Capital Allocation CAS Capital Management Seminar July 8, 2002

1. Capital AllocationCAS Capital Management SeminarJuly 8, 2002 Glenn Meyers Insurance Services Office, Inc.

2. Overview of Methodology • Directed toward making capital management decisions. • For example • How much capital do you need? • How much reinsurance do you buy? • How much premium do you need to recover cost of capital? • All three decisions are linked

3. Focus on Recovering the Cost of Capital • Investors contribute capital to entire insurance company. • Cost of capital is recovered by profit loadings on individual insurance policies. • Investors recover the cost of capital from the entire insurance company – not individual policyholders.

4. Focus on Recovering the Cost of Capital • The amount of capital needed depends on the insurer’s total risk. • Individual insurance policies differ in their contribution to the insurer’s total risk. • Insurance prices are market driven. • The insurer’s problem – Develop an underwriting strategy to maximize its return on its investment.

5. Capital Allocation • Is it the only way to solve the insurer’s underwriting problem? – No • Is it a good idea? – Yes • The reason is motivational. CEO’s like to get the entire company focused on the insurer’s overall objective. • Allocating capital will work only if the method of doing it is fair and economically sound.

6. Coherent Measures of RiskDefined by Axioms • Subadditivity – For all random losses X and Y, r(X+Y)  r(X)+r(Y) • Monotonicity – If X  Y for each scenario, then r(X)  r(Y) • Positive Homogeneity – For all l  0 and random loss X r(lX) = lr(Y) • Translation Invariance – For all random losses X and constants a r(X+a) = r(X) + a

7. Examples of Coherent Measures • Easiest r(X) = Maximum(X) • Next easiest – Tail Value at Risk (TVaR) r(X) = Average highest (1-a)% of X’s • Most general – Risk adjusted probabilities

9. How to Use Coherent Mesures • Determine total assets needed to support insurer’s potential losses. • Capital = r(X) – E[X] • Do not use directly on individual insurance policies to determine risk load!

10. Subadditivityr(X+Y)  r(X)+r(Y) • Means diversification is good • An insurer wants to maximize the differences between r(X+Y) and r(X)+r(Y). • In doing this the insurer makes more efficient use of its capital.

11. How do people use allocated capital? • Use it to set profitability targets. • Really allocating the cost of capital.

12. Why Allocate the Cost of Capital? • Allocating the cost of capital is an internal management tool that relates an underwriting division’s financial goal to the insurer’s corporate financial goal. • Any method that makes economic sense is OK.

13. Economic Sense ??? • Let P = Profit and C = Capital. Then adding a line/policy makes sense if:  Marginal return on new business  return on existing business.

14. OK - Set targets so that marginal return on capital equal to insurer return on Capital? • Before I answer this, let’s discuss a property of capital requirements. • Let C[X] = Capital Required to Insure X Getting C[X] is the hard problem! • C[X] should satisfy the subadditivity axiom: C[X+Y] < C[X] + C[Y] • The subadditivity axiom means that diversification is good.

15. OK - Set targets so that marginal return on capital equal to insurer return on Capital? By Subadditivity • The sum of marginal capitals is less than the total capital!

16. Conclusions: • Marginal cost of capital provides a floor on the allocated cost of capital. • At least one underwriting division must have an allocated cost of capital greater that that floor. • Deciding who pays more by how much is a problem. • Look at business plans.

17. One Way of Allocating the Cost of Capital • The Gross-Up Solution • Multiply the marginal cost of capital times a factor so that sum of allocated cost of capital equals the total capital. • Is this solution fudging?

18. An Insurer Business Strategy • An insurer chooses to write the risks that yields the greatest return on marginal capital. • If the insurer stays in business over the long run in a stable underwriting environment, two things will happen. • The insurer will make an adequate return on capital. • The insurer’s return on marginal capital will be equal for all risks.

19. An Insurer Business Strategy • An insurer chooses to write the risks that yields the greatest return on marginal capital. • The long-run effect of this strategy is the same as the Gross-Up solution. • I originally derived the Gross-Up solution using Lagrange multipliers in a risk load setting. http://www.casact.org/pubs/proceed/proceed91/91163.pdf

20. Other Business Strategies • Game theory – Shapley • Build up insurer portfolio one policy at a time. • Average over all possible policy orders. • Mango – 1998 http://www.casact.org/pubs/proceed/proceed98/980157.pdf • Zanjani Example • In handout • Situation that forces you to allocate in proportion to a predetermined risk load.

21. Duration • You may have to hold capital for several years in the long-tailed lines. • In an ongoing insurance business, you need capital to support uncertain loss reserves. • You allocate capital to reserves. • Anticipate the cost of holding capital over time in pricing. http://www.casact.org/pubs/forum/01spforum/meyers/index.htm

22. Review of Methodology • Directed toward making capital management decisions. • For example • How much capital do you need?  • How much reinsurance do you buy? • How much premium do you need to recover cost of capital? 

23. Reinsurance • Buying reinsurance allows a company to reduce its cost of capital. • Compare cost of capital with the transaction cost of reinsurance. • Cost of financing is the sum of the cost of capital plus the transaction cost of reinsurance. • Insurer’s problem – Minimize the cost of financing.

24. Capital Management • Constantly changing insurance environment • Market driven insurance prices • Changing reinsurance prices • The cost of financing analysis provides a framework to devise a strategy to work in this environment. • Will the insurance market ever reach an equilibrium? • It hasn’t yet!

25. The George Zanjani Example • Division A • Expected return of 30 • Requires capital of 120 as a standalone • Division B • Expected return of 15 • Requires capital of 120 as a standalone • Combine A and B • Expected return of 45 • Requires total capital of 150

26. The George Zanjani Example • Division A • Expected return of 30 • Requires capital of 120 as a standalone • Division B • Expected return of 15 • Requires capital of 120 as a standalone

27. The George Zanjani Example • It makes sense to combine A and B. • ROE for A = 30/120 = 25% • ROE for B = 15/120 = 12.5% • ROE for A+B = 45/150 = 30%

28. The George Zanjani Example • Marginal capital for A and B is 30 • Gross-Up allocated capital = 75 for both A and B • A’s ROE = 30/75 = 40% • B’s ROE = 15/75 = 20% • B does not meet overall target of 30% • Do we “fire” B?

29. The George Zanjani Example • A capital allocation leading to “correct” economic decision • Allocate capital of 100 to A • Allocate capital of 50 to B • Both allocations are above the marginal capital “floor.” • ROE = 30% for both A and B

30. Does this example apply to insurance? • Not really – Insurance decisions are made in smaller chunks. • Suppose the Divisions A and B consist of a bunch of individual insurance policies. • You can devise a more profitable strategy where you write a few more polices in Division A, and fewer in Division B. • The Zanjani example turns capital allocation upside down by forcing you to allocate capital in proportion to the risk load.