Advancements in Territorial RatemakingAllocating Cost of Catastrophe Exposure

May 2006

CAS Spring MeetingStephen Fiete

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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.

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The Components of Cost

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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?

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Cost of Concentration

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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.

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Cat Models

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Masonry/Frame Relativty vs. Frame AAL

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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.

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Cat Models

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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:

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How Capital Cost Varies by RiskA Super-Simple Example

Risk AAL Marginal Effect on 250 PML

Cost of Capital Premium Indication

A $5,000 $200,000 $30,000 $50,000

B $5,000 $0 0 $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.

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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.

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Cost of Capital by Risk

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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.

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Catastrophe Reinsurance Cost Allocation

ReinsExpected Reins

Limit x Attach Premium Margin Margin($m) ($m) ($m) ($m) Factor

Retained 1,000 5100 900 5 30 25 5.0100 800 15 68 53 3.5200 600 50 160 110 2.2200 400 100 250 150 1.5300 100 130 300 170 1.3

Retained 100 0 80

Rei

nsu

red

L

ayer

sLayer

AALwithin layer

($m)

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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 .0003. 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*1.3+200*1.5)

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Catastrophe Reinsurance Cost Allocation

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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.

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Catastrophe Capital Cost Allocation

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Catastrophe Capital Cost Allocation

ReinsExpected Reins

Limit x Attach Premium Margin Margin($m) ($m) ($m) ($m) Factor

Retained 1,000 5 y100 900 5 30 25 5.0100 800 15 68 53 3.5200 600 50 160 110 2.2200 400 100 250 150 1.5300 100 130 300 170 1.3

Retained 100 0 80 x

Rei

nsu

red

L

ayer

sLayer

AALwithin layer

($m)

Find x,y such that y>5, x<1.3, and x+y balance to targeted net profit margin.

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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.

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Putting it Together

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Homeowners AAL Miami-Dade County

Contour Ranges: $2.50 to $15 per $1000 of Coverage A

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Homeowners Total Cost Contours

Contour Ranges: $7.50 to $45 per $1000 of Coverage A