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On Optimal Reinsurance Arrangement

Yisheng Bu

Liberty Mutual Group

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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 RemarksR

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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)

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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.

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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

1. Related Literature and Introduction (cont.)

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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

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Reinsurance Pricing

The reinsurance pricing rule of the aggregate reinsurance portfolio for insurer i:

where can be considered as the market price of risk determined by the industry’s existing reinsurance portfolio, or , (i) referring to all risks excluding contract i.

)],,;([)],;([),;(iii

i

RRiii

i

Riii

i baxLVarbaxLEbaxZ

R

][/])[( )()()( i

R

i

R

i

RRLVarLEZ

2. A Simple Model (cont.)

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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)

2. A Simple Model (cont.)

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The Insurer

)],;([)],;([ :,

baxLVarbaxLEZMINSSSba

BZts ..

-Choose a and b to minimize the sum of reinsurancecosts and expected claim payments net of reinsurancerecovery-Include a penalty term for the variation of net claimpayments-Z: Reinsurance costs -B: Budget constraint for reinsurance purchase

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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

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Optimality Conditions

)/()(])}[(])[({)(

)(])}[(])[({ ])(1[

)(])}[(])[({])(1[

0 aZxdFLExLExdF

xdFLEaLEaxxdF

xdFLEabxLEabxdF

a

a

SSRR

b

aSSRR

a

bSSRR

a

)/()(])}[(])[({()(

)(])}[(])[({)(

)(])}[(])[({])(1[

0 bZxdFLExLExdF

xdFLEaLEaxxdF

xdFLEabxLEabxdF

a

SSRRb

b

aSSRR

b

bSSRR

b

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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.

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3. Numerical Simulations Methodology

a

Eq. (5) Eq. (6)

b

a0 a0

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Figure 1. Claim Payment Distribution (Gamma Distribution)

)1(

)/(

!)(

xexxf

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Figure 2. Reinsurance Premium as a Function of a and b

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Figure 3. The Value of the Insurer’s Objective Function as a Function of a and b

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Table 1. Numerical Simulation Results with 2R, 2S Parameter Values, Expected Value and Variance of Underlying Loss Distribution

(1) (2) (3) (4)alpha beta E(x) Var(x)

0 1 1 1 case 10 2 2 4 case 21 1 2 2 benchmark2 1 3 3 case 3

Simulation Results

(5) (6) (7) (8) (9) (10) (11) (12)a b limit Z E(Lr) Obj ROL Z/E(Lr)

1.018 2.611 1.594 0.806 0.288 2.180 0.506 2.798 case 12.035 5.222 3.187 2.648 0.576 6.719 0.831 4.597 case 21.805 3.813 2.008 1.531 0.497 4.367 0.762 3.078 benchmark2.631 4.982 2.351 2.218 0.670 6.556 0.943 3.311 case 3

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Results

1. 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.

2. The results suggest that the retention be set to be comparable to the expected ground-up claim payments.

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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 againstmore severe events, or higher limit and higherretention. As a result, the optimal reinsurance layerin the above situations also has higher ROL andhigher ratio of premium to expected losses.

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Results (cont.)

4. 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.

5. 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 low-probability layers.

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Varying the value of R. Figure 4. Retention and Limit As Reinsurance Load Changes (assuming 2S)

0.000

0.500

1.000

1.500

2.000

2.500

3.000

3.500

4.000

4.500

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

Retention/Limit

Retention

Retention + Limit

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4. Discrete Loss Distribution

Scenario 1 Scenario 2 Scenario 3Total Cat Loss s1 s2 s3

Probability f1 f2 f3

Scenario 1: little or no loss occurrenceScenario 2: moderate lossesScenario 3: most severe losses

3210 sss 0

321 fff

1321 fff 0

1s

Loss Scenarios:

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It can be shown that: At the optimum,

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

3

*

2

*0 sbsa

SR

Rs

a

2*

SR

SR

SR

Ssss

ab

323**

4. Discrete Loss Distribution (cont.)

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The layer limit is independent of the probability with which each event occurs (not intuitive) and satisfies that

The minimum (optimal) value of the insurer’s value function is equal to the rate on line of the reinsurance contract, or

SR

SRss

abl

32***

)/( ***

minabZObj

4. Discrete Loss Distribution (cont.)

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5. The Value of and Contingent Capital Calls

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.

R

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5. The Value of and Contingent Capital Calls (cont.)

b

b

zaRR

xdFZabg

xdFZaxgLVarMAXR

)())((

)())((][:

where the function g(*) is the capital call charge function and satisfies and . 0)(' g 0)(" g

The objective function of the reinsurer is formulated as

R

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5. The Value of and Contingent Capital Calls (cont.)

For the purpose of simulation, g(*) is specified as

2)( cg

where ( ) is the amount of capital calls and ( ) is the rate at which the marginal cost of capital calls increases. With higher values of , it ismore costly for the reinsurer to underwrite more severe cat events.

0c 0c

c

R

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Figure 5. The Choice of and the Value of the Reinsurer’s Objective Function

R

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

2.00

2.50

3.00

3.50

4.00

4.50

5.00

5.50

6.00

6.50

7.00

7.50

8.00

8.50

9.00

9.50

10.00

GammaR

Val

ue

of

Rei

nsu

rer'

s O

bje

ctiv

e F

un

ctio

n

c=5

c=8

c=4

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5. The Value of and Contingent Capital Calls (cont.)

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.

R

R

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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 .R