Boot Camp on Reinsurance Pricing Techniques Loss Sensitive Treaty Provisions

1
Boot Camp on Reinsurance Pricing Techniques
Loss Sensitive Treaty Provisions

• July 2005

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Introduction to Loss Sensitive Provision

• Definition A reinsurance contract provision that
  varies the ceded premium, loss, or commission
  based upon the loss experience of the contract
• Purpose Client shares in ceded experience
  could be incented to care more about the
  reinsurers results
• Typical Loss Sharing Provisions
• Profit Commission
• Sliding Scale Commission
• Loss Ratio Corridors
• Annual Aggregate Deductibles
• Swing Rated Premiums
• Reinstatements

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Simple Profit Commission Example

• A property pro-rata contract has the following
  profit commission terms
• 50 Profit Commission after a reinsurers margin
  of 10.
• Key Point Reinsurer returns 50 of the
  contractually defined profit to the cedant
• Profit Commission Paid to Cedant 50 x
  (Premium - Loss - Commission - Reinsurers Margin)
• If profit is negative, reinsurers do not get any
  additional money from the cedant.

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Simple Profit Commission Example

• Profit Commission 50 after 10 reinsurers
  Margin
• Ceding Commission 30
• Loss ratio must be less than 60 for us to pay a
  profit commission
• Contract Expected Loss Ratio 70
• 1 Premium - 0.7 Loss - 0.3 Comm - 0.10 Reins
  Margin minus 0.10
• Is the expected cost of profit commission zero?

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Simple Profit Commission Example

• Answer The expected cost of profit commission is
  not zero
• Why Because 70 is the expected loss ratio.
• There is a probability distribution of potential
  outcomes around that 70 expected loss ratio.
• It is possible (and may even be likely) that the
  loss ratio in any year could be less than 60.

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Cost of Profit Commission Simple Quantification

• Earthquake exposed California property pro-rata
  treaty
• LR 40 in all years with no EQ
• Profit Comm when there is no EQ 50 x (1 of
  Premium - 0.4 Loss - 0.30 Commission - 0.1
  Reinsurers Margin)
• 10 of premium
• Cat Loss Ratio 30.
• 10 chance of an EQ costing 300 of premium, 90
  chance no EQ loss
• Expected Cost of Profit Comm
• Profit Comm Costs 10 of Premium x 90
  Probability of No EQ
• 0 Cost of PC x 10 Probability of EQ Occurring
  9 of Premium

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Basic Mechanics of Analyzing Loss Sensitive
Provisions

• Build aggregate loss distribution
• Apply loss sensitive terms to each point on the
  loss distribution or to each simulated year
• Calculate a probability weighted average cost (or
  saving) of the loss sensitive arrangement

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Example of Basic Mechanics PC 50 after 10,
30 Commission, 65 Expected LR
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Determining an Aggregate Distribution - Two
Methods

• Fit statistical distribution to on level loss
  ratios
• Reasonable for pro-rata treaties.
• Determine an aggregate distribution by modeling
  frequency and severity
• Typically used for excess of loss treaties.

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Fitting a Distribution to On Level Loss Ratios

• Most actuaries use the lognormal distribution
• Reflects skewed distribution of loss ratios
• Easy to use
• Lognormal distribution assumes that the natural
  logs of the loss ratios are distributed normally.

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Skewness of Lognormal Distribution
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Fitting a Lognormal Distribution to Projected
Loss Ratios

• Fitting the lognormal
• s2 LN(CV2 1)
• m LN(mean) - s2/2
• Mean Selected Expected Loss Ratio
• CV Standard Deviation over the Mean of the
  loss ratio (LR) distribution.
• Prob (LR X) Normal Dist(( LN(x) - m )/ s)
  i.e.. look up (LN(x) - m )/ s) on a standard
  normal distribution table
• Producing a distribution of loss ratios
• For a given point i on the CDF, the following
  Excel command will produce a loss ratio at that
  CDFi
• Exp (m Normsinv(CDFi) x s)

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Sample Lognormal Loss Ratio Distribution
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Is the resulting LR distribution reasonable?

• Compare resulting distribution to historical
  results
• Focus on level LRs, but dont completely ignore
  untrended ultimate LRs.
• Potential for cat or shock losses not captured
  within historical experience
• Degree to which trended past experience is
  predictive of future results for a book
• Actuary and underwriter should discuss the above
  issues
• If the distribution is not reasonable, adjust the
  CV selection.

