On Optimal Reinsurance
Arrangement
Yisheng Bu
Liberty Mutual Group
1
Agenda
1. Related Literature and Introduction
2. A Simple Model
3. Numerical Simulations
4. Discrete Loss Distribution
5. The Value of  and Contingent Capital Calls
6. Concluding Remarks
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2
1. Related Literature and
Introduction

Related Literature
 Optimality in Reinsurance Arrangement
• Optimal portfolio sharing through quota-share
contracts among insurers (Borch, 1961)
• Optimal proportional reinsurance (Lampaert and
Walhin, 2005)
• Optimal stop-loss reinsurance contracts for
minimizing the probability of ruin (Gajek and
Zagrodny, 2004)
3
1. Related Literature and
Introduction (cont.)

Related Literature
 Value of Reinsurance (Venter, 2001)
• Value of reinsurance comes from stability provided
 Aggregate Profile of Reinsurance Purchases
(Froot, 2001)
• Reinsurance contracts had been more often used to
cover lower catastrophic risk layers rather than
more severe but lower-probability layers.
• Reinsurance contracts had been priced in such a
way that higher reinsurance layers had higher ratios
of premium to expected losses.
4
1. Related Literature and
Introduction (cont.)

Introduction: This Paper
 Standpoints:
• From the ceding company’s perspective
• Focus on the aggregate reinsurance portfolio of the
insurer instead of individual reinsurance contracts
 Objectives:
• Optimal Excess-of-Loss Reinsurance Arrangement
for profit and stability maximization
• Provide justifications for the profile of reinsurance
purchases that had been observed for industry
5
2. A Simple Model
Assumptions:




The reinsurance market consists of one insurer and
one reinsurer
The insurer has no control over the pricing of its
(aggregate) reinsurance portfolio
The insurer knows about the reinsurance pricing rule
and chooses the reinsurance layer for full coverage
The insurer and reinsurer have access to the same
information on the underlying loss distribution
6
2. A Simple Model (cont.)
Reinsurance Pricing

The reinsurance pricing rule of the aggregate reinsurance
portfolio for insurer i:
Z ( x ; a , b )  E[ L ( x ; a , b )]   Var[ L ( x ; a , b )],
i
i
i
i
i
R
i
i
i
i
R
R
i
i
i
where  can be considered as the market price of risk
determined by the industry’s existing reinsurance portfolio,
or   ( Z  E[ L ]) / Var[ L ] , (i) referring to all risks
excluding contract i.
R
R
(i)
(i)
(i)
R
R
R
7
2. A Simple Model (cont.)
Discussions:





Addition of stochastically independent risks and
additive property of reinsurance contracts
No parameter uncertainty is considered
The “down-side” variance vs. total variance
Skewness and higher moments of the claim
payments distribution under reinsurance contracts
Supported by many empirical findings on
reinsurance pricing (Kreps and Major, 2001; Lane,
2004)
8
The Insurer
MIN : Z  E[ L ( x; a, b)]    Var[ L ( x; a, b)]
a ,b
S
S
S
s.t.Z  B
-Choose a and b to minimize the sum of reinsurance
costs and expected claim payments net of reinsurance
recovery
-Include a penalty term for the variation of net claim
payments
-Z: Reinsurance costs
-B: Budget constraint for reinsurance purchase
9
The Insurer (cont.)


For a given volume of premium from the underlying
insurance contracts, this formulation virtually maximizes
the expected value of the net income and its stability.
How much the insurer values stability (  )
depends on the degree of its risk aversion.
S
10
Optimality Conditions
[1   dF ( x)] { (b  a  E[ L ])   ( x  b  a  E[ L ])}dF ( x) 


a
b
R
R
S
S
[1   dF ( x)]  { ( x  a  E[ L ])   (a  E[ L ])}dF ( x)

b
a
a
R
R
S
S
  dF ( x)  { ( E[ L ])   ( x  E[ L ])}dF ( x)    (Z / a )

a
a
0
R
R
S
S
[1   dF ( x)] { (b  a  E[ L ])   ( x  b  a  E[ L ])}dF ( x)


b
b
R
R
S
S
  dF ( x)  { ( x  a  E[ L ])   (a  E[ L ])}dF ( x)

b
b
a
R
R
S
S
  dF ( x)  {( ( E[ L ])   ( x  E[ L ])}dF ( x)    (Z / b)

a
b
0
R
R
S
S
11
Optimality Conditions (cont.)

In essence, by choosing the optimal reinsurance
coverage, the insurer attempts to achieve the optimal
balance between the reduction in the cost of claim
payment variation via reinsurance coverage and the
price for shifting such variation to the reinsurer.
12
3. Numerical Simulations

Methodology
b
Eq. (5)
a0
Eq. (6)
a0
a
13
Figure 1. Claim Payment Distribution
(Gamma Distribution)
xe
f ( x) 
 !

