Commercial Property Size of Loss Distributions

Glenn Meyers

Insurance Services Office, Inc.

Casualty Actuaries in Reinsurance

June 15 , 2000

Boston, Massachusetts

Outline• Data

• Classification Strategy– Amount of Insurance– Occupancy Class

• Mixed Exponential Model– “Credibility” Considerations

• Limited Classification Information

• Program Demonstration

• Goodness of Fit Tests

• Comparison with Ludwig Tables

Separate Tables For• Commercial Property (AY 1991-95)

• Sublines – BG1 (Fire and Lightning)– BG2 (Wind and Hail)– SCL (Special Causes of Loss)

• Coverages– Building– Contents– Building + Contents– Building + Contents + Time Element

Exposures

• Reported separately for building and contents losses

• Model is based on combined building and contents exposure – Even if time element losses are covered

Classification Strategy

• Amount of Insurance– Big buildings have larger losses– How much larger?

• Occupancy Class Group– Determined by data availability

• Not used – Construction Class– Protection Class

Potential Credibility Problems

• Over 600,000 Occurrences

• 59 AOI Groupings

• 21 Occupancy Groups

• The groups could be “grouped” but:– Boundary discontinuities– We have another approach

The Mixed Exponential Size of Loss Distribution

i’s vary by subline and coverage

• wi’s vary by AOI and occupancy group in addition to subline and coverage

F x w eix

i

i( ) /

1

1

6

The Mixed Exponential Size of Loss Distribution

i = mean of the ith exponential distribution

• For higher i’s, a higher severity class will tend to have higher wi’s.

F x w eix

i

i( ) /

1

1

6

The Fitting Strategyfor each Subline/Coverage

• Fit a single mixed exponential model to all occurrences

• Choose the wi’s and i’s that maximize the likelihood of the model.

• Toss out the wi’s but keep the i’s

• The wi’s will be determined by the AOI and the occupancy group.

Back to the Credibility ProblemRaw Mixed Exponential Fits

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1 10 100 1,000 10,000 100,000 1,000,000

AOI (All Occupancy Groups Combined)

W1

W2

W3

W4

W5

W6

Back to the Credibility ProblemFitted Excess Severities

-10000

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

1 10 100 1,000 10,000 100,000 1,000,000

AOI (All Occupancy Groups Combined)

Exc

ess

Sev

erit

y o

ver

$10,

000

Varying Wi’s by AOI

Prior expectations• Larger AOIs will tend to have higher

losses

• In mixed exponential terminology, the AOI’s will tend to have higher wi’s for the higher i’s.

• How do we make this happen?

Solution

• Let W1i’s be the weights for a given AOI.

• Let W2i’s be the weights for a given higher AOI.

• Given the W1i’s, determine the W2i’s as follows.

Step 1Choose 0 d11 1

W11 W21 = (1-d11) W11

W12 W22 = W12+d11W11

W13 W23 = W13

W16 W26 = W16

Shifting the weight from 1st exponential to the 2nd exponential increases the expected claim cost.

Step 2Choose 0 d12 1

W11 W21 = (1-d11) W11

W12 W22 = (1-d12) (W12+d11W11)

W13 W23 = W13+d12 (W12+d11W11)

W16 W26 = W16

Shifting the weight from 2nd exponential to the 3rd exponential increases the expected claim cost.

Step 3 and 4 SimilarStep 5 — Choose 0 d15 1

W11 W21 = (1-d11) W11

W12 W22 = (1-d12) (W12+d11W11)

W13 W23 = (1-d13)(W13+d12(W12+d11W11))

W16 W26 = 1 21

5

Wii

Shifting the weight from 5th exponential to the last exponential increases the expected claim cost.

Several AOI GroupsChoose W’s for lowest AOI Group

W11

W12

W13

W14

W15

W16

Then choose d’s to

Construct W’s for the 2nd AOI Group

W11 W21

W12 W22

W13 W23

W14 W24

W15 W25

W16 W26

Then choose d’s to

Construct W’s for the 3rd AOI Group

W11 W21 W31

W12 W22 W32

W13 W23 W33

W14 W24 W34

W15 W25 W35

W16 W26 W36

Then choose d’s to

Construct W’s for the 4th AOI Group

W11 W21 W31 W41

W12 W22 W32 W42

W13 W23 W33 W43

W14 W24 W34 W44

W15 W25 W35 W45

W16 W26 W36 W46

Continue choosing d’s and constructing W’s until the end.

