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 – Applied to DFA Insurance Company – Additional realistic examples –Primary insurer –Reinsurer – 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 doesn’t pay to diversify.

Strategy – Reinsurance Examples – Excess of Loss –Coinsurance provisions –Treatment of ALAE –Stacked contracts with various inuring provisions Reinsurance costs money You can buy too much reinsurance. There are often a lot of messy details to be worked out. Pretty Good

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 E[b] = 1 and Var[b] = b. For each line h, multiply each loss by b.

Multiple Line Parameter Uncertainty A simple, but nontrivial example E[b] = 1 and Var[b] = 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 Insurer’s 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  (X+Y) ≤  (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 insurer’s entire portfolio of losses.  (X) = Total Required Assets Capital determined by the risk measure. C = r(X) - E[X]

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 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 prospective Reserve for one year settled Reserve for two years settled Reserve for 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

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.

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 InsurerSmall 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.