Insurance Abstract
A method is disclosed for determining the prior net benefit of obtaining
data relating to an individual risk in an insurance portfolio, via
a survey or similar procedure. A risk model is developed at the
individual risk level for mathematically estimating the probability
of expected loss given a set of information about the risk. The
risk model is incorporated into a profitability model. A probability
distribution relating to the type of survey information to be obtained
is developed, which is used for determining the gross value of obtaining
the information. The method produces as an output a quantitative
estimation (e.g., dollar value) of the net benefit of obtaining
survey data for the risk, calculated as the gross value of the survey
less the survey's cost, where the benefit of the survey relates
to a quantitative increase in predictive accuracy resulting from
incorporating the survey data into the predictive model.
Insurance Claims
1. A method for determining the a priori benefit of an insurance
survey, said method comprising the steps of: obtaining at least
one probability distribution of information to be obtained by a
survey, said information relating to at least one insurance risk;
and determining a gross value of the information based at least
in part on said at least one probability distribution and a profitability
model associated with said information.
2. The method of claim 1 further comprising: determining a cost
associated with the survey; and determining a net value of the information
as the gross value less the cost.
3. The method of claim 2 further comprising: determining whether
to conduct the survey based at least in part on the net value.
4. The method of claim 3 wherein the gross value is further based
at least in part on a retention model for said at least one insurance
risk, said retention model incorporating information relating to
terms of an insurance contact for said at least one insurance risk.
5. The method of claim 3 further comprising: conducting the survey
if the net value is a positive monetary value.
6. The method of claim 1 wherein the profitability model incorporates
a risk model associated with said at least one insurance risk, said
risk model setting forth an expected insurance loss of said at least
one insurance risk given one or more characteristics of said at
least one insurance risk.
7. The method of claim 6 wherein the risk model is a Bayesian predictive
model.
8. An insurance survey method comprising the steps of: calculating
the net value of obtaining information through a survey prior to
carrying out the survey, said information relating to at least one
insurance risk; and determining whether to conduct the survey based
at least in part on the net value.
9. The method of claim 8 further comprising: obtaining at least
one probability distribution of the information to be obtained by
the survey; and calculating a gross value of the information based
at least in part on said at least one probability distribution and
a profitability model associated with said information, wherein
the net value is based at least in part on the gross value.
10. The method of claim 9 further comprising: determining a cost
associated with the survey, wherein the net value is calculated
as the gross value less the cost.
11. The method of claim 10 wherein the gross value is further based
at least in part on a retention model for said at least one insurance
risk, said retention model incorporating information relating to
terms of an insurance contact for said at least one insurance risk.
12. The method of claim 9 wherein the profitability model incorporates
a risk model associated with said at least one insurance risk, said
risk model setting forth an expected insurance loss of said at least
one insurance risk given one or more characteristics of said at
least one insurance risk.
13. The method of claim 12 wherein the risk model is a Bayesian
predictive model.
14. The method of claim 8 further comprising: conducting the survey
if the net value is a positive monetary value.
15. An insurance survey method comprising the steps of: calculating
a monetary value associated with information to be obtained by way
of a survey procedure, prior to conducting the survey procedure,
wherein the information relates to at least one insurance risk;
and determining whether to conduct the survey based at least in
part on the calculated monetary value.
16. The method of claim 15 further comprising: obtaining at least
one probability distribution of said information; and calculating
a gross value of the information based at least in part on said
at least one probability distribution and a profitability model
associated with said information, wherein the monetary value is
a net value calculated based at least in part on the gross value.
17. The method of claim 16 further comprising: determining a cost
associated with the survey procedure, wherein the net value is calculated
as the gross value less the cost.
18. The method of claim 17 wherein the gross value is further based
at least in part on a retention model for said at least one insurance
risk, said retention model incorporating information relating to
terms of an insurance contact for said at least one insurance risk.
19. The method of claim 16 wherein the profitability model incorporates
a risk model associated with said at least one insurance risk, said
risk model setting forth an expected insurance loss of said at least
one insurance risk given one or more characteristics of said at
least one insurance risk.
20. The method of claim 19 wherein the risk model is a Bayesian
predictive model.
Insurance Description
CROSSREFERENCE TO RELATED APPLICATIONS
[0001] This is a continuationinpart of U.S. patent application
Ser. No. 11/323,252, filed Dec. 30, 2005, which claims the benefit
of U.S. Provisional Application Ser. No. 60/709,634, filed Aug.
19, 2005.
FIELD OF THE INVENTION
[0002] The present invention relates to data processing and, more
particularly, to an automated electrical financial or business practice
or management arrangement for insurance.
BACKGROUND OF THE INVENTION
[0003] Generally speaking, commercial insurance is a form of risk
allocation or management involving the equitable transfer of a potential
financial loss, from a number of people and/or businesses to an
insurance company, in exchange for fee payments. Typically, the
insurer collects enough in fees (called premiums) from the insured
to cover payments for losses covered under the policies (called
claims), overhead, and a profit. Each insured property or item,
such as a plot of land, a building, company, vehicle, or piece of
equipment, is typically referred to as a "risk." A grouping
of risks, e.g., all the properties insured by an insurer or some
portion thereof, is called a "portfolio."
[0004] At any particular point in time, each portfolio of risks
has an associated set of past claims and potential future claims.
The former is a static, known value, while the latter is an unknown
variable. More specifically, for a given portfolio in a given time
period, e.g., one year, there may be no claims or a large number
of claims, depending on circumstances and factors largely outside
the insurer's control. However, to set premiums at a reasonable
level, it is necessary to predict or estimate future claims, e.g.,
from the insurer's perspective it is beneficial to set premiums
high enough to cover claims and overhead but not so high as would
drive away potential customers due to uncompetitive pricing. The
process of mathematically processing data associated with a risk
portfolio to predict or estimate future loss is called "risk
modeling." Traditionally, this has involved using actuarial
methods where statistics and probability theory are applied to a
risk portfolio as a whole (i.e., with the risks grouped together),
and taking into consideration data relating to overall past performance
of the risk portfolio.
[0005] While existing, actuarialbased methods for risk modeling
in the insurance industry are generally effective when large amounts
of data are available, they have proven less effective in situations
with less onhand data. This is because the data curves generated
with such methods, which are used to estimate future losses, are
less accurate when less data is presentin estimating a curve to
fit discreet data points, the greater the number of data points,
the more accurate the curve. Also, since portfolios are considered
as a whole, there is no way to effectively assess individual risks
using such methods.
[0006] Risk assessment surveys are sometimes used as part of the
process of risk modeling or management, for purposes of collecting
data relating to an insurance portfolio. In a general or nonmathematical
sense, risk assessment surveys may be used to identify risk management
strengths and weaknesses of individual risks and/or risk portfolios.