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Process and Parameter Uncertainty

• Process Uncertainty Random fluctuation of
  results around the expected value.
• Parameter Uncertainty Do you really know the
  true mean of the loss ratio distribution for the
  upcoming year?
• Are your trend, loss development rate change
  assumptions correct?
• For this book, are past results a good indication
  of future results?
• Changes in mix and type of business
• Changes in management or philosophy
• Is the book growing, shrinking or stable
• Selected CV should usually be above indicated
• 5 to 10 years of data does not reflect full range
  of possibilities

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Modeling Parameter Uncertainty One Suggestion

• Select 3 equally likely expected loss ratios
• Assign weight to each loss ratio so that the
  weighted average ties to your selected expected
  loss ratio
• Example Expected LR is 65, assume 1/3
  probability that true mean LR is 60, 1/3
  probability that it is 65, and 1/3 probability
  that it is 70.
• Simulate the true expected loss ratio (reflects
  Parameter Uncertainty)
• Simulate the loss ratio for the year modeled
  using the lognormal based on simulated expected
  loss ratio above your selected CV (reflects
  Process Variance)

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Example of Modeling Parameter Uncertainty
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Common Loss Sharing Provisions for Pro-rata
Treaties

• Profit Commissions
• Already covered
• Sliding Scale Commission
• Loss Ratio Corridor
• Loss Ratio Cap

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Sliding Scale Comm

• Commission initially set at Provisional amount
• Ceding commission increases if loss ratios are
  lower than expected
• Ceding commission decreases if losses are higher
  than expected

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Sliding Scale Commission Example

• Provisional Commission 30
• If the loss ratio is less than 65, then the
  commission increases by 1 point for each point
  decrease in loss ratio up to a maximum commission
  of 35 at a 60 loss ratio
• If the loss ratio is greater than 65, the
  commission decreases by 0.5 for each 1 point
  increase in LR down to a minimum comm. of 25 at
  a 75 loss ratio
• If the expected loss ratio is 65 is the expected
  commission 30?

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Sliding Scale Commission - Solution
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Loss Ratio Corridors

• A loss ratio corridor is a provision that forces
  the ceding company to retain losses that would be
  otherwise ceded to the reinsurance treaty
• Loss ratio corridor of 100 of the losses between
  a 75 and 85 LR
• If gross LR equals 75, then ceded LR is 75
• If gross LR equals 80, then ceded LR is 75
• If gross LR equals 85, then ceded LR is 75
• If gross LR equals 100, then ceded LR is ???

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Loss Ratio Cap

• This is the maximum loss ratio that could be
  ceded to the treaty.
• Example 200 Loss Ratio Cap
• If LR before cap is 150, then ceded LR is 150
• If LR before cap is 250, then ceded LR is 200

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Loss Ratio Corridor Example

• Reinsurance treaty has a loss ratio corridor of
  50 of the losses between a loss ratio of 70 and
  80.
• Use the aggregate distribution to your right to
  estimate the expected ceded LR net of the corridor

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Loss Ratio Corridor Example Solution
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Modeling Property Treaties with Significant Cat
Exposure

• Model non-cat cat LRs separately
• Non Cat LRs fit to a lognormal curve
• Cat LR distribution produced by commercial
  catastrophe model
• Combine (convolute) the non-cat cat loss ratio
  distributions

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Convoluting Non-cat Cat LRs - Example
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Truncated Loss Ratio Distributions

• Problem To reasonably model the possibility of
  high LR requires a high lognormal CV
• High lognormal CV often leads to unrealistically
  high probabilities of low LRs, which overstates
  cost of PC
• Solution Dont allow LR to go below selected
  minimum, e.g.. 0 probability of LRlt30
• Adjust the mean loss ratio used to calculate the
  lognormal parameters to cause the aggregate
  distribution to probability weight back to
  initial expected LR

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Summary of Loss Ratio Distribution Method

• Advantage
• Easier and quicker than separately modeling
  frequency and severity
• Reasonable for most pro-rata treaties
• Usually inappropriate for excess of loss
  contracts
• Does not reflect the hit or miss nature of many
  excess of loss contracts
• Understates probability of zero loss
• May understate the potential of losses much
  greater than the expected loss

