The Cost of Financing Insurance with Emphasis on Reinsurance

1. The Cost of Financing Insurancewith Emphasis on Reinsurance Glenn Meyers ISO CAS Ratemaking Seminar March 10, 2005

2. Fifth 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 – Additional realistic examples • Primary insurer http://www.casact.org/pubs/forum/03sforum/03sf015.pdf • Reinsurer http://www.casact.org/pubs/forum/03spforum/03spf069.pdf • 2004 – No new papers • 2005 – Emphasis on Reinsurance

3. 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.

4. 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.

5. 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.

6. 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.

7. Volatility Determines Capital NeedsLow Volatility

8. Volatility Determines Capital NeedsHigh Volatility

9. Additional Considerations • Correlation • If bad things can happen at the same time, you need more capital. • We will come back to this shortly.

10. 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:

11. 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.

12. Multiple Line Parameter UncertaintyA simple, but nontrivial example E[b] = 1 and Var[b] = b

13. Low Volatilityb = 0.01 r= 0.50

14. Low Volatilityb = 0.03 r= 0.75

15. High Volatilityb = 0.01 r= 0.25

16. High Volatilityb = 0.03 r= 0.45

17. 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.

18. Correlation and Capitalb = 0.00

19. Correlation and Capitalb = 0.03

20. 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.

21. 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

22. The Cost of Financing Insurance • Includes • Cost of capital • Net cost of reinsurance • Net Cost of Reinsurance = Total Cost – Expected Recovery

23. 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?

24. 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.

25. 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.

26. 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]

27. Step 2Allocate 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.

28. 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.

29. 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.

30. 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:

31. 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.

32. 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.

33. 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

34. 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

35. Step 3Reinsurance • Skip this for now

36. Step 4The 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.

37. Step 4The 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

38. Step 4The Cost of Financing Insurance

39. Step 4The Cost of Financing Insurance

40. Back to Step 3Reinsurance 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.

41. Step 4 with Risk TransferThe Cost of Financing Insurance The Allocated $$ should be reduced with risk transfer.

42. Step 4 Without Risk TransferThe Cost of Financing Insurance

43. 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

44. Expected Loss for small insurer is 10 times less,

45. Various Risk Measures

46. Various Risk Measures

47. Different measures of risk imply different amounts of capital

48. 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.

49. Diversification Benefit

50. Note capital is allocated to loss reserves