On Optimal Reinsurance Arrangement

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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 Remarks

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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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1. Related Literature and Introduction (cont.)

• Introduction This Paper
• Standpoints
• From the ceding companys 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

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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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2. A Simple Model (cont.)

• 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 industrys
  existing reinsurance portfolio, or
  , (i) referring to all
  risks excluding contract i.
• 
  

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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)

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The Insurer

• 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

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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.

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Optimality Conditions
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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

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Figure 1. Claim Payment Distribution (Gamma
Distribution)
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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 Insurers Objective
Function as a Function of a and b
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Results

• 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.

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Results (cont.)

• 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.

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Results (cont.)

• 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
  low-probability layers.

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4. Discrete Loss Distribution
Loss Scenarios
Scenario 1 little or no loss occurrence Scenario
2 moderate losses Scenario 3 most severe losses
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4. Discrete Loss Distribution (cont.)

• 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

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4. Discrete Loss Distribution (cont.)

• The layer limit is independent of the probability
  with which each event occurs (not intuitive) and
  satisfies that
• The minimum (optimal) value of the insurers
  value function is equal to the rate on line of
  the reinsurance contract, or

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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 firms
  expected net income after adjusting for the
  capital costs in the unprofitable states.

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5. The Value of and Contingent Capital
Calls (cont.)
The objective function of the reinsurer is
formulated as
where the function g() is the capital call
charge function and satisfies and
.
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5. The Value of and Contingent Capital
Calls (cont.)

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

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 is more costly for
the reinsurer to underwrite more severe cat
events.
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Figure 5. The Choice of and the Value of the
Reinsurers Objective Function
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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.

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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 .