Properly Pricing the
Catastrophe Exposure
David Chernick, FCAS, MAAA
Michael Devine, FCAS, MAAA
Sara Drexler, FCAS, MAAA
CAS Ratemaking Seminar
March, 2004
1
7/26/2016
Agenda


Introduction
History of Methods
– Terminology
– Exposure Base
– Capping

Discussion of Several Alternative Approaches
– Traditional
– Methods Using Cat/AIY – State Based
– Methods Using Cat/AIY – Countrywide Based

Wrap-up
2
7/26/2016
Introduction
The Panelists
 The Data
 The Issue

3
7/26/2016
Introduction

The Panelists
Introductions
Sara Drexler
Michael Devine
David Chernick
4
7/26/2016
Introduction

The Data
The base data we are using in this presentation is
included in the handouts.
Real data
Large Catastrophe in 1998
343.2% loss ratio
9.67 Ratio of Cat/AIY
5
7/26/2016
Introduction

The Issue
– Operating results
$19.1 Million profit prior to 1998
 $41.4 Million loss in 1998

– Statistics
Mean:
 1998 was 22.7 Standard Deviations from mean.

6
7/26/2016
Introduction

The Issue
– A rate should include all costs associated with
the transfer of risk.
– 20 or 30 or even 40 years of data is not
sufficient to properly quantify the tail of the
distribution
– What is the true prospective average (mean)
catastrophe provision?
7
7/26/2016
Introduction

The Issue
– Perspective of this presentation is from large
insurers without reinsurance coverage.
– Reinsurance covering some portion of the
catastrophe exposure would most likely be an
upper bound of the true mean.
– What is the true prospective average (mean)
catastrophe provision?
8
7/26/2016
Terminology
AIY – Amount of Insurance years
1AIY=$1,000 of dwelling coverage
 Losses/AIY – Damage Ratios or Cat/AIY

9
7/26/2016
History of Methods
Data
Category
Category # 1
ISO Excess Wind &
Variations
A. Ratio of wind losses
to ex-wind losses
Averaging Methods
Base: Non-wind incurred
losses
Cat Data: Annual wind
incurred losses over
threshold
B. Ratio of cat losses to
ex-cat losses
Base: Non-cat incurred
losses
Cat Data: Incurred event
losses over fixed external
threshold
Category #2
Cat/AIY Methods
Base: AIY
Cat Data: Incurred event
losses over fixed external
threshold
A. Utilize state Cat/AIY
with no direct use
of countrywide cat
experience
Adjustment for
Extreme Cats
Arithmetic average of excess
wind factors and average wind
to ex-wind ratios. For each
additional year, give 95% weight
for long-term average, 5%
weight for additional year.
Can give less weight
to unusual years

Can adjust
catastrophes for
unusual years
Straight long-term average
Confidence interval
approach
Average adjusted catastrophes and add increment for
extreme
Trended exponential
smoothing
Balance state provisions to
countrywide expectation
May or may not
include adjustments for
extreme catastrophes
Average annual losses are based
on the stochastic event set
Extreme events are
given their appropriate
statistical weight




