We provide a model-free framework for studying the dynamics of the state vector and its risk prices. Specifically, we derive a frequency domain decomposition of the unconditional asset return premium in a general setting with a log-affine stochastic discount factor (SDF). Importantly, we show that the cospectrum between returns and the SDF only displays frequency dependencies through the state vector and that its dynamics and risk prices can be inferred from covariances between asset (portfolio) returns, that is, from the cross-section. Empirically, we find low and high-frequency state vector risk to be differentially priced for US equities.

中文翻译：

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频率相关风险

我们提供了一个无模型的框架来研究状态向量及其风险价格的动态。具体来说，我们在一般情况下使用对数仿射随机贴现因子（SDF）得出无条件资产收益溢价的频域分解。重要的是，我们证明了收益和SDF之间的共同谱仅通过状态向量显示了频率依赖性，并且可以从资产（投资组合）收益之间的协方差（即从横截面）推断出其动态和风险价格。从经验上看，我们发现低频和高频状态向量风险对于美国股票而言具有不同的价格。