Casualty Exposure Rating

Casualty Exposure Rating

CARe Boot Camp 2007H. Smosna

2

Reinsurance XOL Pricing

Per occurrence XOL treaties provide a limit of coverage in excess of a ceding company’s retention.

Reinsurance pricing actuaries must calculate expected loss and ALAE in the layer

The expected loss & ALAE must be loaded for internal expense, C&B, profit, contingencies, loss sensitive features

This reinsurance ceded premium is usually expressed as percent of the Ceding Company’s Prospective Subject Premium

3

What is Exposure Rating? Exposure rating estimates expected loss to

the layer for a prospective period Exposure rating uses your client’s 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

Policy Limit A

Policy Limit B

Policy Limit C

9

Visualization – Reinsurance layer

Top of Reinsurance Layer

Bottom of Reinsurance Layer

Policy Limit B

Policy Limit C

Policy Limit A

10

Visualization – Ceded Loss



Policy Limit C

Policy Limit A

Policy Limit B

11

Visualization – Ceded Loss as % of Gross Loss – Policy A



(1) Ceded Loss/Gross Loss = 20.0%(2) Gross Loss Ratio = 50.0%

(1) * (2) = Ceded Loss/Gross Premium = 10.0%SP = 1,000,000

Expected Loss to layer = 100,000

Ceded Loss Gross Loss

Policy Limit A Policy Limit A

12

Visualization – Ceded Loss as % of Gross Loss – Policy B



(1) Ceded Loss/Gross Loss = 8.0%(2) Gross Loss Ratio = 50.0%

(1) * (2) = Ceded Loss/Gross Premium = 4.0%SP = 1,000,000

Expected Loss to layer = 40,000

Gross Loss

Policy Limit BPolicy Limit B

Ceded Loss

13

Where’s 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 = frequency*severity 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!

14

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

16

LEV vs. ILF – cont’d 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 company’s 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 don’t like the parameters you can change them too (make the tail thicker)

17

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

18

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

19

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

20

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)dx+k[1–F(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

0

k

21

What is an LEV? Limited Expected Value

• LEV(k)=∫xf(x)dx+k[1–F(k)]

• For any random variable x, one of two things can happen:

1. x is <= the limitation k2. x is > the limitation k

• The first part of the equation tackles (1) by calculating the expected loss limited to k when x <= k.

• The second part of the equation tackles (2). For any x>k, 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.

0

k

22

Empirical LEV Example

Individual Loss Experience

Claim# 1 2 3 4 5 6 7 8 9 10

Ground Up Loss $$$ 141,000 16,000 46,000 40,000 351,000 259,000 317,000 1,511,000 107,000 567,000

Limited gu loss at 500k 141,000 16,000 46,000 40,000 351,000 259,000 317,000 500,000 107,000 500,000

Limited gu loss at 1000k 141,000 16,000 46,000 40,000 351,000 259,000 317,000 1,000,000 107,000 567,000

Loss in 500K to 1000K layer 0 0 0 0 0 0 0 500,000 0 67,000

LEV@500K = (141,000+16,000+46,000+40,000+351,000+259,000+317,000+500,000+107,000+500,000)/10 = 227,700

LEV@1000K = (141,000+16,000+46,000+40,000+351,000+259,000+317,000+1,000,000+107,000+567,000)/10 = 284,400

LEV@1000K - LEV@500K Just take the difference of 284,400 and 227,700 56,700

LEV@1000K - LEV@500K (0+0+0+0+0+0+0+500,000+0+67,000)/10 = 56,700

LEV@1000K - LEV@500K = 56,700 = 20%

284,400

If you want to know what percent of loss is in the 500x500K layer on a 1M policy take the ratio:

LEV@1000K

23

The Concept - again

Exposure Factor based on LEVs = Ceded Loss/Gross Loss


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

24


Ground Up Loss Ratio for the subject business

55%

25


The Layer you are pricing:1st Layer

Reinsurance Limit 750,000 Reinsurance Retention: 1,250,000 Effective Limit: 750,000 Effective Retention: 1,250,000

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 policy’s 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%)

28


Limits Profile

SIR Limit Premium100,000 1,000,000 0.2%250,000 1,000,000 6.2%300,000 1,000,000 5.1%750,000 1,000,000 72.5%

1,000,000 1,000,000 6.5%100,000 2,000,000 0.0%250,000 2,000,000 0.7%300,000 2,000,000 0.6%500,000 2,000,000 7.8%

