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September 2008 This publication was produced for review by the United States Agency for International Development. It was prepared under the auspices of CoReform


September 2008 This publication was produced for review by the United States Agency for International Development. It was prepared under the auspices of CoReform

This document was prepared with financial assistance from USAID Contract No. GHS-1-00-03-00039-00 Task Order 800 Authors: Guram Mirzashvili Alexander Omanadze Khatuna Jishiashvili Giorgi Makharadze Zurab Tsigroshvili

Abt Associates Inc.

CARE International

Curatio International Foundation


DISCLAIMER The author’s views expressed in this publication do not necessarily reflect the views of the United States Agency for International Development or the United States Government.

Table of Contents_________________________________________________________________ 0. EXECUTIVE SUMMARY ............................................................................................................................ 2 1. PROBLEM STATEMENT............................................................................................................................ 4 2. DATA DESCRIPTION AND INITIAL ANALYSIS..................................................................................... 5 3. INSURANCE COVERAGE ......................................................................................................................... 7 4. ASSUMPTIONS ........................................................................................................................................... 9 5. PROGRAM BUDGET CALCULATION ................................................................................................... 11 6. INSURANCE PREMIUM ........................................................................................................................... 14 ANNEX 1. MATHMATICAL DESCRIPTION OF THE MODEL................................................................ 16

0. EXECUTIVE SUMMARY Partnership for Healthcare Reform – the Coreform project – is being implemented with the financial support of USAID/Georgia. As a result of a TA provided by the Coreform in the field of healthcare financing MoLHA commenced the implementation of the Poor’s program based on the insurance principles in 2007. It brought into agenda the need to make respective actuarial estimations of the insurance premium of the state program in order to provide for appropriate allotments in the state budget. In 2007 insurance companies started the implementation of the medical insurance program for the population below the poverty line in Tbilisi and Imereti region. In autumn 2007 GoG made a public announcement about expanding such programs gradually to cover around 1.2 million people’s medical insurance through public finances. During 2008 the number of insured people will reach 800 000. To ensure the provision of adequate funding in the 2008 state budget, it became necessary to make actuarial estimation of insurance premium respective to the above mentioned state program. To this end, MoLHSA applied to the USAID and CoReform Project – “Transformation of Healthcare System in Georgia” to provide TA in calculating the healthcare section of the 2008 state budget. The project contracted 5 local consultants to implement the technical assignment as agreed with the ministry. The assignment implied calculation of respective insurance premiums for 2008 and 2009 state programs for people below the poverty line as well as investigation of medical insurance data collection and recording systems used by insurance companies (the goal of the latter assignment is to streamline reporting from private companies, which are implementing these state programs; the information received through this reporting is the main basis for calculating the insurance premium). This report presents recommendations on the appropriate parameters for the poor’s medical insurance program component of the 2008 state budget. To fulfill the objective, the working group obtained necessary statistical information and carried out the initial processing and the analysis of the data. Based on the statistical information, respective actuarial models were developed, the basic insurance premium was calculated, and recommendations on tariff coefficients were elaborated. The annual basic insurance premium for the 2008 poor’s program amounts to 132.15 GEL and its respective monthly payments equal to 11.01 GEL. The structure of the premium is as follows:

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Table 1. The structure of the montly basic insurance premium

Inpatient Service Outpatient Service Combined Margin Medical Price Increase Administrative Expenses, Acquisitions, Profit

Component’s Cost in GEL


5.88 1.20 0.88 1.40

53% 11% 8% 13%



Coefficients for the different tariff categories were defined with respect to the basic insurance premium. If only two tariff categories exist in 2008 (as in the 2007 pilot poor’s program), the respective coefficients will be: 0.839 (the monthly premium 9.24 GEL) for 0-64 age group and 1.363 (the monthly premium 15.01 GEL) for 65+ age group. The working group considered also a more detailed tariff structure providing for 18 tariff categories. The explanation of tariff factors along with the respective table of coefficients is given in Chapter 6 of this report.