For example, if a particular risk weakness is identified through
a survey, e.g., an outdated fire suppression system in a manufacturing
plant, the insured may be encouraged to make appropriate changes
to reduce the problem. Alternatively, premiums may be increased
to compensate for the increased risk factor. Risk assessment surveys
may also be used to harvest data for increasing the overall data
available for risk modeling.
[0007] Risk assessment surveys are typically developed and carried
out by risk management specialists, and may involve a series of
specially selected questions both directly and indirectly related
to the insurance coverage carried by the insured party. The survey
may also involve direct inspections or observations of buildings,
operations, etc. Accordingly, the costs associated with risk assessment
surveys are typically not insignificant. However, it is difficult
to determine (especially beforehand) if the costs associated with
risk assessment surveys are "worth it," i.e., if they
will provide meaningful information as to significantly impact risk
management decisions and/or risk modeling calculations. Heretofore,
prior quantitative determinations of the value associated with surveys
have not been possible, leaving insurers without an accurate tool
to determine when to proceed with surveys for a risk or portfolio.
SUMMARY OF THE INVENTION
[0008] An embodiment of the present invention relates to a method
for determining the prior net benefit of obtaining data relating
to an individual risk in an insurance portfolio, via a survey or
similar procedure, for use in a predictive model or otherwise. (By
"individual risk," it is meant a single insured property,
e.g., a building, item of equipment, vehicle, company, person, or
parcel of land, as well as a grouping of such insured properties.)
The method produces as an output a quantitative estimation (e.g.,
dollar value) of the net benefit of obtaining survey data for a
risk, calculated as the benefit of the survey less the cost of the
survey, where the benefit of the survey relates to a quantitative
increase in predictive accuracy resulting from incorporating the
survey data into the predictive model. With prior knowledge of a
survey's net benefit, either positive or negative, it is possible
to make a more informed decision as whether or not to carry out
a survey for a particular risk.
[0009] The survey benefit method may be implemented in conjunction
with a risk/loss model such as a Bayesian predictive model that
combines historical data, current data, and expert opinion for estimating
frequencies of future loss and loss distributions for individual
risks in an insurance portfolio. The purpose of the Bayesian model
is to forecast future losses for the individual risk based on the
past losses and other historical data for that risk and similar
risks. In addition to the Bayesian predictive model, the method
utilizes a revenue model, a model for the cost of obtaining additional
data (e.g., survey cost), and probability distributions of population
characteristics.
[0010] Initially, a risk model (e.g., a Bayesian predictive model)
is developed at the individual risk level for mathematically estimating
the probability of expected loss given a set of information about
the risk. The risk model is incorporated into a profitability model
for the risk, which also includes premium and expense models for
the risk. (Generally speaking, the profitability model is a statistical
"expansion" of the following insurance truism: profit=premiumslossesmarginal
expenses.) Subsequently, a probability distribution relating to
the type or category of information possibly to be obtained by way
of a survey is developed or determined, which is used as a basis
for determining the gross value of obtaining the information. In
particular, the gross value is the projected profitability of the
best action or outcome (e.g., of insuring or not insuring the risk)
given the additional information obtained from the survey, less
the projected profitability of the best action or outcome to be
expected without knowing the additional information. From the gross
value, the net benefit of conducting the survey is determined, e.g.,
net benefit=gross valuesurvey cost. If the net benefit is positive,
that is, if the benefit of conducting a survey outweighs the survey's
cost, rational insurers will carry out the survey. If not, insurers
may opt not to conduct the survey.
BRIEF DESCRIPTION OF THE DRAWINGS
[0011] The present invention will be better understood from reading
the following description of nonlimiting embodiments, with reference
to the attached drawings, wherein below:
[0012] FIG. 1 is a schematic diagram of a system and method of
predictive modeling for estimating frequencies of future loss and
loss distributions for individual risks in an insurance portfolio;
[0013] FIGS. 2 and 5 are flow charts showing the steps of the method
in FIG. 1;
[0014] FIGS. 3A3F and 4A4C show various equations used in carrying
out the method;
[0015] FIG. 6 is a schematic view of a system for determining the
prior net benefit of obtaining data relating to an individual risk
in an insurance portfolio, via a survey or similar procedure, according
to an embodiment of the present invention; and
[0016] FIG. 7 is a flow chart showing the steps of a method carried
out by the system in FIG. 6.
DETAILED DESCRIPTION
[0017] An embodiment of the present invention relates to a method
for determining the prior net benefit of obtaining data relating
to an individual risk in an insurance portfolio, via a survey or
similar procedure, for use in a predictive model or otherwise. (By
"individual risk," it is meant a single insured property,
e.g., a building, item of equipment, vehicle, company, person, or
parcel of land, as well as a grouping of such insured properties.)
The method produces as an output a quantitative estimation (e.g.,
dollar value) of the net benefit of obtaining survey data relating
to a risk, calculated as the benefit of the survey less the cost
of the survey, where the benefit of the survey relates to a quantitative
increase in predictive accuracy resulting from incorporating the
survey data into the predictive model. With prior knowledge of a
survey's net benefit, either positive or negative, it is possible
to make a more informed decision as whether or not to carry out
a survey for a particular risk.
[0018] The survey benefit method may be implemented in conjunction
with a risk model such as a Bayesian predictive model that combines
historical data, current data, and expert opinion for estimating
frequencies of future loss and loss distributions for individual
risks in an insurance portfolio. The purpose of the Bayesian model
is to forecast future losses for the individual risk based on the
past losses and other historical data for that risk and similar
risks. In addition to the Bayesian predictive model, the method
utilizes a revenue model, a model for the cost of obtaining additional
data (e.g., survey cost), and probability distributions of population
characteristics.
[0019] Initially, a risk model (e.g., a Bayesian predictive model)
is developed at the individual risk level for mathematically estimating
the probability of expected loss given a set of information about
the risk. The risk model is incorporated into a profitability model
for the risk, which also includes premium and expense models for
the risk. (Generally speaking, the profitability model is a statistical
"expansion" of the following insurance truism: profit=premiumslossesmarginal
expenses.) Subsequently, a probability distribution relating to
the type or category of information possibly to be obtained by way
of a survey is developed or determined, which is used as a basis
for determining the gross value of obtaining the information. In
particular, the gross value is the projected profitability of the
best action or outcome (e.g., of insuring or not insuring the risk)
given the additional information obtained from the survey, less
the projected profitability of the best action or outcome to be
expected without knowing the additional information. From the gross
value, the net benefit of conducting the survey is determined, e.g.,
net benefit=gross valuesurvey cost. If the net benefit is positive,
that is, if the benefit of conducting a survey outweighs the survey's
cost, rational insurers will carry out the survey. If not, insurers
may opt not to conduct the survey.
