Casualty Exposure Rating

1
Casualty Exposure Rating

• CARe Boot Camp 2007 H. Smosna

2
Reinsurance XOL Pricing

• Per occurrence XOL treaties provide a limit of
  coverage in excess of a ceding companys
  retention.
• Reinsurance pricing actuaries must calculate
  expected loss and ALAE in the layer
• The expected loss ALAE must be loaded for
  internal expense, CB, profit, contingencies,
  loss sensitive features
• This reinsurance ceded premium is usually
  expressed as percent of the Ceding Companys
  Prospective Subject Premium

3
What is Exposure Rating?

• Exposure rating estimates expected loss to the
  layer for a prospective period
• Exposure rating uses your clients limits
  profile, hazard characteristics, severity curves
  and gross loss ALAE ratio to estimate expected
  loss to the layer
• Severity distributions based on industry data are
  used to calculate LEVs (Limited Expected Values
  also know as LASs or Limited Average Severities)
• The LEVs are used to estimate losses to the
  reinsurance layer by spreading ground up loss
  into the desired layer

4
Why Exposure Rate?

• Complement of credibility for experience rate
• Price for free cover (when the top of your
  layer exceeds the largest trended loss in your
  data)
• Experience Rate is not credible
• Can adjust experience burns for limits drift
• Can use exposure burns to determine relativity
  based burns for higher layers

5
Advantages of Exposure Rating over Experience
Rating

• The current risk profile is modeled
• For a new book of business (so no experience
  available) pro forma profiles can be used to
  project expected loss to the layer
• The exposure rating exercise is often easy to
  perform so UWs can determine an exposure rating
  based reinsurance rate as part of their triage
  process

6
Exposure Rating what info do you need?

• Prospective Ground Up / Gross Loss Ratio for
  subject business
• Prospective Subject Premium
• Limit Attachment Point Profile with Premium
• Prospective limit and attachment profile
• i.e. 50 of premium is written at a 1 million
  per occurrence limit, 25 is written at a 5
  million p.o. limit, 25 is written at a 10
  million p.o. limit. All limits attach excess of a
  100K SIR.
• Severity distribution/LEVs for the line of
  business reflecting hazard level of underlying
  risks (Table 123ABC, Auto) see your UW
• Submission data is rarely provided in the full
  detail corresponding to the ISO Table definitions
• The layer you are pricing
• No loss experience to the layer required

7
The Concept

• Exposure Factor based on LEVs Ceded Loss/Gross
  Loss
• Gross Loss Ratio X Exposure Factor
• Gross Loss X Ceded Loss
• Gross Premium Gross Loss
• Ceded Loss Burn, Loss Cost
• Gross Premium

8
Visualization limits profile
9
Visualization Reinsurance layer
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Visualization Ceded Loss
11
Visualization Ceded Loss as of Gross Loss
Policy A
12
Visualization Ceded Loss as of Gross Loss
Policy B
13
Wheres the frequency?

• Again - Severity distributions based on industry
  data are used to calculate LEVs (Limited Expected
  Values also know as LASs or Limited Average
  Severities)
• Again - The LEVs are used to estimate losses to
  the reinsurance layer by spreading ground up loss
  into the desired layer
• Expected loss frequencyseverity
• Exposure Factor based on LEVs Ceded Loss/Gross
  Loss
• Ceded loss LEV(top) LEV(bottom) frequency
• Gross loss LEV(Policy limit) LEV(0)
  frequency
• Frequency cancels out!