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1. PROBLEM STATEMENT The main goal of this project was to calculate the amount of the insurance premium for the 2008 state health programs to be implemented based on the insurance principles. On one hand, it means setting budget parameters, and on the other hand – defining the adequate insurance premium. To achieve the main goal, it has been necessary to carry out the following intermediate tasks: 1. Obtaining detailed statistical data on the implementation of the 2007 program for the poor from the Health and Social Programs Agency and those private insurers, which participated in this program. 2. Structuring the obtained information, putting it into a standard form and perfomning its initial processing. 3. Consulting with the Ministry of Labour, Health and Social Affairs of Geogia to specify the expected changes in the design (a list of services envisaged by the state program) of the insurance product; analyzing the obtained statistical data in terms of these changes. 4. Elaborating on actuarial models and calculating the basic premium using available statistical data on the ground of these models. 5. Correcting the basic premium considering different factors. 6. Categorizing levels of claims in the target population (among beneficiaries) considering different a priori risk-factors and analyzing claims in these groups. 7. Defining equalizing coefficients for the different riskgroups with regard to the basic premium.

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2. DATA DESCRIPTION AND INITIAL ANALYSIS The following data was obtained and used for this project: 1. Ministry of Labour, Health and Social Affairs of Georgia: • Description of the 2006 and the 2007 State Health Programs • The draft of the 2008 State Medical Insuranse Program for the population below the poverty line • The results of the Household Survey on Health Expenditure and Health Service Utilization (Final Report, December 2007) 2. Health and Social Programs Agency: • Statistical and financial data on the implementation of medical assistance programs for the population below the poverty line (from June 1, 2006 to December 31, 2007) • The pricing schedule used for funding medical services according to the state standards. • The pricing schedule used for funding medical services according to the contracts signed by health service providers • The register of citizens below the poverty line considered as beneficiaries of the poor’s program in 2007 (the entire list as of the end of 2007). • The dynamics of the increase in the number of beneficiaries from June 2006 to the end of 2007. 3. Private insurers participating in the 2007 pilot program for the poor: • Statistical and financial data on the implementation of the program during SeptemberOctober 2007, in the format set by Health and Social Programs Agency. • Statistical and financial data on the implementation of the program during SeptemberDecember 2007, in the format set by Health and Social Programs Agency1 • The entire list of insured beneficiaries by insurance companies. The obtained statistical data covers information on each insured accident as well as data on each beneficiary of the program. To ensure privacy, these data didn’t contain beneficiaries’ names, last names, personal ID numbers, and other types of individually identifiable information. Unique ID codes assigned by State Subsidy Agency were used to identify beneficiaries under this project. Actuarial evaluation was based on the collective risk model, which implies separate analysis of the frequency of insured accidents and the costs of provided services with the following incorporation of the resultant data into the integrated model. This model could be used only for hospital and so called second level outpatient services, so far as funding principles for the primary level outpatient services used by Health and Social Programs Ageny as well as private insurers don’t make it possible to explore an individual insured accident. In order to analyse claims for medical services and expenditures on insurance claims, it was found necessary to split up the data into several groups; it was done essentially to perform statistical processing of the data. The HESPA database was devided into the following four groups: • Obstetrics 1

This report reflects the work carried out from December 2007 to February 2008. In December 2007 budgetary parameters were set for the 2008 poor’s program. For the moment, only the data for September and October were available from the private insurers. The basic premium given in this report is based exactly on these data along with the data provided by Health and Social Program Agency. In January statistical data from the insurance companies were repeatedly requested to correct the basic premium if necessary

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• • •

Elective operations and related outpatient services Urgent cases with a length of stay not more than 12 days Urgent cases with a length of stay more than 12 days

As for the claim statistics of the private insurers, it was relatively difficult to merge them with each other and with respective HESPA data, since notwithstanding the fact that the reporting format of statistical data was defined by HESPA, there still were some inconsistencies in the received data. In some cases monthly claims implied only reimbursed (rather than recognized) claims; on other occasions denials weren’t separated off clearly and so on. Additionally, there were insured accidents with regard to which ultimate decisions on recompense / denial weren’t yet made. A share of such cases turned out to vary quite significantly by insurance companies. Considering all these factors it became necessary to make some methodological assumptions, which essentially implied the presumption that insurance companies exercised more or less uniform recompense policies. After this, payoff statistics of the private insurance companies also was divided into four groups.