[0020] The process for developing a Bayesian predictive model is
described below with reference to FIGS. 15. The method for determining
the prior net benefit of obtaining data relating to an individual
risk in an insurance portfolio via a survey or similar procedure
is described further below with respect to FIGS. 6 and 7.
[0021] FIGS. 15 illustrate a method or system 10 of predictive
modeling for generating a forecast of expected loss 12 for individual
risks 14a, 14b, 14c, 14d, etc. in an insurance portfolio 16. Typically,
this will be done for insurancerelated purposes, for determining
premium levels and the like. By "individual risk," it
is meant a single insured property, e.g., a building, item of equipment,
vehicle, company/business, person, operation/manufacturing line,
or parcel of land. For generating the loss forecast 12, the method
uses a Bayesian procedure 18 that incorporates historical data 20
relating to the individual risk 14b in question. The historical
data 20 will typically comprise information somehow relevant or
related to the risk, and may include any of the following: recorded
losses for the risk, with date, amount and type of loss, for a given
loss basis (such as paid or incurred); the period during which the
risk was exposed to recorded losses, namely, the effective and expiration
dates of any policies applying to the risk; the terms and conditions
of the policies applying to the risk, principally deductible, limit,
and coinsurance; and various characteristics of the risk. For example,
for a building such characteristics could include value, occupancy,
construction type, and address/location.
[0022] The Bayesian procedure 18 also utilizes historical data
20 relating to similar risks 22. By "similar risk," it
is meant a risk other than the individual risk 14b that has some
logical connection or relationship thereto, such as features or
characteristics in common, at least in a general sense. For example,
if the individual risk 14b is a hospital, then similar risks could
include other hospitals, other medical facilities, or even other
buildings within a relevant (i.e., the same or similar) geographical
area. The similar risks may be risks within the portfolio 16, but
do not have to be. As should be appreciated, the historical data
from the similar risks provides a significantly larger data pool
than just the historical data for the individual risk 14b by itself.
It is relevant to the loss forecast for the individual risk 14b
because data from a similar risk will typically tend to have some
bearing on the individual risk, i.e., from a statistical or probabilistic
standpoint, similar risks will likely experience similar losses
over time. For example, if all the hospitals over a certain size
in a particular area experience at least a certain amount of loss
in a given period, such information will tend to increase the probability
that a similar hospital in the same area will also experience at
least the same loss.
[0023] Expert opinion 24 relating to the individual risk 14b is
also obtained and utilized as part of the Bayesian procedure 18
calculations. The expert opinion 24 acts as a baseline for calculating
the loss forecast 12 when little or no historical data 20 is available.
Thus, where historical data is unavailable, the expert opinion 24
dominates the predictive calculation. The expert opinion 24 is typically
provided as (or expressed as part of) a mathematical function or
model that defines an estimated probability distribution of some
aspect of the individual risk 14b or a related or similar risk 24.
As its name implies, the expert opinion 24 may be obtained from
professionals in the field who have studied some aspect of the individual
or similar risks in question. Expert opinion may also be obtained
from reference works. For a particular portfolio, the expert opinion
may collectively include input from a number of professional sources,
each of which relates to one or more aspects of the individual or
similar risks. In other words, when implementing the method 10,
it may be the case that a number of different functions/models are
obtained and utilized as expert opinion, to more fully characterize
the individual or similar risks in the Bayesian procedure 18.
[0024] As an example, in a simple case where all the risks in a
portfolio are generally the same except for value, the frequency
of loss for such risks might be characterized as the following probability
distribution: frequency of loss=c(v/v.sub.0).sup.b, where
[0025] "c" and "b" are system parameters
[0026] v=value
[0027] v.sub.0=reference size/value
[0028] Here, the equation itself might be considered expert opinion,
i.e., obtained from a professional/expert or reference work, as
might the range of values for the system parameters "c"
and "b". For example, given the equation and system parameters,
an expert might be consulted to provide values for "c"
and "b" that give the highest probability to fit the data.
Thus, expert opinion might be solicited for selecting the best model
based on the type of data to be modeled, as well as the best system
parameters given a particular model.
[0029] For the Bayesian procedure 18, current data 26 may also
be obtained and utilized. "Current" data 26 is the same
as historical data but is instead newly obtained as the method 10
is carried out over time. For example, if an individual risk 14b
experiences a loss after the method/system 10 has been implemented
initially, then information about this loss may be entered into
the system 10 as current data 26.
[0030] FIG. 2 summarizes the steps for carrying out the method
10 for forecasting the future losses 16 for an individual risk 14b.
As discussed further below, these steps may be performed in a different
order than as shown in FIG. 2, e.g., it will typically be the case
that expert opinion is obtained after first establishing a predictive
model. At Step 100, the expert opinion 24 relating to the individual
risk 14b and/or similar risks 22 is obtained. Then, at Step 102,
the historical data 20, again relating to the individual risk and/or
similar risks 22 is obtained. If historical data 20 is not available,
then this step will be bypassed until historical and/or current
data become available. In such a case, the Bayesian procedure 18
is carried out with the expert opinion 24 only, which, as noted
above, acts as an estimation or baseline.
[0031] At Step 104 in FIG. 2, the historical data 20, any current
data 26, and expert opinion 24 are combined using the Bayesian procedure
18. The effect of the Bayesian procedure 18 is to forecast the future
losses 12 for the individual risk 14b based on the past losses and
other historical data 20 for that risk 14b and similar risks 22.
Typically, the Bayesian procedure 18 will utilize a Bayesian predictive
model as shown by equation 28 in FIG. 3A. In equation 28, a predictive
conditional probability distribution "f (yx)" of forecast
losses ("y") for the individual risk 14b, given all historical
data ("x"), is represented in terms of: (i) a probability
distribution "f(y.theta.)" of the forecast losses y given
a system parameter set (".theta."), i.e., a forecast losses
likelihood function; (ii) a probability distribution "f(x.theta.)"
for the historical data, i.e., an historical data likelihood function;
and (iii) a prior probability density function of the parameter
set "f(.theta.)."
[0032] Equation 28 in FIG. 3A is generally applicable in carrying
out the method 10. This equation is derived with reference to Steps
106112 in FIG. 2, provided for informational purposes. To derive
equation 28, at Step 106, the conditional probability distribution
f(yx) of forecast losses y for the individual risk 14b, given all
historical data x, is represented as a weighted sum of , probability
distributions, as shown by equation 30 in FIG. 3B. The weighted
sum may be an integral of the probability distribution f(y.theta.)
of the forecast losses y given the system parameter set .theta.
times a parameter set weight "f(.theta.x)." Here, the
parameter set weight f(.theta.x) is a posterior probability density
function of the system parameters .theta. given the historical data
x. Equation 30 is a standard equation for the predictive distribution
of a random variable of interest y given observed data x.