1,000,000 2,000,000 0.7%

29

Exposure Rating – what info do you need? ….Severity distribution/LEVs for the line of business reflecting hazard

level of underlying risks

5 Parameter Pareto2002 Premops 1 - Multistate2002 Premops 2 - Multistate2002 Premops 3 - Multistate2002 Premops 1 - CA2002 Premops 2 - CA2002 Premops 3 - CA2002 Premops 1 - FL2002 Premops 2 - FL2002 Premops 3 - FL2002 Premops 1 - IL2002 Premops 2 - IL2002 Premops 3 - IL2002 Premops 1 - NJ2002 Premops 2 - NJ2002 Premops 3 - NJ2002 Premops 1 - NY2002 Premops 2 - NY2002 Premops 3 - NY2002 Premops 1 - OH2002 Premops 2 - OH2002 Premops 3 - OH2002 Premops 1 - PA2002 Premops 2 - PA2002 Premops 3 - PA2002 Premops 1 - TX2002 Premops 2 - TX2002 Premops 3 - TX2002 Products A2002 Products B2002 Products C2002 CCA Lt Grp 12002 CCA Hvy Grp 12002 CCA XHvy Grp 1

Mixed Exponential2003 ISO CCA Liab Lt - Grp 12003 ISO CCA Liab Hvy - Grp 12003 ISO CCA Liab XHvy - Grp 12003 ISO CCA Liab All Other - Grp 12003 ISO CCA Liab Lt - Grp 22003 ISO CCA Liab Hvy - Grp 22003 ISO CCA Liab XHvy - Grp 22003 ISO CCA Liab All Other - Grp 22003 ISO CCA Liab Lt - Grp 32003 ISO CCA Liab Hvy - Grp 32003 ISO CCA Liab XHvy - Grp 32003 ISO CCA Liab All Other - Grp 32003 ISO CCA Liab Lt - Grp 42003 ISO CCA Liab Hvy - Grp 42003 ISO CCA Liab XHvy - Grp 42003 ISO CCA Liab All Other - Grp 42003 ISO CCA Liab Lt - Grp 52003 ISO CCA Liab Hvy - Grp 52003 ISO CCA Liab XHvy - Grp 52003 ISO CCA Liab All Other - Grp 52003 ISO CCA Liab Lt - Grp 62003 ISO CCA Liab Hvy - Grp 62003 ISO CCA Liab XHvy - Grp 62003 ISO CCA Liab All Other - Grp 62003 ISO CCA Liab Zone Rated2006 ISO Premops 1 - Multistate2006 ISO Premops 2 - Multistate2006 ISO Premops 3 - Multistate2006 ISO Premops 1 - Group A2006 ISO Premops 2 - Group A2006 ISO Premops 3 - Group A2006 ISO Premops 1 - Group B2006 ISO Premops 2 - Group B2006 ISO Premops 3 - Group B

30

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 = ∑ wjλj [1 – e –(x/λ)]

Mixed Exponential Parameters ISO Weight Given to Mixed Exponential ParametersWeight 1 2 3 4 5 6 7 8

100 1,366 6,823 31,157 98,452 500,542 2,074,148 9,146,627 1 Selected ILF Table

2006 ISO Premops 2 - Group B

ISO Weight Given to Mixed Exponential ParametersWeight 1 2 3 4 5 6 7 8

100 0.4927620 0.3169920 0.1130270 0.0565070 0.0182380 0.0020360 0.0004380 - Selected ILF Table

2006 ISO Premops 2 - Group B

J=1

8

31

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 = P*S + [(1-P)/(Q-1)] * [(B+Q*T)-

(B+x)*{(B+T)/(B+x)}^Q] for Q>1

Weight B Q P S T XOL ALAE%

100 57,584 1.39 0.97 5,131 58,557 23.00%

5 Parameter Pareto Parameters

2002 Products ASelected ILF Table

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

Weight B Q P S T XOL ALAE%

15 15,020 1.38 0.97 4,813 58,557 14.00%

15 186,831 1.68 0.96 7,058 58,557 14.00%

20 378,277 1.56 0.98 6,814 18,178 6.00%

20 431,825 1.55 0.98 7,688 18,178 6.00%

15 271,585 1.65 0.93 10,474 58,557 23.00%

15 313,990 1.64 0.88 13,479 58,557 23.00%

5 Parameter Pareto Parameters

2002 CCA XH - Grp 2 (Big Grp)2002 Products B2002 Products C

Selected ILF Table

2002 Premops 1 - Multistate2002 Premops 2 - Multistate2002 CCA Hvy - Grp 2 (Big Grp)

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 @ 500,000 = 10,518 LEV @ 2,000,000 + 500,000 = 12,630 LEV @ 750,000 = 11,149 LEV @ 1,500,000 + 750,000 = 12,527

34

How does the reinsurance apply?

STOP: Read your contract! Who wrote the underlying policy?

• Is the Umbrella supported (over the ceding company’s own primary)?

• Is the Umbrella unsupported (over another company’s primary policy)?

