Systematic risk in pools

In the realm of portfolio management, the focus lies on constructing a well-diversified portfolio to mitigate unsystematic risk, allowing for the identification and measurement of systematic risk for example through uni-factor, such as CAPM, and multi-factor, such APT, models. This approach is rooted in the belief that, with a sufficiently diversified portfolio, unsystematic risk in theory can be eliminated, making the remaining systematic risk more apparent. While diversification is the means to diversify the unsystematic risk in a portfolio management problem, pooling strategies, with a limited strategy of just expanding the pool members, necessitate a distinct approach to systematic risk. In such scenarios, the challenge lies in disentangling the impact of systematic factors from idiosyncratic influences within a pool. This paper explores the methodologies and considerations unique to pooling situations, shedding light on the complexities involved in identifying and quantifying systematic risk in a pool. In our effort to assess the concept of systematic risk in a pool, we adopt an approach that identifies the defining characteristics of systematic risks, which remain invariant regardless of the number of losses or any manipulations within a finite set of losses. To explore these principles, we find a framework of risk management on sequences in Banach lattices to be particularly suitable. In establishing these principles, we introduce the notion of “systematic compatibility”, signifying invariance to variations in finite changes within a sequence of losses. Consequently, we observe that while systematic risk often possesses an implicit representation in the risk space, it exhibits an explicit representation in the bi-dual space. Moreover, we introduce systematic compatible risk measures and establish their dual characterization. We demonstrate that risk measurement can naturally be represented as a split into a summation of systematic and unsystematic components. In practical applications, we employ these measures to address risk management problems, with a specific emphasis on risk pooling scenarios. In revisiting the traditional “principle of insurance” (POI), we propose an extension called the “principle of pooling” (POP). By showing that the principle of pooling holds if and only if the systematic risk is secure, we investigate this novel concept.