INTRODUCTION:
With
a population of almost 70 million, Iran is one of the most populous countries
in the Middle East. Total healthcare spending is expected to rise from $24.3
billion in 2008 to $50 billion by 2013 that reflects the increasing demand on
medical services. Total health spending was equivalent to 4.2% of GDP in Iran
in 2005. 73% of all Iranians have health care coverage.
On
the other hand, according to declaration of the central insurance, loss rate of
health insurance was 46.12% and 39.12% in 1388 and 1389 respectively. Central Insurance of Iran has also declared
in its official website that sum of health insurance produced premium by
Commercial insurance companies was 9850 billion Rials
in 1389 and the compensation was about 9875 billion Rials
which means loss ratio is 1.003 in 1389.Furthermore, sum of health insurance
produced premium by commercial insurance companies was 15834 billion Rials in 1390 and the compensation was about 14661 billion Rials. Therefore, loss ratio is 0.926 in 1390. Loss ratio
for health insurance in Iran insurance company was 0.94 and 0.9 in 1389 and
1390 respectively. In comparison to some
insurance lines in our country, health insurance has higher loss ratio.
This
higher ratio may increase the potential loss of insurance company so that it is
not an economical cost-effective activity for an insurance company. Regarding
this high loss rate, the insurers have to estimate and forecast their loss
distribution with more accurate modeling tools in order to compensate their
insured and continue their activity. These estimations aid insurers to be able
to set premiums fully align with their losses. Although detecting the extreme
losses in this line occurs with lower probability, it has considerable
importance and effects on estimation of future losses.
Recently,
fat tailed distributions and extreme value theory are used by Iranian risk
managements in modeling large claims for different lines such as fire insurance
and catastrophe claims in reinsurance. These studies in general show importance
of these models in estimating insurance claims.
The
specification of a loss distribution is a key ingredient of any modeling
approach in actuarial science and financial risk management. For insurers, a
proper assessment of the size of a single claim is of most importance.
Traditional methods in the actuarial literature use parametric specifications
to model single claims.
A
method which does not require the specification of a parametric model is
nonparametric kernel smoothing. This method provides valid inference under a
much broader class of structures than those imposed by parametric models.
The
detailed analysis of the available models leads to many unsolved problems of
theoretical and practical importance, and this field of research always
generates new challenges. The present contributions are further piece of this
big puzzle.
Health
insurance is one of the insurance services. Humankind has always been in danger
of lots of diseases, so people need to sponsor the charge of all this
treatments. In order to help people in those situations, insurance companies
represent various kinds of health insurance. Also, government employee or
industrial and production units in most of the countries use the group health
insurance.
According
to official data, more than 90% of Iranian people are under the coverage of at
least one kind of health insurance. The boundaries of providing health services
for patients is so much expanded that it is not at least an economical
cost-effective activity in the framework of the health insurances. In many
countries the complementary health insurances have been used to provide these
services.
In
complementary health insurance coverage, the only factor in terminating the
contract of insurance is failure to pay premiums. The coverage is based on the
fact that insured is able to pay premiums.
So
that at first, the insured pays the total costs and then the insurance company
refund the amount to the insured up to entire coverage.
Using insurances in terms of governmental
and private complementary health insurance and creating the competition among
them might have an important role on improvement of health insurance quality,
raising level of customers' satisfactions, and finally improvement of public
health (Vafaee et al., 2007).
LITERATURE:
Standard statistical methodology such as
integrated error and likelihood does not weigh small and big losses differently
in the evaluation of an estimator. Thus, these evaluation methods do not
emphasize on important part of the error: the error in the tail.
Practitioners often decide to analyze large
and small losses separately because no single, classical parametric model fits
all claim sizes. This approach leaves some important challenges: choosing the
appropriate parametric model, identifying the best way of estimating the
parameters and determining the threshold level between large and small losses.
A method which does not require the
specification of a parametric model is nonparametric kernel smoothing. This
method provides valid inference under a much broader class of structures than
those imposed by parametric models.
