Pooling and Systematic Risk

This paper gives a general formulation of systematic risk in a risk pool and studies its relation with principle of insurance (POI), its extension principle of pooling (POP), and valuation. We will see that the systematic risk is secure if and only if POP holds. We call this proposition the fundamental theorem of pooling. Then, we show how the systematic risk can be measured in a natural way. We present a variety of examples within the family of typological common shock models, covering homogeneous and heterogeneous risk pools, where we can identify the systematic risk and measure it in di erent situations. Then, we take a fresh look at the risk valuation from a systematic risk perspective. First, we study the valuation of the ex-ante policies and see that they are not independent of the pool, and need to be adjusted according to the systematic relative safety loading. Second, we study the ex-post policies and see that while in nite pools the relative safety loading is contingent on the common shock, for in nite pools it vanishes. This yields a universal rule of valuation in in nite pools to be nothing but a fair risk-sharing allocation contingent on the systematic risk. Finally, we make an assessment of our theoretical models with reference to two real-world examples. First, we look at the UK Covid-19 job retention case and second we use our theory to study a catastrophic risk. We propose a novel perspective to analyze systematic events.