[0033] At Step 108, the probability distributions f(y.theta.)
for forecast losses y are arranged to depend on the parameter set
.theta., indexed by an index "i". At Step 110, the probability
distributions f (x.theta.) for historical data are also arranged
to depend on the same parameter set .theta., also indexed by the
index "i". Next, at Step 112, the posterior probability
density function f (.theta.x) is calculated as the probability
distribution of the historical data given the parameter set f (x.theta.),
times the prior probability of the parameter set f (.theta.), obtained
from the expert opinion 24. This is shown as equations 32 in FIG.
3C (these equations are standard representations of Bayes' theorem
for probability densities). Thus, combining equations 30 and 32,
the conditional probability distribution f (yx) of forecast losses
y for the individual risk 14b, given all historical data x, is as
shown by equation 28 in FIG. 3A. This can be further represented
by: f(yx)=f(y.theta..sub.1)p(.theta..sub.1)+f(y.theta..sub.2)p(.theta..sub
.2)+ . . . where each "p" is the probability of the particular
respective system parameter .theta..
[0034] Starting with the predictive model 28 (FIG. 3A), the probability
distributions f (y.theta.) and f (x.theta.) are obtained for the
forecast losses y and historical data x, respectively, using a compound
Poisson process model. Generally speaking, a Poisson process is
a stochastic process where a random number of events (e.g., losses)
is assigned to each bounded interval of time in such a way that:
(i) the number of events in one interval of time and the number
of events in another disjoint (nonoverlapping) interval of time
are independent random variables, and (ii) the number of events
in each interval of time is a random variable with a Poisson distribution.
A compound Poisson process is a continuoustime stochastic process
"Y(t)" represented by equation 34 in FIG. 3D, where Y(t)
represents the aggregate loss, "N(t)" is a Poisson process
(here, the underlying rate of losses), and "X.sub.i" are
independent and identically distributed random variables which are
also independent of "N(t)" (here, X.sub.i represents the
severity distribution of the losses). If full knowledge of the characteristics
of a risk 14b were available, historical and forecast losses for
that risk could be approximated by a compound Poisson process, in
which losses for each type of loss occur according to a Poisson
process, and where the size of "groundup" loss is sampled
from a severity distribution depending on the type of loss (groundup
loss refers to the gross amount of loss occurring to a reinsured
party, beginning with the first dollar of loss and after the application
of deductions). Here, in order to accommodate heterogeneity in a
class of similar risks because full knowledge of a risk's characteristics
may not be available, losses for each risk are modeled as a finite
mixture of compound Poisson processes, as at Step 114 in FIG. 2.
As noted, the parameters of the compound Poisson process will typically
be the underlying rate of losses (N(t)) and the severity distribution
(X.sub.i) of the groundup losses, which depend on the known characteristics
20 of the risk 14b. In the case of a building, such characteristics
will typically include value, occupancy, construction type, and
address, and they may also include any historical claims/losses
for that risk.
[0035] At Step 116, the probability distribution f (yx) is calculated
or approximated to produce the probability distribution of losses
12 for the forecast period for the individual risk 14b. With respect
to equation 28 in FIG. 3A, the expert opinion from Step 100 is incorporated
into the equation as the prior probability density function f (.theta.).
Then, at Step 118, forecasts of paid claims for the individual risk
14b may be obtained by applying limits and deductibles to the forecast
of losses 12 for that risk 14b. Generally, gross loss "Z"
(see equation 36 in FIG. 3E) can be represented as the sum of losses
"x.sub.i" from i=1 to N, where "N" is a frequency
of loss, but where each loss x is reduced by any applicable deductibles.
Thus, the final outcome of the system 10 is represented as shown
in equations 38 and 40 in FIG. 3F. At Step 120, current data 26
may be incorporated into the method/system 10 on an ongoing manner.
[0036] For each individual risk 14a14d, the method 10 may also
be used to produce breakdowns of forecasted expected loss by type
of loss, a forecasted probability distribution of losses, a calculation
of the effect of changing limits, deductibles, and coinsurance on
the loss forecast, and a forecasted expected loss ratio, given an
input premium. The method 10 may also be used to produce joint probability
distributions of losses for a forecast period for risks considered
jointly, as indicated by 42 in FIG. 1.
[0037] The abovedescribed Bayesian procedure for estimating the
parameters of a compound Poisson process for the purpose of predictive
risk modeling will now be described in greater detail.
[0038] For a portfolio 16, the ultimate aim of the predictive model
should be to produce a probability distribution for the timing and
amounts of future claims, by type of claim, given the information
available at the time of the analysis, i.e., the historical data
20. This information 20 will generally include: (i) past claims;
(ii) past coverages, including effective dates, expiration dates,
limits, deductibles, and other terms and conditions; (iii) measurements
on past risk characteristics such as (in the case of property coverage)
construction, occupancy, protection, and exposure characteristics,
values, other survey results, and geographic characteristics; (iv)
measurements on past environmental variables, such as weather or
economic events; (v) future coverages (on a "whatif"
basis); (vi) measurements on current risk characteristics; and (vii)
measurements on current and future environmental variables. Future
environmental variables can be treated on a whatif basis or by
placing a probability distribution on their possible values. For
simplicity, it may be assumed (as herein) that current and future
environmental variables are treated on a whatif basis.
[0039] In the formulas discussed below, the following abbreviations
are used:
cl1=future claims occurring in the period t.sub.0 to t.sub.1
cv1=actual or contemplated future coverages for the period t.sub.0
to t.sub.1
rm1=measurements on risk characteristics applicable to the period
t.sub.0 to t.sub.1
ev1=assumed environmental conditions for the period t.sub.0 to
t.sub.1
cl0=future claims occurring in the period t.sub.1 to t.sub.0 (or
more generally, for a specified past period)
cv0=actual past coverages for the period t.sub.1 to t.sub.0
rm0=measurements on risk characteristics applicable to the period
t.sub.1 to t.sub.0
ev0=environmental conditions for the period t.sub.1 to t.sub.0
[0040] The probability distribution for the timing and amounts
of future claims, by type of claim, given the information available
at the time of the analysis, can be written as:
p(cl1cv1, rm1, ev1, cl0, cv0, rm0, ev0)
[0041] where "p" denotes a conditional probability function
or probability density where the variables following the bar are
the variables upon which the probability is conditioned, i.e., a
probability density of cl1 given variables cv1, rm1, ev1, cl0, cv0,
rm0, and ev0. (It should be noted that this is a more detailed rendering
of the more generalized conditional probability distribution "f
(yx)" noted above.) Construction of the predictive model begins
by introducing the set of parameters, collectively denoted by .theta.,
which denote the risk propensities of the risks 14a14d in the portfolio
16. A standard probability calculation results in equation 50 as
shown in FIG. 4A. (Again, it may be noted that equation 50 is a
more detailed equivalent of equation 28 in FIG. 3A.) Equation 50
is true regardless of the assumptions of the model.