The detailed analysis of the available
models leads to many unsolved problems of theoretical and practical importance,
and this field of research always generates new challenges. The present
contributions are further piece of this big puzzle.
Paul Embrechts et
al. (1997) provided a textbook of modeling external events for insurance and
finance. This book is a very comprehensive textbook of probabilistic models for
large and external values of sequences of random variables and of the
statistical problems involved in fitting appropriate distributions to empirical
data of such a type. The presentation of the statistical methods is amply
illustrated by a wealth of concrete examples of data analysis from insurance,
finance, hydrology etc. recent results concerning ruin probabilities when the
claim distributions have heavy tails are presented.
McNeil (1997) provided an extensive
overview of the role of extreme value theory in risk management, as a method
for modeling and measuring extreme risks. Bolace et
al. (2003) also suggested using the kernel density estimation for actuarial
loss functions. They estimated actuarial loss functions based on a symmetric
version of the semi parametric transformation approach to kernel smoothing.
They applied this method to an actuarial study of automobile claims. The method
gives a good overall impression for estimating actuarial loss functions, and it
is capable of estimating both the initial mode and the heavy tail that is so
typical for actuarial and other economic loss distribution. They studied
properties of the transformation kernel density estimation and showed the
differences with the multiplicative bias corrected estimator with variable
bandwidth. They also added insight into the kernel smoothing transformation
method through an extensive simulation study with a particular view to the
performance of the estimation at the tail.
Bolace
et al (2005) studied kernel density estimation for heavy-tailed distributions
using the Champernowne Transformation. In this study,
a unified approach to the estimation of loss distributions was presented. They
proposed an estimator obtained by transforming the data set with a modification
of the Champernowne cdf and
then estimated the density of the transformed data by using the classical
kernel density estimator. They investigated the asymptotic bias and variance of
the proposed estimator. In a simulation study, the proposed method showed a
good performance. They also presented two applications dealing with claims
costs in insurance.
Yamada and Primbs
(2004) presented a Value-at-Risk and Conditional Value-at-Risk estimation
technique for dynamic hedging and investigated the effect of higher order
moments in the underlying on the hedging loss distributions. At first, they
approximated the underlying stock process through its first four moments
including skewness and kurtosis using a general
parameterization of multinomial lattices, and solved the mean square optimal
hedging problem. Then they plugged the moment information into the generalized
lambda distribution to extract the hedging loss distribution, and estimated its
VaR. Finally, they demonstrated how the hedging error
distribution changes with respect to non-zero kurtosis and skewness
in the underlying through numerical experiments, and examined the relation
between VaR and CVaR of the
hedging loss distributions and kurtosis of the underlying.
Gustafsson et al (2007) developed
a tailor made semiparametric asymmetric kernel
density estimator for the estimation of actuarial loss distributions. The
estimator was obtained by transforming the data with the generalized Champernowne distribution initially fitted to the data.
Then the density of the transformed data was estimated by using local
asymmetric kernel methods to obtain superior estimation properties in the
tails. They have found in a vast simulation study that the proposed semiparametric estimation procedure performs well relative
to alternative estimators. The approach should therefore be useful in applied
work in economics, finance and actuarial science involving non- and
semi-parametric techniques. This point has been demonstrated with an empirical
application to operational loss data.
Chiang Lee
(2009) focused on modeling and estimating tail parameters of commercial fire
loss severity. Using extreme value theory, he centralized on the generalized
Pareto distribution (GPD) and compared with standard parametric modeling based
on Lognormal, Exponential, Gamma and Weibull
distributions. In empirical study, the thresholds of GPD are determined through
mean excess plot and Hill plot. Kolmogorv-Smirnov and
LR goodness-of-fit test was conducted to assess how good the fit was. VaR and expected shortfall were also calculated. He also
took into account bootstrap method to estimate the confidence interval of
parameters. Empirical results show that the GPD method is a theoretically well
supported technique for fitting a parametric distribution to the tail of an
unknown underlying distribution. It can capture the tail behavior of commercial
fire insurance loss very well.
