1. Capital Allocation for Insurance Companies Stewart C. Myers James A. Read, Jr. Casualty Actuarial Society of the American Risk and Insurance Association Marco Island, Florida May 20, 2003 Bentley\General\8060\docs/PRS-PROP/SCM/Marco.Island-FL_5-03

2. 2 Surplus for Insurance Companies Ž Capital = Surplus Ž Insurance companies hold capital (surplus) so that possibility of default is remote. <Surplus equals assets minus default-free liabilities. Ž But surplus is costly: <Double taxation of investment income <Agency and information costs Ž Insurance companies operate in many lines. Need to allocate costs for pricing, performance evaluation, etc. Ž Regulators may also have to allocate costs or to set surplus requirements line-by- line.

3. 3 The Surplus Allocation Problem Ž Conventional wisdom: Surplus can not (should not) be allocated to lines of insurance. For a given configuration, the risk capital of a multi-business firm is less than the aggregate risk capital of the businesses on a stand-alone basis. Full allocation of risk- capital across the individual businesses of the firm therefore is generally not feasible. Attempts at such a full allocation can significantly distort the true profitability of individual businesses. ( Merton, R. C. and A.F. Perold, 1993, “Theory of Risk Capital in Financial Firms,” Journal of Applied Corporate Finance, 6, 16-32.) Ž We show how surplus can (should) be allocated, given the line-by-line composition of business.

4. 4 Surplus Allocation Example Default Value = Cost (PV) of complete credit backup Line 1$10038%$ 38 Line 2$10050%$ 50 Line 3$10063%$ 63 Total$300$150 Default Value$0.93 (0.31%) Marginal Surplus Surplus PV(Losses)Requirement Allocation

5. 5 Surplus Allocation Example Ž Set each line’s surplus requirement so that its marginal contribution to default value is the same (0.31% of PV(Losses)). Ž Suppose PV(Losses) for line 3 increases to $101: Line 1$10038%$38.00 Line 2$10050%$50.00 Line 3$10163%$63.63 Total$301$150.63 Default Value$0.933 (0.31%) Marginal Surplus Surplus PV(Losses)Requirement Allocation

6. 6 Surplus Allocation Example Ž Surplus allocations for diversified firms are generally less than stand-alone surplus requirements. Default Values = 0.31% of PV (losses) Stand-alone requirements cannot be used to allocate surplus requirements in multi- line companies. (Here we agree with Merton and Perold.) Company 1$10043%$43 Company 2$10056%$56 Company 3$10172%$72 Total$171 Surplus Surplus PV(Losses)Percentage Required

7. 7 Surplus Allocation Example Ž The surplus allocations for the three-line company are not correct for a two- or four-line company. Two-line Company (Here we agree with Merton and Perold.) Line 1$10040%$40 Line 2$10052%$52 Line 3——— Total$200$92 Default Value$0.62 (0.31%) Marginal Surplus Surplus PV(Losses)Requirement Allocation

8. 8 Line 1$10038%$38 Line 2$10050%$50 Line 3$10063%$63 Total$300 Default Value$0.93 Default Value/PV(Losses)0.31% Panel A: Marginal Surplus Requirements for Three Lines of Insurance PV(Losses)Marginal Surplus RequirementSurplus Allocation Line 1$100$43 Line 2$100$56 Line 3$100$72 Total$300$171 Panel B: Stand-Alone Surplus Requirements for Each Line PV(Losses)Stand-Alone Surplus Requirements Case 1$0$100$100$115$35 Case 2$100$0$100$104$46 Case 3$100$100$0$ 92$58 Total$200$200$200 Default Value$0.62$0.62$0.62 Default Value/PV(Losses)0.31%0.31%0.31% Panel C: Total Surplus Required for Each Line, Given the Other Two Lines PV(Losses)PV(Losses)PV(Losses)RequiredReduction from Line 1Line 2Line 3SurplusPanel A Examples of Surplus Allocations Ž Panel A shows marginal surplus requirements for three lines of insurance. Surplus allocations based on these marginal requirements add up to the total surplus carried by the firm. Ž Panel B shows the stand-alone surplus requirements for each line. Ž Panel C shows the total surplus required by each line, given the other two lines. Ž In all cases default value is held constant at 0.31% of PV(losses).

9. 9 Preview Given the line-by-line composition of business: Ž Marginal default values add up to firm-wide default value. Ž Set surplus requirements so that every line’s marginal default value is the same. Ž Use these line-by-line surplus requirements for pricing, calculating required overall surplus, etc.

