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The Cost of Financing Insurance with Emphasis on

Reinsurance

- Glenn Meyers
- ISO
- CAS Ratemaking Seminar
- March 10, 2005

Fifth Time at CAS Ratemaking Seminar

- 2001 Proof of concept
- http//www.casact.org/pubs/forum/00sforum/meyers/i

ndex.htm - 2002 Applied to DFA Insurance Company
- http//www.casact.org/pubs/forum/01spforum/meyers/

index.htm - 2003 Additional realistic examples
- Primary insurer http//www.casact.org/pubs/forum/0

3sforum/03sf015.pdf - Reinsurer http//www.casact.org/pubs/forum/03spfor

um/03spf069.pdf - 2004 No new papers
- 2005 Emphasis on Reinsurance

Underlying Themes

- The insurer's risk, as measured by its stochastic

distribution of outcomes, provides a meaningful

yardstick that can be used to set capital

requirements. - Risk ? Capital ? Costs money.
- Develop strategy to make most efficient use of

capital.

Strategy Diversification

- Examples
- Increase volume / Law of large numbers
- Manage concentrations in property insurance
- Decide where to grow and/or shrink
- Costs money to diversify

At some point, it doesnt pay to diversify.

Strategy Reinsurance

- Examples Excess of Loss
- Coinsurance provisions
- Treatment of ALAE
- Stacked contracts with various inuring provisions
- Reinsurance costs money

Pretty Good

- You can buy too much reinsurance.
- There are often a lot of messy details to be

worked out.

Outline of Insurance Strategy

- Grow in lines of business where risk is adequate

rewarded. - Shrink in lines of business where risk is not

adequately rewarded. - Diversify when cost effective.
- Buy reinsurance when cost effective.

Volatility Determines Capital Needs Low Volatility

Volatility Determines Capital Needs High

Volatility

Additional Considerations

- Correlation
- If bad things can happen at the same time, you

need more capital. - We will come back to this shortly.

The Negative Binomial Distribution

- Select ? at random from a gamma distribution with

mean 1 and variance c. - Select the claim count K at random from a Poisson

distribution with mean ???. - K has a negative binomial distribution with

Multiple Line Parameter Uncertainty

- Select b from a distribution with Eb 1 and

Varb b. - For each line h, multiply each loss by b.

Multiple Line Parameter Uncertainty A simple,

but nontrivial example

Eb 1 and Varb b

Low Volatility b 0.01 r 0.50

Low Volatility b 0.03 r 0.75

High Volatility b 0.01 r 0.25

High Volatility b 0.03 r 0.45

About Correlation

- There is no direct connection between r and b.
- Small insurers have large process risk
- Larger insurers will have larger correlations.
- Pay attention to the process that generates

correlations.

Correlation and Capital b 0.00

Correlation and Capital b 0.03

Calculating an Insurers Underwriting Risk

- Use the collective risk model.
- Separate claim frequency and severity analysis
- For each line of insurance
- Select a random claim count.
- Select random claim size for each claim.
- The aggregate loss for all lines sum of all the

random claim amounts for all lines. - Reflect the correlation between lines of

insurance.

Consider the Time Dimension

- How long must insurer hold capital?
- The longer one holds capital to support a line of

insurance, the greater the cost of writing the

insurance. - Capital can be released over time as risk is

reduced. - Investment income generated by the insurance

operation - Investment income on loss reserves
- Investment income on capital

The Cost of Financing Insurance

- Includes
- Cost of capital
- Net cost of reinsurance
- Net Cost of Reinsurance
- Total Cost Expected Recovery

The To Do List

- Allocate the Cost of Financing back each

underwriting division. - Calculate the cost of financing for each

reinsurance strategy. - Which reinsurance strategy is the most cost

effective?

Doing it - The Steps

- Determine the amount of capital
- Allocate the capital
- To support losses in this accident year
- To support outstanding losses from prior accident

years - Include reinsurance
- Calculate the cost of financing.

Step 1 Determine the Amount of Capital

- Decide on a measure of risk
- Tail Value at Risk
- Average of the top 1 of aggregate losses
- Example of a Coherent Measure of Risk
- Standard Deviation of Aggregate Losses
- Expected Loss K ? Standard Deviation
- Both measures of risk are subadditive
- ?(XY) ?(X) ?(Y)
- i.e. diversification reduces total risk.

Step 1 Determine the Amount of Capital

- Note that the measure of risk is applied to the

insurers entire portfolio of losses. - r(X) Total Required Assets
- Capital determined by the risk measure.
- C r(X) - EX

Step 2 Allocate Capital

- How are you going to use allocated capital?
- Use it to set profitability targets.

- How do you allocate capital?
- Any way that leads to correct economic decisions,

i.e. the insurer is better off if you get your

expected profit.

