**Insurance Abstract**
A computerized system and method for estimating insurance loss reserves
and confidence intervals using insurance policy and claim level
detail predictive modeling. Predictive models are applied to historical
loss, premium and other insurer data, as well as external data,
at the level of policy detail to predict ultimate losses and allocated
loss adjustment expenses for a group of policies. From the aggregate
of such ultimate losses, paid losses to date are subtracted to derive
an estimate of loss reserves. Dynamic changes in a group of policies
can be detected enabling evaluation of their impact on loss reserves.
In addition, confidence intervals around the estimates can be estimated
by sampling the policy-by-policy estimates of ultimate losses.
**Insurance Claims**
1. A computerized method for predicting ultimate losses of an insurance
policy, comprising the steps of storing policyholder and claim level
data including insurer premium and insurer loss data in a data base,
identifying at least one external data source of external variables
predictive of ultimate losses of said insurance policy, identifying
at least one internal data source of internal variables predictive
of ultimate losses of said insurance policy, associating said external
and internal variables with said policyholder and claim level data,
evaluating said associated external and internal variables against
said policyholder and claim level data to identify individual ones
of said external and internal variables predictive of ultimate losses
of said insurance policy, and creating a predictive statistical
model based on said individual ones of said external and internal
variables.
2. The method of claim 1, further comprising the steps of creating
individual records in said data base for individual policyholders
and populating each of said records with premium and loss data,
policyholder demographic information, policyholder metrics, claim
metrics and claim demographic information.
3. The method of claim 2, wherein said step of associating said
external and internal variables with said policyholder and claim
level data includes associating at least one of said external and
said internal variables with said individual records based on a
unique key.
4. The method of claim 1, further comprising the step of normalizing
said policyholder and claim level data.
5. The method of claim 4, wherein said step of normalizing said
policyholder and claim level data is effected using actuarial transformations.
6. The method of claim 5, wherein said actuarial transformations
include at least one of premium on-leveling, loss trending, and
capping.
7. The method of claim 5, further comprising the steps of calculating
a loss ratio by age of development based on said normalized policyholder
and claim level data.
8. The method of claim 7, further comprising the steps of calculating
frequency and severity measurements of ultimate losses.
9. The method of claim 7, further comprising the steps of defining
a subgroup from said policyholder and claim level data and calculating
a cumulative loss ratio by age of development for said subgroup.
10. The method of claim 9, further comprising the step of effecting
a statistical analysis to identify statistical relationships between
said loss ratio by age of development and said external and internal
variables.
11. The method of claim 10, wherein said step of effecting a statistical
analysis includes using multiple regression models.
12. The method of claim 1, wherein said at least one external data
source includes external variables for business-level data and household-level
data.
13. The method of claim 1, wherein said step of evaluating said
associated external and internal variables against said policyholder
and claim level data is effected using a binning statistical technique.
14. The method of claim 1, wherein said step of evaluating said
associated external and internal variables against said policyholder
and claim level data further includes the step of examining said
external and internal variables for cross-correlation against one
another and removing at least a portion of repetitive external and
internal variables.
15. The method of claim 1, further comprising the step of dividing
said data in said database into a training data set, a testing data
set, and a validation data set.
16. The method of claim 15, further comprising the step of using
said training data set and said test data set to iteratively generate
an initial statistical model.
17. The method of claim 16, wherein said step of using said training
data set and said test data set to generate an initial statistical
model includes effecting at least one of multiple regression, linear
modeling, backwards propagation of errors, and multivariate adaptive
regression techniques.
18. The method of claim 17, wherein said step of using said testing
data set includes iteratively refining said initial statistical
model against overfitting.
19. The method of claim 18, further comprising the step of using
said validation data set to evaluate the predictiveness of said
initial statistical model.
20. The method of claim 19, further comprising the step of calculating
an estimated loss ratio using said initial statistical model to
yield said predictive statistical model.
21. The method of claim 20, further comprising the step of applying
said predictive statistical model to said data in said data base
to yield an estimate of ultimate losses.
22. The method of claim 21, further comprising the steps of aggregating
estimated ultimate losses and calculating loss reserves.
23. The method of claim 22, further comprising the step of estimating
confidence intervals on said estimated ultimate losses and said
loss reserves using a bootstrapping simulation technique.
24. A system for predicting ultimate losses of an insurance policy,
comprising a data base for storing policyholder and claim level
data including insurer premium and insurer loss data, means for
processing data from at least one external data source of external
variables predictive of ultimate losses of said insurance policy
and at least one internal data source of internal variables predictive
of ultimate losses of said insurance policy, means for associating
said external and internal variables with said policyholder and
claim level data, means for evaluating said associated external
and internal variables against said policyholder and claim level
data to identify individual ones of said external and internal variables
predictive of ultimate losses of said insurance policy, and means
for generating a predictive statistical model based on said individual
ones of said external and internal variables.
25. The system of claim 24, further comprising means for creating
individual records in said data base for individual policyholders
and means for populating each of said records with premium and loss
data, policyholder demographic information, policyholder metrics,
claim metrics and claim demographic information.
26. The system of claim 25, wherein said means for associating
said external and internal variables with said policyholder and
claim level data includes means for associating at least one of
said external and internal variables with said individual records
based on a unique key.
27. The system of claim 24, further comprising means for normalizing
said policyholder and claim level data.
28. The system of claim 27, wherein said means for normalizing
said policyholder and claim level data includes means for effecting
actuarial transformations.
29. The system of claim 28, wherein said actuarial transformations
include at least one of premium on-leveling, loss trending, and
capping.
30. The system of claim 28, further comprising means for calculating
a loss ratio by age of development based on said normalized policyholder
and claim level data.
31. The system of claim 30, further comprising means for calculating
frequency and severity measurements of ultimate losses.
32. The system of claim 30, further comprising means for defining
a subgroup from said policyholder and claim level data and means
for calculating a cumulative loss ratio by age of development for
said subgroup.
33. The system of claim 32, further comprising means for effecting
a statistical analysis to identify statistical relationships between
said loss ratio by age of development and said external and internal
variables.
34. The system of claim 33, wherein said means for effecting a
statistical analysis includes means for utilizing multiple regression
models.
35. The system of claim 24, wherein said at least one external
data source includes external variables for business-level data
and household-level data.
36. The system of claim 24, wherein said means for evaluating said
associated external and internal variables against said policyholder
and claim level data includes means for effecting a binning statistical
technique.
37. The system of claim 24, further comprising means for dividing
said data in said database into a training data set, a testing data
set, and a validation data set.
38. The system of claim 37, further comprising means for iteratively
generating an initial statistical model using said training data
set and said testing data set.
39. The system of claim 38, wherein said means for iteratively
generating an initial statistical model using said training data
set and said testing data set includes means for effecting at least
one of multiple regression, linear modeling, backwards propagation
of errors, and multivariate adaptive regression techniques.
40. The system of claim 39, wherein said means for iteratively
generating an initial statistical model using said training data
set and said testing data set includes means for iteratively refining
said initial statistical model against overfitting using said testing
data set.
41. The system of claim 40, further comprising means for evaluating
the predictiveness of said initial statistical model using said
validation data set.
42. The system of claim 41, further comprising means for calculating
an estimated loss ratio using said initial statistical model to
yield said predictive statistical model.
43. The system of claim 42, further comprising means for applying
said predictive statistical model to said data in said data base
to yield an estimate of ultimate losses.
44. The system of claim 43, further comprising means for aggregating
estimated ultimate losses and calculating loss reserves.
45. The system of claim 44, further comprising means for estimating
confidence intervals on said estimated ultimate losses and said
loss reserves including means for effecting a bootstrapping simulation
technique.
**Insurance Description**
CROSS REFERENCE TO RELATED APPLICATIONS
[0001] This application claims the benefit of U.S. Provisional
Patent Application No. 60/609,141 filed on Sep. 10, 2004, the disclosures
of which is incorporated herein by reference in its entirety.
BACKGROUND OF THE INVENTION
[0002] The present invention is directed to a quantitative system
and method that employ public external data sources ("external
data") and a company's internal loss data ("internal data")
and policy information at the policyholder and coverage level of
detail to more accurately and consistently predict the ultimate
loss and allocated loss adjustment expense ("ALAE") for
an accounting date ("ultimate losses"). The present invention
is applicable to insurance companies, reinsurance companies, captives,
pools and self-insured entities.
[0003] Estimating ultimate losses is a fundamental task for any
insurance provider. For example, general liability coverage provides
coverage for losses such as slip and fall claims. While a slip and
fall claim may be properly and timely brought during the policy's
period of coverage, actual claim payouts may be deferred over several
years, as is the case where the liability for a slip and fall claim
must first be adjudicated in a court of law. Actuarially estimating
ultimate losses for the aggregate of such claim events is an insurance
industry concern and is an important focus of the system and method
of the present invention. Accurately relating the actuarial ultimate
payout to the policy period's premium is fundamental to the assessment
of individual policyholder profitability.