10. 10 The Default Option Ž Limited liability implies equity has option to default Ž Default is an exchange option—an option to exchange assets for liabilities Assets (V) Default Option (D) PV Losses (L) Equity (E)

11. 11 Value of Default Option Consider the default option in a one-period (two-date) setting, assuming distribution of asset/liability values is lognormal * Default value for company (d  D/L) depends on Ž Surplus ratio (s  S/L) Ž Variance of losses (  L 2 ) Ž Variance of asset returns (  V 2 ) Ž Covariance of losses and asset returns (  LV ) * This assumption is convenient but not necessary for our results.

12. 12 Default Risk for Multi-Line Companies For companies that write insurance in more than one line, variance of aggregate losses depends on Ž Variance of losses by line (  i 2 ) Ž Correlation of losses across lines ( )  Ž Composition of business (x i  L i /L)

13. 13 Default Risk for Multi-Line Companies(continued) Covariance of losses with asset returns depends on Ž Variance of losses by line (  i 2 ) Ž Variance of asset returns (  V 2 ) Ž Correlation of losses by line with asset returns ( ) Ž Composition of business (x i )

14. 14 Default Values for Lines of Business Marginal default values (d i   D/  L i ) for lines of business depend on marginal surplus requirements and risk Ž Surplus contribution for line (s i ) Ž Covariance of losses with losses on other lines (  ij ) Ž Covariance of losses with returns on assets (  iV ) Ž Composition of portfolio (x i )

15. 15 Marginal Default Values Add Up Ž Covariances of portfolio components add up: Ž Weighted marginal default values add up to default value for company: Ž Therefore default values can be allocated uniquely to lines of business. Ž “Adding up” result assumes losses and investment assets have well defined market values. If so, result holds for any joint probability distribution of losses and investment returns.

16. 16 Retail Insurance Ž In retail insurance markets, default risk is absorbed by an industry pool Ž Surplus requirements are typically the same for all lines of insurance, so marginal default values vary by line Ž This implies that the pool subsidizes high-risk lines of business <Insurance companies “collect” default insurance with value equal to default value for own portfolio <Companies “pay” default value for pool

17. 17 Surplus Allocation Ž All policy holders bear the same default risk. Ž The correct formula for surplus allocation is obtained by setting marginal default values equal to default value for firm (d i = d). Ž Eliminates intra-firm cross subsidies.

18. 18 Default Value and Surplus Allocations Risky Assets, Base-Case Correlations Panel A: Portfolio Assets & Liabilities Panel B: Line-by-Line Allocations Line 1$10033%10.00%1.000.500.500.0092-0.0030 Line 2$10033%15.00%0.501.000.500.0150-0.0045 Line 3$10033%20.00%0.500.501.000.0217-0.0060 Liabilities$300100%12.36%0.740.810.880.0153-0.0045 Assets$450150%15.00%-0.20-0.20-0.200.0225 Surplus$15050% Lognormal Results Asset/Liability Volatility21.63% Default/Liability Value0.31% Delta-0.0237 Vega0.0838 Normal Results Standard Deviation of Surplus28.18% Default/Liability Value0.43% Delta-0.0380 Vega0.0826 CorrelationsRatio to Liabilities Standard Deviation Line 1Line 2Line 3 Covariance with Liabilities Covariance with Assets Line 10.02%0.18%38%41% Line 20.30%0.42%50%50% Line 30.62%0.68%63%59% Liabilities0.31%0.43%50%50% Default/Liability Value (Uniform Surplus) Surplus/Liability Value (Uniform Default Value) LognormalNormalLognormalNormal

19. 19 Default Value and Surplus Allocations Safe Assets Case Panel A: Portfolio Assets & Liabilities Panel B: Line-by-Line Allocations Line 1$10033%10.00%1.000.500.500.00920.0000 Line 2$10033%15.00%0.501.000.500.01500.0000 Line 3$10033%20.00%0.500.501.000.02170.0000 Liabilities$300100%12.36%0.740.810.880.01530.0000 Assets$450150% 0.00%0.000.000.000.0000 Surplus$15050% Lognormal Results Asset/Liability Volatility12.36% Default/Liability Value0.00% Delta-0.0004 Vega0.0022 Normal Results Standard Deviation of Surplus12.36% Default/Liability Value0.00% Delta0.0000 Vega0.0001 CorrelationsRatio to Liabilities Standard Deviation Line 1Line 2Line 3 Covariance with Liabilities Covariance with Assets Line 1-0.01%0.00%23%29% Line 20.00%0.00%49%49% Line 30.01%0.00%78%72% Liabilities0.00%0.00%50%50% Default/Liability Value (Uniform Surplus) Surplus/Liability Value (Uniform Default Value) LognormalNormalLognormalNormal