B. Utilize state Cat/AIY
with direct use of
countrywide cat
experience


Category #3
Modeled Cat Losses
Base: AIY
Cat Data: Computer application generated losses
Category #4
Reinsurance Cost Based
Cat cover reinsurance cost passed through primary pricing
10
7/26/2016
What Base to Relate Catastrophes To?
 Premium:
Cat provisions impacted by rate changes
Trends in non-catastrophe loss & expense dictate cat provision
11
7/26/2016
What Base to Relate Catastrophes To?
 Premium:
Cat provisions impacted by rate changes
Trends in non-catastrophe loss & expense dictate cat provision
 Non-Cat Loss/Ex-Wind Loss:
Still heavily dictated by trends in Crime, Liability, etc. loss
Ex-wind losses can include catastrophic losses
12
7/26/2016
What Base to Relate Catastrophes To?
 Premium:
Cat provisions impacted by rate changes
Trends in non-catastrophe loss & expense dictate cat provision
 Non-Cat Loss/Ex-Wind Loss:
Still heavily dictated by trends in Crime, Liability, etc. loss
Ex-wind losses can include catastrophic losses
 AIY or Amount of Insurance Years
Definition: $1000 of Building Coverage in force for one year
Inflation sensitive
Direct measurement of exposure – incorporates policy
growth and changes in building costs
13
7/26/2016
Should Individual Catastrophes
Be Capped?
14
7/26/2016
Should Individual Catastrophes
Be Capped?
 Stabilizes provision
 Can serve to more appropriately match experience
period used with event return periods
 Potentially more accurate estimate of expected
value results
15
7/26/2016
What Are Some Problems With
Capping Individual Catastrophes?
16
7/26/2016
What Are Some Problems With
Capping Individual Catastrophes?
 What criteria should be used?
 The “unthinkable” is happening every year somewhere.
 Is the result systematic underestimation of loss costs?
 How do we really know appropriate event return periods?
17
7/26/2016
Insurance Services Office (ISO)
Excess Wind Procedure
Basic Approach





Separate wind & non-wind losses
Examine wind/non-wind ratios
Years where wind/non-wind exceed 1.5 times
median are “excess”
Average factor for excess wind
Factor developed for excess wind applied to non-wind,
non-excess losses
18
7/26/2016
Insurance Services Office (ISO)
Excess Wind Procedure
Basic Approach





Separate wind & non-wind losses
Examine wind/non-wind ratios
Years where wind/non-wind exceed 1.5 times
median are “excess”
Average factor for excess wind
Factor developed for excess wind applied to non-wind,
non-excess losses
Characteristics