[0004] As discussed in greater detail hereinafter, "internal
data" include policy metrics, operational metrics, financial
metrics, product characteristics, sales and production metrics,
qualitative business metrics attributable to various direct and
peripheral business management functions, and claim metrics. The
"accounting date" is the date that defines the group of
claims in terms of the time period in which the claims are incurred.
The accounting date may be any date selected for a financial reporting
purpose. The components of the financial reporting period as of
an accounting date referenced herein are generally "accident
periods" (the period in which the incident triggering the claim
occurred), the "report period" (the period in which the
claim is reported), or the "policy period" (the period
in which the insurance policy is written); defined herein as "loss
period".
[0005] Property/casualty insurance companies ("insurers")
have used many different methods to estimate loss and ALAE reserves.
These methods are grounded in years of traditional and generally
accepted actuarial and financial accounting standards and practice,
and typically involve variations of three basic methods. The three
basic methods and variations thereof described herein in the context
of a "paid loss" method example involve the use of losses,
premiums and the product of claim counts and average amount per
claim.
[0006] The first basic method is a loss development method. Claims
which occur in a given financial reporting period component, such
as an accident year, can take many years to be settled. The valuation
date is the date through which transactions are included in the
data base used in the evaluation of the loss reserve. The valuation
date may coincide with the accounting date or may be prior to the
accounting date. For a defined group of claims as of a given accounting
date, reevaluation of the same liability may be made as of successive
valuation dates.
[0007] "Development" is defined as the change between
valuation dates in the observed values of certain fundamental quantities
that may be used in the loss reserve estimation process. For example,
the observed dollars of losses paid associated with a claim occurring
within a particular accident period often will be seen to increase
from one valuation date to the next until all claims have been settled.
The pattern of accumulating dollars represents the development of
"paid losses" from which "loss development factors"
are calculated. A "loss development factor" is the ratio
of a loss evaluated as of a given age to its valuation as of a prior
age. When such factors are multiplied successively from age to age,
the "cumulative" loss development factor is the factor
which projects a loss to the oldest age of development from which
the multiplicative cumulation was initiated.
[0008] For the loss development method, the patterns of emergence
of losses over successive valuation dates are extrapolated to project
ultimate losses. If one-third of the losses are estimated to be
paid as of the second valuation date, then a loss development factor
of three is multiplied by the losses paid to date to estimate ultimate
losses. The key assumptions of such a method include, but may not
be limited to: (i) that the paid loss development patterns are reasonably
stable and have not been changed due to operational metrics such
as speed of settlement, (ii) that the policy metrics such as retained
policy limits of the insurer are relatively stable, (iii) that there
are no major changes in the mix of business such as from product
or qualitative characteristics which would change the historical
pattern, (iv) that production metrics such as growth/decline in
the book of business are relatively stable, and (v) that the legal/judicial/social
environment is relatively stable.
[0009] The second basic method is the claim count times average
claim severity method. This method is conceptually similar to the
loss development method, except that separate development patterns
are estimated for claim counts and average claim severity. The product
of the estimated ultimate claim count and the estimated ultimate
average claim severity is estimated ultimate losses. The key assumptions
of such a method are similar to those stated above, noting, for
example, that operational metrics such as the definition of a claim
count and how quickly a claim is entered into the system can change
and affect patterns. Therefore, the method is based on the assumption
that these metrics are relatively stable.
[0010] The third basic method is the loss ratio method. To estimate
ultimate losses the premium corresponding to the policies written
in the period corresponding to the component of the financial reporting
period is multiplied by an "expected loss ratio" (which
is a loss ratio based on the insurer's pricing methods and which
represents the loss ratio that an insurer expects to achieve over
a group of policies). For example, if the premium corresponding
to policies written from 1/1/.times..times. to 12/31/.times..times.
is $100 and the expected loss ratio is 70%, then estimated ultimate
losses for such policies is $70. The key assumption in this method
is that the expected loss ratio can reasonably be estimated, such
as through pricing studies of how losses appear to be developing
over time for a similar group of policies.
[0011] There are also variations of the foregoing basic methods
for estimating losses such as, for example, using incurred losses
versus paid losses to estimate loss development or combining methods
such as the loss development method and the loss ratio method. The
methods used to estimate ALAE are similar to those used to estimate
losses alone and may include the combination of loss and ALAE, or
ratios of ALAE to loss.
[0012] The conventional loss and ALAE reserving practices described
above evolved from an historical era of pencil-and-paper statistics
when statistical methodology and available computer technology were
insufficient to design and implement scalable predictive modeling
solutions. These traditional and generally accepted methods have
not considerably changed or evolved over the years and are, today,
very similar to historically documented and practiced methods. As
a result, the current paid or incurred loss development and claim
count-based reserving practices take as a starting-point a loss
or claim count reserving triangle: an array of summarized loss or
claim count information that an actuary or other loss reserving
expert attempts to project into the future.
[0013] A common example of a loss reserving triangle is a "ten-by-ten"
array of 55 paid loss statistics. TABLE-US-00001 TABLE A Age Year
0 1 2 3 4 5 6 7 8 9 0 C.sub.0, 0 C.sub.0, 1 C.sub.0, 2 C.sub.0,
3 C.sub.0, 4 C.sub.0, 5 C.sub.0, 6 C.sub.0, 7 C.sub.0, 8 C.sub.0,
9 1 C.sub.1, 0 C.sub.1, 1 C.sub.1, 2 C.sub.1, 3 C.sub.1, 4 C.sub.1,
5 C.sub.1, 6 C.sub.1, 7 C.sub.1, 8 -- 2 C.sub.2, 0 C.sub.2, 1 C.sub.2,
2 C.sub.2, 3 C.sub.2, 4 C.sub.2, 5 C.sub.2, 6 C.sub.2, 7 -- -- 3
C.sub.3, 0 C.sub.3, 1 C.sub.3, 2 C.sub.3, 3 C.sub.3, 4 C.sub.3,
5 C.sub.3, 6 -- -- -- 4 C.sub.4, 0 C.sub.4, 1 C.sub.4, 2 C.sub.4,
3 C.sub.4, 4 C.sub.4, 5 -- -- -- -- 5 C.sub.5, 0 C.sub.5, 1 C.sub.5,
2 C.sub.5, 3 C.sub.5, 4 -- -- -- -- -- 6 C.sub.6, 0 C.sub.6, 1 C.sub.6,
2 C.sub.6, 3 -- -- -- -- -- -- 7 C.sub.7, 0 C.sub.7, 1 C.sub.7,
2 -- -- -- -- -- -- -- 8 C.sub.8, 0 C.sub.8, 1 -- -- -- -- -- --
-- -- 9 C.sub.9, 0 -- -- -- -- -- -- -- -- --
[0014] The "Year" rows indicates the year in which a
loss for which the insurance company is liable was incurred. The
"Age" columns indicates how many years after the incurred
date an amount is paid by the insurance company. C.sub.i,j is the
total dollars paid in calendar year (i+j) for losses incurred in
accident year i.
[0015] Typically, loss reserving exercises are performed separately
by line of business (e.g., homeowners' insurance vs. auto insurance)
and coverage (e.g., bodily injury vs. collision). Therefore, loss
reserving triangles such as the one illustrated in Table A herein
typically contain losses for a single coverage.
[0016] The relationship between accident year, development age
and calendar year bears explanation. The "accident year"
of a claim is the year in which the claim occurred. The "development
age" is the lag between the accident's occurrence and payment
for the claim. The calendar year of the payment therefore equals
the accident year plus the development age.
[0017] Suppose, for example, that "Year 0" in Table A
is 1994. A claim that occurred in 1996 would therefore have accident
year i=2. Suppose that the insurance company makes a payment of
$1,000 for this claim j=3 years after the claim occurred. This payment
therefore takes place in calendar year (i+j)=5, or in 1999. In summary,
accident year plus development age (i+j) equals the calendar year
of payment. It should be noted that this implies that the payments
on each diagonal of the claim array fall in the same calendar year.
In the above example, the payments C.sub.9,0, C.sub.8,1, . . . ,
C.sub.0,9, all take place in calendar year 2003.
[0018] The payments along each row, on the other hand, represent
dollars paid over time for all of the claims that occurred in a
certain accident year. Continuing with the above example, the total
dollars of loss paid by the insurance company for accident Year
1994 is: L 0 = j = 0 9 .times. C 0 , j
[0019] It should be noted that this assumes that all of the money
for accident Year 1994 claims is paid out by the end of calendar
year 2003. An actuary with perfect foresight at December 1994 would
have therefore advised that $R be set aside in reserves where: R
= j = 1 9 .times. C 0 , j
[0020] Similarly, given the earned premium associated with each
policy by year, such premium can be aggregated to calculate a loss
ratio which has emerged as of a given year. This "emerged loss
ratio" (emerged losses divided by earned premium) can be calculated
on either a paid loss or incurred loss basis, in combination with
ALAE or separately.
