1. Risk Loads How they started Where they are Where they’re going Presentation to CANE by Glenn Meyers September 18, 1998

2. General Idea of Risk Loads Less risk is better For greater risk –Greater demand for transfer risk –Greater reluctance to accept risk –Higher price to transfer The problem — Quantify the risk

3. An Attempt to Quantify risk Buhlmann - 1970 Premium Calculation Principles Standard deviation principle –Risk Load =  Std. Dev[Loss] Variance principle –Risk Load =  Var[Loss] Expected utililty principle –U(Equity) = E[U(Equity + Premium - Loss)]

4. An Early (Late 70’s) Use of a Mathematical Formula ISO Increased Limits Ratemaking Used the Variance Principle –Reference — Miccolis (PCAS 1977) –Replaced judgmental risk loads.

5. Problem Answers were too high for high limits.

6. Response Variance  Standard Deviation

7. C apital A sset P ricing M odel Designed for pricing securities Where:

8. Interpretation of CAPM If Cov[R,R M ] = 0, then E[R] = R f The market does not reward one for taking diversifiable risks. Systematic risk is taken when Cov[R,R M ] > 0 The market only rewards those who take on systematic risk. My reaction --

9. Another Interpretation of CAPM H.H. Müller (ASTIN — Nov. 1987) If a security is independent of all other securities in the market, then its price is given by the variance principle! This apparently contradicts prior interpretation. ????

10. The Math Behind of Müller’s Assertion Let:

11. The Math Behind of Müller’s Assertion

12. Hard to Completely Diversify Risk Any security is part of the market. –Almost always some covariance –Usually in practice, Cov[R k,R i ] > 0, so this is a theoretical oddity In practice, there are transaction costs and investors will retain some risk rather than pay the cost of transferring risk. But the CAPM finally got my attention.

13. Assumptions Underlying CAPM Investors: Choose from a fixed set of securities with annual rates of return R i Have a wealth constraint, but they can borrow at risk-free rate R f. Choose how much to invest in each security to maximize expected utility –Utility function depends upon variance –Ignore transaction costs –Can solve by Lagrange multipliers.

14. Assumptions Underlying CAPM If all investors choose securities according to the above rules, what price would get charged in the marketplace? Answer given by the CAPM formula. I call this the “market clearing mechanism” or “the competitive market equilibrium” argument.

15. Back to ISO Increased Limits Ratemaking Problems with standard deviation principle -- for example: –Commercial Auto risk load was higher than the Products Liability risk load. –No risk load reduction by layering –Risk load increase by layering Motivated by CAPM, we decided to adopt a constrained optimization approach to develop a risk load formula.

16. Considerations in Adopting CAPM Methodology Previous attempts at applying CAPM to insurance pricing involved an artificial “allocation of surplus.” Insurance management works within its own constraints. More on “allocating surplus” below.

17. Insurer Management Objectives Maximize Expected Return Subject to a constraint on the variance of the insurer’s total book of business Compare with CAPM assumptions –Maximize investor utility –Subject to constraint on total investor wealth

18. Systematic Risk The original idea -- Generate “systematic risk” by parameter uncertainty More generally -- Covariance Risk Other generators of covariance risk –Catastrophes –Contagion -- e.g. environmental liability –General economic conditions

19. Parameter Uncertainty and Systematic Risk Select random  from a distribution with E[  ] = 1 and Var[  ] = c Select random claim count K from a Poisson distribution with mean  n Think of a Claim Count Simulation

20. Parameter Uncertainty and Systematic Risk

21. Pure premium derivation omitted Modified “Law of Large Numbers” As n increases, the pure premium variance approaches a positive limit.

22. Parameter Uncertainty and Systematic Risk Random parameter modifications applied independently to lines of insurance Done in the paper “An Introduction to the Competitive Market Equilibrium Risk Load Formula”

23. Parameter Uncertainty and Systematic Risk Random parameter modifications applied simultaneously to claim count and claim severity distributions One example of the covariance as a function of exposure. Done in the paper “The Competitive Market Equilibrium Formula for Increased Limits Ratemaking”

24. The Insurer Optimization Express total variance as a function of the exposures of each line. –m = number of lines of insurance –X i = random loss for ith line of insurance

25. The Insurer Optimization Subject to the constraint that: Maximize total expected return:

26. The Insurer Optimization Set up a LaGrange multiplier problem Find the n j ’s and the that satisfy:

27. Solution for Independent Lines The message: These equations are solvable.

28. The Market Clearing Mechanism Our solution to the insurer optimization problem shows how insurers will react to a given market price. The Next Problem Predict the market price –Make assumptions –Following the CAPM analogy