Straightforward application
Definition of “excess wind” can change as median changes
Assumes stable relationship between wind & non-wind losses
Doesn’t consider variability of wind losses
19
7/26/2016
Doesn’t consider non-wind catastrophes
The Fix ‘Em Up Insurance Group
Homeowners
The State of Mich-con-ota
20-Year Average Approach
Year
Amount of
Insurance Years
(AIY)
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
4,276,135
4,306,815
4,540,913
4,774,783
5,001,164
5,193,190
5,367,566
5,574,506
5,745,344
6,223,199
Cat Incurred
Loss
$
CAT/AIY
307,946
0.0720
259,784
0.0603
4,378
0.0010
1,529,513
0.3203
2,736,486
0.5472
50,241,886
9.6746
8,141,594
1.5168
6,676,296
1.1976
12,086,512
2.1037
6,091,053
0.9788
Provision (20-Year Average )
Running
Provision
0.2148
0.2102
0.1719
0.1862
0.2128
0.6948
0.7114
0.7566
0.8465
0.8929
0.8929
20
7/26/2016
Confidence Interval Approach
Step 1 – Establish Company Objective:
 Factors include risk tolerance, surplus position/
availability of capital, reinsurance
 Determine confidence demands for long-term
companywide cat provision
 Calculate companywide mean cat/aiy
 Calculate standard deviation of mean cat/aiy
21
7/26/2016
Confidence Interval Approach
Step 1 – Establish Company Objective (Cont.):
 Company has established that it would like to
be 90% certain it has an adequate catastrophe
provision over the long-term
 The following have been calculated from the
companywide catastrophe history:
Mean Cat/AIY = .3151
Standard Deviation of Mean Cat/AIY = .0372
22
7/26/2016
Confidence Interval Approach
Step 1 – Establish Company Objective (Cont.):
 The long-run companywide benchmark cat provision
is established as follows:
Provision (Cat/AIY) = Mean + (t) x (Standard Deviation)
= .3151 + (1.323) x (.0372)
= .3643
Where : Mean = average cat/aiy companywide
1.323 = t – stat for 90% and (N-1) degrees of
freedom
.0372 = standard deviation of the mean
23
7/26/2016
Confidence Interval Approach
Step 2 – Establish State Level Objective:
 Goal period becomes interval rates are in effect
 Need to be reasonably certain provision is adequate
 Desire to use cap on individual cats to limit volatility
 Largest 5% of companywide cats exceeded .65/AIY
 Establish required confidence for state capped average
24
7/26/2016
Non-Hurricane Catastrophe Provisions
Confidence
Sum of States
Intervals Uncapped Capped
50%
55
60
65
70
75
80
85
90
95
0.3334
0.3825
0.4328
0.4848
0.5392
0.5988
0.6657
0.7447
0.8452
0.9999
0.2682
0.2998
0.3323
0.3657
0.4008
0.4392
0.4823
0.5332
0.5981
0.6973
Companywide
Uncapped
0.3151
0.3198
0.3247
0.3297
0.3349
0.3406
0.3471
0.3546
0.3643
0.3791
25
7/26/2016
Confidence Interval Approach
Step 2 – Establish State Level Objective (Cont.):
 It’s determined that 65% confidence is required
 Calculate state mean cat/aiy (capped)
 Calculate state standard deviation of cat/aiy (capped)
26
7/26/2016
Confidence Interval Approach
Step 3 – Calculate State Provision:
 The following was calculated from the capped state level
catastrophe history:
Mean Cat/AIY = .3912
Standard Deviation of Cat/AIY = .5450
 The short-run state cat provision is established as follows:
Provision (Cat/AIY) = Mean + (t) x (standard deviation)
= (.3912) + (.389) x (.5450)
= .6032
Where: Mean = average capped cat/aiy for Mich-con-ota
.389 = t – stat for 65% and (N-1) degrees of
freedom
.5450 = standard deviation of the annual capped 27
7/26/2016
cat/aiy
The Fix ‘Em Up Insurance Group
Homeowners
The State of Mich-con-ota
Confidence Interval Approach
Year
Amount of
Ins. Years
(AIY)
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
4,276,135
4,306,815
4,540,913
4,774,783
5,001,164
5,193,190
5,367,566
5,574,506
5,745,344
6,223,199
Cat Incurred
Loss
$
307,946
259,784
4,378
1,529,513
2,736,486
50,241,886
8,141,594