[0021] The goal of a traditional loss reserving exercise is to
use the patterns of paid amounts ("loss development patterns")
to estimate unknown future loss payments (denoted by dashes in Table
A). That is, with reference to Table A, the aim is to estimate the
sum of the unknown quantities denoted by dashes based on the "triangle"
of 55 numbers. This sum may be referred to as a "point estimate"
of the insurance company's outstanding losses as of a certain date.
[0022] A further goal, one that has been pursued more actively
in the actuarial and regulatory communities in recent years, is
to estimate a "confidence interval" around the point estimate
of outstanding reserves. A "confidence interval" is a
range of values around a point estimate that indicates the degree
of certainty in the associated point estimate. A small confidence
interval around the point estimate indicates a high degree of certainty
for the point estimate; a large confidence interval indicates a
low amount of certainty.
[0023] A loss triangle containing very stable, smooth payment patterns
from Years 0-8 should result in a loss reserve estimate with a relatively
small confidence interval; however a loss triangle with changing
payment patterns and/or excessive variability in loss payments from
one period or year to the next should result in a larger confidence
interval. An analogy may help explain this. If the height of a 13
year-old's five older brothers all increased 12% between their 13.sup.th
and 14.sup.th birthdays, there is a high degree of confidence that
the 13 year-old in question will grow 12% in the coming year. Suppose,
on the other hand, that the 13 year-old's older brothers grew 5%,
6%, 12%, 17% and 20%, respectively, between their 13.sup.th and
14.sup.th birthdays. In this case, the estimate would still be that
the 13 year-old will grow 12% (the average of these five percentage
increases) in the coming year. In both scenarios, the point estimate
is 12%. However, in the second scenario, in which the historical
data underlying the point estimate are highly variable, the confidence
interval around this point estimate will be larger. In short, high
variability in historical data translates into lower confidence
on predictions based on that data.
[0024] There are several limitations with respect to commonly used
loss estimation methods. First, as noted above, is the basic assumption
in a loss based method that previous loss development patterns are
indicative of future emergence patterns (stability). Many factors
can affect emergence patterns such as, for example:
[0025] (i) changes in policy limits written, distribution by classification,
or the specific jurisdiction or environment (policy metrics), (ii)
changes in claim reporting or settlement patterns (operational metrics),
(iii) changes in policy processing (financial metrics), (iv) changes
in the mix of business by type of policy (product characteristics),
(v) changes in the rate of growth or decline in the book of business
(production metrics), (vi) claim metrics, and (vii) changes in the
underwriting criteria to write a type of policy (qualitative metrics).
[0026] The difficulties surrounding the above limitations are compounded
when aggregate level loss and premium data are used in the common
methodologies. For example, it is generally recognized in actuarial
science that increasing the limits on a group of policies will lengthen
the time to settle losses on such policies, which, in turn, increases
loss development. Similarly, writing business which increases claim
severity, such as, for example, business in higher rated classifications
or in certain tort environments, may also lengthen settlement time
and increase loss development. Changes in operational metrics such
as case reserve adequacy or speed of settlement also affect loss
development patterns.
[0027] Second, with respect to aggregate level premiums and losses,
the impact of financial metrics such as the rate level changes on
loss ratio (the ratio of losses to premium for a component of the
financial reporting period) can be difficult to estimate. This is,
in part, due to assumptions which might be made at the accounting
date on the proportion and quality of new business and renewal business
policies written at the new rate level.
[0028] Subtle shifts in other metrics, such as policy metrics,
operational metrics, product characteristics, production metrics,
claim metrics or qualitative metrics of business written could have
a potentially significant and disproportionate impact on the ultimate
loss ratio underlying such business. For example, qualitative metrics
are measured rather subjectively by a schedule of credits or debits
assigned by the underwriter to individual policies. An example of
a qualitative metric might be how conservative and careful the policyholder
is in conducting his or her affairs. That is, all other things being
equal, a lower loss ratio may result from a conservative and careful
policyholder than from one who is less conservative and less careful.
Also underlying these credits or debits are such non-risk based
market forces as business pressures for product and portfolio shrinkage/growth,
market pricing cycles and agent and broker pricing negotiations.
Another example might be the desire to provide insurance coverage
to a customer who is a valued client of a particular insurance agent
who has directed favorable business to the insurer over time, or
is an agent with whom an insurer is trying to develop a more extensive
relationship.
[0029] One approach to estimating the impact of changes in financial
metrics is to estimate such impacts on an aggregate level. For example,
one could estimate the impact of a rate level change based on the
timing of the change, the amount of the change by various classifications,
policy limits and other policy metrics. Based on such impacts, one
could estimate the impact on the loss ratio for policies in force
during the financial reporting period.
[0030] Similarly, the changes in qualitative metrics could also
be estimated at an aggregate level. However, none of the commonly
used methods incorporates detailed policy level information in the
estimate of ultimate losses or loss ratio. Furthermore, none of
the commonly used methods incorporates external data at the policy
level of detail.
[0031] A third limitation is over-parameterization. Intuitively,
over-parameterization means fitting a model with more structure
than can be reasonably estimated from the data at hand. By way of
producing a point estimate of loss reserves, most common reserving
methods require that between 10 and 20 statistical parameters are
estimated. As noted above, the loss reserving triangle provides
only 55 numbers, or data points, with which to estimate these 10-20
parameters. Such data-sparse, highly parameterized problems often
lead to unreliable and unstable results with correspondingly low
levels of confidence for the derived results (and, hence, a correspondingly
large confidence interval).
[0032] A fourth limitation is model risk. Related to the above
point, the framework described above gives the reserving actuary
only a limited ability to empirically test how appropriate a reserving
model is for the data. If a model is, in fact, over-parameterized,
it might fit the 55 available data points quite well, but still
make poor predictions of future loss payments (i.e., the 45 missing
data points) because the model is, in part, fitting random "noise"
rather than true signals inherent in the data.
[0033] Finally, commonly used methods are limited by a lack of
"predictive variables." "Predictive variables"
are known quantities that can be used to estimate the values of
unknown quantities of interest. The financial period components
such as accident year and development age are the only predictive
variables presented with a summarized loss array. When losses, claim
counts, or severity are summarized to the triangle level, except
for premiums and exposure data, there are no other predictive variables.
[0034] Generally speaking, insurers have not effectively used external
policy-level data sources to estimate how the expected loss ratio
varies from policy to policy. As indicated above, the expected loss
ratio is a loss ratio based on the insurer's pricing methods and
represents the loss ratio which an insurer expects to achieve over
a group of policies. The expected loss ratio of a group of policies
underlies that group's aggregate premiums, but the actual loss ratio
would naturally vary from policy to policy. That is, many policies
would have no losses, and relatively few would have losses. The
propensity for a loss at the individual policy level and, therefore,
the policy's expected loss ratio, is dependent on the qualitative
characteristics of the policy, the policy metrics and the fortuitous
nature of losses. Actuarial pricing methods often use predictive
variables derived from various internal company and external data
sources to compute expected loss and loss ratio at the individual
policy level. However, analogous techniques have not been widely
adopted in the loss reserving arena.
[0035] Accordingly, a need exists for a system and method that
perform an estimated ultimate loss and loss ratio analysis at the
level of the individual policy and claim level, and aggregate such
detail to estimate ultimate losses, loss ratio and reserves for
the financial reporting period as of an accounting date. An additional
need exists for such a system and method that quantitatively include
policyholder characteristics and other non-exposure based characteristics,
including external data sources, to generate a generic statistical
model that is predictive of future loss emergence of policyholders'
losses, considering a particular insurance company's internal data,
business practices and particular pricing methodology. A still further
need exists for a scientific and statistical procedure to estimate
confidence intervals from such data to better judge the reasonableness
of a range of reserves developed by a loss reserving specialist.
[0036] In view of the foregoing, the present invention provides
a new quantitative system and method that employ traditional data
sources such as losses paid and incurred to date, premiums, claim
counts and exposures, and other characteristics which are non-traditional
to an insurance entity such as policy metrics, operational metrics,
financial metrics, product metrics, production metrics, qualitative
metrics and claim metrics, supplemented by data sources external
to an insurance company to more accurately and consistently estimate
the ultimate losses and loss reserves of a group of policyholders
for a financial reporting period as of an accounting date.
SUMMARY OF THE INVENTION
[0037] Generally speaking, the present invention is directed to
a quantitative method and system for aggregating data from a number
of external and internal data sources to derive a model or algorithm
that can be used to accurately and consistently estimate the loss
and allocated loss adjustment expense reserve ("loss reserve"),
where such loss reserve is defined as aggregated policyholder predicted
ultimate losses less cumulative paid loss and allocated loss adjustment
expense for a corresponding financial reporting period as of an
accounting date ("emerged paid loss") and the incurred
but not reported ("IBNR") reserve which is the aggregated
policyholder ultimate losses less cumulative paid and outstanding
loss and allocated loss adjustment expense ("emerged incurred
losses") for the corresponding financial reporting period as
of an accounting date. The phrase "outstanding losses"
will be used synonymously with the phrase "loss reserves."
