RESERVES, SURPLUS, & UNCERTAINTY
Authors:
Aaron
Halpert
a Manager with Peat Marwick Main & Co. in New
?Ir.
Halpert
is
1983 and
is
a
ACAS designation
in
He received
his
York.
He holds a B.S.
the American Academy of Actuaries.
member of
degree from Brooklyn
College.
Douglas
W. Oliver
Mr. Oliver
is a Consultant
with Peat
Maruick
New York.
Douglas
graduated
from
Rutgers
both
Mathematics
degrees
in
with
B.A.
Administration.
Main
& Co.
in
College
in 1983
Business
and
Abstract:
With
the
statutory
stating
attention.
reflecting
attention.
calculation
accounting
of 1986, the topic
of amending
passage
of
the
Tax Reform
Act
money when
and/or
GAAP accounting
to reflect
the
time
value
of
expenses
has
gained
reserves
for
loss
adjustment
and loss
subject
of
the
In conjunction
with
the
discounting
issue,
a provision
for adverse
claim development
has also gained wider
the
The aim of
for
this
paper
is
to present
a framework
of
the
size
of
this
provision
and to discuss
the associated
issues.
Actual
company loss reserve
information
for
several
lines
of
business
as
well
as companies
Both parameter
risk
and
of various
size was considered.
process
risk
are discussed
and reflected
in the model.
Conclusions
arising
from
these
models include:
The familiar
2 to 1 premium to surplus
ratio
is
not appropriate
for all companies
nor for all lines
of
insurance,
and that
the
difference
between
undiscounted
and discounted
reserves
produces
a
greater
margin than our model suggests.
215
1.
insurers
have
1OSS
and
tax
purposes
once
loss
again
of
still
require
1
GAAP
value
of
is
then
adverse
current
unpaid
provide
it
losses
a
lcurrently,
compensation
the
discounting
of
The determination
the
now be amended
income
the
to a large
ultimate
under
of
reflect
to
GAAP accounting,
discussion
time
between
value
value
way as to
of
for
time
degree,
an
unpaid
statutory
whether
to more fully
be
and
framework
of money is
discolxlced
developments.
accounting
that
developments
2Xuch of
resulting
that
difference
for
mandates
and
and casualty
recognize
the
time
money.2
the
margin
issue
be discounted
reflect
should
accounting
property
the
on
reserves.
statutory
reserves
as regards
discussion
reserves
considerable
Some believe
view
that
law
tax
expense
while
that
There
and
prompted
adjustment
money,
the
in
now requires
value
claim.
changes
recent
The
INTRODUCTION
If
and
this
implicitly
loss
adjustment
for
evaluating
and to present
accounting
some exceptions
and medical
defines
reflected
to this
malpractice.
as
reserves
at ultimate
value of all future
reasoning
the
size
part
exist
values,
the
investment
216
of
reasonable
approaches
rule
and
reserves
as a
is
of
this
the
The aim of
to
in the
this
discussion
has been spawned by the
from the Tax Reform Act of 1986.
3By stating
the present
of
expenses.
a
recognized,
undiscounted
line
automatically
already
implicit
income.
this
this
of
and
for
paper
for
reflect
changes
margin3
liability
margin
area
followed,
is
adverse
margin.
workers'
in
the
margin
is
to
tax
equal
laws
to
Section
reviews
the
insurance
used
2
presents
functions
of
illustrates
regarding
capital
reasonable
how
the
safety
results
If
value)
on a company's
changed
the
then
adverse
the
property
and
casualty
and statistical
models
Section
developments.
models
note,
lead
to
4
conclusions
various
for
margin
for
to
(in
be
unpaid
the
the
to probe
the
would
adverse
view
guise
of
value
to continue
losses.
of
that
currently
reserves
margin
stated
will
and/or
GAAP
money in
setting
to reflect
That
has
reserves
how this
statutory
time
developments
is,
this
the
long
at
an
be reflected
accounting
be
reserves.
margin
liability
as part
of
would
be
as:
course,
uncertainty
for
should
for
Unpaid
Margin
between
data
we need
sheet
Liability
Of
the
subscribe
reflect
approach
liability
established
3 we discuss
implicitly
balance
to explicitly
One
a
of
or
we
a provision
undiscounted
for
on a related
ACCOUNTINGFOR RESERVE MARGINS
a buffer
recognized.
such
and,
margins.
The need for
contain
issues
and surplus
margins
2.
been
accounting
In Section
company.
to evaluate
the
the
discounted
that
margin
Losses
for
need
Adverse
not
and undiscounted
is present
in
- Discounted
reserves,
217
+
Development
necessarily
the book
Reserves
be
but
of business
equal
rather
written
to the
would
difference
reflect
by a company.
the
In
surplus.
margins
three
the
fact,
major
purposes
Of these
approach
1.
for
In
uncertainty
to meet
necessary
of
many
viewed
conservative
begins
as
down
estimate.
5
as
be an even
(unnecessary?)
"remove
the
reserves
has been
the
reserves
latter
as
established
cited
(and
the
as one of
provides
a
consequence
of
estimate
might
be
unbiased
to
part
greater
also
estimate
ostensibly
company's
of
balance
the
the
better
conservatism"
unpaid
sheet
so
losses
were
for
reserves
of
unpaid
that
a less
that
which
become
there
the
the
reflect
would
containing
The natural
all
gets
from
to
losses,
as
are
often
required
unpaid
conservatism.
would
is
basis
some "conservatism"
to view
of
reasoning
acceptable,
of
liability
estimate
this
remove
inclination
element
on an actuarial
"conservative"
a
natural
an insurance
margins
a company's
reasons:
an
wittled
of
4
by non-actuaries
A
as part
loss
with
uncertainties)
we believe
instances,
losses.
If
these
surplus.
following
be reflected
associated
two alternatives,
the
could
margin
this
Alternatively,
impulse
an
to
more prevalent,
4A. E. Hofflander,
"Minimum Capital
and Surplus
Requirements
for Multiple
Line Insurance
Companies:
A New Approach,"
Kimball
and Denenberg,
Insurance,
Government
and Social
Policy,
1969.
Three sources
of drains
on surplus
are presented,
two relating
to losses
and loss reserves
and one related
to asset values.
Given the recent
volatility
in the equity
markets,
one cannot
ignore
this
last
source.
However,
we have focused
on adverse
development
of loss reserves
as the key
role for surplus.
5This issue has very practical
applications
in the audit
of a property
and
casualty
insurance
company.
In most cases a reasonable
range of reserves
is projected
and a company's
reserves
are considered
fairly
stated
if the
balance
sheet liatility
is within
this range,
218
2.
It
is
doubtful
margins
in
the
in
GAAP
confusing
3.
that
and
demonstrate
involved
therefore
be
in
uncertainty
reserve
Including
income.
therefore
section,
selecting
a margin
perpetuate
there
an
to expand
a
of
This
would
in
the
meaning
reserve
for
margins
in response
result
of
considerable
provision
and contract
the marketplace.
interpreting
is
acceptable
selection
Management's
expected
of
calculations
the next
in
any reflection
would
reserves
in
6
allow
taxable
in reserve
deviations.
pressures
of
statutory
subjectivity
can
IRS would
calculation
difference
As we will
adverse
the
in
to
considerable
a company's
balance
sheet.
Alternatively,
reflected
as
regulated
if
part
standards
of
a
by determining
regulations
provide
set
are
company's
evaluate
surplus,
acceptable
for
to
premium
considerable
then
that
margins
these
to
surplus
management
discretion
margins
ratios.
are
can be
Current
in setting
these
provision
for
ratios.
We conclude
adverse
development
premium
to surplus
ratios
should
tail
that
lines,
the
is
appropriate
in setting
ratios.
logically
new companies
place
vs.
a
surplus
requirements
and maximum
3
will
demonstrate
that
among
mature
reflect
minimum
In Section
vary
to
we
companies
companies,
writing
short
tail
these
vs.
etc.
6This is analogous
to the elements
of classical
credibility
theory.
The
choice
of a full
credibility
standard
inevitably
depends on a subjective
choice
of an acceptable
difference
between actual
and expected
results.
219
long
A MODEL FOR REFLECTING UNCERTAINTY
3.
Risk
Loading:
The
Pricing
concept
received
Vs.
of
reflecting
draft
rate
expected
experience.
explicit
provision
A
"should
include
If7
Increased
for
risk
similar
application
has only
recently
gained
exposure
draft
made specific
reserve
for
on
of
a charge
limits
The
time
the
pricing
pricing
Most
now.
risk
of
has
recently,
maintained
Ratemaking"
for
of measuring
a GAS
that
variation
has generally
uncertainty
For
"Principles
to
insurance
in
an
from
the
reflected
an
loading.
attention.
reference
projection.
some
"Principles
regarding
insurance
uncertainty
attention
widespread
exposure
Reserving
example,
Regarding
reflecting
earlier
a
Loss
in reserve
the
and Loss
provision
statement
of
measurement
of
recently
evaluations
issued
Expense
CAS
Reserves,"g
for
uncertainty
in
principles
contained
a
no such
reference.
The models
pricing
purposes
evaluations.
sources
underlying
can,
As in pricing,
of
the
however,
we
need
risk
and
uncertainty
for
be
generally
applied
to
reserve
to
distinguish
between
two
primary
uncertainty.
