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# Competing on umbrella premiums with technical bounds 18 Aug 16 03:28

Premium [risk 1 + risk 2] < Premium [risk 1] + Premium [risk 2]

In our case Premium[risk 1 + risk 2] is the price of the combined commercial umbrella product. Next we convert the single risk premiums to their most basic expected value and standard deviation pricing forms
Premium [risk 1] = Expected Loss[risk1] + Std. Dev[risk1]
Premium [risk 2] = Expected Loss[risk2] + Std. Dev[risk2]

We put them together:
Premium [risk 1] + Premium [risk 2] = Expected Loss[risk1] + Std. Dev[risk1] + Expected Loss[risk2] + Std. Dev[risk2]

Then reorder and combine the linear expected mean loss:
Premium [risk 1] + Premium[risk 2] = Expected Loss[risk1 + risk2] + Std. Dev[risk1] + Std. Dev[risk2]

Now combine standard deviations by the square root of variances principle:
Premium [risk 1] + Premium [risk 2] = Expected Loss[risk1 + risk2] + square root{variance[risk1] + variance[risk2] + 2*COV[risk1: risk2]}

Now this is also the pure technical premium of a combined umbrella product:
Premium [risk 1 + risk 2] = Expected Loss[risk1 + risk2] + square root{variance[risk1] + variance[risk2] + 2*COV[risk1: risk2]}

Finally we express the Covariance (COV) between both premiums with a correlation factor (COR):
Premium [risk 1 + risk 2] = Expected Loss[risk1 + risk2] + square root{variance[risk1] + variance[risk2] + 2*COR* Std. Dev[risk1] + Std. Dev[risk2]}

Two statements are evident here: Full dependence modeled with a 100% correlation supports additivity:
Premium[risk 1 + risk 2] = Premium[risk 1] + Premium[risk 2]

Less than full dependence, let us say expressed with 30% correlation, or full independence expressed with 0% correlation, supports sub-additivity
Premium[risk 1 + risk 2] < Premium[risk 1] + Premium[risk 2]

The underwriting and insurance product structuring conclusion is straight forward from here. When underwriting a combined umbrella product for fully dependent risks, in very close geography, same LOB and same peril - the sum of single risk premiums priced independently should approach the price of the aggregated umbrella product. While underwriting an umbrella product for risks which are independent by distance, geography, LOB and peril, the price of the combined product could be less than the sum of single risk premiums. There is an opportunity for premium cost savings with independence.  Independence could be also interpreted as diversification in market share geography, LOB, and insured peril.  So with diversification, lower management cost, and the statistics of premium accumulation, insurers are well justified to be flexible on aggregate product pricing. However the reality of pricing is not always sub-additive. It would have been much easier and simpler if it was. Sometimes it is super-additive:
Premium[risk 1 + risk 2] > Premium[risk 1] + Premium[risk 2]
Super-additivity states that the price of risk of a combined product needs to be larger than the sum of unit risk prices of its components. What are the conditions for such pricing philosophy to be adopted is clearly a topic for further research.

Category: Climate/natural disasters: Climate change, Earthquakes, Floods/storms

Location: New York, NY, United States