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3. INSURANCE COVERAGE The Government Decree #166 of July 31, 2007 defines the amount of insurance coverage for the 2007 poors’ program: The terms of medical insurance financed by insurance voucher provide for covering the following health service costs, which are disbursed by an insurance company: a. Coverage of costs for outpatient services, which are not provided for by the State PHC Program: a. a. Emergency outpatient services; a. b. Services provided by family doctors and nurses as well as specialist consultations and other medical services, including medical services at home, as needed; a. c. Ultrasound and X-ray examination as prescribed by a doctor, laboratory and instrumental examinations in connection with planned hospitalization; b. Coverage of inpatient services: b. a. Emergency inpatient services, including hospitalization due to complicated pregnancy; b. b. Planned surgical operations, with the yearly insurance limit of 12 000 GEL on each insured person; b. c. Expenditures on co-payments, which aren’t covered by the state program for inpatient services b. d. Costs of chemotherapy and radiation therapy; the insurance limit of 12 000 GEL c. Labour and delivery costs – the insurance limit of 400 GEL on each insured person. 2. According to the terms of medical insurance, costs of the following medical services aren’t covered by the insurance voucher: a. Costs and services covered by other State (including municipal) Health Programs; b. Planned therapeutic inpatient services; c. Treatment and self-treatment without medical indications or doctor’s prescription; d. Costs of medical services received outside the country; e. Sanatorium-and-spa treatment; f. Aesthetic surgery and cosmetic treatment; g. Sexual disorders and infertility treatment costs; h. AIDS and chronic Hepatitis treatment costs; i. If a need in medical service arises from a self-injury, participation in terrorist or criminal activities, or substance abuse; j. Transplantation and exophrostesis costs. Such an insurance product is mainly a tool for covering inpatient expenditures and this is reflected in the respective insurance premium: the intpatient component of the premium exceeds that of the outpatient one several times. The inpatient services are almost fully covered, the only exemption is hospital theraphy.As for the outpatient part, it is actually a basic package of such services supplemented by medical examinations to assess the need of an admission or tests which are necessary during a surgical operation. By nature, this insurance product helps to protect the population below the poverty line from “catastrophic” expenditures on health. The terms of the 2007 poors’ program envisages certain links with other State Health Programs. Outpatient service is provided to an insured primarily through the PHC State Program allocations and an insurer has to finance only those outpatient services, which are not covered by the PHC State Program. In addition, beneficiaries of the poors’ program have to cover co-payments if there is a need in cardiosurgical operation; the latter is also financed through the special State Program

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and the insurer has to cover only the rest of this expenditure – 30% of the full cost of an operation. In practice, these two links between the poor’s program and other State Health Programs give rise to various problems. The funding of PHC from two different organizations frequently leads to ambiguity in attribution of costs and, at the same time, it’s clear that such a system implies some duplication in budgetary allocation. As for cardiosurgical operations, pursuant to the governmental decree they are falling into the category of planned surgical interventions and, respectively, a requirement of so-called “two-month waiting period” is applicable to them. This requirement means that there must be maximum of a two-month period between the date of performing surgerical operation and the date of applying to the insurance company. With regard to the fact that a performance of such an operation is linked to the schedule of other State Health Program, meeting the two-month waiting period requirement on this occasion turned out problematic. It’s further confirmed by unusually low share of cardiosurgical operations in the private insurers’ claim statistics. In this case, covering the same service through two different sources of funding limits the accessibility of this service to some extent. Insurance coverage, envisaged for the 2008 Poors’ Program, will remain principally the same. The following components will be added to the described one: • Cardiosurgical operations will be fully covered, i.e. a co-payment from other State Program will no longer exist for beneficiaries of the Poors’ Program. • Outpatient services will be expanded with a component of examinations necessary to grant a patient the status of a disabled person.

