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
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-byline.
2
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 riskcapital 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.
3
Surplus Allocation Example
Line 1
Line 2
Line 3
Total
Default Value
PV(Losses)
Marginal Surplus
Requirement
Surplus
Allocation
$100
$100
$100
38%
50%
63%
$ 38
$ 50
$ 63
$300
$0.93
(0.31%)
$150
Default Value = Cost (PV) of complete credit backup
4
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
Line 2
Line 3
Total
Default Value
PV(Losses)
Marginal Surplus
Requirement
Surplus
Allocation
$100
$100
$101
38%
50%
63%
$38.00
$50.00
$63.63
$301
$0.933
(0.31%)
$150.63
5
Surplus Allocation Example

Surplus allocations for diversified firms are generally less than stand-alone surplus
requirements.
Default Values = 0.31% of PV (losses)
Company 1
Company 2
Company 3
Total
PV(Losses)
Surplus
Percentage
Surplus
Required
$100
$100
$101
43%
56%
72%
$43
$56
$72
$171
Stand-alone requirements cannot be used to allocate surplus requirements in multiline companies. (Here we agree with Merton and Perold.)
6
Surplus Allocation Example

The surplus allocations for the three-line company are not correct for a two- or
four-line company.
Two-line Company
PV(Losses)
Line 1
Line 2
Line 3
Total
Default Value
$100
$100
—
$200
Marginal Surplus
Requirement
Surplus
Allocation
40%
52%
—
$40
$52
—
$92
$0.62
(0.31%)
(Here we agree with Merton and Perold.)
7
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 A: Marginal Surplus Requirements for Three Lines of Insurance
PV(Losses)
Marginal Surplus Requirement
Line 1
$100
38%
Line 2
$100
50%
Line 3
$100
63%
Total
$300
Default Value
$0.93
Default Value/PV(Losses)
0.31%
Panel B: Stand-Alone Surplus Requirements for Each Line
PV(Losses)
Line 1
$100
Line 2
$100
Line 3
$100
Total
$300
Panel C: Total Surplus Required for Each Line, Given the Other Two Lines
PV(Losses)
PV(Losses)
Line 1
Line 2
Case 1
$0
$100
Case 2
$100
$0
Case 3
$100
$100
Total
$200
$200
Default Value
$0.62
$0.62
Default Value/PV(Losses)
0.31%
0.31%
Surplus Allocation
$38
$50
$63
Stand-Alone Surplus Requirements
$43
$56
$72
$171
PV(Losses)
Line 3
$100
$100
$0
$200
$0.62
0.31%
Required
Surplus
$115
$104
$ 92
Reduction from
Panel A
$35
$46
$58
8
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.
9
The Default Option

Limited liability implies equity has option to default
Assets (V)
PV Losses (L)
Default Option (D)
Equity (E)
~
~ ~
~ ~ ~
E  max 0, V  L   V  L  D
 

~
~ ~
D  max 0,  L  V 
 


Default is an exchange option—an option to exchange assets for liabilities
10
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.
11
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 (  ij )
Composition of business (xi  Li/L)
M
2
  
L i 1
M
 x i x j  i  j  ij
j 1
12
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 (  iV )
Composition of business (xi)
M
σ LV   xi σ i σ V ρiV
i 1
13
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)


 d 
 d  1

2


di  d  
s

s

 i

  iL   L    iV   LV  
 s 
   

14
Marginal Default Values Add Up

Covariances of portfolio components add up:
M
 x
i 1
i
iL
  L2
iV
  LV
M
 x
i 1

i
Weighted marginal default values add up to default value for company:
M
xd
i 1


i
i
d
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.
15
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
 d  1

2
di  d  
  iL   L    iV   LV 
   


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
16
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).
1
 d   d  1