'Final
Exposure
Document - Statement
of
Casualty
Insurance
Ratemaking,
October
Principles
5, 1987.
Regarding
8Final
Exposure
Document - Statement
of Principles
Regarding
Casualty
Loss and Loss Adjustment
Expense Reserves,
October
220
Property
and
Property
5, 1987.
and
Parameter
Risk
The
arises
from
exactly.
arises
first
source
our
In
inability
the
often
of uncertainty,
to
context
of
referred
project
expected
evaluating
loss
to as parameter
future
payments
claim
this
reserves
risk,
uncertainty
because:
1.
Various
projection
losses,
paid
losses,
contradictory
This
methods
only
is
often
estimates.
after
have
counts
and
applied
to
may
etc.)
averages,
incurred
yield
results.
uncertainty
reserve
methods
(e.g.,
the
been
difference
Berquist
in results
between
explained
by
render
one or both
data
of
is
by
this
these
and
in
adjusted
to
have
company
technique
shown
that
projections
operations
inappropriate
accommodate
of
be established
incurred
projections
range
projection
Sherman8
paid
a
should
each
and
changes
establishing
range
underlying
tested.
be
with
However,
assumptions
often
underlying
dealt
can
that
unless
these
the
may
the
changes
operations.
2.
There
is
considerable
development
factors
The
of
level
factors
9Berquist,
J.
Comprehensive,
will
at each maturity
variation
depend
variation
present
in
observed
cumulative
level.
in
a
given
set
of
development
on:
R. and Sherman, R. E.,
Systematic
Approach."
"Loss Reserve
PCAS, LXIV,
221
Adequacy
1977.
Testing:
A
in
a
Sample
l
The
size
lines
of
associated
.
The
with
random
exposure
The model
presented
reserve
projections
(and
business
the
environment
regulatory
the
line)
occurrence
of
for
losses
large
a particular
period
below
attempts
to measure
resulting
from
the
the
level
of
variation
in
uncertainty
loss
in
development
factors.
Process
Risk
Process
risk
expected
from
process.
results
Empirical
projections,
parameter
Data
this
from
the
because
tests
fact
of
the
presented
source
of
risk
measure
the
uncertainty
of
accident
that
random
below
is
results
actual
nature
show that
relatively
with
minor
will
of
the
regard
when
differ
insurance
to reserve
compared
to
risk.
Collection
To
data
results
base
year
in
reserve
loss
characteristics:
Size
-
of
Small Company
Large Company
Composite
of Many Companies
Lines
-
Company
of
Insurance
Homeowners
Auto Liability
General
Liability
Workers Compensation
222
projections
experience
we have
with
the
assembled
following
a
Data
-
Elements
Paid Losses
Incurred
Losses
Reported
Counts
Closed Counts
This
data
was available
schedules
0 and P of
Parameter
Risk
this
case
in size
review
is
all
Accident
the
respective
determine
normal
year
if
the
mean
points
reviewing
were
each
of
random
levels.
for
column
appear
that
samples
Each
the
including
CV at earlier
For
CV
later
in
year
these
in
reflect
a complete
for
the
were
of
age to ultimate
of
this
test
can
is
from
last
ages,
of
the
then
factor
where
variation)
which
less
at
tested
to
sample
is
than
was
from
a
Appendix
accepted.
five
at
factors
in
be
latest
divided
was drawn
presented
generally
group.
was
sample
and
factors
factors
the
annual
companies
five
losses
the
data
ranging
billion
The resulting
of development.
223
insurers
year.
that
the
(coefficient
stages
of
column
was calculated
column).
a
of
five
each evaluation
at
for
underlying
$1
of ultimate
hypothesis
(not
over
each
the hypothesis
the
column
available,
the
normally
The
for
by accident
estimate
losses
that
to
values
An illustration
indicates
respective
million
losses
one can accept
in the
would
A.
experience
experience
ultimate
maturity
in
in Exhibit
ultimate
considered
are
variance
appear
$10
incurred
distribution.
and
statement.
compensation
available
the projected
age
A
workers
to be an unbiased
each
than
annual
model
selected
the
considered
into
this
approximately
The
of
of
is
from
premium.
a company's
detail
Model
The elements
in
in finer
The
points
considered
five
selected
sample
by
If
uncertainty
is
uncertainty
each
measured
accident
year
means
losses
for
each
percentiles
the
the
mean)
appear
all
years
combined.
If
we
and the
illustrated
on
of
surplus
that
results
are
later
for
is
the
sample,
year.
as
C.
similar
results
all
company
has written
well
as
Key results
at
is
clear
that
and decreases
as
assumption
of
in
this
for
the
as of
between
12
companies
are
displayed
the
group.
line
Also
some
that
as
portion
displayed
has written
this
and in
company,
are
years
The
combined,
largest
reproduced
at
deviations.
the
are
for
ratios
to
sample
the
all
that
percentile
in our
for
for
refers
of
and
losses
various
adverse
for
months
ultimate
context,
a margin
Various
as a percentage
we can calculate
a company
90th
year
of ultimate
combined.
expressed
in this
five
years
results
losses
the
distribution
all
difference
to provide
with
probability
for
Surplus,
for
that
year
an accident
the
needed
smallest
a company
B for
displayed
exhibits
accident
(with
mean ultimate
Exhibit
mature
(along
and
distributions
calculate
percentile
it
the
year
on Exhibit
now
variance,
variances
and
accident
of
least
to construct
used
were
the
of
10
matures.
The estimated
normality)
at
greatest
is
function
as a
the
results
present
line
below
for
All
Large
Small
Company
Company
l"Analogous
formance
conclusions
Standards",
22.2
3.9
Years
Reserve
to Surplus
Premium
to Surplus
32.6
7.7
our
only
one
in Table
are drawn in Khury,
PCAS, LXVII,
1980.
One Year
Reserve
to Surplus
27.3
4.4
C. K.,
"Loss
15.3
2.8
Reserves:
for
in
Table 1
WORKERS COMPENSATION
PARAMETER RISK MODEL
90th Percentile
Premium
to Surplus
and
Per
1.
It
should
cushion
not
that
be stressed
for
adverse
be viewed
surplus
associated
Various
conclusions
-
familiar
The
for
all
measures
with
but
adverse
to
companies
1
(nor
other
ratios
rather
than
presented
relative
as
providing
a
should
therefore
measures
of
the
developments.
can be drawn
2
purposes
The
developments.
as absolute
uncertainty
serves
from
premium
this
table.
to surplus
ratio
as we demonstrate
later
is
not
for
all
appropriate
lines
of
insurance).
-
If
premium
then
it
to
appears
afford
to
somewhat
the
difference
example,
the
income
percentile
are
four
is
in
Exhibits
the
results
of
the
the
of
lines
estimated
reserves.
The
other
lines
K
through
the model
tested.
for
to
surplus
leverage,
given
a
a mature
model
and
company,
were
$2.8
than
this
discounted
small
only
entering
of
line
company.
ratios
can
This
is
appear
to
leverage.
from
between
undiscounted
detail
leveraged
Reserve
calculated
for
investment
of
used as an indicator
newly
a company
measure
margin
for
are
counter-intuitive.
The
Results
that
ratios
be more highly
be a better
-
surplus
is
considerably
undiscounted
reserves
to
be
than
reserves.
For
discounted
for
million
less
than
the
90th
$4.3
estimated
less
surplus
at
future
million.
of
AN.
insurance
Also
a large
The model
independent.
225
studied
enclosed
and
a
assumes
appear
in exhibits
small
the
company
lines
are
in
similar
A0 through
writing
statistically
AR
all
Process
Risk
The
Model
design
used
model.
frequency-severity
estimating
reserve
with
claims
to
close
mean of
a
Poisson
reserve
Given
the
large
the
number
following
distribution
to
be
the
of
claims
mean
we
Var(T)
"N" and "X" are
and the
random
The
displayed
expected
claims
and
the
via
addition
number
of
severity
a
to
future
to be the
corresponding
average
distribution.
the
analyzed,
modeled
on
in
was assumed
a pareto
have
distribution
a normal
of
distribution
with
- E (N).E(X)
in
the
of
Exhibit
+
frequency
and
"T"
represents
this
model
~.ll
associated
with
E(N).var(X)
-
variable
results
uncertainty
associated
year,
of
of
based
is
moments:
E ('0
where
number
can be adequately
reserves
aggregate
This
payment.
assumed
was
the
estimated
we
frequency
accident
each
For
amounts,
risk
process
evaluate
to
for
parameter
2
respectively,
variables,
reserves.
the
selected
indicate
process
E(X)
severity
total
The results
with
var(N).
risk
group
that
is
at
of
each
considerably
5 companies
are
percentile
the
smaller
than
that
risk.
Compound Model
We have
surplus
also
levels
The results
of
employed
when
this
both
Monte
Carlo
elements
simulation
simulation
of
technique
risk
are
are
also
methods
considered
summarized
to
evaluate
simultaneously.
in
Exhibits
H, X and AH.
llFor
is
this
model,
we have
assumed
that
the
3.0.
226
CV of
the
severity
distribution
D,
4.
results
displayed
1.
