Capital Allocation for Insurance Companies

Stewart C. MyersJames A. Read, Jr.

Casualty Actuarial Society of the

American Risk and Insurance AssociationMarco 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

Surplus Allocation Example

Default Value = Cost (PV) of complete credit backup

Line 1 $100 38% $ 38

Line 2 $100 50% $ 50

Line 3 $100 63% $ 63

Total $300 $150

Default Value $0.93

(0.31%)

Marginal Surplus SurplusPV(Losses) Requirement Allocation

5

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:

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%)

Marginal Surplus SurplusPV(Losses) Requirement Allocation

6

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.)

Company 1 $100 43% $43

Company 2 $100 56% $56

Company 3 $101 72% $72

Total $171

Surplus SurplusPV(Losses) Percentage Required

7

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.)

Line 1 $100 40% $40

Line 2 $100 52% $52

Line 3 — — —

Total $200 $92

Default Value $0.62

(0.31%)

Marginal Surplus SurplusPV(Losses) Requirement Allocation

8

Line 1 $100 38% $38Line 2 $100 50% $50Line 3 $100 63% $63Total $300Default Value $0.93Default Value/PV(Losses) 0.31%

Panel A: Marginal Surplus Requirements for Three Lines of InsurancePV(Losses) Marginal Surplus Requirement Surplus Allocation

Line 1 $100 $43Line 2 $100 $56Line 3 $100 $72Total $300 $171

Panel B: Stand-Alone Surplus Requirements for Each LinePV(Losses) Stand-Alone Surplus Requirements

Case 1 $0 $100 $100 $115 $35Case 2 $100 $0 $100 $104 $46Case 3 $100 $100 $0 $ 92 $58Total $200 $200 $200Default Value $0.62 $0.62 $0.62Default Value/PV(Losses) 0.31% 0.31% 0.31%

Panel C: Total Surplus Required for Each Line, Given the Other Two LinesPV(Losses) PV(Losses) PV(Losses) Required Reduction from

Line 1 Line 2 Line 3 Surplus Panel A

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).

