Advancements in Territorial Ratemaking Allocating Cost of Catastrophe Exposure May 2006 CAS Spring Meeting Stephen Fiete

2  Cost of risk transfer = Expected Gross Loss + Transaction Expenses + Net Cost of Capital + Cost of Reinsurance  Cost of capital = expected ROE * (required capital)  Cost of reinsurance = Reinsurance premium – Ceded AAL  Territorial Ratemaking usually allocates loss then uniformly loads rates for expense and profit provision. 1 The Components of Cost

3  Need methods to to geographically allocate cost of capital and net cost of reinsurance.  Concentration of risk increases both Reinsurance and Capital Cost.  “The Concentration Charge: Reflecting Catastrophe Exposure Accumulation in Rates” by Mango gives one method of rating for risk concentration.  Focus on social need to replenish capital after a large event.  This presentation bases charge on reinsurance costs and allocation of capital to layers.  How does the industry manage concentration costs now?  Underwriting restrictions in cat prone areas.  High minimum deductibles.  Reinsurance.  The first two result in supply shortage. What will induce a change? 1 Cost of Concentration

4  Give us a probability distribution of loss cost for any portfolio of risks  Initial, and still most common, questions answered:  Do we have enough assets to stay solvent? (PML’s)  Are we charging the right rate? (AAL’s)  User still needs to develop a rating plan using these results.  Reinsurance and capital cost allocation varies with concentration and hazard level.  Consider other interactions too. 1 Cat Models

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6  Two steps to geographical rating.  Cat Model.  Portfolio loss by event.  Loss by event for a given risk with a location and construction characteristics defined.  Calculate an allocation of net reinsurance and capital cost for the given risk. 1 Cat Models

7 A hypothetical company insures only wind loss and sets its capital requirement to be the 250 year PML. Consider cost of two individual risks both are $200,000 of TIV: 1 How Capital Cost Varies by Risk A Super-Simple Example RiskAALMarginal Effect on 250 PML Cost of CapitalPremium Indication A$5,000$200,000$30,000$50,000 B$5,000$00$7,143 Risk A is right in the track of the defining 250 year event. Risk B is located remotely. Cost of capital is assumes a 15% ROE target. Premium indication assumes a 30% expense ratio.

8  Other methods to determine capital requirement exist, but any good method should account for correlation of loss between risk (concentration in this case).  The previous calculation was for the marginal cost on top of an existing portfolio. We can also do an average allocation method.  The cost of the extra $200k of PML is allocated to other risks in the portfolio.  We will still get a very high capital cost for risk A.  Because losses between different risks are correlated adding or removing one effects cost of all of the others in the portfolio. 1 Cost of Capital by Risk

9  Net cost of reinsurance can by allocated to layers.  Useful to think of cost in proportion to expected loss.  “Margin Factor” is the unit for net reinsurance cost. 1 Catastrophe Reinsurance Cost Allocation

10  Underlying premise: Reinsurance cost is proportional to AAL within layer.  To the extent a given risk contributes to AAL in a layer, it contributes to reinsurance cost in that layer.  AAL contribution is measured for each simulated event and then summed together.  Example. Say an event has a $500M portfolio loss and a $1000 loss for a given risk and has a frequency of The event loss allocates to retained, first level, and second level reinsurance as 20%, 60% and 20% respectively.  Allocate the individual risk loss with $600 and $200 in the first two reinsurance layers.  Cost of reinsurance for this event is: .0003*(600* *1.5) 1 Catastrophe Reinsurance Cost Allocation

11  Capital and Reinsurance serve the same purpose – Solvency protection.  Need to allocate “catastrophe” surplus.  Not a trivial matter.  Does not have an “accounting reality”, but still useful.  Multi-line companies which set profit targets by line are making similar allocations – implicitly or explicitly.  Useful for ERM.  Method of allocation is similar.  Return on capital is the same in every layer, but the amount of capital required in proportion to expected loss increases at higher layers. 1 Catastrophe Capital Cost Allocation

12 1 Catastrophe Capital Cost Allocation Find x,y such that y>5, x<1.3, and x+y balance to targeted net profit margin.

13  How much does a given risk contribute to AAL in each layer?  Loss from given risk contributes to portfolio loss by event.  Use Margin Factors to calculate contributed margin by layer by (simulated) event for a given risk.  The cat model directly gives gross AAL for the risk.  This is an allocation of cost for a prospective risk, not necessarily an allocation of cost to existing risks.  “Allocation” is forward looking.  Reinsurance premium does not change during the year due to exposure redistributions.  Over time reinsurance rates change to reflect exposure shifts.  Lag between occurrence of reinsurance cost and recovery in rates introduces some risk in the process of reinsurance cost recovery.  Exact match works when there is no exposure distribution shift, and reinstatement premiums come in at EV.  Reinsurance cost and amount of cat exposure are both quantities with risk related to loss cost. Recovering costs is an actuarial, not accounting, practice. 1 Putting it Together

14 Homeowners AAL Miami-Dade County Contour Ranges: $2.50 to $15 per $1000 of Coverage A

15 Homeowners Total Cost Contours Contour Ranges: $7.50 to $45 per $1000 of Coverage A