[0042] The model assumptions now introduced are as follows. Firstly,
p (cl1cv1, rm1, ev1, cl0, rm0, ev0, .theta.)=p (cl1cv1, rm1,
ev1, .theta.)
[0043] which expresses the assumption that if the loss/risk propensities
.theta. are known, the future claims for the portfolio depend only
on the current and future coverages, risk measurements, and environmental
variables, and not on the past claims and other aspects of the past.
The validity of this assumption depends on the ability to construct
a model that effectively captures the information from the past
in terms of knowledge about risk propensities. Secondly,
p (cl0cv1, rm1, ev1, cv0, rm0, ev0, .theta.)=p (cl0cv0, rm0,
ev0, .theta.)
[0044] which expresses the assumption that, provided past coverages,
risk measurements, and environmental variables are known, knowing
future values for these quantities is irrelevant when considering
the likelihood of past claims data. This assumption does not exclude
the case in which present risk measurements can shed light on past
risk characteristics, for example when a survey done more recently
sheds light on risk characteristics further in the past. Thirdly,
p (.theta.cv1, rm1, ev1, cv0, rm0, ev0)=p (.theta.)
[0045] which expresses the assumption that the prior probability
distribution for the risk propensities p(.theta.) does not depend
on additional information. The risk propensities can be expressed
in such a way that this assumption is valid, for example by assigning
prior probability distributions of risk propensity to classes and
types of risks, rather than to individual risks.
[0046] Given these three assumptions, the predictive model can
be written as equation 52 in FIG. 4B.
[0047] The Bayesian model estimation process includes the following
steps, as shown in FIG. 5. Starting with the model from equation
52 in Step 130, the future claims (losses) likelihood function p(cl1cv1,
rm1, ev1, .theta.) is constructed at Step 132. At Step 134, the
past claims (historical data) likelihood function p(cl0cv0, rm0,
ev0, .theta.) is constructed. At Step 136, expert opinion is obtained
for the prior distribution for risk propensities p(.theta.). Next,
at Step 138, the Bayesian predictive model is solved or approximated.
Step 140 involves model criticism and checking.
[0048] The past and future claims likelihood functions may be constructed
as follows (in the basic case). Conditional on a fixed and known
value for .theta., claims are considered to be generated by a multivariate
compound Poisson process, in which groundup losses occur according
to a Poisson process with rate .lamda. (i, j) for risk "i"
and type of loss "j" (as noted above, the risk 14a14d
could be a building, an establishment, or any other specific entity
within the portfolio 16). The groundup loss amounts are considered
to be generated independently from a loss distribution F (i, j)
again depending on risk i and type of loss j. Both A (i, j) and
F (i, j) depend on risk measurements for risk i and environmental
variables, in such a way that
.lamda..sub.Past (i, j)=g.sub.j (past risk measurements for i,
past environmental variables, .theta.)
.lamda..sub.Future (i, j)=g.sub.j (current risk measurements for
i, current environmental variables, .theta.)
F.sub.Past (i, j)=h.sub.j (past risk measurements for i, past environmental
variables,.theta.)
F.sub.Future(i,j)=h.sub.j (current risk measurements for i, current
environmental variables, .theta.)
[0049] The functions g.sub.j and h.sub.j are known functions that
are designed to produce a flexible set of representations for the
way in which the loss process for a risk depends on the characteristics
of a risk and environmental variables. A hypothetical example could
be
g.sub.j (past risk measurements for i, past environmental variables,
.theta.)=exp(a.sub.0+a.sub.1 ln(x.sub.1)+a.sub.2x.sub.2+ . . .)
for occupancy=A, region=X, . . . =exp(b.sub.0+b.sub.1 ln(x.sub.1)+b.sub.2x.sub.2+
. . . ) for occupancy=B, region=X, . . .
where x,=square footage, x.sub.2=mean winter temperature for location,
[0050] In this case a.sub.0, a.sub.1, a.sub.2, b.sub.0, b.sub.1,
b.sub.2, . . . are all elements of the collection of parameters
that is denoted by .theta..
[0051] The basic model makes the assumption that the past risk
propensities equal the future risk propensities, and the functions
linking the risk propensities to the loss process are the same in
past as in the future, so that all the differences in frequency
and severity between past and future are explained by changes in
risk measurements and environmental variables. Extensions to the
model allow for risk propensities and risk characteristics to evolve
according to a hiddenMarkov model. Another extension is to allow
timedependent rates for the Poisson processes generating the groundup
losses. This may be necessary if forecasts of total claims for partialyear
periods are required in order to deal with seasonality issues. Allowing
for seasonally changing rates also allows for slightly more precision
in estimating the claims process. It should be noted that the existing
model allows for the predicted claims for a risk (i) to be influenced
by the number and amount of past claims for that same risk if coverage
existed on that risk in the past.
[0052] In practice, loss distributions are parameterized by a small
number of parametersfor example, F may be lognormal with parameters
.mu. and .sigma., in which case
.mu..sub.Past(i, j)=h.mu..sub.j(past risk measurements for i, past
environmental variables, .theta.)
.mu..sub.Future(i, j)=h.mu..sub.j(current risk measurements for
i, current environmental variables, .theta.)
.sigma..sub.Past(i, j)=h.sigma..sub.j(past risk measurements for
i, past environmental variables, .theta.)
.sigma.Future(i, j)=h.mu..sub.j(current risk measurements for i,
current environmental variables, .theta.)
[0053] The model uses finite mixtures of lognormal distributions
in order to approximate a wider range of loss distributions than
a single lognormal distribution can. In this case there are several
values for .mu. and .sigma., one for each component, as well as
a set of mixing parameters. The extension to the model is that now
there are more functions, but each is still a known function with
unknown parameters that are part of the collection of parameters
.theta..
[0054] The method described does not specify the functions linking
the risk measurements and environmental variables to the parameters
of the compound process. Functions that have been shown to work
well in practice include linear, loglinear and power functions,
and nonlinear functions that are piecewise continuous such as piecewise
linear functions and natural splines. Useful are functions of linear
or nonlinear combinations of several variables, such as the ratio
of value to square footage, or contents value to building value
in the case of property risks.
