The Cost of Financing Insurance Glenn Meyers Insurance Services Office Inc. CAS Ratemaking Seminar March 28, 2002

Third Time at CAS Ratemaking Seminar 2001 – Proof of concept http://www.casact.org/pubs/forum/00sforum/meyers/index.htm 2002 – Applied to DFA Insurance Company http://www.casact.org/pubs/forum/01spforum/meyers/index.htm 2003 – More examples available Live demo Primary insurer Reinsurer

Set Profitability Targets for an Insurance Company The targets must reflect the cost of capital needed to support each division's contribution to the overall underwriting risk. The insurer's risk, as measured by its statistical distribution of outcomes, provides a meaningful yardstick that can be used to set capital requirements.

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

Additional Considerations Reinsurance Reduces the need for capital Is the cost of reinsurance less than the cost of capital it releases? How long the capital is to be held 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.

Additional Considerations 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. Express the result in terms of a “Target Combined Ratio” Is reinsurance 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. r(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.

Sum of Marginal Capitals is  Capital 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. Appropriate for stock insurers. It is also easy.

Ways to Allocate Capital #2 Average marginal capital, where average is taken over all entry orders. Shapley Value Economic justification - Game theory Appropriate for mutual insurers

Ways to Allocate Capital #3 Line headed by CEO’s kid brother gets the marginal capital. Gross up all other lines. Economic justification - ???

Reference The Economics of Capital Allocation The paper: 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.

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

Demonstration of Software OK – Now that we see that the “Cost of Financing” can be quickly implemented, lets look at screen shots.

Demo Will Cover Aggregate loss calculation Capital allocation Evaluating Reinsurance Programs Cat reinsurance Other reinsurance Show the effect of the size of insurer

Input Separate input by contract, line of business, state, branch office etc. Expected Loss Per claim limit, retention and coinsurance Input supplied by ISO Claim severity distributions Claim count distribution parameters Covariance generators User can replace ISO information with its own information.

ISO Severity Distributions User Severity Distribution

Adjust Severity By Scale Factor User Supplies Expected Loss

We need to know how long allocated capital will be held. Up to 7 Years

Output Insurer aggregate loss distribution Calculates mean and standard deviation Calculates Value at Risk (VaR) Calculates Tail Value at Risk (TVaR) Used to derive needed capital Compare needed capital for different reinsurance strategies

Various Risk Measures Cat Re over $30M XS Re over $1M

Cat Re over $30M No Other Re

No Reinsurance

Allocate Capital In this demo we allocate capital in proportion to marginal TVaR99% Calculate TVaR99% with each line/reserve removed Adjust by constant of proportionality

Note capital is allocated to loss reserves

Cost of Financing Insurance = Cost of Capital + Net Cost of Reinsurance User input Target return on equity Cost of reinsurance Return on investments Insurer expense factors Objectives Evaluate reinsurance strategy Set underwriting targets

List of Reinsurance Strategies

Allocated capital in current and future accident years

Schedule for returning capital

Cat Limit - $30M Aggregate Other Line Limits - $1M Occurrence

Cat Limit - $30M Aggregate Other Line Limits - None

Other Line Limits - None Cat Limit - None Other Line Limits - None

Standard Ratemaking Exhibit Scroll to end –>

Cost of Financing Target Combined Ratio

The Effect of Insurer Size Multiply all exposures by 20 Non-cat lines → multiply all expected claim counts by 20 and keep same limits Cat lines → multiply all claim amounts by 20, including the limit Examine reinsurance strategy No reinsurance Only cat reinsurance Cat + other reinsurance

Small - $35,387,970

Small - $25,658,154 Best strategy for big

Best strategy for small

Note Differences Cost of financing is not proportional ( x 20) No Re Small - $35,387,970 Big - $672,790,192 Cat Re Small - $25,658,154 Big - $294,513,255 All Re Small - $25,176,198 Big - $497,733,092 Best reinsurance strategies are different.

Reinsurance Capacity Charges Generally speaking, the same principles apply Capacity charge is proportional to marginal cost of capital over a reference portfolio. Reference – “The Aggregation and Correlation of Reinsurance Exposure” By Glenn Meyers, Fred Klinker, and David Lalonde To be presented at CARe meeting in June

Establish a Reference Portfolio Represents current business Use as a base for calculating Marginal Capital. Marginal Capital = Capital needed for reference portfolio + new contract less the capital needed for the reference portfolio

Capacity Charge Proportional to marginal capital For long-tailed contracts, capital is released over time. Earn reinsurer’s target return on capital as long as capital is being held.

Casualty Insurance Examples

Explain Differences Capacity charges increase with Higher limits More volatility How long you have to hold capital

Capacity Charge for Cat Covers

Explain Differences Contract that pays when rest of contracts also pay are less desirable. Correlation

Scatter Plot for Contract A Capacity Charge = 102% of Expected Loss

Scatter Plot for Contract C Capacity Charge = 33% of Expected Loss

Summary We have demonstrated How to calculate required capital How to evaluate reinsurance strategies How to calculate target combined ratios that take capital management strategies into account How to calculate capacity charges for reinsurers.