Ihe
in the
difference
income
and
margin
than
tested,
at
90th
Generally,
also
2.
considerably
As
Lines
o
Different
Age
The
slightly
levels,
of
or
lower
than
of
or
calculated
undiscounted
we
margin
reserves.
more investment
reserve
at
to
in regulating
to
discounted
be
income
surplus)
vary
values,
these
recognized
leverage
in evaluating
insignificant
the
cases
Insurance
need
uncertainty
incorporating
most
uncertainty.
reflected
will
to be almost
model
be
ratios
a greater
of:
Different
to
In
generating
to surplus
o
in
and the
investment
provide
and the
the
than
of
Size
source
appears
level
Different
are
model
future
to
model.
reserves
lines
o
margin
a
the
(premium
reserves
our
less
among insurers
differences
3.
were
a greater
ratios
our
for
appears
by
discounted
percentile
generate
reserve
of
from
discounted
reserves
be implied
sum
drawn
exhibits:
undiscounted
would
be
reserves
however,
Leverage
If
attached
between
the
the
can
conclusions
important
Three
CONCLUSIONS
both
ratios
to
yields
associated
with
setting
ratios.12
reserves,
relative
sources
in
process
parameter
leverage
the
ratios
parameter
risk
risk.
only
risk
model
12Such differences
are reflected
in the
promulgated
in California
in 1985.
221
"Analysis
of
Surplus
Quality"
tests
FUTURE STUDY
As
recognize
the
issues
.
is
the
that
case
facets
that
of
require
the
projections
situation
independent
general
further
the model.
Testing
many actuarial
the
Incorporating
into
.
with
problem
analysis
determine
where
the
from
each
to
remain
arising
uncertainty
The range
our
model
be
has helped
us
Among
studied.
are:
in
can be considerable
to
analyses,
whether
individual
other.
228
from
between
results
especially
a
for
multi-line
lines
results
contradictory
paid
a small
writer
can
be
and
incurred
company.
truly
faces
modeled
a
as
Parultr
-f-dibit
lli5k We1
selwtd
8orknr’ Colplution
lncurrd-Lo88
Five Insurrs
bunts
Dewlopmt
nga of Davrlopmnt
.
..,
Tears
1977
1978
1979
19W
1981
1982
1983
12
24
688,793
829.778
Bb8.661
BM,874
923,005
953,OM
W2.652
826,074
937,983
1,026,005
1.039,585
1,1%,4W
1,145,93b
1.101,094
36
845,3a
951,679
1,078,335
1,093,7bO
l,lSS,lS3
1,192.317
1,235,?57
a
8s7.429
974,468
1,102,l38
1,113,759
1,176,402
1,213,034
1,271,SOO
60
72
84
%
8b4,28b
abe*?
M8,555
173,221
976,349
978,391
979,753 983,192
l,lOS,%2 IJIll,
1,121,973 1.125,774
1,128,3b5 1.132,5% 1.139,6%2
1,181,8bS 1,193,MS
1.234.559
c
1984 1,111,915 lJa4,930 1.481,%9
1905 l,S11,157 1,876,605
1986 1,913,271
in~ltwaads
102
m,u1
9&098
120
2.
‘
i
Cumlatiw
wars
12
24
1.2952
1.2191
1.3212
1.34%
1.3999
Ia24
1.3572
1.4bw
l..%Jl
1.2949
1.0799
1 .O?M
1.1189
1.1289
1.1670
1.1497
1.1407
1.1601
1.0959
Sslwtod
1.2949
1.0959
1.1035
l.bS95
1.Ob72
Standard
&aviation
0.0457
0.0206
0.0222
0.0273
0.0255
0.0353
l.OlaB
0.0202
0.0258
0.0239
1977
1971
1979
1980
1981
1982
1983
1984
1985
19%
Coat iciat
of
Variation
after
rsviev
108
120
1.0216
1.0205
1.0197
l.OtW
1.0237
1.6146
l.Wb
1.0297
1.0197
1.0237
1.014b
8
0.0225
:
0.0220
t
0.0210
8
0.0208
:
0.0200
t
fdactcd
96
of previous values
229
4
SELECTED
UlTIlMlE
892,116
1,011,549
1,147,952
1,173,5b3
1,292,157
1,317,4bb
1,347,225
1,b34,397
2,on,MS
2,477,407
m,m
I.
Loss Dwelapmnt Factors
:.:
nga of Devslopwt
36
49
60
72
M
.‘,:
1.0553
1.04OS
1.0322
1.0269
1.0271
I.0629
1.0381
I.0361
.I.6339
1.0325
l.Ob4b
1.0410
1.0381
1.0265
1.0232
1.0730
1.0537
1.0401
1.0362
1.0297
1.1186
1.0984
1.0933
1.0826
1.1050
l.Oebl
1 .Ob72
I.0982
1.0595
I. 1035
I
h
Expoctad
Losses at rlatrd
hfcmtila
uahrcmtofthoE#pmctdllur
Sthctcl fiw
Insurers
nccidmt
PerCmtila
.-_....-.
111 bars
C&id
... ... ...
Year pb
-...-...-
99.%t
so
55
99.99a
100.4a
lOO.lO#
60
1oo.m
lOO.ZZt
65
101 .34\
l&sst
70
lOl.MI
100.4u
75
lb2.S7\
100.58t
80
102.?78
in.m
b
103.64a
lW.848
90
1WZt
101.tot
95
105.79t
101.41t
230
Mbit
I
Paramtr
Yorkers
Exhibit
Rish lbdel
Llectad
Capmath
Fiva Insurers
ona Tclr
_______---._________.---..~-------.-.-----
Rll Years
Pafcmtils
--.---_-.-
Prwiun
to Surplus
----------
C
Reserve
to Surplus
..________
DiscoMud
Resarm 6
Surplus to
Undiscounted
Rawva
.._.____----
Praium
to Surplus
_..-------
Rmwva
to Surplus
----------
Discwntmd
Rewrvesb
Surplus to
Ibdiscountcd
Reserw
___----_____
75th
39.9
61.0
0.816
54.3
32.2
0.839
%1b
20.9
31.9
0.831
29.5
16.9
0.867
95th
lb.3
24.9
0.840
23.0
13.2
0.884
231
Exhibit
IilHKERS COIPERSNIO(I
PROCESSRISR
Selected Five Insurers
________________________________________------------ALL YEARS
_________________--.-------E(R):
VnR(R):
E(X):
W(X):
298.466
290,466
16,903
2,571.402,681
E(H):
vnR(n):
E(X):
VAA(X):
E(T):
[vnR(T)1-.5:
5.044,970,790
29,201,908
E(T):
[VAR(T)]^.S:
PREIIUMS
RESERVES
a ’ ILE
ID SURPLUS TO SURPLUS
____________________---------------.---.-.
COIMWIIIMI
----------_
ORE YERFI
_____----.---___--__---.-..184,834
186.834
10.108
919.544.976
1,888,51&072
13,816,346
PRENIUNS
RESERVES
TO SURPLVS TO SUAPLUS
--__-___________-_--________
75 th
168.8
257.9
356.8
204.0
90 tll
08.4
135.0
186.8
106.8
95 th
69.0
105.3
145.8
83.3
OF PRRRNETER
RISK t PROCESSRISK MJDELS
RLL YEARS
____________________________
PREIIIWIS
RESERVES
’ ILE
TO SURPLUS TO SURPLUS
__________________--........--------------
t
OREYEAR
__-_-__--__.-___--__-------PREMJllS
RESERVES
10 SLWLUS
TO SURPLUS
_--___-_-__---_---__--------
15 th
39.5
60.3
54.1
M.9
90 th
19.1
30.1
28.3
16.2
95 th
15.0
22.8
22.1
13.0
232
D
Exhibit
Paraeeter Risk Hedel
Yorkers’
Compenution
Lupl
lncerred
cosrrtr
haunts
Loss Cmveloprsnt
E
in ThoesaA
8ge of Developwit
SLECIED
Years
24
12
36
48
60
72
84
96
108
120
uLTIMIE
613,300 614,900 615,200 612.400 615,800 618,100 620,9UO
w,700
643,600
609,soo
1978 646,100 708,400 717,800 729,OQO 726,100 724,500 726,5W 727.600 730,400
822,700 824,500 833,IW 836,200 839,400
1979 673,400
771,600 807,500
750,500
865,wo
1984
1981
1982
1983
927,ooo
903,ooo
923,ooo
iv77