[0055] To model the claims process, given a model for the groundup
loss process, it is necessary to apply terms of coverage, limits
and deductibles to the modeled groundup loss process. If no coverage
is in effect over an interval of time for a given risk, all losses
generated by the groundup loss process during that interval of
time are not observed. Any losses below the deductible are not observed
and any losses above the limit are capped at the limit. Because
of the characteristics of the compound Poisson process, the claims
process is also a compound Poisson process (during periods of coverage),
with the rate of claims for risk i and loss type j being
.lamda.(i, j)*Pr(X.sub.i,j>deductible.sub.i) where X.sub.i,j
has the distribution given by F.sub.i,j
and the size of the claims for risk i and loss type j having the
same probability distribution of that of
min(X.sub.i,jdeductible.sub.i, polmit.sub.i) conditional on this
quantity being positive.
[0056] Once the past claims process and the future claims process
have both been specified in terms of two (related) compound Poisson
processes, it is straightforward to write the likelihood functions
for past claims and future claims using standard formulas. The function
can be expressed in simple mathematical terms although the formula
is lengthy when written. A single compound Poisson process has a
likelihood function as shown by equation 54 in FIG. 4C, where "N"
is the number of claims (above deductible), "x.sub.i"
are the sizes of the claims (after deductible), ".lamda."
is the annual rate of the Poisson process, "z" is the
number of years exposed to losses, and "f" is the probability
density of the claim distribution (there is a simple modification
for distributions with masses at a single point which occur when
there is a limit).
[0057] Once .theta. is known, .lamda. and f can be calculated for
each combination of risk and loss type for past claims. It is assumed
that losses occur independently at each risk, conditional on .theta.,
so the past likelihood for the whole portfolio is just the product
of factors, one factor for each combination of risk (i) and loss
type (j), where each factor has the form given above, except that
z, .lamda., and f depend on (i, j) and N is replaced by N(i, j)
which is the number of past claims for risk (i) and loss type (j).
The same process produces the likelihood for future claims (z, .lamda.,
and f may be different in the future likelihood function than in
the past likelihood function even for the same risk and loss type).
[0058] The remaining portion of the general formula involves the
prior probability distribution p(.theta.). This is obtained through
expert elicitation, as at Step 100. Where there is sufficient loss
data, the effect of the prior probability distribution tends to
be small. However, in the collection of parameters given by .theta.
there may be some parameters (such as the frequency for a particular
class of business with a small exposure) for which there is little
claim data, in which case these parameters will be more sensitive
to the expert opinion incorporated in the prior.
[0059] Once the past and future likelihood functions and the prior
distribution have been specified, the probability distribution of
predicted claims can be obtained by solving the predictive model
integral given above. This produces a probability distribution for
the predicted claims for each risk and each type of loss in the
future portfolio, given coverage assumptions. Solving this sort
of integral is a central topic of Bayesian computation and is the
subject of extensive literature. In general, numerical techniques
are required, a popular simulation method being Markov Chain Monte
Carlo. An alternative procedure is to obtain the maximum likelihood
estimate of .theta., which is the value of .theta. that maximizes
the past likelihood function. Since all the quantities besides .theta.
in the past likelihood function are known (these are past claims,
past coverages, past risk measurements, and past environmental variables),
this function, namely
p (cl0cv0, rm0, ev0, .theta.)
[0060] can be maximized as a function of .theta.. It is known that
under most conditions and given enough data, the likelihood, as
a function of .theta., can be approximated by a multidimensional
quadratic surface. Experience using the procedure with real data
reinforces this theoretical finding. If this is the case, then the
probability distribution of .theta., given the past data, can be
approximated as a multivariate Normal distribution. A further approximation
uses the mean of this multivariate Normal distribution as the single
point estimate of .theta. (the Bayes posterior mean estimate).
[0061] Given a single point estimate of .theta., the predictive
distribution of future claims is straightforward to calculate, since
it is the given by the future likelihood. The predicted future groundup
losses are given by a compound Poisson process whose parameters
are given in the simplest case by
.lamda..sub.Future(i, j)=g.sub.j(current risk measurements for
i, current environmental variables, .theta.)
.mu..sub.Future(i, j)=h.mu..sub.j(current risk measurements for
i, current environmental variables, .theta.) .sigma..sub.Future(i,
j)=h.sigma..sub.j(current risk measurements for i, current environmental
variables, .theta.) where .theta. is set to the Bayes posterior
mean estimate, and the claims compound Poisson process is obtained
by applying deductible and limit adjustments as described previously.
[0062] If the predicted annual average loss (after deductible and
limit) is desired for risk (i) and loss type (j), and if the posterior
mean estimate is being used, then the average annual loss is given
by .lamda.(i,j)*Pr(X.sub.i,j>d.sub.i)*E(min(X.sub.i,jd.sub.i,l.sub.i)X.
sub.i,j>d.sub.i) where "d" and "l" refer
to deductible and limit respectively. If the severity distributions
are given by mixtures of lognormals, then this formula can be easily
calculated. If a single point estimate of .theta. is not desirable,
then the posterior distribution of .theta. can be approximated by
a finite distribution putting probability on a finite set of points.
In this case the average annual loss is given by a weighted sum
of terms like that above. In either case, the predictive modeling
procedure produces a calculation for that can be done quickly by
a computer, and does not require simulation. Calculation of average
annual losses by layer is also straightforward.
[0063] The method/system 10 may be implemented using a computer
or other automated data processing or calculation device.
[0064] FIGS. 6 and 7 show in more detail the method/system for
determining the prior net benefit of obtaining survey data relating
to an individual risk or category of risks 58 in an insurance portfolio.
[0065] Initially, at Step 150 a risk model 60 is developed at the
individual risk level, if needed. If one or more models have already
been developed, then an existing model may be used. The risk model
may be a Bayesian predictive model developed according to the above.
Generally speaking, the risk model is a mathematical model of the
expected loss for a risk having certain characteristics:
E (lossbasic risk info, additional risk info, offer terms, contract
accepted, loss prevention plan)
[0066] In other words, the risk model looks at the probability
or expectation of loss given a set of information including (in
this example) basic risk information (e.g., location), additional
risk information (e.g., building characteristics), offer terms (e.g.,
policy terms, insurance limits, deductible), whether the insurance
contract has been accepted, and whether a loss prevention plan is
in place and/or the characteristics of such a loss prevention plan.