The process and system according to the present invention focus
on performing such predictions at the individual policy or risk
level. These predictions can then be aggregated and analyzed at
the accident year level.
[0038] In addition, the system and method according to the present
invention have utility in the development of statistical levels
of confidence about the estimated ultimate losses and loss reserves.
It should be appreciated that the ability to estimate confidence
intervals follows from the present invention's use of non-aggregated,
individual policy or risk level data and claim/claimant level data
to estimate outstanding liabilities.
[0039] According to a preferred embodiment of the method according
to the present invention, the following steps are effected: (i)
gathering historical internal policyholder data and storing such
historical policyholder data in a data base; (ii) identifying external
data sources having a plurality of potentially predictive external
variables, each variable having at least two values; (iii) normalizing
the internal policyholder data relating to premiums and losses using
actuarial transformations; (iv) calculating the losses and loss
ratios evaluated at each of a series of valuation dates for each
policyholder in the data base; (v) utilizing appropriate key or
link fields to match corresponding internal data to the obtained
external data and analyzing one or more external variables as well
as internal data at the policyholder level of detail to identify
significant statistical relationships between the one or more external
variables, the emerged loss or loss ratio as of agej and the emerged
loss or loss ratio as of age j+1; (vi) identifying and choosing
predictive external and internal variables based on statistical
significance and the determination of highly experienced actuaries
and statisticians; (vii) developing a statistical model that (a)
weights the various predictive variables according to their contribution
to the emerged loss or loss ratio as of age j+1 (i.e., the loss
development patterns) and (b) projects such losses forward to their
ultimate level; (viii) if the model from step vii(a) is used to
predict each policyholder's ultimate loss ratios, deriving corresponding
ultimate losses by multiplying the estimated ultimate loss ratio
by the policyholder's premium (generally a known quantity) from
which paid or incurred losses are subtracted to obtain the respective
loss and ALAE reserve or IBNR reserve; and (ix) using a "bootstrapping"
simulation technique from modern statistical theory, re-sampling
the policyholder-level data points to obtain statistical levels
of confidence about the estimated ultimate losses and loss reserves.
[0040] The present invention has application to policy or risk-level
losses for a single line of business coverage.
[0041] There are at least two approaches to achieving step vii(a)
above. First, a series of predictive models can be built for each
column in Table A. The target variable is the loss or loss ratio
at age j+1; a key predictive variable is the loss or loss ratio
at age j. Other predictive variables can be used as well. Each column's
predictive model can be used to predict the loss or loss ratio values
corresponding to the unknown, future elements of the loss array.
[0042] Second, a "longitudinal data" approach can be
used, such that each policy's sequence of loss or loss ratio values
serves as a time-series target variable. Rather than building a
nested series of predictive models as described above, this approach
builds a single time-series predictive model, simultaneously using
the entire series of loss or loss ratio evaluations for each policy.
[0043] Step vii(a) above accomplishes two principal objectives.
First, it provides a ratio of emerged losses from one year to the
next at each age j. Second, it provides an estimate of the loss
development patterns from age j to age j+1. The importance of this
process is that it explains shifts in the emerged loss or loss ratio
due to policy, qualitative and operational metrics while simultaneously
estimating loss development from age j to age (j+1). These estimated
ultimate losses are aggregated to the accident year level; and from
this quantity the aggregated paid loss or incurred loss is subtracted.
Thus, estimates of the total loss reserve or the total IBNR reserve,
respectively, are obtained.
[0044] Accordingly, it is an object of the present invention to
provide a computer-implemented, quantitative system and method that
employ external data and a company's internal data to more accurately
and consistently predict ultimate losses and reserves of property/casualty
insurance companies.
[0045] Still other objects and advantages of the invention will
in part be obvious and will in part be apparent from the specification.
[0046] The present invention accordingly comprises the various
steps and the relation of one or more of such steps with respect
to each of the others and the system embodies features of construction,
combinations of elements and arrangement of parts which are adapted
to effect such steps, all as exemplified in the following detailed
disclosure and the scope of the invention will be indicated in the
claims.
BRIEF DESCRIPTION OF THE DRAWINGS
[0047] For a fuller understanding of the invention, reference is
made to the following description, taken in connection with the
accompanying drawings, in which:
[0048] FIGS. 1A and 1B are flow diagrams depicting process steps
preparatory to generating a statistical model predictive of ultimate
losses in accordance with a preferred embodiment of the present
invention;
[0049] FIGS. 2A-2C are flow diagrams depicting process steps for
developing a statistical model and predicting ultimate losses at
the policyholder and claim level using the statistical model in
accordance with a preferred embodiment of the present invention,
as well as the process step of sampling policyholder data to obtain
statistical levels of confidence about estimated ultimate losses
and loss reserves in accordance with a preferred embodiment of the
present invention;
[0050] FIG. 3 shows a representative example of statistics used
to evaluate the statistical significance of predictive variables
in accordance with a preferred embodiment of the present invention;
[0051] FIG. 4 depicts a correlation table which can be used to
identify pairs of predictor variables that are highly correlated
with one another in accordance with a preferred embodiment of the
present invention; and
[0052] FIG. 5 is a diagram of a system in accordance with a preferred
embodiment of the present invention.
DETAILED OF THE PREFERRED EMBODIMENTS
[0053] Reference is first made to FIGS. 1A and 1B which generally
depict the steps in the process preparatory to gathering the data
from various sources, actuarially normalizing internal data, utilizing
appropriate key or linkage values to match corresponding internal
data to the obtained external data, calculating an emerged loss
ratio as of an accounting date and identifying predictive internal
and external variables preparatory to developing a statistical model
that predicts ultimate losses in accordance with a preferred embodiment
of the present invention.
[0054] To begin the process at step 100, insurer loss and premium
data at the policyholder and claim level of detail are compiled
for a policyholder loss development data base. The data can include
policyholder premium (direct, assumed, and ceded) for the term of
the policy. A premium is the money the insurer collects in exchange
for insurance coverage. Premiums include direct premiums (collected
from a policyholder), assumed premiums (collected from another insurance
company in exchange for reinsurance coverage) and "ceded"
premiums (paid to another insurance company in exchange for reinsurance
coverage). The data can also include (A) policyholder demographic
information such as, for example, (i) name of policyholder, (ii)
policy number, (iii) claim number, (iv) address of policyholder,
(v) policy effective date and date the policy was first written,
(vi) line of business and type of coverage, (vii) classification
and related rate, (viii) geographic rating territory, (ix) agent
who wrote the policy, (B) policyholder metrics such as, for example,
(i) term of policy, (ii) policy limits, (iii) amount of premium
by coverage, (iv) the date bills were paid by the insured, (v) exposure
(the number of units of insurance provided), (vi) schedule rating
information, (vii) date of claim, (viii) report date of claim, (ix)
loss and ALAE payment(s) date(s), (x) loss and ALAE claim reserve
change by date, (xi) valuation date (from which age of development
is determined), (xii) amount of loss and ALAE paid by coverage as
of a valuation date by claim (direct, assumed and ceded), (xiii)
amount of incurred loss and ALAE by coverage as of a valuation date
by claim (direct, assumed and ceded), and (xiv) amount of paid and
incurred allocated loss adjustment expense or (DCA) expense as of
a valuation date (direct, assumed and ceded), (C) claim demographic
information such as claim number and claimant information, and (D)
claim metrics such as time of day of incident, line of business
and applicable coverage, nature of injury or loss (for example bodily
injury vs. property damage vs. fire), type of injury or loss (for
example, burn, fracture) cause of injury or loss diagnosis and treatment
codes, and attorney involvement.
[0055] Next, in step 104, a number of external data sources having
a plurality of variables, each variable having at least two values,
are identified for use in appending the data base and for generating
the predictive statistical model. Examples of external data sources
include the CLUE data base of historical homeowners claims; the
MVR (Motor Vehicle Records) data base of historical motor claims
and various data bases of both personal and commercial financial
stability (or "credit") information. Synthetic variables
are developed which are a combination of two or more data elements,
internal or external, such as a ratio of weighted averages.
[0056] Referring to FIG. 5, all collected data, including the internal
data, may be stored in a relational data base 20 (as are well known
and provided by, for example, IBM, Microsoft Corporation, Oracle
and the like) associated with a computer system 10 running the computational
hardware and software applications necessary to generate the predictive
statistical model. The computer system 10 preferably includes a
processor 30, memory (not shown), storage medium (not shown), input
devices 40 (e.g., keyboard, mouse) and display device 50. The system
10 may be operated using a conventional operating system and preferably
includes a graphical user interface for navigating and controlling
various computational aspects of the present invention. The system
10 can also be linked to one or more external data source servers
60. A stand-alone workstation 70, including a processor, memory,
input devices and storage medium, can also be used to access the
data base 20.