2
si  s    
  iL   L    iV   LV 
 s     


Eliminates intra-firm cross subsidies.
17
Default Value and Surplus Allocations
Risky Assets, Base-Case Correlations
Panel A: Portfolio Assets & Liabilities
Line 1
Line 2
Line 3
Liabilities
Assets
Surplus
Ratio to
Liabilities
Standard
Deviation
Line 1
33%
33%
33%
100%
150%
50%
10.00%
15.00%
20.00%
12.36%
15.00%
1.00
0.50
0.50
0.74
-0.20
$100
$100
$100
$300
$450
$150
Lognormal Results
Asset/Liability Volatility
Default/Liability Value
Delta
Vega
21.63%
0.31%
-0.0237
0.0838
Normal Results
Standard Deviation of Surplus
Default/Liability Value
Delta
Vega
28.18%
0.43%
-0.0380
0.0826
Correlations
Line 2
0.50
1.00
0.50
0.81
-0.20
Line 3
0.50
0.50
1.00
0.88
-0.20
Covariance
with Liabilities
Covariance
with Assets
0.0092
0.0150
0.0217
0.0153
-0.0030
-0.0045
-0.0060
-0.0045
0.0225
Panel B: Line-by-Line Allocations
Default/Liability Value
(Uniform Surplus)
Line 1
Line 2
Line 3
Liabilities
Surplus/Liability Value
(Uniform Default Value)
Lognormal
Normal
Lognormal
0.02%
0.30%
0.62%
0.31%
0.18%
0.42%
0.68%
0.43%
38%
50%
63%
50%
Normal
41%
50%
59%
50%
18
Default Value and Surplus Allocations
Safe Assets Case
Panel A: Portfolio Assets & Liabilities
Line 1
Line 2
Line 3
Liabilities
Assets
Surplus
$100
$100
$100
$300
$450
$150
Lognormal Results
Asset/Liability Volatility
Default/Liability Value
Delta
Vega
12.36%
0.00%
-0.0004
0.0022
Normal Results
Standard Deviation of Surplus
Default/Liability Value
Delta
Vega
12.36%
0.00%
0.0000
0.0001
Ratio to
Liabilities
Standard
Deviation
Line 1
33%
33%
33%
100%
150%
50%
10.00%
15.00%
20.00%
12.36%
0.00%
1.00
0.50
0.50
0.74
0.00
Correlations
Line 2
Line 3
0.50
1.00
0.50
0.81
0.00
0.50
0.50
1.00
0.88
0.00
Covariance
with Liabilities
Covariance
with Assets
0.0092
0.0150
0.0217
0.0153
0.0000
0.0000
0.0000
0.0000
0.0000
Panel B: Line-by-Line Allocations
Default/Liability Value
(Uniform Surplus)
Line 1
Line 2
Line 3
Liabilities
Surplus/Liability Value
(Uniform Default Value)
Lognormal
Normal
Lognormal
-0.01%
0.00%
0.01%
0.00%
0.00%
0.00%
0.00%
0.00%
23%
49%
78%
50%
Normal
29%
49%
72%
50%
19
Default Value and Surplus Allocations
Geographic Diversification Case
Panel A: Portfolio Assets & Liabilities
Line 1
Line 2
Line 3
Liabilities
Assets
Surplus
Ratio to
Liabilities
Standard
Deviation
Line 1
33%
33%
33%
100%
150%
50%
15.00%
15.00%
15.00%
9.49%
15.00%
1.00
0.10
0.10
0.63
-0.20
$100
$100
$100
$300
$450
$150
Lognormal Results
Asset/Liability Volatility
Default/Liability Value
Delta
Vega
20.12%
0.20%
-0.0172
0.0639
Normal Results
Standard Deviation of Surplus
Default/Liability Value
Delta
Vega
27.04%
0.34%
-0.0322
0.0722
Correlations
Line 2
0.10
1.00
0.10
0.63
-0.20
Line 3
0.10
0.10
1.00
0.63
-0.20
Covariance
with Liabilities
Covariance
with Assets
0.0090
0.0090
0.0090
0.0090
-0.0045
-0.0045
-0.0045
-0.0045
0.0225
Panel B: Line-by-Line Allocations
Default/Liability Value
(Uniform Surplus)
Line 1
Line 2
Line 3
Liabilities
Surplus/Liability Value
(Uniform Default Value)
Lognormal
Normal
Lognormal
0.20%
0.20%
0.20%
0.20%
0.34%
0.34%
0.34%
0.34%
50%
50%
50%
50%
Normal
50%
50%
50%
50%
20
Default Value and Surplus Allocations
Long Tail Case
Panel A: Portfolio Assets & Liabilities
Line 1
Line 2
Line 3
Liabilities
Assets
Surplus
Ratio to
Liabilities
Standard
Deviation
Line 1
33%
33%
33%
100%
150%
50%
15.00%
15.00%
15.00%
14.49%
15.00%
1.00
0.90
0.90
0.97
-0.20
$100
$100
$100
$300
$450
$150
Lognormal Results
Asset/Liability Volatility
Default/Liability Value
Delta
Vega
22.91%
0.43%
-0.0298
0.1014
Normal Results
Standard Deviation of Surplus
Default/Liability Value
Delta
Vega
29.18%
0.52%
-0.0433
0.0919
Correlations
Line 2
0.90
1.00
0.90
0.97
-0.20
Line 3
0.90
0.90
1.00
0.97
-0.20
Covariance
with Liabilities
Covariance
with Assets
0.0210
0.0210
0.0210
0.0210
-0.0045
-0.0045
-0.0045
-0.0045
0.0225
Panel B: Line-by-Line Allocations
Default/Liability Value
(Uniform Surplus)
Line 1
Line 2
Line 3
Liabilities
Surplus/Liability Value
(Uniform Default Value)
Lognormal
Normal
Lognormal
0.43%
0.43%
0.43%
0.43%
0.52%
0.52%
0.52%
0.52%
50%
50%
50%
50%
Normal
50%
50%
50%
50%
21
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.
22
Two-Line Company
Surplus Allocations
60.00%
Marginal Surplus Requirement
50.00%
40.00%
30.00%
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
23
Four-Line Company
Surplus Allocations
60.00%
Marginal Surplus Requirement
50.00%
40.00%
30.00%
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
Line 4 Liabilities
Company
Line 4
Lines 1 to 3
24
Ten-Line Company
Surplus Allocations
60.00%
Marginal Surplus Requirement
50.00%
40.00%
30.00%
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
25
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.
26
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
27
The Benefit-Cost Trade Off
of
Marginal
Diversification
Increased
and
Reduction
Efficient
Business
administrative
Composition
operating
inSurplus
cost
costs
TheRequired
administrative
As
but
Figure
Present
costs
more
operating,
Trade
2Value
lines
off
of
are
administrative,
costs
reduced
added,
determines
diversification
surplus
and
the
requirements
perhaps
efficient
decreases
agency
composition
against
costs
required
operating
increase.
ofsurplus,
business.
and
Present
Value
of Costs
Marginal Operating
and Administrative Costs
Reduction in Cost
of Required Surplus
Efficient Composition
of Business
Increased
Diversification
28
Conclusions

Surplus can be allocated uniquely.

Allocations appear robust for multi-line companies.

Computational challenges remain.

What about other financial intermediaries?
29
30