[0067] The risk model 60 is incorporated into a profitability model
62, as at Step 152. This may involve developing a premium model,
as at Step 154, and an expense model, as at Step 156. Generally,
the profitability model for a risk may be expressed as the following:
(Uoffer terms_a, acceptance)=E (premium)E (loss)E (marginal
expense)
[0068] Here, "U" is the profitability, "E (premium)"
is the premium model (e.g., expected premium as defined by the insurance
contract), "E (loss)" is the risk model, and "E (marginal
expense)" is the expense model, e.g., the expected value of
marginal expenses of the insurance carrier as relating to this insurance
contract, as possibly determined from expert opinion. Overall, the
profitability model sets forth the expected profit "U"
given certain terms "a" and acceptance by the insured.
In other words, given that an insured party has accepted the offer
for an insurance contract having certain terms "a," the
profitability model sets forth the expected profit.
[0069] At Step 158, a retention model 64 is developed, that is,
a model of the probability of a potential insured party accepting
a particular offer. The retention model 64 is incorporated into
the calculation for determining the gross value associated with
obtaining additional information by survey. The retention model
(probability of acceptance) is given as:
E (Ua)=E (Uoffer terms_a, acceptance)Pr (acceptanceoffer terms_a)+E
(Uoffer terms_a, decline)Pr (declineoffer terms_a)
[0070] Here, "E (Ua)" is the expectation of profitability
U given an action "a," e.g., offering an insurance contract.
"Pr" is the probability, e.g., the probability of a potential
insured party accepting the offer given certain offer terms "a."
As should be appreciated, the second half of the equation (relating
to a party declining the offer) reduces to a 0 (zero) value, because
there is no expected profitability in the case where a party declines
the offer for insurance.
[0071] At Step 160, the gross value 66 of the additional information
to be obtained by way of a survey is determined. Generally speaking,
the gross value of the information is calculated as the profitability
68 of the best action given additional information "X"
less the profitability 70 of the best action without knowing X.
In other words, if more profit is expected from knowing information
X than from not knowing information X, then obtaining the information
X has a positive gross value. This can be expressed more precisely
as follows: [Gross value]=E.sub.x [max.sub.aE (UX, a)]max.sub.aE.sub.x(E
(UX, a)) where: X=additional information E.sub.x=expectation function
max_aE (UX, a)=profitability of best action given additional information
X max_aE.sub.X(E (UX, a))=max_aE (Ua)=profitability of best action
w/o X max_a=payoff for best possible action As part of this determination,
it will typically also be necessary to obtain the probability distribution
72 of the additional information X, that is, the marginal distribution
of the additional information. The probability distribution may
be obtained from expert opinion and/or historical data.
[0072] As a simple example of the above, suppose that an insurance
carrier insures warehouses 58 within a certain geographical area,
e.g., the manufacturing district of a city. Additionally, suppose
that all the warehouses either have a flat roof or a sloped or pitched
roof. Further suppose that past insurance contracts have resulted
in an average of $40 profit (per time period) for warehouses with
pitched roofs, and an average loss of $100 for warehouses with
flat roofs. The relevant issue is whether it is "worth it"
to determine beforehand, via a survey procedure 73, if a prospective
warehouse has a sloped roof or a flat roof 74, prior to the insurance
carrier agreeing to insure the warehouse.
[0073] From expert opinion and/or historical data, the probability
of the additional information is determined or estimated in advance.
Here, for example, suppose 20% of all warehouses have flat roofs,
and 80% have pitched roofs. Without a survey 73, and thereby without
knowing whether a particular warehouse has a flat or pitched roof
74, the insurance carrier will insure all proffered warehouses,
e.g., the insurer has no reason for declining any particular warehouse.
(Additionally suppose that the warehouses accept the offered insurance
under a standard contract.) In this case, the expected profitability
of the best action (e.g., insuring all warehouses) without knowing
the additional information is given as the following: (20%)($100)+(80%)(+$40)=+$12
In other words, out of 100 warehouses seeking insurance, 100 are
offered and accept insurance. Out of these, 20 will have flat roofs
with a total expected loss of $2000, and 80 will have pitched roofs
with a total expected profit of +$3200. This results in a net profit
of +$1200, or $12/warehouse.
[0074] If a survey 73 is conducted, the insurer will know in advance
that a particular warehouse has a flat or pitched roof 74. In such
a case, knowing that a flat roof results in an average loss, a rational
insurer will decline all flatroofed warehouses. Thus, the profitability
of the best action (e.g., insuring only pitchedroof warehouses)
given the additional information as to roof type is as follows:
(20%)($0.fwdarw.insurance is declined, therefore no profit or loss)+(80%)(+$40)=$32
In other words, out of 100 warehouses seeking insurance, the 20
having flat roofs are denied insurance, while the 80 having sloped
roofs are granted insurance, resulting in $3200 profit, or $32/warehouse
among all 100 warehouses.
[0075] The gross value of the additional data=$32$12=$20/warehouse.
In other words, the profit for insuring 100 randomly selected warehouses
would be $1200, while the profit for only insuring the 80 of those
warehouses having pitched roofs (as determined from a survey) would
be $3200. The gross benefit of conducting the survey is $2000, or
$20/warehouse.
[0076] From the gross value of the additional information, the
net value 76 is obtained, as at Step 162. The net value 76 is calculated
as the gross value 66 less the expenses 78 associated with obtaining
the additional information, e.g., the cost of the survey: net value=gross
valuecost/survey The cost per survey can be a standard value, or
a value otherwise obtained by consulting with experts or survey
firms or professionals. For example, there might be a general cost
associated with developing/writing the survey, and a cost associated
with carrying out the survey for each property/risk, e.g., labor
costs for a worker to carry out the survey at each property/risk.
[0077] The net value will inform the decision of whether to carry
out a survey 73. If the net value is negative, then it is more likely
that a survey will not be carried out. If the net value is positive,
that is, if the gross value exceeds the associated survey costs,
then it is more likely that a survey will be carried out to obtain
survey data 74 before contracting to insure a particular risk.
[0078] The types of information to consider for possibly obtaining
by survey will depend on the nature of the risk. Examples include
credit characteristics, prior loss history, location characteristics,
construction characteristics, and the age and condition of building
fixtures. Additionally, it will typically be the case that the survey
information is correlated to some other characteristic or set of
characteristics of the property, e.g., location, occupancy, age,
and size, which act as the basic drivers for especially the risk
model.