637,400
680,400
685.208
6a5,IOO
1984 803,sw
761,300
806,100
791,Ow
790,600
847,100
840,300
824,300
827,000
819,800
831,200
833.400
838,900
=%5oQ
855,700 860,600 869,500
838.000 850,500
847,600
979,500 1,041,900
1,161,OOO
l,SI7,ooo
1,897,Ow
l%S 1,122,800 1,368,600
1986 1,514,300
Cuwlative
12
Tears
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1.2429
I.1616
1.2845
1.3845
1 A624
1.3179
1.3472
1.4449
1.3511
1.2527
24
it4
48
Lees kvelopumt
Factors
nge of Deve1opent
60
72
96
108
120
1.03%
1.04%
1.0712
1.0934
1.1032
1.0955
1.1161
1.1145
1.0289
I.0262
1.0257
1.0304
1.0247 1.0209
1.029s
1.0514
1.0765
1.0833
1.0776
1.0890
1.0328
1.0491
1.0617
1.0772
1.0620
1.0359
1.0330
1.0341
1.0520
1.031s
1.0305
1.0275
1.0661
1.0520
1.0305
1.0275
1.0163
8
0.0200
8
0.0200
t
0.0183
t
0.0175
t
0.0160
1.2527
1.1084
1.1143
l.M%
1.0620
Stwdard
Dwirtiam
0.0478
0.0168
0.0164
0.0227
0.0208
0.0381
0.0152
0.0147
0.0208
0.01%
1.0375
1.0589
1.0661
t
Selected after
84
1.0454
1.0594
1.1210
1.1592
l.lsca
1.1416
1.1675
1.1853
1.1084
8elwted
Coeficient
of
Variation
rtriea
of previws
631,000
values
233
1.0163
Paramter
Expcted Lesses at selected Percentiles
a.9 a Percent of the Expected Hean
Risk lbdel
8erkers’ Cqansatiom
Percentile
---- -----
kcidwd
Year 1966
_--_-----
111 Years
Co&id
--------99.990
so
99.w
55
lOO.(6t
loo. lot
60
100.95t
1w.21t
65
101.4sa
1w.m
70
101.988
100.448
75
102.551
100.57~
80
103.2ot
tw.in
8.5
103.938
1W.880
90
104.88l
101.m
95
106.2u
1Ol.W
234
Exhibit
F
Peramter
-
Percmtile
----_-----
Exhibit 4
Risk Hodel
a11 Years
---------_-____---------------------------
Prmiup
to Surplus
------_-__
We Year
-----------.-------;---------------;-r---
wserve
to Surplus
-_----_--_-_
Prmiua
to Surplus
---_-_-___
Reserve
to Swplws
-----_------
75th
42.4
62.3
52.2
29.2
90th
22.2
32.6
27.3
15.3
95th
17.3
25.5
21.3
11.9
235
Parutw
Risk lb&l
Me&d
Campmution
Exhibit
Incurrd
%a11 cqny
Loss hwlopmt
II
Wnents in Ilwwds
agaofrmwlopmnt
YWS
1977
1978
1979
1900
1pIl
1902
1983
1984
1985
12
24
2,500 3.m
2.50
3,~
2,542
2,168
2,815
4,012
4,529
5.4@8
3,853
3.816
4,963
4,127
4,530
4,930
5.532
6.m
6,386
36
48
68
72
84
4.648
4,464
w6
5,315
5.859
S&l
7.01
7,602
4,225
4,675
5,WS
5,653
6.075
s;929
7.987
4,477
4,9Q8
6,051
5,770
6.165
6;307
4,526
5,120
6,110
6,052
6.451
-
4.610
5,194
6,161
6,326
%
4,610
sJ90
6.215
1w
120
4,576
5,340
4,736
8.%9
9,116
9,335
8,075
Loss Wehpmnt Factors
m of Dm1oo#lt
72
48
60
84
12
24
36
1.91%
2.1242
1.9483
2.5535
3.0489
2.41%
2.2355
2.0121)
1.7252
2.0952
1.410(
1.43M
1.3112
1.5728
1.4592
1 A793
1.6213
1.3049
1.4615
1.1855
1.2224
1.1613
1.2213
1.1282
1.2498
l-1973
1.1992
1.1359
1.1673
1.1075
1.1482
l.wBl
1.1469
1.1229
1.0719
1.1119
l.Q(l2
1.1234
1.0722
1.0782
1.0603
1.0632
1.0522
1.0723
1.0246
1.0410
1.0506
l.o43!i
1.0261
selected
2.6952
1.4615
1.1992
1.1229
1.0782
1.0246
Standard
Deviatim
0.4974
0.1314
0.0481
0.0326
0.6276
0.2374
0.0899
O.o(ol
0.0291
0.0236
a
0.0230
1977
1978
1979
19W
1901
19B2
1903
1914
1985
1986
Coaf icint
of
Variation
StbctedJftar
rwir
4,799
5,157
6,429
6,491
6,610
69.
Cuwlatiw
YWS
SELEUED
LWWTE
ef previous VJ~JJJ
236
%
1w
120
1.0410
1.0316
1.0344
1.0487
1.0219
1.0133
1.0261
1.0344
1.0219
1.0133
t
0.0225
1
0.0208
a
0.0208
t
o.o2w
hramter ai*
hrkers’
Vtiee
fxpnctedLe5ses
at selectd Perceetiles
a5 a Perceet of the Expected nean
We1
SJBll CoornY
Percentile
so
55
60
Accident
YeJr 19s
----_--_-
All Years
Cedeed
____-----
94.76t
w.97:
102.0%
lW.S6t
les.9R
lW.7U
65
1W.024
101.14\
70
112.34:
101.w
75
115.908
lOZ.Ol\
80
119.94:
102.533
85
124.458
103.1w
w
lSO.Jcn
lOLa5a
95
11.93a
104.931
231
Exhibit
I
Parader
Yorkers’
Mbit
Risk IMel
sull
Coymatim
Percentile
______-___
J
co&any
111 Years
________________--------------------------
One Year
-_____-____--_---______________________^--
Prmim
to Surplus
__--______
Prmiup
to Surplus
----------
ltesnrve
to Surplus
--_---------
temrve
to Surplus
75th
7.4
14.8
8.4
5.4
90th
3.9
7.7
4.4
2.8
95th
3.0
6.0
3.4
2.2
236
Purrtr
Exhibit
Risk lbdel
fmwnts
Incurred Loss Davelopmt
Selected Five Insurers
General liability
X
in lbousands
ngo of oew1opmnt
YeJrs
1977
1978
1979
1980
1981
1982
1983
1984
1985
19W
12
24
36
48
60
72
84
‘96
108
120
64,297
58,423
60,111
65,841
71,945
74,727
77,028
83,548
12s,sO3
153,931
120,407
114,671
122,306
127,156
140,792
157,474
145,956
1112,876
254,768
168,806
155,571
172,945
184,976
193.854
197,145
2wol
254,511
197,781
180,828
199,381
213,158
218,443
226,322
245,844
200.092
189,WS
205,872
225,Wl
234,666
210,656
201,796
19i,723
208,532
2j6.296
236.%2
201,878
188,580
209,144
237,231
202.343
190,421
2W.455
202,104
191,084
201.826
Cuulatiw
Years
1971
1978
1979
1980
1981
1982
1983
1984
1985
1986
Standard
Wviatim
Coeficisnt
of
Variatioa
Lm Davelopwnt Factors
fqe of Developlant
48
60
72
12
24
36
3.1929
3.m7
3.5108
3.6425
3.3427
3.4528
S.SS41
3.9642
3.5549
3.3901
1.7050
1.6791
1.7255
1.8861
I. 7081
1.6385
1.8757
I.8110
1.7512
1.2161
1.2377
1.2203
1.2965
1.2406
1.3088
1.3264
I.30013
1.0380
1.0648
1 .os85
1.1251
1.1009
1.1401
1.1136
1.0269
1.0187
1.0251
1.0659
1.0248
1.0294
3.3901
1.7512
1.3013
I.1134
1.0294
0.2351
0.1078
0.0457
0.0360
0.0191
0.0694
0.0616
0.0351
0.0323
0.0186
Selected after
reviw
84
96
108
120
‘1.0173
l-W43
1.0120
1.0149
1.0149
1.0169
1.0210
1.0091
1.0109
1.0146
1.0112
1.0124
1.0158
1.0077
1.0172
1.0149
1.0109
1.0124
1.0077
1.0172
*
O.OlW
2
0.0175
t
0.0175
1
0.0150
*
0.0150
of previous values
239
SELECKD
UlIWiE
205,291
192,546
211,037
239,826
240,490
258,019
273,765
331) 197
446,153
521,835
Paraater
heral
Expcted Lasses at selahd
Percemtilm
IS a Percent ef tbe Expected Bean
Risk H&l
Liability
Selected Five Insurers
Percentile
_________
kcidmt
YeJr 1906
------_ --
Lll Years
Ceabined
_- - - - - - - -
50
w.934
99.981
55
100.83%
lW.2Oa
68
101.7SI
loo.421
65
102-w
1w.63t
70
103.61t
lW.87*
75
104.69
101.121
88
105.83?
