，我们表明规定的多尺度和多重分形相关性会传递到最优权重，且符号保留比聚合波动阶数更有助于降低尾部风险。应用于金融多资产时，该准则在每个要求的收益率下，样本内和样本外均相对于均值-方差基准降低了最大回撤、风险价值（VaR）和预期缺口，且实现的投资组合收益率无损失。该构造将带符号的多尺度相互作用结构映射到资源分配决策，适用于任何组件跨异尺度、振幅依赖耦合的复杂系统。
英文摘要
Cross-correlations between financial signals are neither scale-free nor amplitude-independent: they vary with the time scale over which they are measured and with the magnitude of the fluctuations that dominate the average. We exploit this structure to construct a portfolio allocation model in which the risk functional is the signed fluctuation function of multifractal cross-correlation analysis (MFCCA), indexed by a scale $s$ and a fluctuation order $q$. Unlike MFDCCA-type criteria, which rectify local detrended covariances before aggregation, MFCCA retains their sign, so that co-moving and counter-moving components contribute to risk with opposite signs; for $q=2$ the resulting quadratic form coincides with the detrended fluctuation function of the portfolio series itself, recovering the mean--variance criterion as a scale-dependent limit. Using two-component ARFIMA and Markov-switching multifractal processes, we show that prescribed multiscale and multifractal dependence is transmitted into the optimal weights, and that sign preservation contributes more to the reduction of tail risk than aggregation over fluctuation orders. Applied to financial multi-assets, the criterion lowers drawdown, Value-at-Risk, and expected shortfall relative to the mean--variance benchmark at every required return, in and out of sample, without any loss in realized portfolio return. The construction maps signed multiscale interaction structures onto resource-allocation decisions, and applies to any complex system whose components interact across heterogeneous scales with amplitude-dependent coupling.