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Limit

• GL policies have 2 kinds of limits
• Per occurrence
• Annual aggregate
• In this presentation we ignore the effects of
  aggregate limits.
• Assume that the primary policy aggregate limit is
  high enough and ground up frequency low enough
  that agg limit has minimal effect
• Frequency from previous slide would not cancel
  out if considering aggregate limits

15
LEV vs. ILF

• ISO creates severity distributions and from these
  they generate ILFs (Increased Limits Factors)
  Tables 1,2,3,A,B,C for example
• ILFs are used to rate primary policies (at 100K
  basic limit) to higher limits. They represent the
  ratio of all per occurrence costs at policy limit
  to all p.o. costs at basic limit
• ILFs include all per occurrence costs at a
  specific policy limit
• Expected limited p.o. loss cost
• All ALAE
• ULAE
• Risk Load

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LEV vs. ILF contd

• Do not use ISO ILFs in exposure rating
• Risk load and ULAE get in to your calculations
• The expected ALAE per claim for years was assumed
  to be the same for all policy limits so was
  loaded 100 into the basic limit. So when taking
  ratios of ILFs the ALAE would cancel out
• ISO has a new ALAE methodology
• Sometimes the ceding companys ILFs are the only
  info available (ie for an esoteric LOB). May need
  to use them.
• Use LEVs from the severity distributions
• If you are pricing to a different average DOL
  than that underlying the ISO parameters you can
  adjust the parameters
• If you dont like the parameters you can change
  them too (make the tail thicker)

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More on ISO ILFs

• The data underlying ISO ILFS comes from 2
  sources
• individual loss info from primary polices
  included in the standard statistical reporting to
  ISO
• Data from a special call for individual loss
  info from Umbrella and excess policies
• ISO fits a mixture of exponential distributions
  to empirical distributions derived from the data
• Because the data is sparse in the tail ISO uses a
  distribution above some point to smooth the
  empirical distribution
• The mixed exponential (ME) curves are less severe
  than the Pareto curves
• Curve history Truncated pareto, Pareto Soup, ME
• ISO data falls off at a relatively low level and
  is very sparse over 5M

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Proceed with Caution when using ISO Mixed
Exponential curves

• Not a lot of data above 1M
• M/E fits data closely up to 1M, but excess of 1
  million tail appears to thin
• ME maxes out at 10M
• M/E 4x1 and 5x5 factors are inconsistent, for
  example, Premops 2 4x1 factors are lower than
  Premops 1

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Clash not captured in exposure rating

• Casualty treaties often cover losses for which
  exposure rating does not provide an answer
• A Long-haul trucker collides with an auto
• WC loss from trucker
• CAL loss from individual in the auto
• Treaty will define this as one occurrence
• The WC claim is below the treaty retention
• The CAL claim is below the treaty retention
• Combined the WC CAL claims pierce the treaty
  retention Clash loss
• Exposure rating also does not estimate for ECO/XPL

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What is an LEV?

• Limited Expected Value
• The average size of loss when all losses are
  limited to a particular value
• The expected loss from ground up to some limit
  (k)
• LEV(k)?xf(x)dxk1F(k)
• x is the severity of an individual claim
• f(x) is the pdf of the severity
• F(x) is the cdf of the severity

k
0
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What is an LEV?

• Limited Expected Value
• LEV(k)?xf(x)dxk1F(k)
• For any random variable x, one of two things can
  happen
• x is lt the limitation k
• x is gt the limitation k
• The first part of the equation tackles (1) by
  calculating the expected loss limited to k when x
  lt k.
• The second part of the equation tackles (2). For
  any xgtk, you cap x at k.
• The sum of (1) and (2) gives you the average
  ground up loss when all losses are limited to k.

k
0
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Empirical LEV Example
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The Concept - again

• Exposure Factor based on LEVs Ceded Loss/Gross
  Loss
• Gross Loss Ratio X Exposure Factor
• Gross Loss X Ceded Loss
• Gross Premium Gross Loss
• Ceded Loss Burn, Loss Cost
• Gross Premium
• Credibility weight exposure burn with experience
  burn and then load up Z weighted burn for
  expenses, profit, etc to derive reinsurance rate
• Reinsurance rate Subject Premium Ceded Premium

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Exposure Rating what info do you need?

• Ground Up Loss Ratio for the subject business
• 55

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Exposure Rating what info do you need?

