capital allocation for insurance companies stewart c. myers james a. read, jr. casualty actuarial...
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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
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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.
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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.
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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
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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
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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).
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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.
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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.
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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
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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
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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
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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
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.
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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
29
Conclusions
Surplus can be allocated uniquely.
Allocations appear robust for multi-line companies.
Computational challenges remain.
What about other financial intermediaries?
30