THEORETICAL
PRINCIPLES:
In finance and non-life insurance,
estimation of loss distributions is a fundamental part of the business. In most
situations, losses are small, and extreme losses are rarely observed but the
number and the size of extreme losses can have a substantial influence on the
profit of the company. In fact, for estimating loss distributions in insurance,
large and small losses are usually split because it is difficult to find a
simple parametric model that fits all claim sizes.
In This sector, we present parametric and
nonparametric model to estimate the claims distribution. For each model we
employ these steps: choosing the appropriate parametric model, identifying the
best way of estimating the parameters and determining the threshold level
between large and small losses.
KERNEL DENSITY
ESTIMATION:
Kernel density estimators belong to a class
of estimators called non-parametric density estimators. In comparison to
parametric estimators where the estimator has a fixed functional form
(structure) and the parameters of this function are the only information we need
to store, Non-parametric estimators have no fixed structure and depend upon all
the data points to reach an estimate. This estimator is widely used in practice
despite the known fact that smoothing can produce efficiency gains in finite
samples.
In kernel density estimation, the shape of
the estimated density is determined by the data, and in principle, given a
sufficiently large data set, the technique is capable of estimating an
arbitrary densityfairly
accurate. It is a nonparametric method that does not make any distributional
assumption about the underlying density. Kernel density estimation has
attracted the attention of many researchers; a good introduction to the subject
is given in Silverman (1986).
Let be
a random sample from an unknown density function Then the kernel density estimator of is
given by
Where the kernel function is
generally a unimodal probability density function and
h(> 0) is a smoothing parameter often called the bandwidth.
The basic properties of at
interior points are well-known, and under some smoothness assumptions these
include
The most common optimality criterion used
to select this parameter is the expected L2 risk function,
also known as the mean integrated squared error
The
semiparametric transformed kernel density estimator
Transformed kernel density estimator
presents a systematic approach to the estimation of loss distributions which is
suitable for heavy tailed situations. The proposed estimator is obtained by
transforming the data set with a parametric estimator and afterwards estimating
the density of the transformed data set using the classical kernel density
estimator.
Let ,
be
positive stochastic variables with an unknown cdfand
density .
The following describes in detail the transformation kernel density estimator.
(i) Estimate the parameters of by
maximizing
(ii)
Where
(iii)
Transform
the data set ,
by
the estimated cdf,
The transformation function transforms data
into the interval,
and the parameter estimation is designed to make the transformed data as close
to a uniform distribution as possible.
(iv) Calculate the classical kernel density
estimator on the transformed data, Yi, i= 1,
. . . , N:
Where and
is
the kernel function. The boundary correction, ,is
required because the is in the interval so
that we need to divide it by the integral of the part of the kernel function
that lies in this interval. The boundary correction is
defined as
.
(iv) Obtain the kernel density for the
original data, by
back transformation:
We will call this method a semi
parametric estimation procedure because a parameterized transformation
family is used.
Champernowne
distribution
We use a transformation based on the
little-known Champernowne cdf,
because it produces good results in all the studied situations and it is
straightforward to apply.
The original Champernowne
distribution has the density
Where is
a normalizing constant and and
M are
parameters. The distribution was mentioned for the first time in 1936 by D.G.
When equals 1 and the normalizing constant c equals,
the density of the original distribution is simply called the Champernowne distribution
With cdf
The Champernowne
distribution converges to a Pareto distribution in the tail, while looking more
like a lognormal distribution near 0 when.
Its density is either 0 or infinity at 0 (unless).
Parametric
models
Although the empirical distribution
function can be a useful tool in understanding claims data, there is often a
natural desire to “fit” a probability distribution with reasonably tractable
mathematical properties to claims data.
The claims actuary will also want to
consider the impact of deductibles, reinsurance arrangements and inflation on
that part of a claim which will be handled by the base insurance company. This
involves a good understanding of conditional probabilities and distributions.