1Ol.M
8!i
107.1u
101.7%
90
lW.Wa
102.lU
95
lll.JBt
102.7sI
240
Exhibit
L
Exhibit
Parameter Risk lbdel
marl1 Liability
Selected Five Insurers
he Year
111 YeJrs
____________________----------------------
Percentile
_--_------
N
Prailfl
to Surplus
--_-_--_--
Reserve
to Surplus
------____
Discounted
ReservesL
Surplus to
LMisceunted
RKarvK
____________
Prsliur
te Surplus
-___-__-_-
Reswve
to Surplus
-_--_-----
Oiscounted
Resews 6
Surplus to
Wiscantsd
Resvr ves
--------_---
75th
22.9
48.4
0.742
30.7
20.6
0.804
wtb
12.0
25.3
0.761
16.1
10.8
0.848
9w
9.3
19.8
0.772
12.6
8.4
0.874
241
Exhibit
RERERALLIR8ILIlY
PROCESSRISX
Selected Five Insurers
____________________------------------.------------------------ALL YEARS
____________________--------
WE YEAR
____________-_______--------
E(R):
VAR(I):
E(X):
vnR(x):
61.435
61,435
25.699
5.943,947,409
E(H):
VAR(I):
E(X):
VAR(X):
E(T):
(VnR(T)]“.S:
l.S78,818.065
20,143,000
E(I):
[vn!J(r)J-.s:
PREMUHS
RESERVES
a ’ ILE TO SURPLUS TO SURPLUS
____________________----------------
29,692
29,692
16,865
2,559,854.025
500,755,580
9,189,800
RESERVES
PREllIuns
TO SURPLUS 10 SURPLUS
__--____.__._--_____--------
75 th
55.2
117.0
121.1
81.3
90 th
28.9
61.2
63.4
42.6
95 th
22.6
47.0
49.5
33.2
COl!RlRITIOR OF PARMETER RISX 6 PROCESSRISX MDELS
ML YEMIS
____________________________
PREIIIUNS
RESERVES
a ’ ILE TO SURPLUS TO SURPLUS
___-________________------~---------
ME YEM
---_________-.______________
PREAIMS
RESERVES
10 SURPLUS TO sunPlus
----_______-_--_____--------
75 th
19.9
42.1
29.1
19.5
90 th
11.6
24.5
14.9
10.0
95 th
8.5
18.0
12.1
8.2
242
I
Ptruoter
Gm8rtl
&bit
Risk Ilode.1
nge of Dwalopm?le
rears
1977
1978
1979
1980
lrnl
1982
1983
1984
1965
1986
12
40,900
36,ooo
37,700
39,soD
42,9OD
41,300
46,700
47,500
70,600
97,eOD
24
36
78,sDD
74,900
74.100
WQQ
%9QQ
=,9oQ
%9QQ
107,4DO
159,9aD
115,000,
99,300
112,200
125,500
124.600
114,900
123,500
1%.500
hunts
Incurtad LossDewlopKd
LJW cDlpur
1iJility
I
131,700
119.000
lg3.600
145,000
140,204
129,!iOO
148,000
60
132,BOO
134.100
13a,OOO
148,000
152,000
145,200
72
in lhomads
‘I74
%
108
120
134,300
127,700
134,900
132,404
128.700
132,500
84
96
loa
120
134,000
134,900
129.700 * 128,000
13e,loo
136,900
151.600
156,800
150,800
Loss Oevalopwnt &tars
llgsofD@veloplant
48
60
72
Cuwlrtive
12
24
36
3.3007
3.6111
3.6074
3.9544
3.4%5
3.5109
3.4261
4.1053
3.6260
3.6278
1.7197
1.73%
I.8280
1.9525
I.7261
1.6310
1 .a412
1.81%
1.7824
1.1739
1.3092
1.2121
1.2446
1.2039
1.2620
1.2955
1.2460
I .0251
1.0924
I .OlEo
1.0772
1.0699
1.1197
1.0811
1.0166
0.9694
0.9855
l.OSs4
0.9868
0.9986
1.0075
1 A023
0.9a48
1.0103
0.9947
1.0007
1.01%
0.9934
0.9962
I.0052
1.0180
1.0082
1.01%
1.0101
1.0189
klected
3.6278
1.7824
1.2460
1.0811
0.9986
0.9947
0.9962
I.6082
1.0101
1.0189
!Standard
Owirtion
0.2736
0.1214
0.0374
0.0374
0.0340
0.0754
0.0681
0.0300
0.0346
0.0340
:
0.0340
F
0.032s
F
0.0300
F
0.0275
F
0.0250
Yurr
1917
1976
1979
1980
19nl
I982
1983
1984
1985
19%
Coaficisnt
of
Vafirtian
F
Selected after
reriw
of previous values
243
0
SELECTED
UrlmlE
135,000
130,000
13d,WO
156.200
150,000
145,000
160,ooo
195,000
285,ooo
3n,EDD
Puwtw
mm1
Ilid
L-cd
Losam at rlatod
Iermmtilos
~srhrcmntof
tbmfr#ctdnBm
Rod01
LirLility
brcmtilr
_----_-_so
LIl ymws
C&id
--_____--
hccidant
You 1966
____---- -
W.rr\
99.928
s5
1oo.M
l&3.23\
60
to1.m
100.4a
65
102.87\
100.7;u
70
103.928
101 .Ol\
7s
105.esa
10l.W
m
106.U
101.628
8s
107.77%
LOl.99t
90
109.6U
102.4n
9s
112.373
103.17t
244
Exhibit
P
bll mrr
-.-----.-*-_._.--.-*...--*--*----.-.---.--
k voar
-.--------.------“--*-----‘--..----.-~.-.-*-
Prnim
to srr*lsa
--F--r----
to 8srpln
----.---y--.
75th
21.2
41.5
28.3
mtb
11.1
21.7
14.0
9.9
ntr
8.7
17.0
I1.b
7.7
245
19.0
Paramter
&nerd
/,
Risk llodrl
Sal1
Liability
t.*@it
Mounts in 7fmsds
Incurred loss Demlapment
CopcnY
I
ngs of DawlWt
12
24
36
48
60
72
84
96
-366
418
5111
463
283
372
643
729
890
646
471
700
675
866
737
8%
711
669
964
I.922
,789
351’
720
1,028
965
802
1.361
770
1,023
b%
IA57
1,241
948
791
l,,Ul
760
1,204
1,~
8Xi
1,2b2
934
1,292
970
1,344
1,026
1.001
1.428
1,14a
72
84
%
100
120
Years
1977
1970
1979
1980
l%l
1982
1983
1984
1985
19M
b!il
6a4
548
621
740
1,216
1,405
Cuwlatiw
selected
Factors
24
36
2.0224
JAW
2.0212
s.0648
s.%w
3.5457
2.9176
4.8569
4.1715
4.5139
2.1932
2.cm
lS9S6
2.0746
2.1759
2.1240
2.i351
2.8758
2.522b
1.5344
1.6813
1.4206
1.6994
2.2166
1.9716
1.946L
1 .a195
1.3@93
1.5214
1.4542
1.3M4
1.6332
1.6446
1.3784
1.341b
1.4233
l.SOiJ
1.22&4
1.26h
1.nir
1.3Oow
1.4122
1.3776
l.I7@
1.0234
1.2371
1.1537
1.1210
1.0983
1.0649
1.w.I
1.0205
1.m
1.01%
o.l99t
4.5139
2.5226
1.8195
1.3784
1.3914
1.0234
1.0983
1.0205
1.01%
0.8998
1.0462
0.3893
0.3023
0.1144
0.1128
F
8
a
0.2318
0.154s
a.1661
0.0827
0.0011
O.@7SO 0.0700
0.06oo
standd
Deviation
Conficint
of
Vrrirtia
F
0.08oo
F
Selactd
after
raiaa
of previous values
246
F
0.0775
SELECTED
UTIIWE
I
12
Years
1977
1978
1979
19a
1981
1982
IpAl
1984
1905
1986
Loss -1-t
hoe of bmwmmt
40
&o
120
II
1N
..1,45(
1.047
1,419
1,576
1,319
1,876
3,497
3,746
2.916
Pmmtw
Expected Losses at ShCtsd PerCeittik5
as a Percant of thr Expected (ban
Risk IMel
Gsneral Lirbility
111 Years
Combined
.__... a.-
Rrmtilr
.-- . . . . . .
ficci&mt
Year 1%
. . . ..--.-
so
99.77:
99.94:
5s
102.7w
100.66t
60
105.798
101.m
65
lOSAlt
102.m
70
112.05t
102Am
75
115.53t
103.71\
RO
119.47t
104&t
85
123.87;
105.711
90
129.673
107.m
95
138.01a
109.088
247
Exhibit
S
Parader
fsnerrl
Exbibit
Risk Hod01
I
Liability
1111Years
_______..____^..__._..--..----.-.---..-.-.
!Jorcmtile
.._.______
Prmiw
to Surplus
__--___...
he Year
-____.___.._-_..__._--...---.-.-...-.-.-.-
Reserw
to Surplus
.-----------
Prmiw
to Surplus
--___-.._.
Rewrva
to Surplus
75th
5.6
14.4
9.2
6.0
90th
3.0
7.6
4.0
3.1
95th
2.3
5.9
3.8
2.4
248
Puamter
Exhibit
Risk Rode1
Selected
Six
Ret0 liability
n# of
Years
12
24
34
48
huunts
Incurred Loss Developeent
Insurers
in lhousands
oeve!opemt
60
72
84
96
1977 3%0,314 510,217
548,131 572.049 576,854
1974 399,225
535,596 6W,O33 628,090
641,340
1979 426,428 607,100 686.294
718,556
725,433
19W 401,396
6117.403 763,750
797,a.M 811,974
1981 569,897 804,DDS 901,118
925,486 943,228
1982 651,365
903.294 988,193 1,025,499 1,010.947
1983 725,010 992,073 1,102,BlO 1,170,169
1984 765,490 1,130,481 1,2%,113
1985 887,020 1,362,646
1986 1,085,914
578,156
643,142
724.766
810.207
940,671
579,966
644,749
726,454
820.137
581.691
653,443
725,457
582,536
653,675
583,503
n4
%
108
120
1977
1918
1979
1980
19Dl
19D2
1983
1984
1985
19%
%lKted
Loss Developrent factors
Age of Developeent
40
60
72
I#
120
Cuwlative
Years
12
24
36
l.54W
1 AD45
1.7094
1.7043
I.6784
1.6171
1.6S43
1.7921
1 JO39
1.6743
I.1185
1.2242
1.2007
1.1936
1.1885
1.1661
1.2090
1.2135
1.1743
1.06%
I. 0927
1.0622
1.0743
1.0615
1.0659
1 .W76
1.0585
1.02u
1.0439
1.0145
1.0284
1.0335
1.0272
1.0250
l.OlSd
1.0223
1.0049
1.0104
1.0141
1.0119
1.013s
1.0195
1.0058
1.w2a
1.0083
1.0104
1.0169
l.WM
l.OOD4
1.0074
1.0034
I.0048
1.0059
1.0030
I.0042
1.6143
1.1743
1.0585
1.0250
1.0119
l.OWJ
l.ODO4
1.004a
l.WSO
1.0042
0.0841
0.0188
0.0109
0.0107
0.0065
0.0502
0.0160
0.0103
0.0104
0.0064
F
0.0060
F
0.0055
o.wso
Stahrd
Deviatim
Ceoficieet
of
Variation
F
Selected after
reviw
U
of previous values
249
F
F
0.#50
F
o.wso
SELECTED
ULlIlMTE
585,979
655,663
72D.953
820,459
956,511
1,053,35D
I,lW.394
1,371,872
1,6W,107
1,8la,lw
Pmmter
Ri*
I\uto Liability
Expected Lossa at selected Percentiles
as a Percent of the Expected lkae
Wodel
Selected Six Insurers
Percentile
_- - - _- _-.