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

The Default Option

Limited liability implies equity has option to default

Default is an exchange option—an option to exchange assets for liabilities

VLD

DLVLVE

~~,0max

~

~~~~~,0max

~

Assets (V)

Default Option (D)

PV Losses (L)

Equity (E)

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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)

ijjix jxiM

j

M

iL 11

2

ij

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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)

ρσσxσ iVVii

M

iLV

1

iV

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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)

LViVLiLii

dss

s

ddd

21

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.

LV

M

iiVi

L

M

iiLi

x

x

1

2

1

ddxM

iii

1

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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

LViVLiLi

ddd

21

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.

LViVLiLi

dsd

ss

21

1

18

Default Value and Surplus Allocations Risky Assets, Base-Case Correlations

Panel A: Portfolio Assets & Liabilities

Panel B: Line-by-Line Allocations

Line 1 $100 33% 10.00% 1.00 0.50 0.50 0.0092 -0.0030Line 2 $100 33% 15.00% 0.50 1.00 0.50 0.0150 -0.0045Line 3 $100 33% 20.00% 0.50 0.50 1.00 0.0217 -0.0060Liabilities $300 100% 12.36% 0.74 0.81 0.88 0.0153 -0.0045Assets $450 150% 15.00% -0.20 -0.20 -0.20 0.0225Surplus $150 50%

Lognormal ResultsAsset/Liability Volatility 21.63%Default/Liability Value 0.31%Delta -0.0237Vega 0.0838

Normal ResultsStandard Deviation of Surplus 28.18%Default/Liability Value 0.43%Delta -0.0380Vega 0.0826

CorrelationsRatio toLiabilities

StandardDeviation Line 1 Line 2 Line 3

Covariancewith Liabilities

Covariancewith Assets

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/Liability Value(Uniform Surplus)

Surplus/Liability Value(Uniform Default Value)

Lognormal Normal Lognormal Normal

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Default Value and Surplus Allocations Safe Assets Case

Panel A: Portfolio Assets & Liabilities

Panel B: Line-by-Line Allocations

Line 1 $100 33% 10.00% 1.00 0.50 0.50 0.0092 0.0000Line 2 $100 33% 15.00% 0.50 1.00 0.50 0.0150 0.0000Line 3 $100 33% 20.00% 0.50 0.50 1.00 0.0217 0.0000Liabilities $300 100% 12.36% 0.74 0.81 0.88 0.0153 0.0000Assets $450 150% 0.00% 0.00 0.00 0.00 0.0000Surplus $150 50%

Lognormal ResultsAsset/Liability Volatility 12.36%Default/Liability Value 0.00%Delta -0.0004Vega 0.0022

Normal ResultsStandard Deviation of Surplus 12.36%Default/Liability Value 0.00%Delta 0.0000Vega 0.0001

CorrelationsRatio toLiabilities

StandardDeviation Line 1 Line 2 Line 3

Covariancewith Liabilities

Covariancewith Assets

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/Liability Value(Uniform Surplus)

Surplus/Liability Value(Uniform Default Value)

Lognormal Normal Lognormal Normal

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Panel A: Portfolio Assets & Liabilities

Panel B: Line-by-Line Allocations

Line 1 $100 33% 15.00% 1.00 0.10 0.10 0.0090 -0.0045Line 2 $100 33% 15.00% 0.10 1.00 0.10 0.0090 -0.0045Line 3 $100 33% 15.00% 0.10 0.10 1.00 0.0090 -0.0045Liabilities $300 100% 9.49% 0.63 0.63 0.63 0.0090 -0.0045Assets $450 150% 15.00% -0.20 -0.20 -0.20 0.0225Surplus $150 50%

Lognormal ResultsAsset/Liability Volatility 20.12%Default/Liability Value 0.20%Delta -0.0172Vega 0.0639

Normal ResultsStandard Deviation of Surplus 27.04%Default/Liability Value 0.34%Delta -0.0322Vega 0.0722

CorrelationsRatio toLiabilities

StandardDeviation Line 1 Line 2 Line 3

Covariancewith Liabilities

Covariancewith Assets

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/Liability Value(Uniform Surplus)

Surplus/Liability Value(Uniform Default Value)

Lognormal Normal Lognormal Normal

Default Value and Surplus Allocations Geographic Diversification Case

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Panel A: Portfolio Assets & Liabilities

Panel B: Line-by-Line Allocations

Line 1 $100 33% 15.00% 1.00 0.90 0.90 0.0210 -0.0045Line 2 $100 33% 15.00% 0.90 1.00 0.90 0.0210 -0.0045Line 3 $100 33% 15.00% 0.90 0.90 1.00 0.0210 -0.0045Liabilities $300 100% 14.49% 0.97 0.97 0.97 0.0210 -0.0045Assets $450 150% 15.00% -0.20 -0.20 -0.20 0.0225Surplus $150 50%

Lognormal ResultsAsset/Liability Volatility 22.91%Default/Liability Value 0.43%Delta -0.0298Vega 0.1014

Normal ResultsStandard Deviation of Surplus 29.18%Default/Liability Value 0.52%Delta -0.0433Vega 0.0919

CorrelationsRatio toLiabilities

StandardDeviation Line 1 Line 2 Line 3

Covariancewith Liabilities

Covariancewith Assets

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/Liability Value(Uniform Surplus)

Surplus/Liability Value(Uniform Default Value)

Lognormal Normal Lognormal Normal

Default Value and Surplus Allocations Long Tail Case

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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.

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Two-Line Company

Surplus Allocations

-30.00%

-20.00%

-10.00%

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

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

Line 2 Liabilities

Mar

gin

al S

urp

lus

Req

uir

emen

t

Company

Line 2

Line 1

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Four-Line Company

Surplus Allocations

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

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

Line 4 Liabilities

Mar

gin

al S

urp

lus

Req

uir

emen

t

Company

Line 4

Lines 1 to 3

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Ten-Line Company

Surplus Allocations

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

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

Mar

gin

al S

urp

lus

Req

uir

emen

t

Company

Line 10

Lines 1 to 9

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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.

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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

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The Benefit-Cost Trade Off

PresentValue

of Costs

Efficient Compositionof Business

IncreasedDiversification

Marginal Operatingand Administrative Costs

Reduction in Costof Required Surplus

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Conclusions

Surplus can be allocated uniquely.

Allocations appear robust for multi-line companies.

Computational challenges remain.

What about other financial intermediaries?

30