W11 W21 W31 W41

W12 W22 W32 W42

W13 W23 W33 W43

W14 W24 W34 W44

W15 W25 W35 W45

W16 W26 W36 W46

Estimating W’s (for the 1st AOI Group) and d’s (for the rest)

Let:

• Fk(x) = CDF for kth AOI Group

• (xh+1, xh) be the hth size of loss group

• nhk = number of occurrences for h and k

Then the log-likelihood of data is given by:

n F x F xhk k h k hkh

log ( ) ( )1a f

Estimating W’s (for the 1st AOI Group) and d’s (for the rest)

• Choose W’s and d’s to maximize log-likelihood

• 59 AOI Groups

• 5 parameters per AOI Group

• 295 parameters!

Too many!

Parameter Reduction

• Fit W’s for AOI=1, and d’s for AOI=10, 100, 1,000, 10,000, 100,000 and 1,000,000. Note AOI coded in 1,000’s

• The W’s are obtained by linear interpolation on log(AOI)’s

• The interpolated W’s go into the log-likelihood function.

• 35 parameters -- per occupancy group

On to Occupancy Groups

• Let W be a set of W’s that is used for all AOI amounts for an occupancy group.

• Let X be the occurrence size data for all AOI amounts for an occupancy group.

• Let L[X|W] be the likelihood of X given W i.e. the probability of X given W

There’s No Theorem Like Bayes’ Theorem

• Let be n parameter sets.

• Then, by Bayes’ Theorem:

Wk knk p1

PrPr

Pr

W XX W W

X W Wk

k k

k kk

n

L

Ll q l q k p

l q k p

1

Bayesian Results Applied to an AOI and Occupancy Group

• Let be the ith weight that Wk assigns

to the AOI/Occupancy Group.

• Then the wi‘s for the AOI/Occupancy

Group is:

wik

w wi ik

k Pr W Xl qk=1

n

What Does Bayes’ Theorem Give Us?

• Before– A time consuming search for parameters– Credibility problems

• If we can get suitable Wk’s we can reduce our search to n W’s.

• If we can assign prior Pr{Wk}’s we can solve the credibility problem.

Finding Suitable Wk’s

• Select three Occupancy Class Group “Groups”

• For each “Group”– Fit W’s varying by AOI– Find W’s corresponding to scale change

• Scale factors from 0.500 to 2.000 by 0.025

• 183 Wk’s for each Subline/Coverage

Graph of Log-Likelihoods

1.81.821.841.861.88

1.91.921.941.961.98

2

0.000 0.500 1.000 1.500 2.000 2.500Scale Factor

Ne

g. A

ve

rag

e L

og

Lik

elih

oo

d High

Medium

Low

Prior Probabilities

• Set:

• Final formula becomes:

Pr / /Wk nk p 1 1 183

PrPr

Pr

W XX W W

X W W

X W

X Wk

k k

k kk

nk

kk

n

L

L

L

Ll q l q k p

l q k pl ql q

1 1

• Can base update prior on Pr{Wk |X}.

The Classification Data Availability Problem

• Focus on Reinsurance Treaties– Primary insurers report data in bulk to

reinsurers– Property values in building size ranges– Some classification, state and deductible

information

• Reinsurers can use ISO demographic information to estimate effect of unreported data.

