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

2. Third Time at CAS Ratemaking Seminar
2001 – Proof of concept
2002 – Applied to DFA Insurance Company
2003 – More examples available
Live demo
Primary insurer
Reinsurer

3. 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.

4. Volatility Determines Capital Needs Low Volatility

5. Volatility Determines Capital Needs High Volatility

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

7. 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:

8. 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.

9. Multiple Line Parameter Uncertainty A simple, but nontrivial example
E[b] = 1 and Var[b] = b

10. Low Volatility b = r = 0.50

11. Low Volatility b = r = 0.75

12. High Volatility b = r = 0.25

13. High Volatility b = r = 0.45

14. 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.

15. Correlation and Capital b = 0.00

16. Correlation and Capital b = 0.03

17. 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.

18. Additional Considerations
Investment income generated by the insurance operation
Investment income on loss reserves
Investment income on capital

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

20. 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?

21. 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.

22. 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.

23. 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]

24. 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.

25. 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.

26. 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.

27. 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:

28. 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.

29. 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

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

31. 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.

32. 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

33. Step 3 Reinsurance
Skip this for now

34. 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.

35. 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

36. Step 4 The Cost of Financing Insurance

37. Step 4 The Cost of Financing Insurance

38. 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.

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

40. Step 4 Without Risk Transfer The Cost of Financing Insurance

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

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

43. 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.

44. ISO Severity
Distributions
User Severity
Distribution

45. Adjust Severity
By Scale Factor
User Supplies
Expected Loss

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

47. 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

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

49. Cat Re over $30M
No Other Re

50. No Reinsurance

51. 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

52. Note capital is
allocated to
loss reserves

53. 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

54. List of Reinsurance
Strategies

55. Allocated capital in current and
future accident years

56. Schedule for
returning capital

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

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

59. Other Line Limits - None
Cat Limit - None
Other Line Limits - None

60. Standard Ratemaking
Exhibit
Scroll to end –>

61. Cost of
Financing
Target
Combined
Ratio

62. 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

63. Small - $35,387,970

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

65. Best strategy for small

66. 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.

67. 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

68. 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

69. 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.

70. Casualty Insurance Examples

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

72. Capacity Charge for Cat Covers

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

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

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

76. 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.