50
AceideA
Year 1986
-_--- --..
111 Years
Gxbined
--.. ....-
99.958
99.991
55
1#.6QI
100.11%
60
101.26t
1w.23t
65
101.91!
lW.344
70
102.61t
1#.47\
75
103.37a
100.60:
w
104.228
lW.76t
85
105.17\
1w.93:
90
106.43t
101.16t
95
108.241
101.48¶
250
Exhibit
V
Ptramtw
Exbibit
RillL lb&l
f&o Liability
kloctod
Sir Imnrers
One You
411 hrs
_____________-_---..----------------------
Percmtile
---------?%b
9Xb
Prmim
to sllrplw
_---_----37.2
15.2
V
Rowrvo
to Surplus
__-___-__52.7
21.5
Discounted
Rowrws i
surplw to
~Undisounted
&aarws
__----_---__
0.844
0.872
Prriw
-to Surplus
-----_----
Rmrw
to Surplus
-_--_---__
Oisamnted
Rewrvosk
Surplus to
Wiscawtod
Reservw
____________
39.6
26.1
0.909
20.1
13.4
0.944
16.2
10-b
0.964
251
Exhibit
Mm LWILIIY
Pwan
fflss
klocted Six Inrrers
___.____._______________I_______________-.---------------------al6 YELIII
_-___---------______________
ML YEMS
_.___i.______-______----.--243,@7
243,021
14,160
1.004.5sQ.400
E(I):
W(I):
E(X):
VIIR(X):
E(T):
[vm(r))-3:
3.441.262.&JD
22.074.402
E(T):
lvnn(r)l".s:
PMHlws
RaEnYEs
t
' ILE
TOSURPLUS 10 SURPLUS
_____________.__________________I_______--
of PMuElEil
Pt7En1uw
1.594.959,128
L2.440.543
RESERVES
10 SURPLUS
ID SWPLUS
____________-_-___-_--------
15 ttl
163.9
252.7
290.7
191.2
90 til
85.a
L2i.a
152.1
Ma.1
67.0
95.1
1111.7
9s
cwalua11w
164,IY
lb4,Isa
9,716
a49,605,904
E(I):
VM(ll):
((Xl:
van(X):
th
7a.L
RlsT I PROCESSRISK nDDELS
ML 'IEMS
_-__-____________-__-*-----mfnw
PREIIWS
t ' ILE
10 SWIPLUS 10 SURPLUS
__________.____._-_--------------------.--
alf VEM
_______-_-____-_____________
PREIIIUM
LmvfS
10 SWPLUS
TO SURPLUS
-_____*___-____.-_----------
15 tb
Jb.0
51.0
I.3
25.2
90 til
I9.S
21.4
20.3
13.3
95 tb
13.9
19,a
L5.a
10.4
252
X
Exhibit
Paraeeter Risk lbdel
Auto Liability
Law
Incurred Lass Develapeent
Colpsny
hunts
Y
in Thousands
Age of Rvelopmnt
Years
1917
1978
1979
1980
1901
1982
1903
1984
1985
1986
12
24
13t,OOO
150,300
177,800
194,200
226,600
258.800
262,300
215,lOiI
334,100
437,300
197,900
223,400
z53,sDo
289,100
333 ) 200
364,300
389,300
433,500
570.8DO
St.
221,000
252,100
295,400
328,900
376,600
401,100
443,500
521.500
48
60
231,900
267,200
311,900
344,600
385,400
414,200
465,500
234,300
273,700
314,200
351,700
391.540
421,100
72
233,500
273,500
313,400
352,400
393,MIo
84
96
233,900
273,500
314,200
354,300
233,100
214,100
312,200
233,000
274,400
234,400
120
a4
%
108
120
Cueelrtive
Years
Lass Develepeeet Factors
RgsafDevelopcaitt
48
60
72
108
12
24
36
1.7602
1.8204
1.7537
l.S#Ol
1.7255
I .6202
1.8109
1.9949
2.0937
1.8751
1.1784
1.2247
1.2310
I.2096
1.1735
1.1510
1.2201
1.2b67
I .2255
1.0552
1.0840
1.0548
I.0632
1.0382
1.0154
1.0710
I.0547
I.0056
1.0240
0.9991
1.Ol(d
1.0145
1.0123
I.0204
0.9953
0.9996
0.9924
0.9943
0.9987
0.9951
0.9907
1.0004
0.9949
0.9923
0.9934
0.9910
l.ooo4
0.9924
0.9070
0.9979
0.9982
O.Wlll
1.0009
0.9971
0.9949
Selected
1.8751
1.2255
1.0547
1.0204
0.9957
0.9334
0.9870
0.9987
0.9971
0.9949
S&lard
lkviatioe
0.1938
0.0453
0.0132
o.Ow
0.0030
Coef icieet
of
Variation
0.0370
0.0125
0.0085
0.0031
f
0.0030
t
0.0025
:
0.0025
f
0.1034
0.0020
a
0.0020
1971
1978
1979
1980
1961
1982
1983
1984
1985
1986
Selected after
reviee of previous values
253
SELECTED
L!LiIMlE
233,200
213,boQ
311,lJOO
349.100
391,000
419.3OD
415,ooo
SM,OOO
699,500
820,000
Paramtar
&to
Expected Lmes at selected Percmtiles
as a Percent of the Expected lkao
Risk lbdel
Liability
Law howr
Percentile
____- - - __
411 Years
Coebined
____- - ___
Accident
rear 1966
-___---- -
54
W.90?