Database Behind PSOLD

30,000+ records (for each coverage/line combination) containing:

• Severity model parameters

• Amount of insurance group– 59 AOI groups

• Occupancy class group

• State

• Number of claims applicable to the record

Constructing a Size of Loss Distribution Consistent with Available Data Using ISO Demographic Data

• Select relevant data

• Selection criteria can include:– Occupancy Class Group(s)– Amount of Insurance Range(s)– State(s)

• Supply premium for each selection

• Each state has different occupancy/class demographics

Constructing a Size of Loss Distribution for a “Selection”

• Record output - Layer Average Severity

• Combine all records in selection:

LASSelection = Wt Average(LASRecords)

Use the record’s claim count as weights

Constructing a Size of Loss Distribution for a “Selection”

Where:

i = ith overall weight parameter

wij = ith weight parameter for the jth record

Cj = Claim weight for the jth record

i

ij jj

n

jj

n

w C

C

1

1

The Combined Size of LossDistribution for Several “Selections”

• Claim Weights for a “selection” are proportional to Premium Claim Severity

LASCombined = Wt Average(LASSelection)

Using the “selection” total claim weights

• The definition of a “selection” is flexible

The Combined Size of LossDistribution for Several “Selections”

• Calculate i’s for groups for which you have pure premium information.

• Calculate the average severity for jth group

j iji

i

1

6

The Combined Size of LossDistribution for Several “Selections”

• Calculate the group claim weights

• Calculate the weights for the treaty size of loss distribution

j

j

j

Pure emium

Pr

i

ij jj

n

jj

n

1

1

The Deductible Problem

• The above discussion dealt with ground up coverage.

• Most property insurance is sold with a deductible– A lot of different deductibles

• We need a size of loss distribution net of deductibles

Size of Loss Distributions Net of Deductibles

Loss Amount

Rela

tive F

req

uen

cy

• Remove losses below deductible • Subtract deductible from loss amount

Re

lati

ve

Fre

qu

enc

y

Size of Loss Distributions Net of Deductibles

• Combine over all deductibles

LASCombined Post Deductible

Equals Wt Average(LASSpecific Deductible)

• Weights are the number of claims over each deductible.

Size of Loss Distributions Net of Deductibles

For an exponential distribution:

Net severity

Excess Pure emiumExcess Frequecy

e

eGround up severity

d

d

Pr

/

/

Need only adjust frequency -- i.e. wi’s

Adjusting the wi’s

• Dj jth deductible amount

ij

• Wi

w e

w e

iD

iD

i

j i

j i

/

/

1

6

C jded

ijj

Goodness of Fit - Summary

• 16 Tables

• Fits ranged from good to very good

• Model LAS was not consistently over or under the empirical LAS for any table

• Model unlimited average severity– Over empirical 8 times– Under empirical 8 times

A Major Departure from Traditional Property Size of

Loss Tabulations

• Tabulate by dollars of insured value

• Traditionally, property size of loss distributions have been tabulated by % of insured value.

Fitted $ Average Severity against Insured Value

0

20,000

40,000

60,000

80,000

100,000

120,000

0 200,000 400,000 600,000 800,000 1,000,000 1,200,000

Amount of Insurance

Exp

ecte

d L

oss

at

AO

I

Fitted Average Severity as % of Insured Value

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

0 200,000 400,000 600,000 800,000 1,000,000 1,200,000

Amount of Insurance

E[L

oss

] as

a %

of

AO

I

Blow up this area

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

0 100 200 300 400 500 600 700 800 900 1,000

Amount of Insurance

E[L

oss

] as

% o

f A

OI

Fitted Average Severity as % of Insured Value

Eventually, assuming that loss distributions based on a percentage of AOI will produce layer costs that are too high.

PSOLD Demonstration

• No Information

• Size of Building Information

• Size + Class Information

• Size + Class + Location Information

Comparison with Ludwig Tables

• Tabulated by % of amount of insurance

• Organized by occupancy class and amount of insurance– Broader AOI classes– Broader occupancy classes

• Fewer occurrances

• No model

• A very good paper

Comparison with Ludwig Tables

• Ludwig — Exhibit 15 (all classes)

• Matched insured value ranges

• Obtained % of insured value distributions from PSOLD– assuming low end of range– assuming high end of range

• Results on Spreadsheet

What’s new for the next review?

• Include data through 1998

• Fewer exclusions of loss information– Recall that we excluded claims if exposure

and class information were missing.– Include claims if we trust the losses and

use Bayesian techniques to spread losses to possible class and exposure groups.

• Include HPR classes