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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

results - Can compensate for differences between reinsurer

and client view of reinsurance program expected

loss - 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

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?

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

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

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

Example of Basic Mechanics PC 50 after 10,

30 Commission, 65 Expected LR

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.

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

CDFi - Exp (m Normsinv(CDFi) x s)

Sample Lognormal Loss Ratio Distribution

Is the resulting LR distribution reasonable?

- Compare resulting distribution to historical

results - 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

issues - 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

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)

Example of Modeling Parameter Uncertainty

Common Loss Sharing Provisions for Pro-rata

Treaties

- 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

80. - 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

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 - 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

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

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

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.

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

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

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

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

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

presentation. - 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

formula - 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

Treaties

- 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

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?

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

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)

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

mandatory - 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

caution - Assumes years are independent (probably

unrealistic) - 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

reinsurer.

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

results

Additional Issues Uses of Aggregate

Distributions

- Correlation between lines of business often

higher than you think due to directives from

upper management influencing multiple 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

Risk Transfer Governing Regulations

- 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. - 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

terms - 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,

stop - 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

testing. - 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

occurred - (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

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