As for the first addition, insurance premium was calculated with a proportionate increase in the cost of cardiac surgery to include the amount of co-payment. A potential rise in claim frequencies owing to improved accessibility was also reflected in the effective premium. With regard to the second addition, the respective yearly cost for the entire population doesn’t exceed 100 000 GEL and it is clear that it would only cause a minor rise in the resultant premium. Considering this, it was decided not to correct the premium relative to the second addition. Additions envisaged in 2008 solve the issue of limited accessibility of cardiac surgery; however, the problem of duplication in financing outpatient services still remains in place. Consultations with the Ministry of Labour, Health and Social Affairs of Georgia were held on this issue. As a result, it was found that a full inclusion of outpatient services into the Poors’ Program is associated with many technical difficulties and such duplication will be retained at this stage.

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4. ASSUMPTIONS Below we present the assumptions, which were used as the basis for the actuarial calculations. Hence, all the results given in the report will be valid only if the assumptions come true. Assumption 1. Under outpatient and planned inpatient care components of the program insurance companies will sign appropriate contracts with individual healthcare providers, who will provide health services for the beneficiaries. Under the emergency care component of the program the delivery of health services will be grounded on the principle of geographic accessibility. It is a common practice for both the insurance companies and the SUSIF. The main goal of such a practice is to keep healthcare prices within particular limits. If the most part of health care services are not delivered through providers contracted by insurance companies, healthcare prices will grow significantly. Consequently, the calculated tariff will require considerable revision. Assumption 2. Administrative and acquisition costs along with the component of planned profit shall be set as 15% of gross premium in the insurance premium. The main goal of this project is to estimate the so-called risk-premium (i.e. net premium). This amount should be increased with certain loadings (e.g. administrative and acquisition costs) to derive the ultimate gross premium. These loadings vary significantly according to different schemes of procurement and to a certain extent fall outside the scope of actuarial calculations. The mentioned loading coincides with the loading provided for in the respective premium of the 2007 program for the population below the porverty line. Assumption 3. While implementing the 2008 Poors’ Program, insurance companies won’t change the claims management policies significantly with regard to those used for carrying out the 2007 Poors’ Program by the private insurance companies and the HESPA. Usually, insurance companies may exercise different claims management policies (e.g. level of liberalism in the process of indemnification) and this may have quite a big impact on the general picture of claims pattern. As a rule, inferences grounded on any statistics will be valid only if factors leading to certain developments do not change significantly. Claims management policy is one of the principal factors in this respect. Assumption 4. There won’t be significant qualitative changes in the statistical structure of losses (“outbreak of losses”). This assumption implies continuation of existing trends in morbidity, healthcare providers’ accessibility, and other relevant indicators during 2008-2009 yy. It is noteworthy because current healthcare reform in the country provides for major renovation of healthcare infrastructure. There is a consideration that as a result of this process the accessibility of healthcare services may imrpove considerably due to a number of factors, including logistical ones. At this stage, it’s difficult to evaluate such an effect; however, if the situation develops this way, it will considerably change the picture of losses and require a major revision of the resultant insurance premium. The second factor, which may lead to the same effect, is better awareness of the population about the provided services.

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Assumption 5. Each company, participating in the 2008 Poors’ Program will have “sufficiently large” insurance pool at its disposal. The amount of an insurance premium will change significantly in accordance with a number of the insured. The number of the 2008 Poors’ Program beneficiaries reaches 800 000 people. If the premium is calculated based on the pool of this size, it will ensure that the virtual pool compiled for the entire insurance market be well-balanced. However, at this moment some companies, participating in the program may still face considerable losses. To accout for the effect of pool fragmentation, a special loading will be introduced in chapter 5 with the aim to offset against emergence of additional risk. However, this loading ensures the stability of losses if the pool is fragmented to a certain level. A “sufficiently large” pool in this asumption implies one consisting of up to 100 000 insured. The least size for which the amount of premium might be considered adequate, is about 50 000 insured.