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Excess of Loss Contracts Separate Modeling of
Frequency and Severity

• Used mainly for modeling excess of loss contracts
• Most aggregate distribution approaches assume
  that frequency and severity are independent
• Different Approaches
• Simulation (Focus of this presentation)
• Numerical Methods
• Heckman Meyers Fast calculating approximation
  to aggregate distribution
• Panjer Method
• Select discrete number of possible severities
  (i.e. create 5 possible severities with a
  probability assigned to each)
• Convolutes discrete frequency and severity
  distributions.
• A detailed mathematical explanation of these
  methods is beyond the scope of this session.
• Software that can be used for simulations
• _at_Risk
• Excel

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Common Frequency Distributions

• Poisson
• f(xl) exp(-l) lx / x!
• where l mean of the claim count distribution
  and x claim count 0,1,2,...
• f(xl) is the probability of x losses, given a
  mean claim count of l
• x! x factorial, i.e. 3! 3 x 2 x 1 6
• Poisson distribution assumes the mean and
  variance of the claim count distribution are
  equal.

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Fitting a Poisson Claim Count Distribution

• Trend claims from ground up, then slot to
  reinsurance layer.
• Estimate ultimate claim counts by year by
  developing trended claims to layer.
• Multiply trended claim counts by frequency trend
  factor to bring them to the frequency level of
  the upcoming treaty year.
• Adjust for change in exposure levels, i.e..
• Adjusted Claim Count year i
• Trended Ultimate Claim Count i x
• (SPI for upcoming treaty year / On Level SPI year
  i)
• Poisson parameter l equals the mean of the
  ultimate, trended, adjusted claim counts from
  above

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Example of Simulated Claim Count
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Modeling Frequency- Negative Binomial

• Negative Binomial Same form as the Poisson
  distribution, except that it assumes that l is
  not fixed, but rather has a gamma distribution
  around the selected l
• Claim count distribution is Negative Binomial if
  the variance of the count distribution is greater
  than the mean
• The Gamma distribution around l has a mean of 1
• Negative Binomial simulation
• Simulate l (Poisson expected count)
• Using simulated expected claim count, simulate
  claim count for the year.
• Negative Binomial is the preferred distribution
• Reflects some parameter uncertainty regarding the
  true mean claim count
• The extra variability of the Negative Binomial is
  more in line with historical experience

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Algorithm for Simulating Claim Counts Using a
Poisson Distribution

• Poisson
• Manually create a Poisson cumulative distribution
  table
• Simulate the CDF (a number between 0 and 1) and
  lookup the number of claims corresponding to that
  CDF (pick the claim count with the CDF just below
  the simulated CDF) This is your simulated claim
  count for year 1
• Repeat the above two steps for however many years
  that you want to simulate

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Negative Binomial Contagion Parameter

• Determine contagion parameter, c, of claim count
  distribution
• (s2 / m) 1 c m
• If the claim count distribution is Poisson, then
  c0
• If it is negative binomial, then cgt0, i.e.
  variance is greater than the mean
• Solve for the contagion parameter
• c (s2 / m) - 1 / m

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Additional Steps for Simulating Claim Counts
using Negative Binomial

• Simulate gamma random variable with a mean of 1
• Gamma distribution has two parameters a and b
• a 1/c b c c contagion parameter
• Using Excel, simulate gamma random variable as
  follows Gammainv(Simulated CDF, a, b)
• Simulated Poisson parameter
• l x Simulated Gamma Random Variable Above
• Use the Poisson distribution algorithm using the
  above simulated Poisson parameter, l, to simulate
  the claim count for the year

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Year 1 Simulated Negative Binomial Claim Count
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Year 1 Simulated Negative Binomial Claim Count
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Year 2 Simulated Negative Binomial Claim Count
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Year 2 Simulated Negative Binomial Claim Count
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Modeling Severity Common Severity Distributions

• Lognormal
• Mixed Exponential (currently used by ISO)
• Pareto
• Truncated Pareto.
• This curve was used by ISO before moving to the
  Mixed Exponential and will be the focus of this
  presentation.
• The ISO Truncated Pareto focused on modeling the
  larger claims. Typically those over 50,000

