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Boot Camp on Reinsurance Pricing Techniques Loss Sensitive Treaty Provisions


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Title: Boot Camp on Reinsurance Pricing Techniques Loss Sensitive Treaty Provisions

Boot Camp on Reinsurance Pricing Techniques
Loss Sensitive Treaty Provisions
  • August 2007
  • Jeffrey L, Dollinger, FCAS

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
  • Purposes
  • Client shares in ceded experience could be
    incented to care more about the reinsurers
  • Can compensate for differences between reinsurer
    and client view of reinsurance program expected
  • Typical Loss Sharing Provisions
  • Profit Commission
  • Sliding Scale Commission
  • Loss Ratio Corridors
  • Annual Aggregate Deductibles
  • Swing Rated Premiums
  • Reinstatements

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.

Simple Profit Commission Example
  • Profit Commission 50 after 10 Reinsurers
  • 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?

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.
  • Giving back some profits below a 60 loss ratio
    has a cost

Cost of Profit Commission Simple Quantification
  • Earthquake exposed California property pro-rata
  • 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

Basic Mechanics of Analyzing Loss Sensitive
  • 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

Example of Basic Mechanics PC 50 after 10,
30 Commission, 65 Expected LR
Determining an Aggregate Distribution - Two
  • Fit statistical distribution to on level loss
  • Reasonable for pro-rata treaties.
  • Determine an aggregate distribution by modeling
    frequency and severity
  • Typically used for excess of loss treaties.

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.

Skewness of Lognormal Distribution
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
  • Exp (m Normsinv(CDFi) x s)

Sample Lognormal Loss Ratio Distribution
Is the resulting LR distribution reasonable?
  • Compare resulting distribution to historical
  • On level LRs should be the focus, 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
  • If the distribution is not reasonable, adjust the
    CV selection.

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

Modeling Parameter Uncertainty One Suggestion
  • Select 3 (or more) possible true expected loss
  • 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)

Example of Modeling Parameter Uncertainty
Common Loss Sharing Provisions for Pro-rata
  • Profit Commissions
  • Already covered
  • Sliding Scale Commission
  • Loss Ratio Corridor
  • Loss Ratio Cap

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

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?

Sliding Scale Commission - Solution
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 ???

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

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

Loss Ratio Corridor Example Solution
Modeling Property Treaties with Significant Cat
  • 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
  • Alternate easier method Simulate non-cat loss
    ratio, then simulate cat loss ratio

Convoluting Non-cat Cat LRs - Example
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

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
  • 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

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
  • A detailed mathematical explanation of these
    methods is beyond the scope of this session.
  • Software that can be used for simulations
  • _at_Risk
  • Excel

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

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
  • Poisson parameter l equals the mean of the
    ultimate, trended, adjusted claim counts from

Example of Simulated Claim Count
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

Algorithm for Simulating Claim Counts Using a
Poisson Distribution
  • Poisson
  • Manually create a Poisson cumulative distribution
  • 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

Negative Binomial Contagion Parameter
  • Determine contagion parameter, c, of claim count
  • (s2 / m) 1 c m
  • If the claim count distribution is Poisson, then
  • 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

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

Year 1 Simulated Negative Binomial Claim Count
Year 1 Simulated Negative Binomial Claim Count
Year 2 Simulated Negative Binomial Claim Count
Year 2 Simulated Negative Binomial Claim Count
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
  • The ISO Truncated Pareto focused on modeling the
    larger claims. Typically those over 50,000

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

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
  • x Expln(tb) - ln(1-TCDF) - ln(1-p)/Q - b
  • From simulated ground up loss calculate loss to
    the layer

Year 1 Loss 1 Simulated Severity to the Layer
Year 1 Loss 2 Simulated Severity to the Layer
Simulation Summary
Common Loss Sharing Provisions for Excess of Loss
  • Profit Commissions
  • Already covered
  • Swing Rated Premium
  • Annual Aggregate Deductibles
  • Limited Reinstatements

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?