• The Layer you are pricing

If ALAE is included in the limit the effective
reinsurance limit and retention should be reduced
by the ALAE load
26
ALAE

• ALAE in reinsurance contracts
• Pro-rata
• Included with loss
• ALAE in primary policies
• Unlimited (ISO) or outside the limit
• Included in limit

27
ALAE

• Assume the primary policys ALAE is unlimited (as
  per ISO)
• When reinsurance treatment of ALAE is prorata
• Estimate ratio of ALAE to loss and load in (use a
  gross loss alae ratio when deriving your burn)
• When ALAE is included with loss
• Correct way is complicated
• Can reduce attachment point and limit by dividing
  both by (1 XOL ALAE)

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Exposure Rating what info do you need?

• Limits Profile

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Exposure Rating what info do you need?
.Severity distribution/LEVs for the line of
business reflecting hazard level of underlying
risks
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Severity Distributions Mixed Exponential LEVs

• Send ISO a check
• For the mixed exponential you will receive the
  parameters, weights and closed form formula - all
  you need to build your own exposure model
• ME closed form formula
• LEV ? wj?j 1 e (x/?)

8
J1
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Severity Distributions 5 Parameter Pareto LEVs

• For the 5 Parameter Pareto you need the
  parameters and closed form formula and you can
  build your own exposure model
• 5PP closed form formula
• LEV PS (1-P)/(Q-1) (BQT)-(Bx)(BT)/
  (Bx)Q for Qgt1

32
Appropriate LEVs not that simple.

• Rare to receive data from the ceding company that
  maps perfectly to ISO severity distributions
• In the example below we are deriving LEVs for an
  Umbrella profile
• Umbrella policies are exposed by underlying GL,
  AL, EL policies
• In exposure rating umbrella we want our LEVs
  based on a weighting of the underlying , varied
  exposures
• Your exposure rating model should have the
  capability of deriving LEVs based on weighting
  together the exposure factors from the various
  severity distributions

33
An Example Select a Limit band from the sample
profile

• Underlying Policy 500,000
• Policy Limit 2,000,000
• This represents 7.8 of the premium
• Reinsurance Attachment Point 750,000
• Reinsurance Limit 1,500,000
• You will use LEVs from a severity distribution to
  allocate ground up expected loss to layers
• LEV _at_ 500,000 10,518
• LEV _at_ 2,000,000 500,000 12,630
• LEV _at_ 750,000 11,149
• LEV _at_ 1,500,000 750,000 12,527

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How does the reinsurance apply?STOP Read your
contract!

• Who wrote the underlying policy?
• Is the Umbrella supported (over the ceding
  companys own primary)?
• Is the Umbrella unsupported (over another
  companys primary policy)?
• How does the reinsurance respond?
• If supported (over their own) typically the
  reinsurance attaches from the ground up
• If unsupported (over others primary) the
  reinsurance attaches on top of the underlying
• How is Subject Premium defined?

35
First some simple examples Umbrella Policy
36
First some simple examples Umbrella policy with
a 4M ground up loss
37
First some simple examples Reinsurance payment
when over own
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First some simple examples Reinsurance payment
when over other underlying
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A bit trickier The Policy
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A bit trickier The Reinsurance Layer
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The Policy the Reinsurance Layer OVER THEIR
OWN/reinsurance attaches from the ground up
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Exposure Factor/OVER THEIR OWN

• Your exposure factor ceded loss/gross loss
• Your ceded loss is the expected ground up loss at
  2.25M less the expected ground up loss at 750K
• Your gross loss is the expected ground up loss at
  2.5M less the expected loss at the 500K
  underlying
• Using LEVs the exposure factor
• LEV(2.25M) LEV (0.75M)/LEV(2.5M)
  LEV(0.5M)

43
The Policy the Reinsurance Layer OVER
OTHERS/reinsurance attaches on top of the
underlying
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Exposure Factor/OVER OTHERS

