Transcript Capital Allocation for Property-Casualty Insurers: A Catastrophe Reinsurance Application CAS Reinsurance Seminar

Capital Allocation for Property-Casualty
Insurers: A Catastrophe Reinsurance
Application
CAS Reinsurance Seminar
June 6-8, 1999
Robert P. Butsic
Fireman’s Fund Insurance
Yes, Capital Can Be Allocated!
 Outline of Presentation:
 General approach: Myers-Read model
– Joint cost allocation is a common economics problem
– Another options-pricing application to insurance
– Extensions, simplification and practical application of MR method
 Reinsurance (and primary insurance) application:
the layer as a policy
 Semi-realistic catastrophe reinsurance example
 Results and conclusions
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Economic Role of Capital in Insurance
 Affects value of default when insolvency occurs
 Default = expected policyholder deficit (market value)
 More capital implies smaller default value (good)
 But more capital implies higher capital cost (bad)
 Equilibrium:
Capital Cost
Solvency Benefit
Capital Amount
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Fair Premium Model
 For all an insurer’s policies:
P  L  D T C
 Important Points:
–
–
–
–
Shows cost and benefit of capital
All quantities at market values (loss includes risk load)
Loss can be attributed to policy/line
But C and D are joint
 Single policy model :
Pi  Li  Di  T  Ci
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Allocation Economics
 Capital ratios to losses are constant: C   ci Li
 Premiums are homogeneous:
 Implies that
Pi Li  Pi Li
di  Di Li  D L  d
 And marginal shift in line mix doesn’t change
default ratio:
D Li  d
 Solve this equation for ci
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Lognormal Model
 To solve for ci we need to specify relationship between
L, C and D
 Assume that loss and asset values are lognormal
 D is determined from Black-Scholes model
 Final result (modified Myers-Read):
(1  c) n( y)  iL   L2  iA   AL 
ci  c 



2
N ( y)  (1   L ) (1   AL ) 
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Simplifying the Myers-Read Result
 Assume that loss-asset correlation is small
 Define Loss Beta:  i   iL /  L2
 Result:
ci  c  (  i  1) Z
 Implications:
–
–
–
–
Relevant risk measure for capital allocation is loss beta
Capital allocation is exact; no overlap
Allocated capital can be negative
Z value is generic for all lines
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Numerical Example
Table 1.1
Loss Beta and Capital Allocation for Numerical Example
Liability
Loss
Loss
Capital/
Value
CV
Beta
Liability
Capital
Line 1
500
0.2000
0.8463
0.3957
197.87
Line 2
400
0.3000
1.3029
0.7055
282.19
Line 3
100
0.5000
0.5568
0.1993
19.93
Total
1000
0.2119
1.0000
0.5000
500.00
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Negative Capital Example
Assumptions:
• losses are
independent
•no asset risk
•total losses are
lognormal
Number of Policies
Policy 1
Policy 2
Loss CV
Policy 1
Policy 2
Total
Expected Loss
Capital
Default
Default Ratio
Capital/Loss, Policy 2
Original
Case
Add LowRisk Policy
Adjust
Capital
1000
0
1000
1
1000
1
10.0
0.3162
10.0
1.0
0.3159
10.0
1.0
0.3159
1000.000
500.000
16.594
1001.000
500.000
16.586
1001.000
499.660
16.610
1.6594%
1.6569%
1.6594%
-0.340
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Reinsurance Application
 For policy/treaty, capital allocation to layer depends on:
– covariance of layer with that of unlimited loss
– covariance of unlimited loss with other risks
 Layer Beta is analogous to loss beta
 Capital ratio for policy/layer within line/policy:
 

ci  ck   i  ik  1  k Z
 ik  k

 Point beta for layer is limit for narrow layer width:
 ( x) 

1  E1 ( x)



1
2 
s  X E0 ( x) 
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Point Betas for Some Loss Distributions
25.00
20.00
15.00
10.00
5.00
0.00
0
50
100
150
200
250
300
350
400
x
Legend: right hand side (x = 400), top to bottom
Pareto
Lognormal
Exponential
Gamma
Normal
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Market Values and Risk Loads
 Layer Betas depend on market values of losses
 Market values depend on risk loads
 Modern financial view of risk loads
– Adjust probability of event so that investor is indifferent to the expected
outcome or the actual random outcome
– Risk-neutral valuation