6,676,296
12,086,512
6,091,053
CAT/AIY
Capped
CAT/AIY
Running
Provision
0.0720
0.0603
0.0010
0.3203
0.5472
9.6746
1.5168
1.1976
2.1037
0.9788
0.0720
0.0603
0.0010
0.3203
0.5472
1.6175
1.5168
1.1976
1.9537
0.9788
0. 2983
0.2911
0.2825
0.2865
0.3016
0.3934
0.4612
0.5004
0.5835
0.6032
Provision (Confidence Interval Approach)
28
7/26/2016
0.6032
Issues With Confidence Interval Approach
Pluses
 Recognizes individual state variability
 Stable provision
 Provides means to assure companywide sufficiency
29
7/26/2016
Issues With Confidence Interval Approach
Pluses
 Recognizes individual state variability
 Stable provision
 Provides means to assure companywide sufficiency
Drawbacks
 Not particularly responsive to distributional changes,
coverage changes, etc. (data back to 1971)
 Capping can result in less responsiveness
 Recognition of variability interpreted as risk margin
30
7/26/2016
The Fix ‘Em Up Insurance Group
Homeowners
The State of Mich-con-ota
Extreme Events Adjustment
Year
Amount of
Ins. Years
(AIY)
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
4,276,135
4,306,815
4,540,913
4,774,783
5,001,164
5,193,190
5,367,566
5,574,506
5,745,344
6,223,199
Cat Inc.
Loss
$
Non
Extreme
Cat Inc.
Loss
Extreme
Cat Inc.
Loss
307,946 $
259,784
4,378
1,529,513
2,736,486
50,241,886 45,217,697
8,141,594
6,676,296
12,086,512
6,091,053
307,946
259,784
4,378
1,529,513
2,736,486
5,024,189
8,141,594
6,676,296
12,086,512
6,091,053
Extreme
CAT/AIY
Contrib.
From
Extreme
$
20 Year Average
8.7071
1.7414
0.0871
Provision (Extreme plus Non-Extreme)
Contrib.
From
Non
Extreme
Running
Provision
0.0720
0.0603
0.0010
0.3203
0.5472
0.9675
1.5168
1.1976
2.1037
0.9788
0.2148
0.2102
0.1719
0.1862
0.2128
0.3465
0.3631
0.4083
0.4982
0.5446
0.4576
0.5446
31
7/26/2016
Extreme Events Adjustment
Pluses
 Relatively stable
 As opposed to censoring, reflects events fully
Drawbacks
 Accurate determination of event return period
difficult
 Can be viewed as arbitrary and difficult to
explain
32
7/26/2016
95% / 5% Trended Approach:
Methodology:
 All years used
 Exponential smoothing
 Trend factor applied – recognizes static cat definition
 10% annual cap to change in provision
33
7/26/2016
The Fix ‘Em Up Insurance Group
Homeowners
The State of Mich-con-ota
95/5 Trended
Year
Amount of
Insurance Years
(AIY)
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
4,276,135
4,306,815
4,540,913
4,774,783
5,001,164
5,193,190
5,367,566
5,574,506
5,745,344
6,223,199
Cat Incurred
Loss
$
307,946
259,784
4,378
1,529,513
2,736,486
50,241,886
8,141,594
6,676,296
12,086,512
6,091,053
Provision (95/5 Trended)
CAT/AIY
0.0720
0.0603
0.0010
0.3203
0.5472
9.6746
1.5168
1.1976
2.1037
0.9788
Running
Provision
0. 1972
0.1930
0.2054
0.2259
0.2485
0.2733
0.3007
0.3307
0.3638
0.4002
0.4002
34
7/26/2016
95% / 5% Trended Approach:
Advantages:
 Not volatile, yet responsive for non-extreme events
 Simple to understand
 Trend factor to compensate for static definition of cats
 Reduced data complications
35
7/26/2016
Summary of Results So Far:
Approach
Mich-con-ota
Cat/AIY
Stability
Rank
Cat/Ex-Cat
20-Year Average
Confidence Interval Approach
Extreme Events
95% / 5% Trended
All Years Weighted Average
.7292
.8929
.6032
.5446
.4002
.9940
4
5
2
3
1
6
36
7/26/2016
Pivotal Question:
Can Countrywide or Regional Data
Help Quantify the True Prospective
Mean Catastrophe Loss in a Given
State?
37
7/26/2016
Pivotal Question:
Can Countrywide or Regional Data
Help Quantify the True Prospective
Mean Catastrophe Loss in a Given
State?
Issues:
 Provisions need to reflect adequacy and stability
 All company surplus is generally available and at risk
 Are regional or sub state provisions appropriate?
 Perceived cost sharing will be scrutinized
38
7/26/2016
Goals of Relativity Method