[0057] Referring back to FIG. 1A, in step 108, the policyholder
premium and loss data are normalized using actuarial transformations.
The normalized data ("work data") including normalized
premium data ("premium work data") and normalized loss
data ("loss work data") are associated with the data sources
to help identify external variables predictive of ultimate losses.
[0058] In step 112, the normalized loss and loss ratio that have
emerged as of each relevant valuation date are calculated for each
policy. The data are aggregated by loss period to determine the
relative change in aggregate emerged loss or loss ratio from one
valuation age to the next. That is, each policy's losses are aggregated
by accident year and age of development. For example, if policy
k had a claim or claims which occurred in accident year i, the losses
recorded by accident year i at age j=0 would be the losses as they
emerged in the first twelve months from the date of occurrence.
The losses for that same accident year at age j=1, that is, in the
next 12 months of development, would be the aggregate of losses
occurring in accident year i as of age j=1. For paid losses, the
aggregate equals the sum of all losses paid for claims reported
in accident year i through age j=0, 1. For incurred losses, it equals
the sum of all losses paid for claims reported for accident year
i through age j=0, 1 plus the outstanding reserve at the end of
the age j=1. This aggregation is done policy-by-policy across accident
year and valuation dates.
[0059] In step 116, a cumulative loss and loss ratio is then calculated
by age of development for a defined group of policyholders.
[0060] In step 120 the internal and external data are analyzed
for their predictive statistical relationship to the normalized
emerged loss ratio. For example, internal data such as the amount
of policy limit or the record of the policyholder's bill paying
behavior or combination of internal data variables may be predictive
of ultimate losses by policy. Likewise, external data such as weather
data, policyholder financial information, the distance of the policyholder
from the agent, or combination of these variables may be predictive
of ultimate losses by policy. It should be noted that, in all cases,
predictions are based on variable values that are historical in
nature and known at the time the prediction is being made.
[0061] In step 124 predictive internal and external variables are
identified and selected based on their statistical significance
and the determination of highly experienced actuaries and statisticians.
Taking a linear model such as C.sub.ij=a+bX.sub.1+cX.sub.2 for example,
there are standard statistical tests to evaluate the significance
of predictive variables X.sub.1, which could represent an internal
data variable and X.sub.2, which could represent an external data
variable. These tests include the F and t statistics for X.sub.1
and X.sub.2, as well as the overall R.sup.2 statistic, which represents
the proportion of variation in the loss data explained by the model.
[0062] After the individual external variables have been selected
by the analyst as being significant, these variables are examined
by the analyst in step 128 against one another for cross-correlation.
To the extent cross-correlation is present between, for example,
a pair of external variables, the analyst may elect to discard one
external variable of the pair of external variables showing cross-correlation.
[0063] Referring now to FIGS. 2A and 2B, the steps in the process
for generating the predictive statistical model based on internal
and external data are generally depicted. In step 200, the data
are split into multiple separate subsets of data on a random or
otherwise statistically significant basis that is actuarially determined.
More specifically, the data are split into a training data set,
test data set and validation data set. The training data set includes
the data used to statistically estimate the weights and parameters
of a predictive model. The test data set includes the data used
to evaluate each candidate model. Namely, the model is applied to
the test data set and the emerged values predicted by the model
are compared to the actual target emerged values in the test data
set. The training and test data sets are thus used in an iterative
fashion to evaluate a plurality of candidate models. The validation
data set is a third data set held aside during this iterative process
and is used to evaluate the final model once it is selected.
[0064] Partitioning the data into training, test and validation
data sets is essentially the last step before developing the predictive
statistical model. At this point, the premium and loss work data
have been calculated and the variables predictive of ultimate losses
have been initially defined.
[0065] The actual construction of the predictive statistical model
involves steps 204A and 204B, as shown in FIG. 2A. More particularly,
in step 204A, the training data set is used to produce initial statistical
models. Having used the training data set to develop "k"
models of the form c.sub.k=a.sub.k+bx.sub.1k+cx.sub.2k+ . . . ,
the various models are applied to the test data set to evaluate
each candidate model. The models which could be based on incurred
loss and/or ALAE data, paid loss and/or ALAE data, or other types
of data are applied to the test data set and the emerged values
predicted by the models are compared to the actual emerged target
values in the test data set. In so doing, the training and test
data sets are used iteratively to select the best candidate model(s)
for their predictive power. The initial statistical models contain
coefficients for each of the individual variables in the training
data, that relate those individual variables to emerged loss or
loss ratio at age j+1, which is represented by the loss or loss
ratio of each individual policyholder's record in the training data
base. The coefficients represent the independent contribution of
each of the predictor variables to the overall prediction of the
dependent variable, i.e., the policyholder emerged loss or loss
ratio.
[0066] In step 204B, the testing data set is used to evaluate whether
the coefficients from step 204A reflect intrinsic and not accidental
or purely stochastic, patterns in the training data set. Given that
the test data set was not used to fit the candidate model and given
that the actual amounts of loss development are known, applying
the model to the test data set enables one to evaluate actual versus
predicted results and thereby evaluate the efficacy of the predictive
variables selected to be in the model being considered. In short,
performance of the model on test (or "out-of-sample")
data helps the analyst determine the degree to which a model explains
true, as opposed to spurious, variation in the loss data.
[0067] In step 204C, the model is applied to the validation data
set to obtain an unbiased estimate of the model's future performance.
[0068] In step 208, the estimated loss or loss ratio at age j+1
is calculated using the predictive statistical model constructed
according to steps 204A, 204B and 204C. This model is applied to
each record in the validation data set. More explicitly, suppose
the model C.sub.j+1=.beta..sub.j+1,0+.beta..sub.j+1,1*X.sub.1+.beta..sub.j+.sub.1,2-
*X.sub.2+.beta..sub.j+1,3*X.sub.3+ . . . is the model constructed
to predict the value of each policy's loss, evaluated at period
j+1. Each of the quantities {X.sub.1, X.sub.2, X.sub.3, . . . }
are predictive variables, the values of which are known. The .beta.
parameters were estimated as part of the model construction process
and are therefore also known. Estimating the expected loss at age
j+1 (C.sub.j+1) is therefore simply a matter of applying the above
equation to these known quantities.
[0069] In step 212 the emerged loss or loss ratio from years past
is used as a base from which the predicted ultimate losses or loss
ratio can be estimated. The predicted loss ratio for a given year
is equal to the sum of all actual losses emerged plus losses predicted
to emerge at future valuation dates divided by the premium earned
for that year.
[0070] In step 216 the loss ratio is then multiplied by the policy's
earned premium to arrive at an estimate of the policy's ultimate
losses.
[0071] In step 220 the policyholder ultimate losses are aggregated
to derive policyholder estimated ultimate losses. From this quantity,
cumulative aggregated paid loss or incurred loss is subtracted to
obtain respective estimates of the total loss reserve or the total
IBNR reserve.
[0072] In step 224 a technique known as bootstrapping is applied
to the policy-level data base of estimated ultimate losses and loss
reserves to obtain statistical levels of confidence about the estimated
ultimate losses and loss reserves. Bootstrapping can be used to
estimate confidence intervals in cases where no theoretically derived
confidence intervals are available. Bootstrapping uses repeated
"re-sampling" of the data, which is a type of simulation
technique.
[0073] As indicated above and as will be explained in greater detail
hereinafter, the task of developing the predictive statistical model
is begun using the training data set. As part of the same process,
the test data set is used to evaluate the efficacy of the predictive
statistical model being developed with the training data set. The
results from the test data set may be used at various stages to
modify the development of the predictive statistical model. Once
the predictive statistical model is developed, the predictiveness
of the model is evaluated on the validation data set.
[0074] The steps as shown in FIGS. 1A, 1B, and 2A-2C are now described
in more detail. In the preferred embodiment of the present invention,
actual internal data for a plurality of policyholders are secured
from the insurance company in step 100. Preferably, several years
of policyholders' loss, ALAE and premium data are gathered and pooled
together in a single data base of policyholder records. The data
would generally be in an array of summarized loss or claim count
information described previously as a loss triangle with corresponding
premium for the year in which the claim(s) occurred. That is, for
a given year i there are N.sub.i observations for an age of development.
Relating observations of older years from early ages of development
to later years of development provides an indication of how a less
mature year might emerge from its respective earlier to later ages
of development. This data base will be referred to as the "analysis
file."
[0075] Other related information on each policyholder and claim
by claimant (as previously described in connection with step 100)
is also gathered and merged onto the analysis file, e.g., the policyholder
demographics and metrics, and claim metrics. This information is
used in associating a policyholder's and claimant's data with the
predictive variables obtained from the external data sources.