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Truncated Pareto

• Truncated Pareto Parameters
• t truncation point.
• s average claim size of losses below
  truncation point
• p probability claims are smaller than
  truncation point
• b pareto scale parameter - larger b results in
  larger unlimited average loss
• q pareto shape parameter - lower q results in
  thicker tailed distribution
• Cumulative Distribution Function
• F(x) 1 - (1-p) ((t b)/(x b))q
• Where xgtt

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Algorithm for Simulating Severity to the Layer

• For each loss to be simulated, choose a random
  number between 0 and 1. This is the simulated CDF
• Transformed CDF for losses hitting layer (TCDF)
• Prob(Loss lt Reins Att. Pt)
• Simulated CDF x Prob (Loss gt Reins Att. Pt)
• If there is a 95 chance that loss is below
  attachment point, then the transformed CDF (TCDF)
  is between 0.95 and 1.00.
• Find simulated ground up loss, x, that
  corresponds to simulated TCDF
• Doing some algebra, find x using the following
  formula
• x Expln(tb) - ln(1-TCDF) - ln(1-p)/Q - b
• From simulated ground up loss calculate loss to
  the layer

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Year 1 Loss 1 Simulated Severity to the Layer
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Year 1 Loss 2 Simulated Severity to the Layer
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Simulation Summary
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Common Loss Sharing Provisions for Excess of Loss
Treaties

• Profit Commissions
• Already covered
• Swing Rated Premium
• Annual Aggregate Deductibles
• Limited Reinstatements

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Swing Rated Premium

• Ceded premium is dependent on loss experience
• Typical Swing Rating Terms
• Provisional Rate 10
• Minimum/Margin 3
• Maximum 15
• Ceded Rate Minimum/Margin
• Ceded Loss as of SPI x 1.1
• subject to a maximum rate of 15.
• Why did 100/80 x burn subject to min and max rate
  become extinct?

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Swing Rated Premium - Example

• Burn (ceded loss / SPI) 10. Rate 3 10 x
  1.1 14
• Burn 2. Rate 3 2 x 1.1 5.2.
• Burn 14. Calculated Rate 3 14 x 1.1
  18.4. Rate 15 maximum rate

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Swing Rated Premium Example

• Swing Rating Terms Ceded premium is adjusted to
  equal to a 3 minimum rate ceded loss times 1.1
  loading factor, subject to a maximum rate of 15
• Use the aggregate distribution to your right to
  calculate the ceded loss ratio under the treaty

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Swing Rated Premium Example - Solution
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Annual Aggregate Deductible

• The annual aggregate deductible (AAD) refers to a
  retention by the cedant of losses that would be
  otherwise ceded to the treaty
• Example Reinsurer provides a 500,000 xs
  500,000 excess of loss contract. Cedant retains
  an AAD of 750,000
• Total Loss to Layer 500,000. Cedant retains
  all 500,000. No loss ceded to reinsurers
• Total Loss to Layer 1 mil. Cedant retains
  750,000. Reinsurer pays 250,000.
• Total Loss to Layer 1.5 mil. Cedant retains?
  Reinsurer pays?

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Annual Aggregate Deductible

• Discussion Question Reinsurer writes a 500,000
  xs 500,000 excess of loss treaty.
• Expected Loss to the Layer is 1 million (before
  AAD)
• Cedant retains a 500,000 annual aggregate
  deductible.
• Cedant says, I assume that you will decrease
  your expected loss by 500,000.
• How do you respond?

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Annual Aggregate Deductible Example

• Your expected burn to a 500K xs 500K
  reinsurance layer is 11.1. Cedant adds an AAD of
  5 of subject premium
• Using the aggregate distribution of burns to your
  right, calculate the burn net of the AAD.

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Annual Aggregate Deductible Example - Solution
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Limited Reinstatements

• Limited reinstatements refers to the number of
  times that the risk limit of an excess can be
  reused.
• Example 1 million xs 1 million layer
• 1 reinstatement It means that after the cedant
  uses up the first limit, they also get a second
  limit
• Treaty Aggregate Limit
• Risk Limit x (1 number of Reinstatements)

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Limited Reinstatements Example
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Reinstatement Premium

• In many cases to reinstate the limit, the
  cedant is required to pay an additional premium
• Choosing to reinstate the limit is almost always
  mandatory
• Reinstatement premium should simply be viewed as
  additional premium that reinsurers receive
  depending on loss experience

