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Pricing for Profit share in group

business a simple model

7th GCA

Delhi

February 2005

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What is profit share?

A mechanism whereby an insurer allows a group scheme

to share in any positive financial result of the insurance in

a predefined way.

Example profit share formula:

Profit = 50% * ( 90% Premiums Claims)

Premiums must be earned and claims incurred

Losses generally not carried forward

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Expenses + Margin protection

Insurer must be sure that in the event of a profit refund,

sufficient margin exists to meet the expenses and capital

costs of the insurer, in particular:

net premium expenses

- i.e expenses proportionate to risk cost

per mille expenses

- i.e. expenses/capital charges based on sum@risk

- eg solvency margin

gross premium expenses, e.g. commissions

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Profit commission characteristics

PC = PC% * (Premium - Claims - Expenses)

e.g. 50% ( 90% Premium Claim)

Layers - structures are possible where greater levels of

profit include greater levels of return

If the expense deduction in the PC formula is less than the

insurers requirement, there is a dramatic impact on the PC

loading

EXPENSES SHOULD INCLUDE COST OF CAPITAL

Longer loss carry forward periods reduce PC loadings

However, schemes that have losses are not persistent

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Profit Distribution

ProfitDistribution

curve

0

Expected Margin

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Profit curve is a function of

Deviation in premium from that expected

as a result of changes in group demographics

Deviation in the number and amount of claims

Number of claims (k) can be modelled using a poisson

distribution as:

Claim amounts can be modelled as lognormal

Sum of k lognormal claims can be modelled using Box-

Muller approach.

!

)np(np

k

ek

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Profit Commission

ProfitDistribution

curve

0

Expected Margin

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Profit commission

0

Expected Margin

ProfitDistribution

curve

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Profit commission

0

Profit Share

Expense allowance

ProfitDistribution

curve

Profit share

amount

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Profit commission

0

Additional

premium

Revised

Profit

Distribution

curve

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New Profit share

amount

Profit commission

0

Revised

ProfitDistributioncurve

Profit Share

Expense allowance

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Profit commission

0

Revised

ProfitDistributioncurve

Previously paidProfit share

amount

Profit Share

Expense allowance

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Profit commission

0

Revised

ProfitDistributioncurve

Additional premium

previously added

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Profit commission

An iterative processProfitcommission

0

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Profit commission - Size of

portfolio

Profit

commission

0

Profit

commission

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Profit commission - Size of

portfolio

For a larger portfolio

profit commission

has a much smaller

impact on reinsuranceprice

Profit

commission

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A model interfaceSimple Group Profit share calculator

Portfolio Parameters Basic Premium parameters

Reinsured Portfolio Size 2,000 Best Estimate Claim Rate 1.000 per mille

Average Sum Insured 200,000 Per Mille expense 0.30Std Dev Sum Insured 100,000 Net Premium Loading 5.00%

Gross Premium Loading 7.00%

Profit Share Formula a(eP-C) Final Gross Premium 1.826

Percentage Profit (a) 50% Simulation parameters

Percentage Expenses (e) 85%

Number of Simulations 40,000

Calculated WP Loading 25.76% Interest 0.0%

Calculate loading