[0076] According to a preferred embodiment of the present invention
in step 104, the external data sources include individual policy-level
data bases available from vendors such as Acxiom, Choicepoint, Claritas,
Marshall Swift Boeckh, Dun & Bradstreet and Experian. Variables
selected from the policy-level data bases are matched to the data
held in the analysis file electronically based on unique identifying
fields such as the name and address of the policyholder.
[0077] Also included as an external data source, for example, are
census data that are available from both U.S. Government agencies
and third parties vendors, e.g., the EASI product. Such census data
are matched to the analysis file electronically based on the policyholder's
zip code. County level data are also available and can include information
such as historical weather patterns, hail falls, etc. In the preferred
embodiment of the present invention, the zip code-level files are
summarized to a county level and the analysis file is then matched
to the county-level data.
[0078] These data providers offer many characteristics of a policyholder's
or claimant's household or business, e.g., income, home owned or
rented, education level of the business owner, etc. The household-level
data are based on the policyholder's or claimant's name, address,
and when available, social security number. Other individual-level
data sources are also included, when available. These include a
policyholder's or claimant's individual credit report, driving record
from MVR and CLUE reports, etc.
[0079] Variables are selected from each of the multiple external
data sources and matched to the analysis file on a policy-by-policy
basis. The variables from the external data sources are available
to identify relationships between these variables and, for example,
premium and loss data in the analysis file. As the statistical relationship
between the variables and premium and loss data are established,
these variables will be included in the development of a model that
is predictive of insureds' loss development.
[0080] The matching process for the external data are completely
computerized. Each individual external data base has a unique key
on each of the records in the particular data base. This unique
key also exists on each of the records in the analysis file. For
external data, e.g., Experian or Dun & Bradstreet, the unique
key is the business name and address. For the census data, the unique
key is either the county code or the zip code. For business or household-level
demographics, the unique key is either the business name or personal
household address, or social security number.
[0081] The external data are electronically secured and loaded
onto the computer system where the analysis file can be accessed.
One or more software applications then match the appropriate external
data records to the appropriate analysis file records. The resulting
match produces expanded analysis file records with not only historical
policyholder and claimant data but matched external data as well.
[0082] Next, in step 108, necessary and appropriate actuarial modifications
to the data held in the analysis file are completed. Actuarial transformations
are required to make the data more useful in the development of
the predictive statistical model since much of the insurance company
data within the analysis file cannot be used in its raw form. This
is particularly true of the premium and loss data. These actuarial
transformations include, but are not limited to, premium on-leveling
to achieve a common basis of premium comparison, loss trending,
capping and other actuarial techniques that may be relied on to
accurately reflect the ultimate losses potential of each individual
policyholder.
[0083] Premium on-leveling is an actuarial technique that transforms
diversely calculated individual policyholder premiums to a common
basis. This is necessary since the actual premium that a policyholder
is charged is not entirely a quantitative, objective, or consistent
process. More particularly, within any individual insurance company,
premiums for a particular policyholder typically can be written
by several "writing" companies, each of which may charge
a different base premium. Different underwriters will often select
different writing companies even for the same policyholder. Additionally,
a commercial insurance underwriter may use credits or debits for
individual policies further affecting the base premium. Thus, there
are significant qualitative judgments or subjective elements in
the process that complicate the determination of a base premium.
[0084] The premium on-leveling process removes these and other,
subjective elements from the determination of the premium for every
policy in the analysis file. As a result a common base premium may
be determined. Such a common basis is required to develop the ultimate
losses or loss ratio indications from the data that are necessary
to build the predictive statistical model. For example, the application
of schedule rating can have the effect of producing different loss
ratios on two identical risks. Schedule rating is the process of
applying debits or credits to base rates to reflect the presence
or absence of risk characteristics such as safety programs. If schedule
rating were applied differently to two identical risks with identical
losses, it would therefore be the subjective elements which produce
different loss ratios; not the inherent difference in the risk.
Another example is that rate level adequacy varies over time. A
book of business has an inherently lower loss ratio with a higher
rate level. Two identical policies written during different timeframes
at different rate adequacy levels would have a different loss ratio.
Inasmuch as a key objective of the invention is to predict ultimate
loss ratio, a common base from which the estimate can be projected
is first established.
[0085] The analysis file loss data is actuarially modified or transformed
according to a preferred embodiment of the present invention to
produce more accurate ultimate loss predictions. More specifically,
some insurance coverages have "long tail losses." Long
tail losses are losses that are usually not paid during the policy
term, but rather are paid a significant amount of time after the
end of the policy period.
[0086] Other actuarial modifications may also be required for the
loss data. For example, very large losses could be capped since
a company may have retentions per claim that are exceeded by the
estimated loss. Also, modifications may be made to the loss data
to adjust for operational changes.
[0087] These actuarial modifications to both the premium and loss
data produce actuarially sound data that can be employed in the
development of the predictive statistical model. As previously set
forth, the actuarially modified data have been referred to as "work
data," while the actuarially modified premium and loss data
have been referred to as "premium work data" and "loss
work data," respectively.
[0088] In related step 112, the loss ratio is calculated for each
policyholder by age of development in the analysis file. As explained
earlier, the loss ratio is defined as the numerical ratio of the
loss divided by the premium. The emerged loss or loss ratio is an
indication of an individual policy's ultimate losses, as it represents
that portion of the premium committed to losses emerged to date.
[0089] In another aspect of the present invention, emerged "frequency"
and "severity", second important dimensions of ultimate
losses, are also calculated in this step. Frequency is calculated
by dividing the policy term total claim count by the policy term
premium work data. Severity is calculated by dividing the policy
term losses by the policy term emerged claim count. Although the
loss ratio is the most common measure of ultimate losses, frequency
and severity are important components of insurance ultimate losses.
[0090] The remainder of this invention description will rely upon
loss ratio as the primary measurement of ultimate losses. But it
should be correctly assumed that frequency and severity measurements
of ultimate losses are also included in the development of the system
and method according to the present invention and in the measurements
of ultimate losses subsequently described herein.
[0091] Thereafter, in step 116 the loss ratio is calculated for
a defined group. The cumulative loss ratio is defined as the sum
of the loss work data for a defined group divided by the sum of
the premium work data for the defined group. Typical definable groups
would be based on the different insurance products offered. To calculate
the loss ratio for an individual segment of a line of business all
of the loss work data and premium work data for all policyholders
covered by the segment of the line of business are subtotaled and
the loss ratio is calculated for the entire segment of the line
of business.
[0092] In step 120, a statistical analysis on all of the data in
the analysis file is performed. That is, for each external variable
from each external data source, a statistical analysis is performed
that relates the effect of that individual external variable on
the cumulative loss ratio by age of development. Well known statistical
techniques such as multiple regression models may be employed to
determine the magnitude and reliability of an apparent statistical
relationship between an external variable and cumulative loss ratio.
A representative example of statistics which can be calculated and
reviewed to analyze the statistical significance of the predictor
variables is provided in FIG. 3.
[0093] Each value that an external variable can assume has a loss
ratio calculated by age of development which is then further segmented
by a definable group (e.g., major coverage type). For purposes of
illustration, the external variable of business-location-ownership
might be used in a commercial insurance application (in which case
the policyholder happens to be a business). Business-location-ownership
is an external variable, or piece of information, available from
Dun & Bradstreet. It defines whether the physical location of
the insured business is owned by the business owner or rented by
the business owner. Each individual variable can take on appropriate
values. In the case of business-location-ownership, the values are
O=owned and R=rented. The cumulative loss ratio is calculated for
each of these values. For business owner location, the O value might
have a cumulative loss ratio of 0.60, while the R value might have
a cumulative loss ratio of 0.80, for example. That is, based on
the premium work data and loss work data, owners have a cumulative
loss ratio of 0.60 while renters have a cumulative loss ratio of
0.80, for example.
[0094] This analysis may then be further segmented by the major
type of coverage. So, for business-owner-location, the losses and
premiums are segmented by major line of business. The cumulative
losses and loss ratios for each of the values O and R are calculated
by major line of business. Thus, it is desirable to use a data base
that can differentiate premiums and losses by major line of business.
[0095] In step 124, a review is made of all of the outputs derived
from previous step 120. This review is based on human experience
and expertise in judging what individual external variables available
from the external data sources should be considered in the creation
of the statistical model that will be used to predict the cumulative
loss ratio of an individual policyholder.
[0096] In order to develop a robust system that will predict cumulative
losses and loss ratio on a per policyholder basis, it is important
to include only those individual external variables that, in and
of themselves, can contribute to the development of the model (hereinafter
"predictor variables"). In other words, the individual
external variables under critical determination in step 124 should
have some relationship to emerged loss and thus ultimate losses
and loss ratio.
[0097] In the above example of business-location-ownership, it
can be gleaned from the cumulative loss ratios described above,
i.e., the O value (0.60) and the R value (0.80), that business-location-ownership
may in fact be related to ultimate losses and therefore may in fact
be considered a predictor variable.
