Risk Loads How they started Where they are Where they’re going

1. Risk LoadsHow they startedWhere they areWhere 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 riskBuhlmann - 1970Premium 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. ProblemAnswers were too high for high limits.

6. ResponseVariance  Standard Deviation

7. Capital Asset Pricing Model • Designed for pricing securities Where:

8. Interpretation of CAPM • If Cov[R,RM] = 0, then E[R] = Rf • The market does not reward one for taking diversifiable risks. • Systematic risk is taken when Cov[R,RM] > 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[Rk,Ri] > 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 Ri • Have a wealth constraint, but they can borrow at risk-free rate Rf. • 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 Think of a Claim Count Simulation • Select random claim count K from a Poisson distribution with mean n

20. Parameter Uncertainty and Systematic Risk

21. Parameter Uncertainty and Systematic Risk 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 • Xi = random loss for ith line of insurance

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

26. The Insurer Optimization Set up a LaGrange multiplier problem Find the nj’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 • ri is the same for all insurers • uiand vi are the same for all insurers • For insurer j, nij and j may be different • Goal: Predict ri

30. The Market Clearing Mechanism • Obtain solution by adding up all the nij’s for g insurers and solving for ri. The solution: • Many ways to play with the assumptions

31. Reinsurer Risk Loads from Marginal Surplus Requirementsby 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 Traditional Actuarial Risk Load CAPM Müller Competitive Market Equilibrium Marginal Cost of Capital Heckman Strike Gold?

35. Joining Two Rich Traditions Actuarial Theory of Risk Cost of Capital DFA 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

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 Contracts Y - 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[X1,Y] • If Y is the loss on the nth 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.”