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5. PROGRAM BUDGET CALCULATION This chapter includes a brief description of actuarial methods used to calculate the basic insurance premium and the results of their application. While calculating the insurance premium, one should consider its following traditional structure: premium consists of two major components – a risk-premium (net premium) and a loading. The risk-premium part serves for covering insurance liabilities; the markup part accounts for various direct and indirect costs and profit. The goal of carrying out actuarial calculations was essentially evaluation of the risk-premium. As for the second part of the insurance premium – the loading, its amount is generally dependent on tariff policy of a particular insurer and, in this case, on the viewpoint of the government. The standard assumption about the amount of the loading was given in section 4. The insurance premium was calculated considering a one-year insurance period. The first objective to be faced by the working group was defining budgetary parameters of the 2008 Poors’ Program. It essentially implies calculation of a so-called insurance fund rather than an insurance premium; the fund should be sufficient to cover the yearly aggregate insurance claims of the given insurance pool with high probability. The budget of the program is based exactly on this amount. For the purpose of convenience as well as drawing comparisons with the budget of the similar 2007 program, the amount of the resultant insurance fund might be divided by the number of beneficiaries giving the basic premium, i.e the program budget per beneficiary. To achieve the objective, it became necessary to divide the insurance coverage into two components. For convenience we name these components as inpatient service and outpatient service. These are conventional names and their actual content is as follows: Inpatient service – comprises chemotherapy, radiation therapy, and surgical operations as well as all related laboratory and instrumental investigations. Disagregated statistical data with respect to this component were provided by both HESPA and private insurance companies, i.e. separate records were available for each and every case. Outpatient service – comprises so-called first-level outpatient services, i.e. services, which are delivered above the package provided for by the State PHC to the beneficiaries of this program. To finance such services, the HESPA utilizes per capita funding principle (all the risk is placed on the healthcare providers), while the private insurance companies use a certain hybrid approach, which imparts only a partial risk on a medical facility. Both funding principles imply that a financing organization doesn’t require a detailed report about the fulfilled work (under the detailed report we mean a full description of each and every case.) Such division of the insurance coverage was caused by the characteristics of the available statistical data rather than by the purpose of component price analyses. The monthly basic premium calculated this way amounts to 11.01 GEL. The structure of the premium by the components is as follows:

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FIG. 5.1 The Basic premium by components

The formula to calculate the gross premium is:

PG = (( PI * (1 + k ) + Po) * (1 + i )) /(1 − e) Where,

PG PI Po k i e

- is the basic gross premium - is the respective risk-premium of the inpatient services - is the respective risk-premium of the outpatient services - is the coefficient of the combined margin - is the price growth rate for medical services - is the share of administration and acquisition costs in the premium

The risk-premium for inpatient services was calculated based on the collective risk model. To this end, statistical data from the HESPA and the private insurance companies were divided into eight groups. For each group daily claims for medical services and expenditures on insurance claims were analysed, after which we determined the probability distribution of the aggregate yearly loss (using the so-called Panjer Recursive Algorithm). The insurance fund of the respective part of the premium is a 90% quantile of the aggregate yearly loss distribution function. Such confidence level is commensurate with one of the respective premium in the 2007 Poors’ Program. The deterministic model was used to calculate the respective risk-premium for outpatient services so far as the existing statistical data didn’t make it possible to perform a stochastic analysis. Data on payments made by the private insurance companies to the PHC facilities were used as the primary source for calculation. Based on this data the average monthly expenditure incurred by insurance companies per beneficiary was estimated.