W.90t
55
101.24t
100.24t
60
102.583
100.49t
65
103.93t
100.75a
70
105.37t
101.02t
75
106.92t
101.32t
80
108.68t
101.65t
85
110.654
102.03t
98
113.23t
102.523
95
116.95%
103.23t
254
Exhibit
Z
Parmeter
Exhibit
Rist lbdel
Aeto Liability
Lugs -PmY
fill Years
________________________________________--
Perceetile
__________
AA
Prwiw
to Surplus
________--
RWW
to Surplus
-------_----
One Ye4r
____________________----.-----------------
Prmim
to Surplus
__________
Ressr ve
to Surplus
75th
18.3
19.1
19.3
12.7
90th
9.b
10.3
10.1
6.1
95th
7.5
a.0
7.9
5.2
255
Paruter
Risk Rode1
Exhibit
Auto Liability
lnwrred
%a11 ceqasy
m
Years
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
Loss Develcqmnt
Mounts in lhwseeds
of Developeent
12
24
36
48
60
72
84
96
108
2.816
2,359
2,631
2,352
2.417
2,310
3,311
4,502
7,078
4,668
3.710
3.M
3,104
3,103
3,253
3,615
4,693
1,273
9,049
3,954
3,421
3,244
3,505
3,660
4,181
5,491
8,175
3.873
3,303
3,505
3,626
3.766
4.3M
5,664
3,927
3,480
3,690
3,767
3,823
4,334
4,008
3,463
3,690
3,876
3,064
4,078
3,529
3,712
3,963
4,089
3,534
3,713
4,114
3,5bQ
4,137
04
96
108
120
Loss Dewlcqwet Factors
&i of Dwelopmnt
60
12
48
120
Cumlativa
Years
12
24
36
1.4638
1.5032
1.4175
1.6943
1.6384
1.9291
1.8728
2.0548
1.7280
2.2271
1.1111
1.1483
1.1740
1.2520
1.2173
1.2647
1.31%
1.2945
1.3516
1.0425
1.0365
1.1523
1.1369
1.0820
I.0935
1.1293
1.1517
1.0643
1.0482
I.0665
1.0990
1.0515
I.0588
1.0948
1.0497
1 .DIW
1.0130
1.0579
1.0358
1.0559
1.0284
1.0240
1.0130
1.02Bl
1.021
l.OII
I.0018
l.QoM
1.0056
1.0081
1.0034
I.0067
1.0019
0.9961
0.9964
Selected
2.2271
1.3516
1.1517
1.0948
1.0559
1.024a
1.0056
1.0067
0.9961
0.9964
Standard
Deviatim
0.1645
0.0393
0.0298
0.0204
0.0192
Coef iciest
of
variation
0.073)
0.6291
0.0259
0.0186
o.ota2
:
O.OlW
I
0.0175
8
0.0175
I
0.01so
t
0.0154
1977
1978
1979
1980
1981
1962
1983
1984
1985
1986
8
Selected after
reviw
of previws
MI
values
256
SELECTED
UTIMTE
4,122
3,546
3.138
3,985
S,%O
4.572
6.201
9,415
12,231
10,396
Parwtar
Risk Rod81
Expected Losses at sslectxd Percmtila
as a Pefmt of the Expected Man
&to Liability
Accident
Year 1986
______- - _
Ml Yeats
Coabinel
50
w.93t
99.m
55
1oo.m
100.17t
bo
101.051
100.36\
65
102.01:
lOO.ss\
70
103.04\
100.768
75
104.958
100.97:
00
106.20:
101.m
05
107.61%
101.5oa
90
109.45\
lOl.wa
95
112.11t
102.38a
Percmtilr
257
Exhibit
IIC
Puwotsr
Exhibit
Risk lbdd
seal1 -WY
iluto Lilbility
Pemntilc
-_-_--_---
kD
111 Years
____________________----------------------
one YMf
___.____________________________________--
Prwiur
to Surplus
_-------__
Prariw
to Surplus
----______
neserrs
to Sarplas
-------_-_-_
Reserve
to Surplus
-_-_--_-----
75th
22.9
29.0
24.9
15.1
90th
12.0
15.2
14.1
1.9
95tb
9.4
11.8
11.0
6.2
25%
Exhibit
Parameter Risk Nodal
Homemars
S&cted
The
hunts
Incurred Lass Dwslopmnt
Capaniss
Af
in Thw5ands
&e of Development
Years
1977
1978
1979
1980
19111
1902
1983
1984
1985
1986
12
24
36
40
60
12
84
96
100
120
111,276
119,%7
lM.620
178.537
18B,Mb
232,938
214,112
214,924
205,653
183,216
129,084
141,806
186.839
220.615
222.199
266,809
2P,6OS
255,026
242,034
130,105
143,905
f87.340
220,110
222.166
268.93E
269,650
260,956
134,323
144.175
188,096
220.J41
221,490
271,507
271,615
130,480
143,641
187,314
219,809
221,839
273,661
130,669
143.7%
187,324
219,809
221,942
130,407
144,059
187,898
219,445
130,277
144,234
187,724
130,230
144,195
130,144
04
96
108
120
Couiative
Loss Oevsloplsnt Factors
l&e. of Dwalopment
12
48
60
12
24
36
1.~700
1.2021
1.1989
1.2295
1.1766
I.1600
1.2487
1.1801
1.1585
1.1152
1.0024
1.0169
1.0050
0.9950
l.DODD
1.0204
0.9954
0.9945
0.9842
1.0007
1.0021
1.0023
0.9972
1.ooo2
1.0123
0.9915
0.9719
0.9990
l.DDO2
0.9903
0.9971
l-M32
1.0027
a.9843
0.9978
1.0039
l.OD25
0.9986
1.0016
0.9948
0.9943
1.0031
I .0024
0.9986
1.0012
0.9983
1.0010
0.9994
l.ODO3
0.9993
0.99%
1.0003
0.9997
1.0001
l.ooo4
selwt@d
1.1752
0.9842
0.9119
0.9043
0.9948
1.0012
l.DDO3
1.0003
1.0001
I.9004
Standard
Dwirtiaa
0.0350
0.0110
0.0077
0.0027
0.0026
Coeficiant
of
Vu irtim
0.0304
0.0112
o.w79
0.0627
0.0026
t
0.0020
t
0.0020
Years
1977
1978
1979
1900
1981
1902
1903
1904
1985
1986
8
t
I
0.0025
0.0025
0.0020
t
Selected after
rsvisr
of previous values
259
SELECTED
ULlIMTf
130,191
144,207
107,779
219,507
222,201
272,248
267,356
253,634
238,207
215.314
Puwttr
Hmawtrs
Expctcd Losses at selected Parwntilw
1s a Psrcwk of tlw Expected lb&n
Risk lkdsl
Selected Tbrw Cwpanits
Percwtile
---------
&xi&at
You 1986
- _- - -- -- -
Ml Ywrs
Codhad
- - _- - - - - -
50
w.97a
100.00%
55
100.37%
100.04\
60
100.76\
100.09t
65
lOl.lM
100.138
10
lOl.y#
lOO.lst
15
102.04t
100.2st
04
102.56:
loo.298
0.5
103.lW
100.36\
90
103.898
loo.451
95
104.99?
100.57t
260
Exhibit
M
Pmmtcr
Exhibit
Risb tbdel
S&&d
Ho#mCdfS
Tbrw c#panies
Dns Year
All Years
____________________----------------------
Psrcmtile
__________
Prwiup
to Surplus
-_-----___
M
Reserve
to Surplus
_------_--
Discounted
Rssrmes L
Surplus to
Undiscounted
Reserves
_---_--_----
Prwiw
to Surplus
_________-
Reserve
to Surplus
_-_---___-
Discwntsd
Reserves C
Surplus to
Undiscwnted
Reserves
__-_________
75th
66.1
25.5
0.970
75.5
10.7
Wth
34.6
13.4
1.oov
39.5
9.8
1.029
95tb
27.0
10.4
1.033
30.8
7.6
1.057
261
0.W
Exhibit
HMlEWnERS
PROCESSRISX
Selected Three Insurers
__.__.._________________________________---------.-------
t s ILE
_______
DXE YERR
ALL YEARS
_______________.-----------E(R):
VRR(lt):
E(X):
VRR(X):
31,700
31.7oo
4,044
147,185,424
E(H):
VRR(ll):
E(X):
Vlvl(X):
26,604
26.604
3,088
a5.821.696
E(1):
[VRR(I)]^.S:
128,194,aW
2,276.&U
E(T):
[vAR(l)J~.S:
82,153.152
1.592.762
PREllIUNS
TO SURPLUS
RESERVES
PREMUMS
TO SURPLUS TO SURPLUS
__________________..--.-----
RESERVES
10 SURPLUS
__---_______
15 th
2168.9
84.0
310.4
77.0
90 th
1135.3
44.0
162.5
40.3
95 th
886.1
34.3
126.8
31.5
COlEIWATiOWOF PARMETERRISX 6 PROCESSRISX HODELS
RLL YEARS
-------_--_-________________
DRE YERR
___________--__-------------
PREIWHS
RESERVES
t ’ ILE
10 SURPLUS 10 SURPLUS
_.._..-_..______________________________--
PREIIIUHS
RESERVES
10 SURPLUS 10 SURPLUS
____________________________
15 th
63.1
24.4
75.1
18.6
90 th
31.6
12.2
38.3
9.5
95 th
25.2
9.1
21.1
6.9
262
RR
Exhibit
Risk Bode1
Paruster
hwnta
Incurred Loss Developmat
Large Company
8OWWWrS
AI
in Thwsaods
Age of Dwslopnt
Years
1917
1970
1979
1980
1981
1982
1983
1984
1985
1986
12
24
36
18
60
72
84
96
108
120
101,866
110,967
146,165
165,189
177,480
216,952
198,610
196,159
183,209
164.771
119.658
131,869
174,813
205,631
209,401
251,424
250.165
234,017
218,281
119,89e
133,915
175,389
205,DD2
209,224
253,540
251,728
239,820
120,@82
134,215
176,042
204,986
208,820
256,i14
253.443
120,164
133,823
175,070
204,668
209,204
258,462
120,325
133,973
175,120
204,735
209,368
120,175
134,272
175,790
204,459
119,998
134,451
175,758
119,972
134,412
119,891
84
96
IO8
120
Cvwlative
Years
Loss Dsvalopwnt Factors
nge of Oevsloplant
48
60
72
12
24
36
1.1772
1.211s
1.2027
I.2334
1.1803
1.1847
I.2579
1.1878
1.1708
I.1835
1.0021
1.019s
I.oos6
0.9944
l.DQO4
1.0222
0.9987
0.9957
0.9827
1.oool
I.0039
I.0023
0.9974
1.0013
I.0137
0.9925
0.9716
0.9986
I.0017
0.9984
0.9975
1.0032
1.0035
0.9857
0.9919
1.0046
1.0041
0.9991
1.0014
0.9944
0.9966
1.003s
I.0038
0.9987
l.DW6
0.9978
1.0012
l.oooQ
1.0001
0.9993
0.9999
1.0003
0.9995
1.0002
l.OCa2
selected
1.1835
0.9827
0.9716
0.9857
0.9944
l.oOQ6
1.0001
1.0003
1.0002
1.0002
Standard
Dwirtia
O.OSSO
0.0114
0.0079
0.0027
0.0030
0.02%
0.0116
0.0081
0.0028
0.0030
L
0.0030
t
0.0028
I
0.0026
t
0.0025
t
0.0025
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
Coeficient
of
Variation
Selected after
feviee of previoos
values
263
SELECTED
ULTIIMTE
119,915
134,439
17s.m
204,479
209,466
257,013
249,830
233,DW
214,500
195,000
Expected Losses at selected Percentiles
as a Perceet of the Expected lkaa
Law
Cowy
Percentile
Accidmt
Year 19M
_-___--- -
All Years
C&ined
---- -----
so
99.97%
100.00%
55
loo.368
100.04t
60
100.74%
100.08t
65
101.128
100.13\
10
10l.W
1oo.lst
7s
101.98t
100.23:
80
102.4%
100.288
85
103.05?