(  x/ )
(  1 )
14
Figure 2. Reinsurance Premium as
a Function of a and b
15
Figure 3. The Value of the Insurer’s Objective
Function as a Function of a and b
16
Table 1. Numerical Simulation Results with  R  2 ,  S  2
Parameter Values, Expected Value and Variance of Underlying Loss Distribution
(1)
alpha
0
0
1
2
(2)
beta
1
2
1
1
(3)
E(x)
1
2
2
3
(4)
Var(x)
1
4
2
3
(7)
limit
1.594
3.187
2.008
2.351
(8)
Z
0.806
2.648
1.531
2.218
case 1
case 2
benchmark
case 3
Simulation Results
(5)
a
1.018
2.035
1.805
2.631
(6)
b
2.611
5.222
3.813
4.982
(9)
E(Lr)
0.288
0.576
0.497
0.670
(10)
Obj
2.180
6.719
4.367
6.556
(11)
ROL
0.506
0.831
0.762
0.943
(12)
Z/E(Lr)
2.798
4.597
3.078
3.311
case 1
case 2
benchmark
case 3
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Results
1.
2.
The results justify the aggregate profile of
reinsurance purchases observed in Froot (2001):
• To stabilize its book of business and
maximize net income, it is optimal to use
reinsurance protection against risks of
moderate size, but leave the most severe
loss scenarios uncovered or self-insured.
The results suggest that the retention be set to be
comparable to the expected ground-up claim
payments.
18
Results (cont.)
3.
In situations where
•
underlying claim payments are more dispersed
(higher  ),
•
events of higher severity occur with larger
probabilities (higher  ),
the insurer should purchase more protection against
more severe events, or higher limit and higher
retention. As a result, the optimal reinsurance layer
in the above situations also has higher ROL and
higher ratio of premium to expected losses.
19
Results (cont.)
4.
5.
The optimal choices of the reinsurance layer can
be very sensitive to the chosen values of the model
parameters, which implies that parameter
uncertainty is an important consideration in
reinsurance purchase.
To the extent that higher reinsurance layers are
more vulnerable to prediction errors from
engineering models, parameter uncertainty may
well explain the observed high prices for lowprobability layers.
20
Varying the value of  R .
Figure 4. Retention and Limit As Reinsurance Load Changes (assuming  S  2 )
Retention/Limit
4.500
4.000
3.500
Retention + Limit
3.000
2.500
Retention
2.000
1.500
1.000
0.500
0.000
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
Reinsurance Load
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4. Discrete Loss Distribution
Loss Scenarios:
Scenario 1 Scenario 2 Scenario 3
Total Cat Loss
s1
s2
s3
Probability
f1
f2
f3
0s s s
1
2
3
f  f  f 0
1
2
3
f  f  f 1 s  0
1
2
3
1
Scenario 1: little or no loss occurrence
Scenario 2: moderate losses
Scenario 3: most severe losses
22
4. Discrete Loss Distribution
(cont.)
It can be shown that:

At the optimum,
0a s b s
*
*
2

3
which implies that it is advisable for the insurer to
purchase some reinsurance protection against both
moderate and most severe cat loss scenarios rather
than against any one particular scenario only.
Specifically, the optimal reinsurance layer boundaries
are given by
s
a 
 
*
2
R
R
s
s  s
b a 

 
 
*
S
*
3
R
S
2
S
R
R
3
S
S
23
4. Discrete Loss Distribution
(cont.)

The layer limit is independent of the probability with
which each event occurs (not intuitive) and satisfies
that
s  s
l b a 
 
*
*
*
2
R
R

3
S
S
The minimum (optimal) value of the insurer’s value
function is equal to the rate on line of the reinsurance
contract, or
Obj  Z /(b  a )
*
*
*
min
24
5. The Value of  and
Contingent Capital Calls
R

The capital consumption approach to reinsurance
pricing uses the value of potential capital usages as
the risk load (Mango, 2004)

The reinsurer attempts to maximize the firm’s
expected net income after adjusting for the capital
costs in the unprofitable states.
25
5. The Value of  and
Contingent Capital Calls (cont.)
R
The objective function of the reinsurer is formulated as
MAX :   Var[ L ]   g ( x  (a  Z ))dF ( x) 
b
R
R
R
a z
g (b  (a  Z ))   dF ( x)

b
where the function g(*) is the capital call charge function
and satisfies g ' ()  0 and g" ()  0 .
26
5. The Value of  and
Contingent Capital Calls (cont.)
R
For the purpose of simulation, g(*) is specified as
g ( )    c  
2
where  (   0 ) is the amount of capital calls and
c ( c  0 ) is the rate at which the marginal cost of
capital calls increases. With higher values of c , it is
more costly for the reinsurer to underwrite more
severe cat events.
27
Figure 5. The Choice of  and the Value of
the Reinsurer’s Objective Function
R
0.20
c=4
0.10
c=5
0.00
0
.0
10
50
9.
00
9.
50
8.
00
8.
50
7.
00
7.
50
6.
00
6.
50
5.
00
5.
50
4.
00
4.
50
3.
00
3.
-0.10
50
2.
00
2.
Value of Reinsurer's Objective Function
0.30
c=8
-0.20
-0.30
-0.40
Gam m aR
28
5. The Value of  and
Contingent Capital Calls (cont.)
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Results from Figure 5:
 When the marginal cost of capital calls
increases relatively faster for the reinsurer
(higher c), the reinsurer sets higher  and the
insurer tends to purchase reinsurance
protection for moderate losses only and leave
higher layers uncovered.
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6. Concluding Remarks

Summary
 Optimal excess-of-loss reinsurance purchase
when the insurer maximizes net income and
stability for both the discrete and continuous
loss distribution


Optimal Reinsurance Purchase
 Other Considerations
 Can it be optimized?
The cost of reinsurance capital
 Empirical measurement of  .
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