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

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

Swing Rated Premium Example - Solution
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?

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
  • Cedant retains a 500,000 annual aggregate
  • Cedant says, I assume that you will decrease
    your expected loss by 500,000.
  • How do you respond?

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.

Annual Aggregate Deductible Example - Solution
Limited Reinstatements
  • Limited reinstatements refers to the number of
    times that the risk limit of an excess can be
  • 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
  • Treaty Aggregate Limit
  • Risk Limit x (1 number of Reinstatements)

Limited Reinstatements Example
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
  • Reinstatement premium should simply be viewed as
    additional premium that reinsurers receive
    depending on loss experience

Reinstatement Premium Example 1
Reinstatement Premium Example 2
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 of losses to the
    layer to the right, calculate our expected
    ultimate loss, premium, and loss ratio

Reinstatement Example 3 Solution
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

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

Deficit Carry forward Example
  • Solution - Shift Sliding Scale Commission terms.

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
  • Assumes years are independent (probably
  • Treaty terms may change, or treaty may be
    terminated before the benefit of the deficit
    carry forward is felt by the reinsurer. Also,
    reinsurer with deficit could be replaced by new

DCF/Multi-Year Block - Example
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

Additional Issues Uses of Aggregate
  • Correlation between lines of business often
    higher than you think due to directives from
    upper management influencing multiple lines of
  • Reserving for loss sensitive treaty terms
  • Some companies Use aggregate distributions to
    measure risk allocate capital. One hypothetical
  • Capital 99th percentile Discounted Loss x
    Correlation Factor
  • Fitting Severity Curves Dont Ignore Loss
  • Increases average severity
  • Increases variance claims spread as they
  • See Survey of Methods Used to Reflect
    Development in Excess Ratemaking by Stephen
    Philbrick, CAS 1996 Winter Forum

Risk Transfer Governing Regulations
  • FASB 113 A reinsurance contract should be booked
    using deposit accounting unless
  • The reinsurer assumes significant insurance
  • 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.
  • Statutory Statements
  • SSAP 62 is governing document requirements are
    similar to FASB 113.
  • Also requires CEOs and CFOs attestation under
    penalty of perjury that
  • No side agreements exist that alter reinsurance
  • For contracts where risk transfer is not
    self-evident, documentation concerning economic
    intent and risk transfer analysis is available
  • Reporting entity in compliance with SSAP 62
    proper controls in place
  • Recent Developments NY State and FASB proposed
    bifurcation proposals. Very troubling, but seem
    to be going nowhere

Report of 2005 CAS Working Party on Risk Transfer
Key Findings
  • Three step risk transfer testing process
  • Does contract transfer substantially all risk of
    ceding company? If yes, no testing required
  • Is reinsurers risk position the same as the
    ceding companies?
  • Is risk transfer reasonably self evident? If yes,
  • Facultative, Cat XOL, XOL contracts without
    significant loss sensitive features, and
    contracts with immaterial premium (less than 1
    mil of premium or 1 of GEP)
  • Remaining contracts Perform risk transfer
  • Calculate recommended risk metric compare to
    critical thresh-hold
  • Aggregate distribution should contemplate process
    parameter uncertainty
  • Recommend that 10/10 rule be replaced with
    Expected Reinsurer Deficit Calculation (ERD)
  • Above are only CASs working party
    recommendations. Actual procedures and methods
    are determined by company management and
    accounting firm

Exp. Reinsurer Deficit (ERD) Example
  • ERD p T / Premium
  • p Probability of loss to reinsurer 7
  • T Average Severity of Loss given a loss
  • (3.5 35 2 80 1.5 125) / 7 67.1
  • ERD 7 67.1 / 10 47
  • CAS Working Party implied a standard that ERD
    must be above 1, which equates to 10/10 rule,
    although it is less conservative

Example from CAS article by David Ruhm and Paul
Brehm, Risk Transfer Testing of Reinsurance
Contracts A Summary of the Report by the CAS
Research Working Party
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
  • 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