How does the reinsurance respond?• If supported (over their own) typically

the reinsurance attaches from the ground up

• If unsupported (over other’s primary) the reinsurance attaches on top of the underlying

How is Subject Premium defined?

35

First some simple examples: Umbrella Policy

Umbrella

10MUmbrella

Underlying

Reinsurance: 4x1M

11M

1M

36

First some simple examples: Umbrella policy with a 4M ground

up loss

Umbrella

Reinsurance: 4x1MGround Up Loss: 4M

11M

1M

4M

37

First some simple examples: Reinsurance payment when over

ownUmbrella

Reins payment/over other: 3M (excess of 1M)

Reinsurance: 4x1M from guGround Up Loss: 4M

11M

1M

4M

2M

3M

38

First some simple examples: Reinsurance payment when over

other underlyingUmbrella

Ground Up Loss: 4MReins payment/over other: 2M (excess of 2M)

Reinsurance: 4x1Mx1M UL

11M

1M

4M

2M

3M

39

A bit trickier: The Policy

2M policy limit xs 500K UL

250K

250K

250K

250K

2M

250K

250K

250K

250K

250K

500K

250K

40

A bit trickier: The Reinsurance Layer

1.5M reins limit xs 750K client retention

250K

250K

250K

1.5M

250K

250K

250K

250K

750K 250K

250K

41

The Policy & the Reinsurance Layer – OVER THEIR OWN/reinsurance

attaches from the ground up

250K

250K

250K

250K

250K

250K

250K

250K

250K

250K

UL = underlying policy limit or SIR (500,000) PL = Policy Limit (2,000,000)

AP = Attachment Point of Reinsurance (750,000) Lim = Limit of Reinsurance (1,500,000)

500K

750K

2.25M

2.50M

1.5M

42

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)]

250K

250K

250K

250K

250K

250K

250K

250K

250K

250K

500K

750K

2.25M

2.50M

1.5M

43

The Policy & the Reinsurance Layer – OVER OTHER’S/reinsurance

attaches on top of the underlying

250K

250K

250K

250K

250K

250K

250K

250K

250K

250K



500K

1.25M

2.0M

2.50M

750K

1.5M

44

Exposure Factor/OVER OTHER’S 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)]

250K

250K

250K

250K

250K

250K

250K

250K

250K

250K



500K

1.25M

2.0M

2.50M

750K

1.5M

45

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 @ 500,000 = 10,518 LEV @ 2,000,000 + 500,000 = 12,630 LEV @ 750,000 = 11,149 LEV @ 1,500,000 + 750,000 = 12,527

46


1st LayerCeded Loss

as a % ofSIR Limit Premium Gross Loss

100,000 1,000,000 0.2% 15.08%250,000 1,000,000 6.2% 28.75%300,000 1,000,000 5.1% 34.15%750,000 1,000,000 72.5% 100.00%

1,000,000 1,000,000 6.5% 100.00%100,000 2,000,000 0.0% 29.96%250,000 2,000,000 0.7% 43.92%300,000 2,000,000 0.6% 48.10%500,000 2,000,000 7.8% 65.24%

1,000,000 2,000,000 0.7% 78.25% Total: 100% 88.54%

For this 1.5 x 0.75M layer the weighted exposure factor is 88%

47



Gross Loss X Ceded LossGross 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 $490,000Gross 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 doesn’t 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 layerTrended Ground Up

Claims Loss to 250x250K layer Loss to 750x250K layer

500,000 250,000 250,000 400,000 150,000 150,000 300,000 50,000 50,000 200,000 - - 100,000 - -

50

Free Cover – cont’d 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

Layer Experience Burn Exposure Burn Selected Burn250x250K 10.0% 12.0% 11.0%500x500K 0.0% 6.0% 5.5%

750x250K 10.0% 18.0% 16.5%

Where 5.5% = 11% * (6%/12%)

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.

52

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

53

Problems with experience rating – Limits Drift

profile prem weight ILF Expected Loss 1x1 weighted el 1x1 limits drift factor

2006 100,000 2.0% 0.0%

500,000 30.0% 1.10 0.0%

1,000,000 60.0% 1.25 0.0%

2,000,000 8.0% 1.70 26.5% 2.12% 1.625

2007 100,000 2.0% 0.0%

500,000 20.0% 1.10 0.0%

1,000,000 65.0% 1.25 0.0%

2,000,000 13.0% 1.70 26.5% 3.44% 1.000

26.5% = 1-(ILF@1M/ILF@2M) or the ceded loss/gross loss

1.625 = .0344/.0212

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

54

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 limit&attachment 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)?

55

Unique Application of Exposure Rating

- Umbrella Pricing AdequacySee CARe 2006 presentation for more on this topic

56

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 can’t adjust, correct or supplement then ignore or discount your result!

57

The End

Documents

Casualty Exposure Rating