29. The Market Clearing Mechanism Illustrate for independent lines Assumptions –r i is the same for all insurers –u i and v i are the same for all insurers –For insurer j, n ij and j may be different Goal: Predict r i

30. The Market Clearing Mechanism Obtain solution by adding up all the n ij ’s for g insurers and solving for r i. The solution : Many ways to play with the assumptions

31. Reinsurer Risk Loads from Marginal Surplus Requirements by Rodney Kreps Notation –  2 = Variance of current losses –  2 = Variance of loss for new contract –  = Coefficient of Correlation Marginal surplus is proportional to marginal variance.

32. Kreps: Marginal Surplus Proportional to Marginal Variance Kreps makes assumptions –  = 1 (conservative) –  >>  Result:

33. Dilemma Two risk load approaches Both based on sound microeconomic principles Phil Heckman –“Some Unifying Remarks about Risk Loads” CAS Forum, Summer 1993 The two approaches are equivalent!

34. Connections Strike Gold? Müller Traditional Actuarial Risk Load CAPM Heckman Competitive Market Equilibrium Marginal Cost of Capital

35. Joining Two Rich Traditions Stochastic Models Collective Risk Model –Claim Frequency –Claim Severity –Parameter Risk –Process Risk Catastrophe Models Covariance Structures Financial Models Time value of money Make comparisons with non-insurance risks Additional risk financing Instruments –Reinsurance –Securitization Actuarial Theory of Risk Cost of Capital DFA

36. Remainder of Presentation Reinventing risk loads How long must you hold capital? –Duration –Loss reserve risk Capital substitutes –Reinsurance –Catastrophe options Allocating surplus Conclusion

37. The Risk Load and the Insurer’s Financial Environment  An insurer must raise capital to take on additional risk.  An insurer must obtain an adequate return on its capital to remain in business.  The risk load is the cost of maintaining this marginal capital.

38. Calculating Marginal Capital Make the technical assumption that the total capital requirement is a function of the variance. i.e. Capital = C(Variance) The marginal capital requirement is then given by:

39. A Capital Requirement Formula Then:

40. Other capital requirement formulas can be based on:  Probability of Ruin  Expected Policyholder Deficit Where: f(x) is the pdf of insurer’s total loss distribution Selected (usually small) number

41. Marginal Variance {X}’s - Current ContractsY - New Contract

42.  Variance Depends on Existing Business If Y is the loss on the first contract  Variance = Var[Y] If Y is the loss on the second contract  Variance = Var[Y] + 2  Cov[X 1,Y] If Y is the loss on the n th contract  Variance = Var[Y] + 2 

43. Calculating Marginal Capital Combine the above results

44. The Underwriting Process  Writing new contracts with losses that are correlated with losses of existing business is discouraged by the higher cost of marginal capital.  The insurer will diversify exposure to minimize the cost of marginal capital -- with the result that:

45. Combining Above Results

46. Continuing the Math

47. What is K?  As the economic process evolves, K will be determined by the market rate for similar investments.  If K is high enough, the insurer will be able to attract new capital and write more policies.  If K is too low, the insurer will not be able to attract capital to write more policies at this  Expected Return, i.e. risk load.  K will usually be lower than the insurer’s target ROE because the total variance is less than the sum of the marginal variances.

48. Risk Load as an Underwriting Tool The price of insurance is determined externally –By market –By rate regulation If the cost — including the cost of marginal capital — is less than the price, write the policy. A catastrophe example –Policies in concentrated areas will find it hard to get insurance.

49. A Quote from AIR Web Site CATRADER will help you price your contracts. –It will analyze the incremental risk retained or ceded on a particular transaction. –You can easily review both ceded and retained losses and compare the impact of alternative risk transfer strategies on expected loss.

50. A Quote from RMS Web Site RMS “RiskLink seamlessly integrates catastrophe loss analysis results for multiple perils, geographies, and business units, allowing insurers and reinsurers to examine the overall catastrophe risk for their business, and assess the contribution to risk from the underlying components.”

51. The “New Risk Load” is more of a trial and error optimization. –Corresponds to current practices. For example, AIR and RMS, as noted above. The “Old Risk Load” provides a more elegant solution. –Used in a company environment? –I am keeping it in my toolbox. New Risk Load Underwriting Process Old Risk Load Insurer Optimization Heckman Equivalence

52. The Market Clearing Mechanism Y = random losses for insurance contract under consideration. X ij = random losses for the existing contracts of insurer j in group i. R = risk load for the insurance contract, which we assume to be equal for all m insurers. j = risk load multiplier for insurer j.