20. 20 Panel A: Portfolio Assets & Liabilities Panel B: Line-by-Line Allocations Line 1$10033%15.00%1.000.100.100.0090-0.0045 Line 2$10033%15.00%0.101.000.100.0090-0.0045 Line 3$10033%15.00%0.100.101.000.0090-0.0045 Liabilities$300100%9.49%0.630.630.630.0090-0.0045 Assets$450150%15.00%-0.20-0.20-0.200.0225 Surplus$15050% Lognormal Results Asset/Liability Volatility20.12% Default/Liability Value0.20% Delta-0.0172 Vega0.0639 Normal Results Standard Deviation of Surplus27.04% Default/Liability Value0.34% Delta-0.0322 Vega0.0722 CorrelationsRatio to Liabilities Standard Deviation Line 1Line 2Line 3 Covariance with Liabilities Covariance with Assets Line 10.20%0.34%50%50% Line 20.20%0.34%50%50% Line 30.20%0.34%50%50% Liabilities0.20%0.34%50%50% Default/Liability Value (Uniform Surplus) Surplus/Liability Value (Uniform Default Value) LognormalNormalLognormalNormal Default Value and Surplus Allocations Geographic Diversification Case

21. 21 Panel A: Portfolio Assets & Liabilities Panel B: Line-by-Line Allocations Line 1$10033%15.00%1.000.900.900.0210-0.0045 Line 2$10033%15.00%0.901.000.900.0210-0.0045 Line 3$10033%15.00%0.900.901.000.0210-0.0045 Liabilities$300100%14.49%0.970.970.970.0210-0.0045 Assets$450150%15.00%-0.20-0.20-0.200.0225 Surplus$15050% Lognormal Results Asset/Liability Volatility22.91% Default/Liability Value0.43% Delta-0.0298 Vega0.1014 Normal Results Standard Deviation of Surplus29.18% Default/Liability Value0.52% Delta-0.0433 Vega0.0919 CorrelationsRatio to Liabilities Standard Deviation Line 1Line 2Line 3 Covariance with Liabilities Covariance with Assets Line 10.43%0.52%50%50% Line 20.43%0.52%50%50% Line 30.43%0.52%50%50% Liabilities0.43%0.52%50%50% Default/Liability Value (Uniform Surplus) Surplus/Liability Value (Uniform Default Value) LognormalNormalLognormalNormal Default Value and Surplus Allocations Long Tail Case

22. 22 Robustness of Marginal Default Values Ž Marginal default values depend on mix of business as well as line-by-line risk. Are they robust to changes in mix? Ž Experiment: Consider surplus allocations for hypothetical companies with N and N+1 identical lines of business.

23. 23 Two-Line Company

24. 24 Four-Line Company

25. 25 Ten-Line Company

26. 26 Solvency Regulation Ž Define a “base-case” composition of insurance business and asset risk along with marginal surplus requirements consistent with uniform default value. Ž If a company deviates from base-case composition or asset risk, adjust surplus requirements to keep default value constant.

27. 27 The Efficient Composition of Business Ž Diversification provides financial benefits in the form of reduced risk and surplus requirements. Ž Diversification entails real costs. (Diminished focus? Administrative friction?) Ž Efficient composition of business represents a trade-off between financial benefits and real costs. <May not be unique <May not be sharply defined

28. 28 The Benefit-Cost Trade Off The Trade off of reduced surplus requirements against operating andadministrative costs determines the efficient composition of business.As more lines are added, diversification decreases required surplus,but operating, administrative, and perhaps agency costs increase.Figure 2Present Valueof costs Efficient Compositionof BusinessIncreasedDiversificationMarginal operatingand administrative costsReduction in costof Required Surplus The Trade off of reduced surplus requirements against operating andadministrative costs determines the efficient composition of business.As more lines are added, diversification decreases required surplus,but operating, administrative, and perhaps agency costs increase.Figure 2Present Valueof costs Efficient Compositionof BusinessIncreasedDiversificationMarginal operatingand administrative costsReduction in costof Required Surplus Present Value of Costs Efficient Composition of Business Increased Diversification Marginal Operating and Administrative Costs Reduction in Cost of Required Surplus

29. 29 Conclusions Ž Surplus can be allocated uniquely. Ž Allocations appear robust for multi-line companies. Ž Computational challenges remain. Ž What about other financial intermediaries?

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