• Your exposure factor ceded loss/gross loss
• Your ceded loss is the expected ground up loss at
  2.5M less the expected ground up loss at 1.25M
• Your gross loss is the expected ground up loss at
  2.5M less the expected loss at the 500K
  underlying
• Using LEVs the exposure factor
• LEV(2.5M) LEV (1.25M)/LEV(2.5M) LEV(0.5M)

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Pull it all together

• In the over their own scenario the exposure
  factor
• LEV(2.25M) LEV(0.75M)/LEV(2.5M)
  LEV(0.5M)
• (12,527 11,149)/(12,630 10,518)
• 65
• LEV _at_ 500,000 10,518
• LEV _at_ 2,000,000 500,000 12,630
• LEV _at_ 750,000 11,149
• LEV _at_ 1,500,000 750,000 12,527

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Pull it all together
For this 1.5 x 0.75M layer the weighted exposure
factor is 88
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Pull it all together

• Gross Loss Ratio X Exposure Factor
• Gross Loss X Ceded Loss
• Gross Premium Gross Loss
• Ceded Loss Burn
• Gross Premium
• 55 X 88 49
• IF Subject Premium 10M the expected loss to the
  1.5 x 0.75M layer is 490,000
• Gross up for expense, profit, et al and you have
  your reinsurance ceded premium

48
Data Issues

• Limits profile
• Broad ranges
• Limits profile separate from attachment point
  profile
• Count based (to fix, multiply by the LEV at
  corresponding policy limit)
• Premium from profile doesnt reconcile well with
  historical or projected premium
• Do you have all business units, companies?
• PPA split limits (10/20 per person/per occ)
• ILF table breakdown

49
Free Cover

• In your experience rating no losses have trended
  into the highest portion of the layer you are
  pricing
• Example
• You are pricing a 750x250K layer
• Largest trended loss is 500K from ground up
• Your experience burn will be the same for your
  750x250K layer as for a 250x250K layer

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Free Cover contd

• One solution
• Split the layer you are pricing (750x250K) into 2
  pieces
• 250x250K
• 500x500K
• If deemed credible, use the experience burn as
  your selected burn for the lower part of your
  layer OR credibility weight your experience and
  exposure burns to derive your selected burn
• Then use exposure burn relativities to derive the
  burn for the upper part of your layer
• Sum the selected burns for the two parts of the
  layer you are pricing to derive the full layer
  selected burn

51
Problems with experience rating

• Presence or absence of a few large claims drives
  the indicated rates.
• If the book of business has been shifting
• Can be difficult to estimate LDFs, trend, OLFs
• There may be significant data issues
• Trending individual claims past policy limits.
• Impact of current policy limit profile vs.
  historical profile.
• History not reflective of current situation
  reserving practices, type of business, coverage,
  etc.

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Problems with experience rating Limits Drift

• Your limits profile is shifting/drifting upwards
• In 2006 8 of your policies were at limits of 2M
• Now , 13 of your policies are at 2M
• You can use an exposure approach to adjust your
  AY experience burns for limits drift
• Problem need historical limits profiles

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Problems with experience rating Limits Drift
Adjust your experience burn from AY 2006 by a
factor of 1.625 to correct for the fact that the
cedant is writing a higher percentage of 2M
policies than in the past
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How credible is my exposure rating result?

• Is my exposure curve a good fit ?
• Do I have enough information in the submission to
  determine the hazard profile (Tables 1,2,3,A,B,C,
  Auto Tables)
• Is my limitattachment point profile a good
  representation of the prospective treaty year
  exposure?
• What is the basis for my gross loss ratio?
• Is there anything I am not modeling (i.e. clash)?

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Unique Application of Exposure Rating- Umbrella
Pricing Adequacy

• See CARe 2006 presentation for more on this topic

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Caveat

• Any pricing tool is only a first step towards
  determining adequate reinsurance premium
• Note when modeling or data assumptions are not
  met then try to adjust, correct, supplement.
• If you cant adjust, correct or supplement then
  ignore or discount your result!

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The End ?