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
6
F( x)  1   w i  e
 x/  i
i1
i’s vary by subline and coverage
• wi’s vary by AOI and occupancy group
in addition to subline and coverage
The Mixed Exponential
Size of Loss Distribution
6
F( x)  1   w i  e
 x/  i
i1
i = mean of the ith exponential distribution
• For higher i’s, a higher severity class will
tend to have higher wi’s.
The Fitting Strategy
for 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 Problem
Raw Mixed Exponential Fits
1.2
1
0.8
W1
W2
0.6
W3
W4
0.4
W5
W6
0.2
0
1
10
100
1,000
10,000
100,000
-0.2
AOI (All Occupancy Groups Combined)
1,000,000
Back to the Credibility Problem
Fitted Excess Severities
90000
Excess Severity over $10,000
80000
70000
60000
50000
40000
30000
20000
10000
0
1
10
100
1,000
10,000
100,000
-10000
AOI (All Occupancy Groups Combined)
1,000,000
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 1
Choose 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 2
Choose 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 Similar
Step 5 — Choose 0  d15  1
W11
W21
= (1-d11) W11
W12
W22
= (1-d12) (W12+d11W11)
W13
W23
= (1-d13)(W13+d12(W12+d11W11))



5
W16
W26
= 1  W2i
i1
Shifting the weight from 5th exponential to the last
exponential increases the expected claim cost.
Several AOI Groups
Choose 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
W12
W13
W14
W15
W16
W21
W22
W23
W24
W25
W26
Then choose d’s to
Construct W’s for the 3rd AOI Group
W11
W12
W13
W14
W15
W16
W21
W22
W23
W24
W25
W26
W31
W32
W33
W34
W35
W36
Then choose d’s to
Construct W’s for the 4th AOI Group
W11
W12
W13
W14
W15
W16
W21
W22
W23
W24
W25
W26
W31
W32
W33
W34
W35
W36
W41
W42
W43
W44
W45
W46
Continue choosing d’s and
constructing W’s until the end.
W11
W12
W13
W14
W15
W16
W21
W22
W23
W24
W25
W26
W31
W32
W33
W34
W35
W36
W41
W42
W43
W44
W45
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:
Fk ( xh1)  Fk ( xh )f
  nhk  loga
h
k
Estimating W’s (for the 1st AOI
Group) and d’s (for the rest)
• Choose W’s and d’s to maximize loglikelihood
• 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 loglikelihood 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
kp
n
Wk k 1 be
n parameter sets.
• Then, by Bayes’ Theorem:
k
p
l
q
Prl W X q
 LlX W q PrkW p
k
L X Wk  Pr Wk
n
k 1
k
k
Bayesian Results Applied to
an AOI and Occupancy Group
• Let
k
wi
be the ith weight that Wk assigns
to the AOI/Occupancy Group.
• Then the wi‘s for the AOI/Occupancy
Group is:
wi 
n
l q
k
w
 i  Pr Wk X
k=1
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
Neg. Average Log Likelihood
Graph of Log-Likelihoods
2
1.98
1.96
1.94
1.92
1.9
1.88
1.86
1.84
1.82
1.8
0.000
High
Medium
Low
0.500
1.000
1.500
Scale Factor
2.000
2.500
Prior Probabilities
kp
• Set: Pr Wk  1 / n  1 / 183
• Final formula becomes:
LlX W q
k
p
l
q
Prl W X q

 LlX W q PrkW p  LlX W q
k
L X Wk  Pr Wk
k
n
k 1
n
k
k
k 1
• Can base update prior on Pr{Wk |X}.
k
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”
n
 w ij  C j
i 
j1
n
 Cj
j1
Where:
i = ith overall weight parameter
wij = ith weight parameter for the jth record
Cj = Claim weight for the jth record
The Combined Size of Loss
Distribution 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 Loss
Distribution for Several “Selections”
• Calculate i’s for groups for which you
have pure premium information.
• Calculate the average severity for jth
group
6
 j    ij  i
i1
The Combined Size of Loss
Distribution for Several “Selections”
• Calculate the group claim weights
Pure Pr emium j
j 
j
• Calculate the weights for the treaty size
of loss distribution
n
  ij   j
i 
j1
n
 j
j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
Relative Frequency
Relative Frequency
• Remove losses below deductible
• Subtract deductible from loss amount
Loss Am ount
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 Pr emium
Excess Frequecy

e
 d/ 
 d/ 

e
 Ground up severity
Need only adjust frequency -- i.e. wi’s
Adjusting the wi’s
• Dj  jth deductible amount
ij 
wi  e
6
D j /  i
 wi  e
D j /  i
i1
ded

C
• Wi  j   ij
j
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
Expected Loss at AOI
120,000
100,000
80,000
60,000
40,000
20,000
0
0
200,000
400,000
600,000
800,000
Amount of Insurance
1,000,000
1,200,000
Fitted Average Severity as
% of Insured Value
E[Loss] as a % of AOI
60.00%
50.00%
40.00%
30.00%
20.00%
10.00%
Blow up this area
0.00%
0
200,000
400,000
600,000
800,000
Amount of Insurance
1,000,000
1,200,000
Fitted Average Severity as
% of Insured Value
60.00%
E[Loss] as % of AOI
50.00%
Eventually, assuming that loss
distributions based on a percentage
of AOI will produce layer costs that
are too high.
40.00%
30.00%
20.00%
10.00%
0.00%
0
100
200
300
400
500
600
700
Amount of Insurance
800
900
1,000
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