[0098] As might be expected, the critical determination process
of step 124 becomes much more complex as the number of values that
an individual external variable might assume increases. Using a
40 year average hail fall occurrence as an example, this individual
external variable can have values that range from 0 to the historical
maximum, say 30 annual events, with all of the numbers in-between
as possible values. In order to complete the critical determination
of such an individual external variable, it is viewed in a particular
manner conducive to such a critical determination, so that the highly
experienced actuary and statistician can in fact make the appropriate
critical determination of its efficacy for inclusion in the development
of the predictive statistical model.
[0099] A common statistical method, called binning, is employed
to arrange similar values together into a single grouping, called
a bin. In the 40 year average hail fall individual data element
example, ten bins might be produced, each containing 3 values, e.g.,
bin 1 equals values 0-3, bin 2 equals values 4-6 and so on. The
binning process, as described, yields ten surrogate values for the
40 year average hail fall individual external variable. The critical
determination of the 40 year average hail fall variable can then
be completed by the experienced actuary and statistician.
[0100] The cumulative loss ratio of each bin is considered in relation
to the cumulative loss ratio of each other bin and the overall pattern
of cumulative loss ratios considered together. Several possible
patterns might be discernable. If the cumulative loss ratio of the
individual bins are arranged in a generally increasing or decreasing
pattern, then it is clear to the experienced actuary and statistician
that the bins and hence the underlying individual data elements
comprising them, could in fact be related to commercial insurance
emerged losses and therefore, should be considered for inclusion
in the development of the statistical model.
[0101] Likewise, a saw toothed pattern, i.e., one where values
of the cumulative loss ratio from bin to bin exhibit an erratic
pattern when graphically illustrated and do not display any general
direction trend, would usually not offer any causal relationship
to loss or loss ratio and hence, would not be considered for inclusion
in the development of the predictive statistical model. Other patterns,
some very complicated and subtle, can only be discerned by the trained
and experienced eye of the actuary or statistician, specifically
skilled in this work. For example, driving skills may improve as
drivers age to a point and then deteriorate from that age hence.
[0102] Thereafter in step 128, the predictor variables from the
various external data sources that pass the review in prior step
124, are examined for cross correlations against one another. For
example, suppose two different predictor variables, years-in-business
and business-owners-age, are compared one to another. Since each
of these predictor variables can assume a wide range of values,
assume that each has been binned into five bins (as discussed above).
Furthermore, assume that the cumulative loss ratio of each respective
bin, from each set of five bins, is virtually the same for the two
different predictor variables. In other words, years-in-business's
bin 1 cumulative loss ratio is the same as business-owners-age's
bin 1 cumulative loss ratio, etc.
[0103] This type of variable to variable comparison is referred
to as a "correlation analysis." In other words, the analysis
is concerned with determining how "co-related" individual
pairs of variables are in relation to one another.
[0104] All individual variables are compared to all other individual
variables in such a similar fashion. A master matrix is prepared
that has the correlation coefficient for each pair of predictor
variables. The correlation coefficient is a mathematical expression
for the degree of correlation between any pair of predictor variables.
Suppose X.sub.1 and X.sub.2 are two predictive variables; let .mu..sub.1
and .mu..sub.2 respectively denote their sample average values;
and let .sigma..sub.1 and .sigma..sub.2 respectively denote their
sample standard deviations. The standard deviation of a variable
X is defined as: [.SIGMA.(X-.mu..sub.x).sup.2] The correlation between
X.sub.1 and X.sub.2 is defined as: .rho..sub.12=[.SIGMA.(X.sub.1-.mu..sub.1)*(X.sub.2-.mu..sub.2)]/[(.sigma.-
.sub.1*.sigma..sub.2] (The standard "sigma" symbol .SIGMA.
represents summation over all records in the sample.) If there are
N predictive variables X.sub.1, X.sub.2, . . . , X.sub.N the correlation
matrix is formed by quantities .rho..sub.ij where i and j range
from 1 to N. It is a mathematical fact that .rho..sub.ij takes on
a value between 0 and 1. A correlation of 0 means that the two variables
are statistically independent; a correlation of 1 means that the
two variables co-vary perfectly and are therefore interchangeable
from a statistical point of view. The greater the correlation coefficient,
the greater the degree of correlation between the pair of individual
variables.
[0105] The experienced and trained actuary or statistician can
review the matrix of correlation coefficients. The review can involve
identifying those pairs of predictor variables that are highly correlated
with one another (see e.g., the correlation table depicted in FIG.
4). Once identified, the real world meaning of each predictor variable
can be evaluated. In the example above, the real world meaning of
years-in-business and business-owner-age may be well understood.
One reasonable causal explanation why this specific pair of predictive
external variables might be highly correlated with one another would
be that the older the business owner, the longer the business owner
has been in business.
[0106] The experienced actuary or statistician then can make an
informed decision to potentially remove one of the two predictor
variables, but not both. Such a decision would weigh the degree
of correlation between the two predictor variables and the real
world meaning of each of the two predictor variables. For example,
when weighing years in business versus the age of the business owner,
the actuary or statistician may decide that the age of the business
is more directly related to potential loss experience of the business
because age of business may be more directly related to the effective
implementations of procedures to prevent and/or control losses.
[0107] As shown in FIG. 2A, in step 200, the portion of the data
base that passes through all of the above pertinent steps is subdivided
into three separate data subsets, namely, the training data set,
the testing data set and the validation data set. Different actuarial
and statistical techniques can be employed to develop these three
data sets from the overall data set. They include a random splitting
of the data and a time series split. The time series split might
reserve the most recent few years of historical data for the validation
data set and the prior years for the training and testing data sets.
Such a final determination is made within the expert judgment of
the actuary and statistician.
[0108] 1. Training Data Set
[0109] The development process to construct the predictive statistical
model requires a subset of the data to develop the mathematical
components of the statistical model. This subset of data are referred
to as the "training data set."
[0110] 2. Testing Data Set
[0111] At times, the process of developing these mathematical components
can actually exceed the true relationships inherent in the data
and overstate such relationships. As a result, the coefficients
that describe the mathematical components can be subject to error.
In order to monitor and minimize the overstating of the relationships
and hence the degree of error in the coefficients, a second data
subset is subdivided from the overall data base and is referred
to as the "testing data set."
[0112] 3. Validation Data Set The third subset of data, the "validation
data set," functions as a final estimate of the degree of predictiveness
of ultimate losses or loss ratio that the mathematical components
of the system can be reasonably expected to achieve on a go forward
basis. Since the development of the coefficients of the predictive
statistical model are influenced during the development process
by the training and testing data sets, the validation data set provides
an independent, non-biased estimate of the efficacy of the predictive
statistical model.
[0113] The actual construction of the predictive statistical model
involves steps 204A and 204B, as shown in FIG. 2A. More particularly,
in step 204A, the training data set is used to produce an initial
statistical model. The initial statistical model results in a mathematical
equation, as described previously, that produces coefficients for
each of the individual variables in the training data, that relate
those individual variables to emerged loss or loss ratio at age
j+1, which is represented by the loss or loss ratio of each individual
policyholder's record in the training data base. The coefficients
represent the independent contribution of each of the predictor
variables to the overall prediction of the dependent variable, i.e.,
the policyholder emerged loss ratio.
[0114] Several different statistical techniques are employed in
step 204A. Conventional multiple regression is the first technique
employed. It produces an initial model. The second technique employed
is generalized linear modeling. In some instances this technique
is capable of producing a more precise set of coefficients than
the multiple regression technique. A third technique employed is
a type of neural network, i.e., backwards propagation of errors,
or "backprop" for short. Backprop is capable of even more
precise coefficients than generalized linear modeling. Backprop
can produce nonlinear curve fitting in multi-dimensions and as such,
can operate as a universal function approximator. Due to the power
of this technique, the resulting coefficients can be quite precise
and as such, yield a strong set of relationships to loss ratio.
A final technique is the Multivariate Adaptive Regression Splines
technique. This technique finds the optimal set of transformations
and interactions of the variables used to predict loss or loss ratio.
As such, it functions as a universal approximator like neural networks.
[0115] In step 204B, the testing data set is used to evaluate if
the coefficients from step 204A have "overfit" the training
data set. No data set that represents real world data is perfect;
every such real world data set has anomalies and noise in the data.
That is to say, statistical relationships that are not representative
of external world realities. Overfitting can result when the statistical
technique employed develops coefficients that not only map the relationships
between the individual variables in the training set to ultimate
losses, but also begin to map the relationships between the noise
in the training data set and ultimate losses. When this happens,
the coefficients are too fine-tuned to the eccentricities of the
training data set. The testing data set is used to determine the
extent of the overfitting.