For example, if X is a typical claim this year and inflation of size i is expected next year, then what is the distribution of? If the excess of any claim over M is
to be handled by a reinsurer, what is the typical claim distribution for the
base insurer?
Generalized lambda distribution
This distribution was first introduced by Tukey (1960) and later generalized to the four-parameter
case by Ramberg and Schmeiser
(1974). It can produce a wide variety of curve shapes including that of many
standard symmetric and skewed distributions. The GLD fit has been successfully
applied in a variety of disciplines. These include modeling of quantile response in bioassay and economics, meteorology,
and engineering and quality management.
In Ramberg and Schmeiser's, the probability density function of the
generalized lambda distribution with parameters is
given by
With the quantile
function Q(y) given by
Where and are location and scale parameters
respectively, and , are shape parameters
(skewness and kurtosis, respectively).
Generalized Pareto Distribution
From the fundamental Fisher-Tippett (1928) theorem in classical extreme value theory,
we know that if is a sequence of independently and identically
distributed random variables with a common distribution function which
has mean (location parameter) and variance (scale parameter) denote the
sample maxima of by, and let denote the real line. Given a sequence of and
some non-degenerate
distribution function such that
Where, then H must be of one of the
three types of extreme value distributions: Frqechet,
Weibull or Gumbel
distribution.
The extreme value distributions are closely
related to the generalized Pareto distribution, which describes the limit
distribution of excesses over a high threshold. For sufficiently high threshold,
the distribution function of the excess may
be approximated by the generalized Pareto distribution (GPD) (Balkema and de Haan(1974)and Pickands(1975)), because as the threshold gets large, the
excess distribution converges to the GPD.
The GPD in general is defined as:
With
Where is the shape parameter, is the tail index, is the scale parameter, and is the location parameter.
Value
at Risk Estimation in kernel density estimation
In nonparametric approach which used
historical claims such as historical simulation, value at risk is calculated
directly by taking the desired percentile of the distribution of losses. The VaR in this case is estimated by
Whereis
the th quantile of the sample distribution.
Value
at Risk Estimation in GPD distribution
The
data points in the tail are represented by
For
large,
we can estimate by.
Also can
be estimated from the data by
Therefore,
the tail estimate is
This
approximates the distribution of F(x). It can be shown that is GPD which has the same shape parameter
value (k) of F(x). Given a threshold,
an estimate of may be obtained as where
n is the sample size and is
the number of exceeding. The tail estimator, is calculated by
For a given probability q >F (q),
a percentile at the tail is estimated by inverting the tail
estimator:
Where is a threshold, is the estimated scale
parameter, is the estimated shape parameter, is the sample size, is the number
of exceedances and
Empirical work
In
this chapter we are going to
present empirical result of fitting transformed kernel estimation, generalized pareto distribution and generalized lambda distribution.
For practical analysis, we consider medical claims of Iran insurance company in
1389 and 1390. First, we use simulation study for investigating the best model
to use in transformed kernel density estimation. Then we fit the relevant distribution and compare
the estimation distribution to select the distribution which provides best fit
to the claims. For data analysis we use MATLAB (by using packages evim and bounds_matlab) and R
software (by using packages gld, POT and kernlab).
Simulation
study on transformed kernel estimation
In this section we report the results of a
simulation study to investigate the relative performance of some selected cdf transformations used in the transformed kernel density
estimation.
We consider four cdf
transformations, namely, the lognormal (LN), generalized Pareto, Champernowne, and modified Champernowne.
The choice of these transformations is motivated by the fact that the lognormal
and generalized Pareto are commonly used to model insurance loss data (Embrechts, Kluppelberg, and Mikosch 1997; Klugman and Rioux 2006), while the Champernowne
distribution approaches a form of the Pareto distribution for the extreme
values (Fisk 1961) and approximates the lognormal distribution for values near
zero in some cases (Balasooriya and low, 2008).
For this simulation study, data are
generated from GPD and LN distributions with selected parameter values that
represent different shapes of the underlying distributions.