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Reinstatement Premium Example 1
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Reinstatement Premium Example 2
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Reinstatement Example 3

• Reinsurance Treaty
• 1 mil xs 1 mil
• Upfront Premium 400K
• 2 Reinstatements 1st at 50, 2nd at 100
• Using the aggregate distribution to the right,
  calculate our expected ultimate loss, premium,
  and loss ratio

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Reinstatement Example 3 Solution
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Reinstatement Example 4

• Note Reinstatement provisions are typically
  found on high excess layers, where loss tends to
  be either 0 or a full limit loss.
• Assume Layer 10M xs 10M, Expected Loss 1M,
  Poisson Frequency with mean .1

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Deficit Carry forward

• Treaty terms may include Deficit Carry forward
  Provisions, in which some losses are carried
  forward to next years contract in determining
  the commission paid.
• Example

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Deficit Carry forward Example

• Solution - Shift Sliding Scale Commission terms.

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DCF/Multi-Year Block

• Question How much credit do you give an
  account for Deficit Carry forwards, other than
  using the CF from the previous year (e.g.
  unlimited CFs)?
• Can estimate using an average of simulated
  years, but this method should be used with
  caution
• Assumes independence (probably unrealistic)
• Accounts for both Deficit and Credit carry
  forwards
• Deficits are often forgiven, treaty terms may
  change, or treaty may be terminated before the
  benefit of the deficit carry forward is felt by
  the reinsurer.

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DCF/Multi-Year Block - Example
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Technical Summary

• Modeling loss sensitive provisions is easy.
• Selecting your expected loss and aggregate
  distribution is hard
• Steps to analyzing loss sensitive provisions
• Build aggregate loss distribution
• Apply loss sensitive terms to each point on the
  loss distribution or to each simulated year
• Calculate probability weighted average of treaty
  results

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Additional Issues Uses of Aggregate
Distributions

• Correlation between lines of business
• Reserving for loss sensitive treaty terms
• Some companies Use aggregate distributions to
  measure risk allocate capital. One hypothetical
  example
• Capital 99th percentile Discounted Loss x
  Correlation Factor
• Fitting Severity Curves Dont Ignore Loss
  Development
• Increases average severity
• Increases variance claims spread as they
  settle.
• See Survey of Methods Used to Reflect
  Development in Excess Ratemaking by Stephen
  Philbrick, CAS 1996 Winter Forum

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Risk transfer

• FASB 113 A reinsurance contract should be booked
  using deposit accounting unless
• The reinsurer assumes significant insurance
  risk
• Insurance risk not significant if the
  probability of a significant variation in either
  the amount or timing of payments by the reinsurer
  is remote
• It is reasonably possible that the reinsurer may
  realize a significant loss from the transaction.
• 10/10 Rule of Thumb Is there a 10 chance that
  the reinsurer will have a loss of at least 10 of
  premium on a discounted basis
• Calculation excludes brokerage and reinsurer
  internal expense.
• SFAS 62 governs statutory accounting.
  Requirements are similar to FASB 113.
• Recent regulator concerns have centered on
  pro-rata reinsurance.

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Risk Transfer Recent Developments

• New York State Draft Bifurcation Proposal
• Bifurcation applies to any pro-rata treaty that
  contains one of the following features profit
  commissions, sliding scale commissions, loss
  ratio corridors or caps, occurrence limits below
  an unspecificed of premium, etc.
• Excess of loss and facultative contracts are
  excluded
• If above conditions are met, premium must be
  split as follows
• Premium covering exposure in excess of the 90th
  percentile of the loss distribution counts as
  reinsurance.
• The remaining premium should be booked as a
  deposit.
• Rule would be applied retroactively to business
  written 1/1/94 and later.
• Appears unlikely that NAIC will approve this
  proposal, but proposal emphasizes regulators
  concerns.

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Concluding Comment

• Aggregate distributions are a critical element in
  evaluating the profitability of business.
• They are frequently produced by (re)insurers as a
  risk management tool.
• They are being used on a broader spectrum of
  contracts to review risk transfer.
• Some accountants and regulators seem to treat
  these aggregate distributions as if they were
  gospel.
• Critical to effectively communicate the
  difficulties in projecting aggregate
  distributions of future results.
• Need to make regulators and accountants
  understand the degree of parameter uncertainty.