[0116] In more detail, the model coefficients were derived by applying
a suitable statistical technique to the training data set. The test
data set was not used for this purpose. However, the resulting model
can be applied to each record of the test data set. That is, the
values C.sub.j for each record in the data set are calculated (C.sub.j
denotes the model's estimate of loss evaluated at period j). For
each record in the test data set, the estimated value of losses
evaluated at j can be compared with the actual value of losses at
j. For example, the mean absolute deviation (MAD) of the model estimates
can be calculated from the actual values. The MAD is defined as
the average of the absolute value of the difference between the
actual value and the estimated value: MAD=AVG[|actual-estimated|].
[0117] For any model, the MAD can be calculated both on the data
set used to fit the model (the training data set) and on any test
data set. If a model produces a very low (i.e., "good")
MAD value on the training data set but a significantly higher MAD
on the test data set, there is strong reason to suspect that the
model has "over-fit" the training data. In other words,
the model has fit idiosyncrasies of the training data that cannot
be expected to generalize to future data sets. In information-theoretic
terms, the model has fit too much of the "noise" in the
data and perhaps not enough of the "signal".
[0118] The method of fitting a model on a training data set and
testing it on a separate test data set is a widely used model validation
technique that enables analysts to construct models that can be
expected to make accurate predictions in the future.
[0119] The model development process described in steps 204A (fitting
the model on training data) and 204B (evaluating it on test data)
is an iterative one. Many candidate models, involving different
combinations of predictive variables and/or model techniques options,
will be fit on the training data; each one will be evaluated on
the test data. The test data evaluation offers a principled way
of choosing a model that is the optimal trade-off between productiveness
and simplicity. While a certain degree of model complexity is necessary
to make accurate predictions, there may come a point in the modeling
process where the addition of further additional variables, variable
interactions, or model structure provides no marginal effectiveness
(e.g., reduction in MAD) on the test data set. At this point, it
is reasonable to halt the iterative modeling process.
[0120] When this iterative model-building process has halted, further
assurance that the model will generalize well on future data is
desirable. Each candidate model considered in the modeling process
was fit on the training data and evaluated on the test data. Therefore,
the test data were not used to fit a model. Still, the model performance
on the test data (as measured by MAD or another suitable measure
of model accuracy) might be overly optimistic. The reason for this
is that the test data set was used to evaluate and compare models.
Therefore, although it was not used to fit a model, it was used
as part of the overall modeling process.
[0121] In order to provide an unbiased estimate of the model's
future performance, the model is applied to the validation data
set, as described in step 204C. This involves the same steps as
applying the model to the test data set: the estimated value is
calculated by inserting the (known) predictive variable values into
the model equation. For each record, the estimated values are compared
to the actual value and MAD (or some other suitable measure of model
accuracy) is calculated. Typically, the model's accuracy measure
deteriorates slightly in moving from the test data set to the validation
data set. A significant deterioration might suggest that the iterative
model-building process was too protracted, culminating in a "lucky
fit" to the test data. However, such a situation can typically
be avoided by a seasoned statistician with expertise in the subject-matter
at hand.
[0122] By the end of step 204C, the final model has been selected
and validated. It remains to apply the model to the data in order
to estimate outstanding losses. This process is described in steps
208-220 (FIG. 2B). A final step, 224 (FIG. 2C), will use the modern
simulation technique known as "bootstrapping" to estimate
the degree of certainly (or "variance") to be ascribed
to the resulting outstanding loss estimate.
[0123] The modeling process has yielded a sequence of models (referred
to hereinafter as "M.sub.2, M.sub.3 . . . , M.sub.k")
that allow the estimation (at the policy and claim level) of losses
evaluated at period 2, 3, . . . , k. In step 212, these models are
applied to the data in a nested fashion in order to calculate estimated
ultimate losses for each policy. More explicitly, model M.sub.2
is applied to the combined data (train, test and validation combined)
in order to calculate estimated losses evaluated at period 2. These
period-2 estimated losses in turn serve as an input for the M.sub.3
model; the period-3 losses estimated by M.sub.3 in turn serve an
input for M.sub.4 and so on. The estimated losses resulting from
the final model M.sub.k are the estimated ultimate losses for each
policy.
[0124] At this point, two considerations should be made. First,
there will be cases in which the estimated losses arising from M.sub.k
are judged to be somewhat undeveloped despite the fact that the
available data do not allow further extrapolation beyond period
k. In such cases, a selected multiplicative "tail factor"
can be applied to each policy to bring the estimated losses C.sub.k
to ultimate. This use of a tail factor (albeit on summarized data)
is currently in accord with established actuarial practice.
[0125] Second, building and applying a sequence of models to estimate
losses at period k has been described above--it is possible to use
essentially the same methodology to estimate ultimate loss ratios
(i.e. loss divided by premium) at period k. Either method is possible
and justifiable; the analyst might prefer to estimate losses at
k directly, since that is the quantity of interest. On the other
hand, the analyst might prefer to work with loss ratios, deeming
these quantities to be more stable and uniform across different
policies. If the models M.sub.2 . . . M.sub.k have been constructed
to estimate loss ratios evaluated at period k, these loss ratios
for each policy are multiplied by that policy's earned premium to
arrive at estimated losses. This is illustrated in step 216.
[0126] In step 220, the estimated ultimate losses are aggregated
to the level of interest (either the whole book of business or to
a sub-segment of interest). This gives an estimate of the total
estimated ultimate losses for the chosen segment. From this the
total currently emerged losses (paid or incurred, whichever is consistent
with the ultimate losses that have been estimated) can be subtracted.
The resulting quantity is an estimate of the total outstanding losses
for the chosen segment of business.
[0127] At this point, the method described above yields an optimal
estimate of total outstanding losses. But how much confidence can
be ascribed to this estimate?
[0128] In more formal statistical terms, a confidence interval
can be constructed around the outstanding loss estimate. Let L denote
the outstanding loss estimate resulting from step 220. A 95%-confidence
interval is a pair of numbers L.sub.1 and L.sub.2 with the two properties
that (1) L.sub.1, <L.sub.2 and (2) there is a 95% chance that
L falls within the interval (L.sub.1,L.sub.2). Other confidence
intervals (such as 90% and 99%) can be similarly defined. The preferred
way to construct a confidence interval is to estimate the probability
distribution of the estimated quantity L. By definition, a probability
distribution is a catalogue of statements "L is less than the
value .lamda. with probability .PI.." Given this catalogue
of statements it is straightforward to construct any confidence
interval of interest.
[0129] Referring to FIG. 2C, step 224 illustrates estimating the
probability distribution of estimate L of outstanding losses. A
recently introduced simulation technique known as "bootstrapping"
can be employed. The core idea of bootstrapping is sampling with
replacement, also known a "resampling." Intuitively, the
actual population being studied can be treated as the "true"
theoretical distribution. Suppose the data set used to produce a
loss reserve estimate contains 1 million (1M) polices. Resampling
this data set means randomly drawing 1M polices from the data set,
each time replacing the randomly drawn policy. The data set can
be resampled a large number of times (e.g., 1000 times). Any given
policy might show up 0, 1, 2, 3, . . . times in any given resample.
Therefore, each resample is a stochastic variant of the original
data set.
[0130] The above method can be applied (culminating in step 220)
to each of the 1000 resampled data sets. This yields 1000 outstanding
loss reserve estimates L.sub.1, . . . , L.sub.1000. These 1000 numbers
constitute an estimate of the distribution of outstanding loss estimates,
i.e., the distribution of L. As noted above, L can be used to construct
a confidence interval around L. For example, let L.sub.5% and L.sub.95%
denote the 5.sup.th and 95.sup.th percentiles respectively of the
distribution L.sub.1, . . . , L.sub.1000. These two numbers constitute
a 90%-confidence interval around L (that is, L is between the values
L.sub.5% and L.sub.95% with 90% probability 0.9). A small (or "tight")
confidence interval corresponds to a high degree of certainty in
estimate L; a large (or "wide") confidence interval corresponds
to a low degree of certainty.
[0131] In accordance with the present invention, a computerized
system and method for estimating insurance loss reserves and confidence
intervals using insurance policy and claim level detail predictive
modeling is provided. Predictive models are applied to historical
loss, premium and other insurer data, as well as external data,
at the level of policy detail to predict ultimate losses and allocated
loss adjustment expenses for a group of policies. From the aggregate
of such ultimate losses, paid losses to date can be subtracted to
derive an estimate of loss reserves. A significant advantage of
this model is to be able to detect dynamic changes in a group of
policies and evaluate their impact on loss reserves. In addition,
confidence intervals around the estimates can be estimated by sampling
the policy-by-policy estimates of ultimate losses.
[0132] It will thus be seen that the objects set forth above, among
those made apparent from the preceding description, are efficiently
attained and, since certain changes can be made in carrying out
the above method and in the constructions set forth for the system
without departing from the spirit and scope of the invention, it
is intended that all matter contained in the above description and
shown in the accompanying drawings shall be interpreted as illustrative
and not in a limiting sense.
[0133] It is also to be understood that the following claims are
intended to cover all of the generic and specific features of the
invention herein described and all statements of the scope of the
invention which, as a matter of language, might be said to fall
therebetween. |