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# Measuring long tailed and surplus risk

07 Sep 16 03:20

Insurance economists and risk scientists are more familiar with the concepts of risk super and sub additivity in context of portfolio risk measures. In our previous note, we looked at how the statistical concept of sub-additivity is applied in pricing of an aggregate umbrella insurance product: https://openminds.swissre.com/stories/1079/ We can take the discourse one level up to the micro-economics of the (re)insurance firm and the same principles will still hold. In a case study format we will have the following typical story. In a notional economy with {1 … to … N} insurance policy holders at time T, there are {1 … to … N} insurance firms and each insurance firm prices and holds only one insurance policy. In the next time period T+1 there is only one insurance firm left in the economy, which now holds all insurance policies {1 … to … N}. Then if there are no other market regime changes between time (T) and time (T+1) for insurance policies premiums we should have:

The same equation can be expressed for the premiums of the insurance firms at time (T) and (T+1), in a more familiar sub-additive form

The statistical mechanics, which justify these inequalities are the same as we derived in our earlier umbrella pricing example. With the two bounding extremes being full additivity i.e. full dependence, modeled with 100% correlations and full independence modeled with 0% correlations. The business and market mechanics in principle are of similar nature as well. If all insurance premiums in an economy are priced and held by a single firm, the sum of these premiums should be less than the sum of premiums as if each one was priced and held by a different firm. A single insurance firm is more efficient in scale than having as many insurance firms as insurance policy holders. Economies of scale in claims management, administration, and brokerage and particularly diversification of risk across geography justify this market condition.

In this note we look at a less intuitive and a harder to defend proposition. Under certain conditions risk and premium become super-additive, at the level of aggregate products, and for the whole (re)insurance firm’s portfolio. We can express super-additivity for premiums and TVaR (tail value at risk) as:

TVaR [whole Portfolio] > TVaR [Risk 1] + … + TvaR [Risk N]

The market case study statement would be that if a portfolio of (re)insurance risks is priced as one single accumulated risk, and the risk measure of TVaR is taken, super-additivity gives that, it is larger than the sum of single premiums and TVaR(s) measures computed and taken for each single risk separately. Tail Value at Risk is sub-additive when it is a coherent risk measure, while VaR is not guaranteed to be such. There is no shortage of literature deriving the requirements and proofs for both conditions. We are interested in one particular exception called ‘extreme aggregation’ or ‘distorted aggregation’.

Mathematicians have proved that if there is a change of regime in expectation, i.e. ‘distorted expectation’, the aggregation of TVaR and policy premiums can become super-additive. The theoretical studies are important because they frame the fundamental possibility of super additivity. At least as important as the theoretical framework is the best industry practices’ interpretation of change in the expectations’ regime which would lead to a distorted expectation, which in turn would cause the extreme aggregation effect of super-additivity. To be very clear we are viewing distortion within one temporal measure, within one time period. Changes of expectation between economic and underwriting cycles is a different type of problem. The ‘distortion’ or fundamental change in regime is between measuring and pricing risk for each single component individually and aggregating them to the portfolio profile versus aggregating all risk factors and measuring and computing a combined risk metric and a total portfolio premium. Because the risk is static in a single time period the distortion can develop due to the behavior of tail risk after some unknown, but measurable, low likelihood thresholds.

As modelers and practitioners our first task is to identify the industry and market conditions under which such extreme ‘distorted’ effects are possible. Long – tailed lines of business, contingent BI policies, and surplus and specialty lines come to mind - perhaps a topic for another technical note.

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