Develop accurate, stable results by state that
results in an appropriate provision on a
countrywide basis
Systematic approach to handle extreme events so
a single outlying year does not drive the cat
provision for a state
Appropriate application of credibility procedure
Provide result that is responsive to recent
demographic and cat definition shifts
39
7/26/2016
Issues Addressed
How to be responsive to changes in risk
due to population shifts or cat definition
changes while still including an
appropriate number of years
 How does one define an outlying event

– Individual state vs. countrywide

How to incorporate credibility
40
7/26/2016
State Relativity Weighted with Countrywide
Complement – General Outline
I.
II.
III.
IV.
V.
VI.
VII.
VIII.
Develop State Damage Ratios
Calculate Countywide Damage Ratios
Calculate State Relativities
Cap State Relativities
Average Capped Relativities
Credibility Weight with CW Average of 1.000
Balance Back to CW Average of 1.000
Calculate Statewide Catastrophe Provision
41
7/26/2016
State Relativity Weighted with
Countrywide Complement

I.
II.
III.
IV.
V.
VI.
Develop State Damage Ratios
Calculate Countrywide Damage Ratios
Calculate State Relativities
Cap State Relativities
Average Capped Relativities
Credibility Weight with CW Average of
1.000
VII. Balance Back to CW Average of 1.000
VIII. Calculate Statewide Catastrophe
Provision
Develop each state’s damage ratios for years 1981-2000
– State Damage Ratios – Losses/AIY
– Only use years 1981 forward. Data for years 1971 through 1980
is sparse as evidenced by yearly variance.
42
7/26/2016
State Relativity Weighted with
Countrywide Complement

I.
II.
III.
IV.
V.
VI.
Develop State Damage Ratios
Calculate Countrywide Damage Ratios
Calculate State Relativities
Cap State Relativities
Average Capped Relativities
Credibility Weight with CW Average of
1.000
VII. Balance Back to CW Average of 1.000
VIII. Calculate Statewide Catastrophe
Provision
Each year’s Countrywide damage ratio is
calculated as the weighted average of state
damage ratios using latest year AIYs as weights
– Eliminates distortion of state distributional shifts over
time

Countrywide catastrophe provision is the
arithmetic average of the most recent 10 years of
damage ratios
43
7/26/2016
Figure 1
CW Cat damage ratios
including 2000
0.80
0.70
Damage ratio
0.60
0.50
0.40
0.30
0.20
0.10
0.00
1970
1975
1980
1985
1990
1995
2000
2005
Year
Years
Linear trend
1971-1978
0.006
1979-1989
0.000
1990-1999
-0.019
1990-2000
-0.010
44
7/26/2016
State Relativity Weighted with
Countrywide Complement



I.
II.
III.
IV.
V.
VI.
Develop State Damage Ratios
Calculate Countrywide Damage Ratios
Calculate State Relativities
Cap State Relativities
Average Capped Relativities
Credibility Weight with CW Average of
1.000
VII. Balance Back to CW Average of 1.000
VIII. Calculate Statewide Catastrophe
Provision
Calculate state relativities as the ratio of state damage
ratios to countrywide damage ratios
Relativities should be more stable than damage ratios
Trend should not be a problem so we can use more years
of data than the Countrywide Catastrophe Provision
45
7/26/2016
State Relativity Weighted with
Countrywide Complement

I.
II.
III.
IV.
V.
VI.
Develop State Damage Ratios
Calculate Countrywide Damage Ratios
Calculate State Relativities
Cap State Relativities
Average Capped Relativities
Credibility Weight with CW Average of
1.000
VII. Balance Back to CW Average of 1.000
VIII. Calculate Statewide Catastrophe
Provision
Any relativity greater than the mean plus three
standard deviations is capped to the next lowest
relativity (not the cap number)
– Intuitively we are replacing a once in a hundred year
event with a once in 20

Benefit of capping process
– Represents a systematic approach to dealing with
extreme events
– Cap is dynamic and is allowed to shift if exposure in a
state is changing over time
– Censoring at the cap would not have much impact and
therefore would not result in increased stability
46
7/26/2016
State Relativity Weighted with
Countrywide Complement

I.
II.
III.
IV.
V.
VI.
Develop State Damage Ratios
Calculate Countrywide Damage Ratios
Calculate State Relativities
Cap State Relativities
Average Capped Relativities
Credibility Weight with CW Average of
1.000
VII. Balance Back to CW Average of 1.000
VIII. Calculate Statewide Catastrophe
Provision
Calculate arithmetic average of 1981-2000 capped
relativities
– Using a arithmetic average is simple
– No benefit of weighting relativities has been shown since
relationship of variability to exposure level is unclear
– Arithmetic average relativity does not differ significantly from an
AIY weighted average
47
7/26/2016
State Relativity Weighted with
Countrywide Complement