Better Off?

- Let P Profit and C Capital. Then the insurer

is better off by adding a line/policy if

? Marginal return on new business ? return

on existing business.

OK - Set targets so that marginal return on

capital equal to insurer return on Capital?

- If risk measure is subadditive then
- Sum of Marginal Capitals is ? Capital
- Will be strictly subadditive without perfect

correlation. - If insurer is doing a good job, strict

subadditivity should be the rule.

OK - Set targets so that marginal return on

capital equal to insurer return on Capital?

If the insurer expects to make a return, e P/C

then at least some of its operating divisions

must have a return on its marginal capital that

is greater than e. Proof by contradiction If

then

Ways to Allocate Capital 1

- Gross up marginal capital by a factor to force

allocations to add up. - Economic justification - Long run result of

insurers favoring lines with greatest return on

marginal capital in their underwriting.

Reference

- The Economics of Capital Allocation
- By Glenn Meyers
- Presented at the 2003 Bowles Symposium

http//www.casact.org/pubs/forum/03fforum/03ff391.

pdf - The paper
- Asks what insurer behavior makes economic sense?
- Backs out the capital allocation method that

corresponds to this behavior.

Ways to Allocate Capital 2

- Average marginal capital, where average is taken

over all entry orders. - Shapley Value
- Economic justification - Game theory
- Additive co-measures Kreps
- Capital consumption Mango

Remember the time dimension. Allocate capital to

prior years reserves.

- Target Year 2003 - prospective
- Reserve for 2002 - one year settled
- Reserve for 2001 - two years settled
- Reserve for 2000 - three years settled
- etc

Step 3 Reinsurance

- Skip this for now

Step 4 The Cost of Financing Insurance

- The cash flow for underwriting insurance
- Investors provide capital - In return they
- Receive premium income
- Pay losses and other expenses
- Receive investment income
- Invested at interest rate i
- Receive capital as liabilities become certain.

Step 4 The Cost of Financing Insurance

- Net out the loss and expense payments
- Investors provide capital - In return they
- Receive profit provision in the premium
- Receive investment income from capital as it is

being held. - Receive capital as liabilities become certain.
- We want the present value of the income to be

equal to the capital invested at the rate of

return for equivalent risk

Step 4 The Cost of Financing Insurance

Step 4 The Cost of Financing Insurance

Back to Step 3 Reinsurance and Other Risk

Transfer Costs

- Reinsurance can reduce the amount of, and hence

the cost of capital. - When buying reinsurance, the transaction cost

(i.e. the reinsurance premium less the provision

for expected loss) is substituted for capital.

Step 4 with Risk Transfer The Cost of Financing

Insurance

The Allocated should be reduced with risk

transfer.

Step 4 Without Risk Transfer The Cost of

Financing Insurance

Examples

- Use ISO Underwriting Risk Model
- Parameterization based on analysis of industry

data. - Big and small insurer
- Big Insurer is 10 x Small Insurer
- Three reinsurance strategies

Expected Loss for small insurer is 10 times less,

Various Risk Measures

Various Risk Measures

Different measures of risk imply different

amounts of capital

Allocating (Cost of) Capital

- Calculate marginal capital for each profit

center. - Calculate the sum of the marginal capitals for

all capital centers. - Diversification multiplier equals the total

capital divided by the sum of the marginal

capitals. - Allocated capital for each profit center equals

the product of the diversification multiplier and

the marginal capital for the profit center.

Diversification Benefit

Note capital is allocated to loss reserves

Optimizing Reinsurance

- User input
- Target return on capital
- Return on investments (sensitivity analysis on

investment income) - Corporate income tax rate
- Cost of reinsurance
- Insurer expense provisions

List of Reinsurance Strategies

Cost of Financing Insurance Cost of Capital

Net Cost of Reinsurance

- Cost of capital target return x capital
- Net cost of reinsurance
- Premium Expected Recovery
- Minimize the cost of financing.

(No Transcript)

Big Insurer Cost of Financing with No Reinsurance

Small Insurer Cost of Financing with No

Reinsurance

Big Insurer Cost of Financing with Cat

Reinsurance

Small Insurer Cost of Financing with Cat

Reinsurance

Big Insurer Cost of Financing with Cat

Reinsurance and XS of Loss Reinsurance

Small Insurer Cost of Financing with Cat

Reinsurance and XS of Loss Reinsurance

Optimize reinsurance by minimizing the cost of

financing

Big Insurer

Small Insurer

Note Small insurer costs multiplied by 10.

Discussion of Behavioral Issues

- Smooth out earnings Wall Street punishes shock

losses. - Question Cat limit to capital ratio?
- Answer 10 to 15.
- Impairment issues Can you raise additional

capital if you lose 1/3 of capital? - Silos Divisional incentives work against

corporate objectives.