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The combined margin is the combination of two factors conveying different meanings: 1. Diversification margin. The largest part of the basic premium – the risk-premium for inpatient services – was calculated based on the probability distribution of aggregate yearly loss, which corresponds to the placement of all beneficiaries into one risk pool. And in practice, several insurance companies will take part in the implementation of the program; i.e. the above mentioned part of the premium is calculated only based on the pool of hypothetical size; and in practice this pool will be fragmented. To balance the resultant additional risk, it became necessary to introduce the diversification margin, which ensures the stability of losses while dividing the pool approximately into ten even parts (See section 4, Assumption about “Sufficiently Large” Pools). 2. The loadind for the expected growth in demand for cardiac surgery. The problems arising from two different programs to finance cardiac operations were described in Section 3. During 2008 these issues will be resolved leading to increased number of such operations. According to the specialists’ opinions, almost a triple increase in claim frequency should be expected. Proceeding from the prices of cardiosurgical operations, such an increase will have significant impact on the companies’ losses. Due to the lack of relevant information it turned out very difficult to precisely assess the elements of the combined margin. Their values are essentially based on a heuristic judgment. Both elements of the combined margin are about the same size in percentage. The price growth rate for medical services is a price increase for the implementing organizations of the poors’ program rather than general inflation rate of medical prices in the country. The HESPA 2006 and 2007 claim statistics were analysed to calculate this indicator. The most common and expensive services accounting for 10% of aggregate claim frequency and almost 30% of the total expenditure were chosen for the analysis. The rate is the weighed average of the price growth of these services. The loading for the administration and acquisition costs is the same percentage increase as the one provided for in the 2007 Poors’ Program. Besides the mentioned loadings, the working group considered an additional one, which is related with increase in number of claims. It’s natural to suppose that the ongoing healthcare reform will have a serious impact on the accessibility of healthcare services, thus increasing their utilization. Clearly, it implies the escalation of losses for the insurance companies implementing the poors’ program and it must be reflected in the insurance premium. At this stage, it’s difficult to assess such an effect; however, a certain increase in the number of claims is yet noticeable (better awareness of the beneficiaries of the Poors’ Program may be the main reason for this). Probably, the assessment of this effect will be possible in summer 2008, when the working group will carry out the ongoing evaluation of the program implementation and elaborate on recommendations about the budgetary parameters for 2009. The growth in demand may prove to be quite significant (in some assessments from 10% to 20%); in such a case it will be necessary to appropriately adjust the premium for the 2008 Poors’ Program.

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6. INSURANCE PREMIUM The basic premium, as calculated in the previous section, in fact, is the amount of insurance fund per insured person. According to one of the fundamental principles of insurance, the insurance premium should correspond to a particular risk. In practice, the most suitable way to implement this is the introduction of tariff categories, in which more or less uniform risks will be pooled together. The pilot 2007 Poors’ Program provided for only two tariff categories: the first one covered the beneficiaries under 65 and the other one – the rest of insured people. The implementation of the program showed that, considering the apparent nonuniform pattern of risk distribution, the introduction of a more detailed tariff structure is reasonable for this program. For example, the comparison of claim patterns in Tbilisi and Imereti region gives quite different pictures. To introduce the tariff categories, the working group divided the claim statistics into 18 groups. The division took place according to the following factors: Tariff factor

Values 1. Tbilisi

Place of residence

2. Urban

Sex Age

Coment Implies all cities and district centers save Tbilisi

3. Rural 1. Male 2. Female 1. 0-18 yy. 2. 19-64 yy. 3. ≼ 65 y.

Such age grouping corresponds with the terms of refercence of the project

The averages for yealy aggregate claims and their standard deviations were calculated for each of these groups. The insurance funds for the categories were calculated in proportion with the resultant values, according to the following principle: respective averages of yearly aggregate claims by categories were subtracted from the aggregate insurance fund and the remaining part was distributed among categories in proportion with the standard deviations of the corresponding yearly aggregate claims. To derive the insurance premium from the resultant insurance sub-funds, these amounts were divided by the number of insured people attributed to each category. The process could be presented schematically as follows:

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Insurance fund Distribution according to averages and standard deviations

Insurance sub-fund # 1

Insurance sub-fund # 2


Insurance sub-fund # 18

Division by the number of beneficiaries Insurance premium for the first category

Insurance premium for the ..................... second category

Insurance premium for the eighteenth category

This principle was used for the component of inpatient services. Due to the lack of relevant statistical data, findings of the the results of the Household Survey on Health Expenditure and Health Service Utilization were used to divide the outpatient service component of the insurance fund by categories. The loading for administration and acquisition costs was distributed equally among the insured people. In such way the following table of insurance premiums was generated: Tariff category Monthly insurance premium Tariff category Monthly insurance premium Tariff category Monthly insurance premium

Tbilisi Male 0-18 yy. 11.64 Rural Male 0-18 yy. 5.32 Urban Female 0-18 yy. 8.43

Tbilisi Male 19-64 yy. 12.99 Rural Male 19-64 yy. 7.88 Urban Female 19-64 yy. 13.65