53. The Market Clearing Mathematics Summing over the m insurers and dividing by m yields:

54. Some comments about the market clearing concept It came from the CAPM It is probably about as good as the CAPM is in predicting security prices. It fails to account for relevant facts. –Capital must be held longer for some lines of insurance. –Risk can be transferred — at a cost I am keeping it in my toolbox.

55. Risk Loads for Long Tailed Lines The relationship is applicable when insurance liabilities expire in one year. In long-tailed lines: The capital must be held longer At the end of a year, only a portion of the capital can be released Still, the return on marginal capital = K

56. C t = Marginal capital needed to support risk at time t i = interest rate earned on invested capital R 0 = Risk Load collected at time t = 0. Choose R 0 so that C 0 = R 0 + Present Value (E[Capital Returned] ) at rate K

57. One way to calculate the C t ’s Use: Y t = Losses settled after time t. ISO compiles claims by settlement lag for use in increased limits ratemaking. Use a parameter uncertainty model to get Cov[Y t,X i ].

58. Reinsurance and Catastrophe Options When you write a new contract –You add capital –You buy more of the capital substitutes. Define the risk load as the marginal cost of capital plus the marginal cost of the additional capital substitutes. Example - Reflect the cost of reinsurance in catastrophe pricing. Capital Substitutes

59. Question — How much of the capital substitutes should you use? Answer — It depends upon the price. You have to compare the cost of capital with the cost of obtaining the substitute. Define the “transaction cost” of the substitute as: Price - Expected Loss Transferred

60. Current State of Affairs “Balancing Transaction Costs and Risk Load in Risk Sharing Arrangements” by Clive Keatinge Given: Insurer i charges a transaction cost given by  i  E[X] + i  Var[X]. Find the combination of insurers and coverages that minimized the total expected loss + transactions costs. Keatinge solves the problem!

61. Current State of Affairs “A Buyers Guide for Options on a Catastrophe Index” by Glenn Meyers “On the Cost of Financing Catastrophe Insurance” by Glenn Meyers and John Kollar These papers tell how many catastrophe options needed to minimize: (Cost of Capital + Transaction Costs) –Method reflects basis risk –Results are sensitive to cost of capital and transaction costs.

62. A Connection between Capital Substitutes and DFA Treat catastrophe options as an asset. –Price of the options is the investment –The return is correlated with losses Treat reinsurance as an asset –Reinsurance premium is the investment –The return is correlated with losses

63. So far, solutions are incomplete. Keatinge paper is “old style” risk loads. Meyers and Kollar papers do not give a price for individual insurance contracts. Ultimate solution: Marginal cost of capital plus Marginal cost of risk transfer

64. Allocating Capital All capital supports all insureds Russ Bingham’s problem: –Reflecting underwriting results in bonuses. –Keeps line managers focused on the company’s overall results –I did not have a better way Rodney Kreps - “Customers demand it!” 1998 Reinsurance Seminar session: –How to get E in ROE

65. How the Reinsurers Use Capital Allocation From 1998 Reinsurance Seminar Session: How to Get to E in ROE Much has been written about return on equity (ROE) and the allocation of capital to individual deals. Unfortunately, it is not easy to put these ideas into practice when pricing deals. This session will present various approaches to allocating capital to individual deals with an emphasis on putting these concepts to use. Looks like a risk load to me!

66. Call Paper Program for 1999 Reinsurance Seminar The Committee on Reinsurance Research is seeking papers on capital allocation techniques which reflect the impact of ceded and/or assumed reinsurance. Insurers and reinsurers often allocate capital to contracts or books of business using many different risk measures as the basis for making those allocations.

67. (Continued) Possible areas the papers could address include but are not limited to: How long capital is allocated for and how it is released over time Cost of the allocated capital Alternative forms of capital The impact on pricing decisions How correlation is reflected Others

68. Risk Load Capital Allocation Given one, you can get the other. Allocate in proportion to marginal capital. –Reflect settlement lag –Reflect capital substitutes A solution to Russ Bingham’s problem Does this solve other insurer management problems? –Stay tuned.

69. CAPM Actuarial Premium Principles Competitive Market Equilibrium Marginal Cost of Capital Allocated Surplus Actuarial Stochastic Models Time that Capital is Held Capital Substitutes Recapping our Journey DFA Methods