For
each generated sample, four transformed kernel densities are obtained using the
Champernowne, modified Champernowne,
lognormal, and generalized Pareto cdf transformations
as outlined and illustrated in chapter 3. In assessing the goodness-of-fit of
these estimated densities, we employ global distance measure criteria.
Among four cdf
transformations, the GPD performs the best. For all the sample sizes, parameter
values and both underlying data-generating distributions, GPD cdf transformation yields the highest percentages in Table
1. When the data-generating distribution is LN, it gives higher percentages
than the LN cdf transformation. This shows the
robustness of the GPD transformation to the data-generating process.
Empirical
comparisons of value at risk estimators
The value at risk estimation using kernel
density estimator and GPD distribution are presented in table 7. Both models
result close estimation of empirical value at risk for data set 1389 and 1390.
Comparison of value at risk estimation for
GPD model and kernel density estimator with empirical value at risk for two
data sets 1389 and 1390 are shown in figure 4. This figure demonstrates that
two model estimate the empirical value at risk well.
Figure 4: comparison of
value at risk estimation of GPD model and kernel density estimator with
empirical values.
In Iran, the main
public health insurers include the social security organization and the Medical service insurance organization.
The boundaries of providing health services
for patients is so much expanded that it is not at least an economical
cost-effective activity in the framework of the health insurances. In many
countries the complementary health insurances have been used to provide these
services.
The complementary insurance is applied to
fill gaps between cost and based health insurance service. The main advantages
of these systems are in direct proportion to the benefits provided in the basic
health insurance system.
For insurers, a proper assessment of the size of a
single claim is of most importance. Insurance companies need to investigate
claims experience and apply mathematical techniques for many purposes such as
rating, reserving, reinsurance arrangements and solvency. In fact, the total
amount of claims in a particular time period is a quantity of fundamental
importance to the proper management of an insurance company.
In this study we will look at loss distributions,
which are a mathematical method of modeling individual claims. Actually, in
fitting distribution to insurance loss data, several families of distribution
have been proposed. The common characteristic of these distributions are their skewness to the right and their tails to capture occasional
large values that commonly are present in insurance loss data. One fundamental
question conforming actuaries and other researchers, however, is the approach
used to select the best model for a given data set (Balasooriya,
2005).
Then in the second part of this thesis, parametric and
nonparametric specifications to model single claim have been mentioned. The
detailed analysis of the available models leads to many unsolved problems of
theoretical and practical importance, and this field of research always
generates new challenges.
Three models have
been considered that have appropriate characteristic in modeling financial
data. Our models include transformed kernel density, generalized Pareto
distribution and generalized lambda distribution.
Our hypothesis is that the transformed kernel density
estimation as a semiparametric approach has better
performance than GLD and GPD models. It is also more appropriate for heavy
tailed distribution and yields more accurate estimation.
We applied these
models to medical claims amounts
exceeding 250,000 of Iran
insurance company in the year 1389 and compared the estimation result with real
claim of the year 1390.
We have also
utilized simulation study for investigating the best model to use in
transformed kernel density
estimation. The
results of this analysis showed that in general, transformed kernel performs
better than the classical kernel, and GPD is particularly the best model that
can be used in transformed kernel density.
In the next step, we have proposed our models to
medical claims data of Iran insurance company. The results of primary analysis
showed that the claim distribution is strongly skewed to the right.
By considering the whole claims data set, transformed
kernel and GPD models-estimations provide reasonable results. However, GLD
model is not good enough in modeling higher
claims.
With claims above threshold value of 15,000,000, we
can observe that transformed kernel density, generalized Pareto, and
generalized lambda fit the empirical data well.
Finally, comparison of value at risk estimation
indicates that transformed kernel density and generalized Pareto model estimate
the empirical value at risk well. These estimations are also very close to
empirical value at risk for 1390 claims and can apply for forecasting the claim
in the year 1390.
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Risk
Factors for Individual Health Care Insurance |
eHow.comhttp://www.ehow.com/about_5175825_risk-individual-health-care-insurance.html#ixzz1qttNOiDl
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