I.
II.
III.
IV.
V.
VI.
Develop State Damage Ratios
Calculate Countrywide Damage Ratios
Calculate State Relativities
Cap State Relativities
Average Capped Relativities
Credibility Weight with CW Average of
1.000
VII. Balance Back to CW Average of 1.000
VIII. Calculate Statewide Catastrophe
Provision
Uses Buhlmann credibility factor: n/(n+k)
– n = number of years of relativities in average

We use number of years rather than exposures because exposures not
independent, especially past a certain threshold where exposure
concentration increases
– k = average process variance/variance of hypothetical means


The process variance and variance of hypothetical means are
calculated using all available years of capped relativities across all
states.
Complement of credibility of 1.000 is not appropriate when
there is a wide spread of average relativities
– Solution lies in balancing process described on next slide
48
7/26/2016
State Relativity Weighted with
Countrywide Complement

I.
II.
III.
IV.
V.
VI.
Develop State Damage Ratios
Calculate Countrywide Damage Ratios
Calculate State Relativities
Cap State Relativities
Average Capped Relativities
Credibility Weight with CW Average of
1.000
VII. Balance Back to CW Average of 1.000
VIII. Calculate Statewide Catastrophe
Provision
At this point, the individual state relativities result in a
countrywide relativity of less than 1.000. Relativities are
adjusted to achieve an overall adequate level as follows:
– Determined on a countrywide basis what our expected losses
would be based on the countrywide selected catastrophe factor
– Sum the pre-balanced expected losses across all states
– We distribute the difference between 1 and 2 in proportion to
each state’s standard deviation measured in latest year expected
losses.

Using this approach has several benefits:
– Results in an appropriate provision countrywide
– It compensates for high (low) relativity states being
underestimated (overestimated) by the use of a 1.000
complement of credibility.
– Each state’s resulting cat load is a function of its own
size and variability
49
7/26/2016
State Relativity Weighted with
Countrywide Complement


I.
II.
III.
IV.
V.
VI.
Develop State Damage Ratios
Calculate Countrywide Damage Ratios
Calculate State Relativities
Cap State Relativities
Average Capped Relativities
Credibility Weight with CW Average of
1.000
VII. Balance Back to CW Average of 1.000
VIII. Calculate Statewide Catastrophe
Provision
Statewide catastrophe provision is calculated by
multiplying capped, credibility weighted,
balanced relativity by the countrywide
catastrophe provision
Benefits of method:
– Allows use of long term data to determine relativity
while using more responsive data for countrywide
provision
– Adjustments to data are determined objectively with
each state’s characteristics used to determine both
capping and balancing
50
7/26/2016
Issues Addressed

How to be responsive to changes in risk due to
population shifts or cat definition changes while still
including an appropriate number of years
– We use as many years of relativities as possible while using only
the latest 10 years of Countrywide to determine the Countrywide
load.

How do you define an outlying event
– Greater than the mean plus 3 standard deviation in any given
state. On a countrywide basis these “outlying” events occur
fairly regularly (about 2% of relativities have been capped)

How to incorporate credibility
– Uses Buhlmann credibility to account for variability in
relativities. Used a relativity of 1.000 as complement. Balancing
method adjusts for bias in complement for when a 1.000 may not
be an appropriate complement.
51
7/26/2016
Summary of Results:
Approach
Cat/Ex-Cat
20-Year Average
Confidence Interval Approach
Extreme Events
95% / 5% Trended
All Years Weighted Average
Relativity Method
Mich-con-ota
Cat/AIY
.7292
.8929
.6032
.5446
.4002
.9940
.6270
52
7/26/2016
Conclusion
We have made significant progress as a
profession in quantifying the catastrophe
exposure.
 Do our current methods capture the true
mean?
 My purpose will be achieved if this session
helps to keep focus on this issue

53
7/26/2016
Conclusion
Comments
and
Discussion
54
7/26/2016