Capital Allocation for Insurance Companies

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

3. The Surplus Allocation Problem • 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.) • We show how surplus can (should) be allocated, given the line-by-line composition of business.

4. Marginal Surplus Surplus PV(Losses) Requirement Allocation Line 1 $100 38% $ 38 Line 2 $100 50% $ 50 Line 3 $100 63% $ 63 Total $300 $150 Default Value $0.93 (0.31%) Surplus Allocation Example Default Value = Cost (PV) of complete credit backup

5. Marginal Surplus Surplus PV(Losses) Requirement Allocation Line 1 $100 38% $38.00 Line 2 $100 50% $50.00 Line 3 $101 63% $63.63 Total $301 $150.63 Default Value $0.933 (0.31%) Surplus Allocation Example • Set each line’s surplus requirement so that its marginal contribution to default value is the same (0.31% of PV(Losses)). • Suppose PV(Losses) for line 3 increases to $101:

6. Surplus Surplus PV(Losses) Percentage Required Company 1 $100 43% $43 Company 2 $100 56% $56 Company 3 $101 72% $72 Total $171 Surplus Allocation Example • 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.)

7. Marginal Surplus Surplus PV(Losses) Requirement Allocation Line 1 $100 40% $40 Line 2 $100 52% $52 Line 3 — — — Total $200 $92 Default Value $0.62 (0.31%) Surplus Allocation Example • 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.)

8. Panel A: Marginal Surplus Requirements for Three Lines of Insurance PV(Losses) Marginal Surplus Requirement Surplus Allocation Line 1 $100 38% $38 Line 2 $100 50% $50 Line 3 $100 63% $63 Total $300 Default Value $0.93 Default Value/PV(Losses) 0.31% Panel B: Stand-Alone Surplus Requirements for Each Line PV(Losses) Stand-Alone Surplus Requirements Line 1 $100 $43 Line 2 $100 $56 Line 3 $100 $72 Total $300 $171 Examples of Surplus Allocations • 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. • Panel B shows the stand-alone surplus requirements for each line. • Panel C shows the total surplus required by each line, given the other two lines. • In all cases default value is held constant at 0.31% of PV(losses). Panel C: Total Surplus Required for Each Line, Given the Other Two Lines PV(Losses) PV(Losses) PV(Losses) Required Reduction from Line 1 Line 2 Line 3 Surplus Panel A 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%

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

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

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 • Surplus ratio (s  S/L) • Variance of losses (L2) • Variance of asset returns (V2) • Covariance of losses and asset returns (LV) *This assumption is convenient but not necessary for our results.

12. Default Risk for Multi-Line Companies • For companies that write insurance in more than one line, variance of aggregate losses depends on • Variance of losses by line (i2) • Correlation of losses across lines ( ) • Composition of business (xi Li/L)

13. Default Risk for Multi-Line Companies (continued) • Covariance of losses with asset returns depends on • Variance of losses by line (i2) • Variance of asset returns (V2) • Correlation of losses by line with asset returns ( ) • Composition of business (xi)

14. Default Values for Lines of Business • Marginal default values (di D/Li) for lines of business depend on marginal surplus requirements and risk • Surplus contribution for line (si) • Covariance of losses with losses on other lines (ij) • Covariance of losses with returns on assets (iV) • Composition of portfolio (xi)

15. Marginal Default Values Add Up • Covariances of portfolio components add up: • Weighted marginal default values add up to default value for company: • Therefore default values can be allocated uniquely to lines of business. • “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. Retail Insurance • In retail insurance markets, default risk is absorbed by an industry pool • Surplus requirements are typically the same for all lines of insurance, so marginal default values vary by line • 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. Surplus Allocation • All policy holders bear the same default risk. • The correct formula for surplus allocation is obtained by setting marginal default values equal to default value for firm (di = d). • Eliminates intra-firm cross subsidies.

18. Panel A: Portfolio Assets & Liabilities Ratio to Liabilities Standard Deviation Correlations Covariance with Liabilities Covariance with Assets Line 1 Line 2 Line 3 Line 1 $100 33% 10.00% 1.00 0.50 0.50 0.0092 -0.0030 Line 2 $100 33% 15.00% 0.50 1.00 0.50 0.0150 -0.0045 Line 3 $100 33% 20.00% 0.50 0.50 1.00 0.0217 -0.0060 Liabilities $300 100% 12.36% 0.74 0.81 0.88 0.0153 -0.0045 Assets $450 150% 15.00% -0.20 -0.20 -0.20 0.0225 Surplus $150 50% Lognormal Results Asset/Liability Volatility 21.63% Default/Liability Value 0.31% Delta -0.0237 Vega 0.0838 Normal Results Standard Deviation of Surplus 28.18% Default/Liability Value 0.43% Delta -0.0380 Vega 0.0826 Panel B: Line-by-Line Allocations Default/Liability Value (Uniform Surplus) Surplus/Liability Value (Uniform Default Value) Lognormal Normal Lognormal Normal Line 1 0.02% 0.18% 38% 41% Line 2 0.30% 0.42% 50% 50% Line 3 0.62% 0.68% 63% 59% Liabilities 0.31% 0.43% 50% 50% Default Value and Surplus Allocations Risky Assets, Base-Case Correlations