[0079] The following sections provide another simplified example
illustrating the elements of a value of information calculation
for a hypothetical insurance survey. In this example, there is a
class of prospective insurance risks (such as commercial establishments)
with some known characteristics, as determined by information on
an insurance application, for example. However, additional information
may be obtained about these risks using certain measurements that
incur costs, for example, the information may be obtained via a
phone survey or via a more costly onsite survey. Suppose that the
phone survey can accurately classify the age of a building or equipment
system into classes: (A) 010 years, (B) 1025 years, and (C) 25
years or older. Further suppose that the onsite survey can in addition
accurately classify the condition of the system into the classes:
(a) good for its age class, (b) average for its age class, and (c)
poor for its age class. The following calculations give the value
of information for a per phone survey and per site survey. The value
of information is in dollars, for this example, and the net benefit
of the information would be obtained by subtracting the cost of
obtaining the information from the value of the information. The
possible actions of the insurer could include: (1) perform no survey,
or (2) perform a phone survey, or (3) perform a site survey, followed
in all cases by either offering a policy having a lower premium
(rate 1), offering a policy having a higher premium (rate 2), or
declining to offer coverage. Additional strategies might be available
to the insurer, such as performing a phone survey and then performing
a site survey in some cases, depending on the results of the phone
survey. These will not be considered in this example, although the
valueofinformation calculations are similar.
[0080] The following elements are used for the calculation, which
may have been obtained through a combination of historical or sample
data analysis, modelfitting, expert opinion, or the like.
[0081] Table 1 below shows the population breakdown by age and
condition, knowing only that the prospect belongs to the given class
of risks: TABLEUS00001 TABLE 1 Condition a b c Total Age A 6%
21% 3% 30% B 8% 28% 4% 40% C 6% 21% 3% 30% Total 20% 70% 10% 100%
[0082] Tables 2a and 2b below show the expected marginal net revenue
per policy, conditional on the policy being written at either rate
1 or rate 2: TABLEUS00002 TABLE 2a Offered and accepted rate 1
Condition Age a b c A 4000 3000 1000 B 3000 1000 5000 C 1000 1000
10000
[0083] TABLEUS00003 TABLE 2b Offered and accepted rate 2 Condition
Age a b c A 6000 5000 3000 B 5000 3000 3000 C 3000 1000 8000
These values would typically be obtained from a risk model combined
with premium and cost data. Assume that the marginal net revenue
is zero if the policy is not written.
[0084] Tables 3a and 3b show the rate of acceptance by the prospect
of the insurer's offer, conditional on the policy being offered
at either rate 1 or rate 2: TABLEUS00004 TABLE 3a Probability
of Insured Acceptance, conditional on insurer offer, Rate 1 Condition
Age a b c A 0.6 0.6 0.7 B 0.6 0.7 0.7 C 0.7 0.7 0.8
[0085] TABLEUS00005 TABLE 3b Probability of Insured Acceptance,
conditional on insurer offer, Rate 2 Condition Age a b c A 0.2 0.2
0.5 B 0.2 0.5 0.6 C 0.4 0.6 0.6
In the context of policy renewals, this would be termed a retention
model. In either new business or renewal contexts, this model might
be obtained through a combination of price elasticity studies or
expert opinion.
[0086] Given these three elements, the following can be calculated.
[0087] Tables 4a and 4b below show the expected marginal net revenue
per policy, conditional on the policy being offered at either rate
1 or rate 2. In the simplest case, this is obtained by multiplying
Tables 2a/2b and Tables 3a/3b. The calculation may be more complex
if adverse selection or moral hazard is modeled, as in the case
of an insurance prospect that accepts a high premium offer because
it is aware of hazards unknown to the insurer. TABLEUS00006 TABLE
4a Rate 1 Condition Age a b c A 2400 1800 700 B 1800 700 3500 C
700 700 8000
[0088] TABLEUS00007 TABLE 4b Rate 2 Condition Age a b c A 1200
1000 1500 B 1000 1500 1800 C 1200 600 4800
[0089] From this, one can obtain the optimal insurer action for
each combination of age and condition, as shown in Table 5a: TABLEUS00008
TABLE 5a Optimal insurer offer Condition Age a b c A rate 1 rate
1 rate 2 B rate 1 rate 2 decline C rate 2 rate 2 decline
[0090] The expected marginal net revenue can be obtained for each
combination of age and condition, as shown in Table 5b: TABLEUS00009
TABLE 5b Expected net revenue given insurer optimal strategy Condition
Age a b c A 2400 1800 1500 B 1800 1500 0 C 1200 600 0
For example, the optimal insurer offer for (A)(a) is to offer rate
1, whose expected payoff is $2400. The weighted average of the optimal
strategy payoffs, weighted by the prevalence of each class, gives
the overall expected net revenue for a portfolio of risks, randomly
distributed according to Table 1. This quantity is $1329 and is
the expected payoff per prospect under the site survey strategy.
[0091] In comparison, for the nosurvey strategy, the same action
must be applied to all the cells in the above tables, since there
is no information available to classify the risks as above. In this
case, the optimal strategy becomes: TABLEUS00010 TABLE 6a Optimal
insurer offer Condition Age a b c A rate 2 rate 2 rate 2 B rate
2 rate 2 rate 2 C rate 2 rate 2 rate 2
[0092] TABLEUS00011 TABLE 6b Expected net revenue given insurer
optimal strategy Condition Age a b c A 1200 1000 1500 B 1000 1500
1800 C 1200 600 4800
The expected payoff under this strategy is the weighted average
of Table 6a, weighted by Table 1. This quantity is $809, and is
the expected payoff per prospect under the nosurvey strategy. To
check that this is the optimum, replace the rate 2 tables with the
rate 1 tables and perform the same calculation.
[0093] The difference between the two expected payoffs is the value
of information, which in this case is $520 per prospect. If the
marginal cost of a site survey were less than $520, the expected
net benefit criterion would suggest adopting the site survey strategy
and performing a site survey for all prospects in this class, given
a choice between the two strategies.
[0094] The optimum set of actions under the phone survey strategy
can be shown to be: TABLEUS00012 TABLE 7a Optimal insurer offer
Condition Age a b c A rate 1 rate 1 rate 1 B rate 2 rate 2 rate
2 C rate 2 rate 2 rate 2
[0095] TABLEUS00013 TABLE 7b Expected net revenue given insurer
optimal strategy Condition Age a b c A 2400 1800 700 B 1000 1500
1800 C 1200 600 4800
This yields an expected payoff of $1025, and a value of information
of $216 per prospect. If, for example, the marginal cost of a site
survey were $600 and that of a phone survey were $100, the best
of the three (simple) strategies according to the net benefit criterion
would be the phone survey.
[0096] Since certain changes may be made in the abovedescribed
method for determining the prior net benefit of obtaining data relating
to an individual risk in an insurance portfolio via a survey or
similar procedure, without departing from the spirit and scope of
the invention herein involved, it is intended that all of the subject
matter of the above description or shown in the accompanying drawings
shall be interpreted merely as examples illustrating the inventive
concept herein and shall not be construed as limiting the invention.