LOO.351
90
103.793
100.43%
95
104&t
100.55#
264
Exhibit
AJ
Pmmtw
Exhibit
Risk kdfil
M
Rll Years
__________------____----------------------
be hat
____________________----------------------
Premium
to Surplus
----_--__-
Reserve
to Surplus
-_--_-______
Prsriw
to Surplus
_-_-------
75th
66.6
25.1
77.6
19.4
90th
34.8
13.5
40.6
10.1
95th
21.2
10.5
31.7
7.9
Percentile
-_________
265
Rmuvr
to Surpiw
Paramtw
Exhibit
Risk tbdd
InairrsdLuss
II-ars
hunts
Dw8lapmnt
#
in lhwsan&
nge of Dwe1opment
Years
1977
1978
1979
1900
1981
1982
1983
1984
1985
1986
Years
12
24
36
18
60
12
84
96
108
120
520
w
559
945
680
1,W
1,378
1,023
2.081
196
630
649
691
1,102
782
1,471
1.479
1,453
2,296
610
645
688
1,133
822
1,m
1.407
1,595
625
679
697
1,135
816
1,525
1,489
630
680
700
1,137
824
1,547
647
680
119
1,139
0%
647
6ao
694
1,149
675
6ao
694
685
680
68s
a4
96
108
120
Cumulative Loss Dwehpmnt Factors
nge of Devalopwnt
36
40
60
12
12
24
1.3154
1.2409
1.23Oa
1.2307
1.2500
1.1323
1.1074
I.6461
1.2148
1.2205
1.0857
1.0478
0.9957
1.0554
1.0870
1.0530
I.0318
1.1590
1.1010
1.1213
I.0543
1.0000
1.0265
1.0341
1.0403
1.0262
1.0558
1.0944
1.0015
0.9871
1.0247
1.0417
1.0157
1.0248
1.0857
l.OOOa
0.9029
1.0229
1.0316
1.0013
1.0572
l.woO
0.9569
1.0211
0.9930
1.0572
l.OOon
0.9914
1.0122
1.0133
l.OOOo
0.9914
0.9915
l.OOOa
0.9985
s&fad
1.2205
1.1010
I.0558
I.0248
1.0013
0.9930
1.0122
0.9914
l.oow
0.9985
Standard
Dwiatim
0.2181
0.04W
0.0154
0.0210
0.0392
0.1787
0.0452
0.0145
0.0205
0.0391
1977
1978
1979
MO
1981
1902
1903
1984
19as
1986
Cwf iciat
of
Variatim
t
t
1
t
t
0.0350
0.0300
0.0275
0.0250
0.0250
t
Selected after
revia
of previous values
266
SELECTED
ULTIMTE
684
600
68n
1,163
as0
1.549
I.526
1,w
2,528
2,314
Paramtsr
tImemars
Risk tbdol
Expected Losses at sdectad Pwcmtilas
as a Perceat of the Expected lban
-11
CapuY
Rll Years
Coebined
_________
kci&at
Yew 1986
50
W.%n
w.97a
55
102.1u
100.3%
60
104.47t
lOO.SOI
65
106.79t
101.m
70
109.2%
101.67t
75
111.97:
102.1sa
80
115.01;
102.7Ot
as
11a.41t
103.31:
90
122.88%
104.11%
95
129.31:
105.26\
261
Erdtibit RH
Parader
Ri*
Exhibit
Itodd
#
Howomtrs
Parertile
--------_-
Lll VMrs
____________________----------------------
One Year
______________---__------------.----------
Praiua
to Surplus
--_-------
Prmism
to Rurplus
_______---
RISWV8
to Srrplrs
_----__---_-
Reserve
to Surplus
-___________
75th
12.1
0.7
12.8
3.9
90th
6.3
4.6
6.7
2.0
95th
4.9
3.6
5.2
1.4
268
Parmeter
Risk lbdel
nli Lines Co&i&
Expected losses at selected Perceotiles
as a Percent of the Expected Mae
Law Cosw~
Perceotile
______- __
50
Accidaot
Year 1986
____- - - - _
nil Years
Coebined
____- ____
w.wt
99.96t
5s
lW.42t
100.08t
60
100.88t
100.17:
65
101.33#
lW.26t
70
101.83t
IW.Yt
7s
102.35%
1#.47\
80
102.95\
1#.58%
85
103.62t
1w.72t
90
104.sot
1#.89?
9s
105.76%
101.14t
269
Exhibit
110
Parader
Exhibit
Risk llodsl
Lll lines Cabined
Law
ColprnY
One Year
--__----------_---------------------------
All Years
____________________----------------------
Percentile
Prwiw
to Surplus
------__--
1Ip
Reserve
to Surplus
______---___
Prmium
to Surplus
---_______
Reserve
to Surplus
--__________
75th
50.5
6a.3
57.6
33.3
90th
26.4
35.8
30.2
17.4
95th
20.6
27.9
23.5
13.6
270
Paramtsr
All Lines
Expected Losses at selectrd Pwcmtila
as a Percent of the Expected lb?8
Risk lbdel
sull
Cmplny
kcidwit
Year 1986
All Yurs
Ccrbinsd
__- _- _- - -
50
99.918
W.pBt
55
101.12?
loo. 19t
60
102.33t
100.39\
65
103.558
100.60:
70
lO4.E5\
100.828
75
106.25t
101.OSI
a0
107.84t
101.32?
85
109.61\
101.61t
90
Ill.948
102.01\
95
115.30:
102.57;
Petmtile
271
Exhibit
1IQ
Parader
Exhibit
Risk Hod81
Ml lines
seal1 Caprnr
One Year
-----_-_-----___-__-----------------------
All Years
____________________-~--------------------
Percentile
__________
hR
Preaium
to Surplus
__________
Reserve
to Surplus
____________
Prmium
to Surplus
__________
Reserve
to Surplus
------___-_-
15th
18.4
29.2
21 .a
12.5
90th
9.6
15.3
11.4
6.5
95th
7.5
11.9
a.9
5.1
272
Appendix
Sheet
1
Support of Horeal Oistrlbution
of Curuiatlve
Ruto Liability
Selected Six Insurers
Years
i;
Loss Oevelopeent Factors
24
36
48
60
2
a4
?b
108
120
I, 0690
! 0927
I 06::
i.1;43
:.otl15
: .:h,F
1.0244
i -0439
I.0145
:.2x
1.0335
! .0272
I.Oi58
i .xX::
1.0049
; .3104
I.0141
!.0119
1.0135
LO195
I.0058
I .00?8
I.0083
i 0134
1.31b9
1.0034
I .0004
I .0074
L.3034
1.0048
1.0059
i.5030
1.0042
i977
19:ti
1919
i980
1981
i 982
1983
1984
1985
i9flb
I. 5408
1.6845
I. 1094
I. 1043
I.6784
;.b::!
1.6543
I.7921
i 8039
1.6743
1.1485
:.::42
i.2007
; iF30
: it!85
..;30!
I .2090
1.2135
1.174:
:
i.bRS9
;..w
,.3;15
,.0281
1.3132
I.0100
I JOiA
I.0052
1.0045
1 .a042
Selectee.
1 s74:
I l!d.J
l.OS85
1.0250
LO119
i.0083
I.0004
I .0048
LOO.30
1.0042
Variance:
O.GOIi
0 0004
0.0001
0.0001
‘j.00004
0.00004
0.00003
?.00003
0.00003
Statistic.
C.436S
2 b5j8
3.:6X
0 7165
c.504,
c.mii
i.bjbt
C. 1379
3.4089
Accept
Accept
acceDt
f
%iXpt
@ aloha : .s;:.
ficcept
1.067b I .ozso
:.35ti5
Reject
%eJect !f Sta;.;:kc
Accept
abovetbeiow r!- 2.675
273
Scctp:
nccepr
0.00003
O.OGOO
AKEpt
Appendix
Sheet 2
Support
A.
E,
of
H,
Stdndar’J
K.
Specific
Devlatlon
Calculation
as
shown
on
Exhlbrts
etc.
.:..
Examples
taken
Sample
from
Standard
each
age
Of
Exhibit
Workers’
A:
Devlatlons
tram
development
~r.:1nin
most
Compensation
recent
five
points
wlthln
CalCiliated
were
; j’J’iV
Example:
L.5R24
1.357’,
i
t .4699
1.3911
;
Sample
Medn
:
Sample
:x
Variance:
: 1
-
.,,,,,“.,,,,,,,,,,.,“,,
SamDle-Mean
:
n
1
Sanipla
Standard
Oevlatlon
:
Yarldnce
jSample
lc
convert
15
mu!t:olled
divided
the
Sample
by
by
the
the
lnumber
Pouplatlon
Vdrlance
number
ot
to
of
points
Variance:
a
Population
points
In
in
the
Variance.
the
sample
sample
(Sample
less
Standard
Devldtion:
the
Sample
five)
and
1.
-Variance
n-
Population
(here
)
n
1
-’
. _.
,Powulation_Var1ance
274
Variance
then