19. Panel A: Portfolio Assets & Liabilities Ratio to Liabilities Standard Deviation Correlations Covariance with Liabilities Covariance with Assets Line 1 Line 2 Line 3 Line 1 $100 33% 10.00% 1.00 0.50 0.50 0.0092 0.0000 Line 2 $100 33% 15.00% 0.50 1.00 0.50 0.0150 0.0000 Line 3 $100 33% 20.00% 0.50 0.50 1.00 0.0217 0.0000 Liabilities $300 100% 12.36% 0.74 0.81 0.88 0.0153 0.0000 Assets $450 150% 0.00% 0.00 0.00 0.00 0.0000 Surplus $150 50% Lognormal Results Asset/Liability Volatility 12.36% Default/Liability Value 0.00% Delta -0.0004 Vega 0.0022 Normal Results Standard Deviation of Surplus 12.36% Default/Liability Value 0.00% Delta 0.0000 Vega 0.0001 Panel B: Line-by-Line Allocations Default/Liability Value (Uniform Surplus) Surplus/Liability Value (Uniform Default Value) Lognormal Normal Lognormal Normal Line 1 -0.01% 0.00% 23% 29% Line 2 0.00% 0.00% 49% 49% Line 3 0.01% 0.00% 78% 72% Liabilities 0.00% 0.00% 50% 50% Default Value and Surplus Allocations Safe Assets Case

20. Panel A: Portfolio Assets & Liabilities Ratio to Liabilities Standard Deviation Correlations Covariance with Liabilities Covariance with Assets Line 1 Line 2 Line 3 Line 1 $100 33% 15.00% 1.00 0.10 0.10 0.0090 -0.0045 Line 2 $100 33% 15.00% 0.10 1.00 0.10 0.0090 -0.0045 Line 3 $100 33% 15.00% 0.10 0.10 1.00 0.0090 -0.0045 Liabilities $300 100% 9.49% 0.63 0.63 0.63 0.0090 -0.0045 Assets $450 150% 15.00% -0.20 -0.20 -0.20 0.0225 Surplus $150 50% Lognormal Results Asset/Liability Volatility 20.12% Default/Liability Value 0.20% Delta -0.0172 Vega 0.0639 Normal Results Standard Deviation of Surplus 27.04% Default/Liability Value 0.34% Delta -0.0322 Vega 0.0722 Panel B: Line-by-Line Allocations Default/Liability Value (Uniform Surplus) Surplus/Liability Value (Uniform Default Value) Lognormal Normal Lognormal Normal Line 1 0.20% 0.34% 50% 50% Line 2 0.20% 0.34% 50% 50% Line 3 0.20% 0.34% 50% 50% Liabilities 0.20% 0.34% 50% 50% Default Value and Surplus Allocations Geographic Diversification Case

21. Panel A: Portfolio Assets & Liabilities Ratio to Liabilities Standard Deviation Correlations Covariance with Liabilities Covariance with Assets Line 1 Line 2 Line 3 Line 1 $100 33% 15.00% 1.00 0.90 0.90 0.0210 -0.0045 Line 2 $100 33% 15.00% 0.90 1.00 0.90 0.0210 -0.0045 Line 3 $100 33% 15.00% 0.90 0.90 1.00 0.0210 -0.0045 Liabilities $300 100% 14.49% 0.97 0.97 0.97 0.0210 -0.0045 Assets $450 150% 15.00% -0.20 -0.20 -0.20 0.0225 Surplus $150 50% Lognormal Results Asset/Liability Volatility 22.91% Default/Liability Value 0.43% Delta -0.0298 Vega 0.1014 Normal Results Standard Deviation of Surplus 29.18% Default/Liability Value 0.52% Delta -0.0433 Vega 0.0919 Panel B: Line-by-Line Allocations Default/Liability Value (Uniform Surplus) Surplus/Liability Value (Uniform Default Value) Lognormal Normal Lognormal Normal Line 1 0.43% 0.52% 50% 50% Line 2 0.43% 0.52% 50% 50% Line 3 0.43% 0.52% 50% 50% Liabilities 0.43% 0.52% 50% 50% Default Value and Surplus Allocations Long Tail Case

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

23. Surplus Allocations 60.00% 50.00% 40.00% 30.00% Marginal Surplus Requirement 20.00% 10.00% 0.00% 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250 0.275 0.300 0.325 0.350 0.375 0.400 0.425 0.450 0.475 0.500 -10.00% -20.00% -30.00% Line 2 Liabilities Company Line 2 Line 1 Two-Line Company

24. Surplus Allocations 60.00% 50.00% 40.00% 30.00% Marginal Surplus Requirement 20.00% 10.00% 0.00% 0.000 0.013 0.025 0.038 0.050 0.063 0.075 0.088 0.100 0.113 0.125 0.138 0.150 0.163 0.175 0.188 0.200 0.213 0.225 0.238 0.250 Company Line 4 Liabilities Line 4 Lines 1 to 3 Four-Line Company

25. Surplus Allocations 60.00% 50.00% 40.00% 30.00% Marginal Surplus Requirement 20.00% 10.00% 0.00% 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 0.055 0.060 0.065 0.070 0.075 0.080 0.085 0.090 0.095 0.100 Line 10 Liabilities Company Line 10 Lines 1 to 9 Ten-Line Company

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

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

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

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