We show that when asset returns satisfy a location-scale property (possibly conditionally as e.g. for a multivariate generalized hyperbolic distribution) and the investor has law-invariant and increasing preferences, the optimal investment portfolio always exhibits two-fund or three-fund separation. As a consequence, we recover many of the three-fund (and two-fund) separation theorems that have been derived in the literature under very specific assumptions on preferences or distributions. These are thus merely special cases of the general characterization result for optimal portfolios that we provide.

中文翻译：

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二基金或三基金分离定理何时成立？

我们表明，当资产回报满足位置尺度属性（可能有条件地，例如对于多元广义双曲线分布）并且投资者具有不变性和递增偏好时，最佳投资组合总是表现出两基金或三基金的分离。因此，我们恢复了许多三基金（和两基金）分离定理，这些定理是在关于偏好或分布的非常具体的假设下在文献中推导出来的。因此，这些只是我们提供的最佳投资组合的一般特征结果的特例。