– General formula:
Xˆ  X (1   )   xfˆ ( x) dx
0
 In finance, standard risk process is GBM lognormal
– Risk load equals location parameter shift:
1    exp( )  exp(     )
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Reinsurance Risk Loads
 Risk-neutral valuation insures value additivity of layers
 Risk load for a layer
– integrate R-N density instead of actual density, giving
pure premium loaded for risk
– risk load is difference from unloaded pure premium
 Point risk load
– load for infinitesimally small layer
– parallel concept to point beta
 Simple formula:
Gˆ ( x)
 ( x) 
 1.
G ( x)
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General Layer Risk Load Properties
 Monotonic increasing with layer
 Generally unbounded
 Zero risk load at lowest point layer
 Lognormal example:
location PS
7.0
6.0
5.0
4.0
3.0
2.0
1.0
0.0
0
200
400
600
800
1000
x
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PRL and the Generalized PH Transform
 Location parameter shift may not be “risky” enough
 Wang’s Proportional Hazard transform
0  q 1
Gˆ ( x)  [G ( x)]q
 More general form:
Gˆ ( x)  [G ( x)]q ( x )
 Gives all possible positive point risk loads
 Fractional transform:
– No economic basis
– But it works
qx
q ( x) 
xm
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Parameter Estimation
 Market valuation requires modified statutory data
 Representative insurer concept necessary for capital
requirements
– particular insurer could have too much/little capital, risk, line mix, etc.
– industry averages can be biased
 Overall capital ratio
 CV estimates
– losses: reserves and incurred losses, cat losses
– assets
 Catastrophe beta
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Catastrophe Pricing Application
 Difficult, since high layers significantly increase
estimation error
 But, made easier because cat losses are virtually
independent of other losses
 Present value pricing model has 3 parts:
– PV of expected loss:
X (a, b) /(1  r )
– PV of risk load:
X (a, b)  (a, b) /(1  r )
– PV of capital cost:
X (a, b)[1   (a, b)] c(a, b) rt
(1  r )(1  t )
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Example: Annual Aggregate Treaty
a
b
0
100
100
200
200
300
300
400
400
500
500
600
600
700
700
800
800
900
900
1000
1000 Infinity
Expected
Loss
31.51
6.83
3.01
1.66
1.03
0.69
0.48
0.35
0.27
0.21
1.13
Capital
Cost
0.74
0.73
0.58
0.46
0.38
0.32
0.27
0.23
0.20
0.18
2.33
Risk
Load
8.78
4.73
2.64
1.67
1.14
0.82
0.62
0.47
0.37
0.30
2.03
Fair
Premium
41.04
12.29
6.24
3.79
2.55
1.83
1.37
1.06
0.84
0.69
5.48
Expected
Loss Ratio
0.814
0.589
0.512
0.463
0.427
0.398
0.374
0.354
0.336
0.321
0.219
Implied
ROE
0.272
0.213
0.180
0.161
0.149
0.140
0.133
0.127
0.123
0.119
0.094
0 Infinity
47.17
6.42
23.58
77.18
0.648
0.137
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Return on Equity for Treaty
 Look at point ROE
 Varies by layer
 Equals risk-free interest rate at zero loss size
0.400
0.350
0.300
0.250
0.200
0.150
0.100
0.050
0.000
0
20
40
60
80
100
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Summary
 How to allocate capital to line, policy or layer
– Key intuition is to keep a constant default ratio
– Relevant risk measure is loss or layer beta
– Allocated capital is additive
 Reinsurance and layer results
– Layer betas are monotonic, zero to extremely high
– Layer risk loads are monotonic, zero to extremely high
 ROE pricing method has severe limitations
– ROE at fair price will vary by line and layer
– capital requirement can be negative
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Conclusion
 Capital allocation is essential to an ROE pricing model
– capital is the denominator
– but this model has severe problems
 It’s less (but still) important in a present value pricing
model
– capital determines the cost of double taxation
– this model works pretty well (cat treaty example)
 The real action is in understanding the risk load process
– knowing the capital requirement doesn’t give the price
– because the required ROE is not constant
 We’ve got a lot of work to do!
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