Tbilisi Male ≥ 65 y. 36.29 Rural Male ≥ 65 y. 17.12 Urban Female ≥ 65 y. 15.80

Urban Male 0-18 yy. 6.85 Tbilisi Female 0-18 yy. 11.64 Rural Female 0-18 yy. 7.08

Urban Male 19-64 yy. 11.83 Tbilisi Female 19-64 yy. 19.72 Rural Female 19-64 yy. 9.00

Urban Male ≥ 65 y. 11.46 Tbilisi Female ≥ 65 y. 24.07 Rural Female ≥ 65 y. 11.57

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ANNEX 1. MATHMATICAL DESCRIPTION OF THE MODEL I. Introduction As it is known, an insurance premium consists of two major components – a risk-premium and a markup. Our objective was to assess the amount of the risk-premium. To this end, we used the socalled collective risk model in each of the above mentioned disease categories; in this model the total expenditure (“losses”) is as follows:

S N (t ) =

N (t )

∑X k =1




where N(t) denotes the number of claims in the particular category in [0;t] time interval, Xk _ the expenditure (costs) to cover requested services (cost) on k-th request and SN(t) is the total expenditure in [0;t] time interval. Since the number of claims and the expenses per requested services aren’t known in advance in a given time period (year, half-year, month or day), N(t) and Xk, k ≥ 1, are considered to be random variables with a certain unknown probability distribution and the principal objective of the statistical analysis is to assess the very unknown distributions and their parameters. Our model implies that N(t) is independent from Xk, k ≥ 1, random variables, which, on their own part, are independent and equally distributed according to the function of distribution F. Let’s assume that the distribution of N(t) variable is described by pn(t) = P{N(t) = n} probabilities. Then, as it is known, the function of distribution of total losses SN(t) has the following pattern: ∞

n =1

n =1

FS ( x; t ) ≡ P{S N ( t ) ≤ x} = p0 (t ) + ∑ p n (t ) ⋅ P{S n ≤ x} = p 0 (t ) + ∑ p n (t ) ⋅ F n* ( x)


Where n

Sn = ∑ X k ,


k =1

and x

F ( x) = F ( x) and F ( x) = ∫ F ( n −1)* ( x − z )dF ( z ) , 1*




is the n-multiple “band” of the F distribution. Thus, to assess the distribution of total losses SN(t) it’s necessary to know the distributions of N(t) and Xk random variables, we talked about above. However, it should also be noted that knowing these distributions is not enough to idendify analytical form of the FS(x;t) distribution: firstly, due to the fact that the identification of analytical form of the “band”, as defined in (3), is possible only in case of the so-called infinitely divisible distribution and secondly, had the distribution of certain costs been of this kind, the calculation of the total expenditure, as defined in (1), could have been carried out only for the very specific pn(t) probabilities coinciding with the F(x) distribution function (it’s unclear why our assessed distributions will turn out to be so concording with each other!). Therefore, we concluded that it is simply not possible to identify the analytical form of the FS(x;t) distribution.

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Nevertheless, it turned out that it is possible to give the values of FS(x;t) distribution quite a wide range of pn(t) probabilities (binomial, negative binomial, puasson distributions, etc.) in the tabular form (i.e. calculating FS(x;t) value in each point x). In literature, this procedure is known as Panjer’s Recursion (see H.Panjer, 1981) to be discussed in the next section.

II. Panjer’s formula to calculate compound distributions Since we’ll talk about SN(t) aggregate loss in the fixed point t (e.g., t = 1 corresponds with the yearly aggregate loss), we toss out “t” from all time dependent variables to simplify notations, i.e. suppose thedistribution {pn} of the number of terms in the compound total S N =


∑X j =1


for any arbitrary a and

b satisfies the following recurrent ratio: pn = pn – 1⋅ (a + b / n ) , n =1,2,... (4) and Xj terms have f(x) density of distribution. Then f(x) density of distribution of random variable SN satisfies the following integral equation: x

f S ( x) = p1 ⋅ f ( x) + ∫ (a + b ⋅ y / x) ⋅ f S ( x − y ) ⋅ f ( y )dy .



This is exactly the Panjer’s Formula. However, its integral form (5) is not suitable for calculations. Therefore we give so-called discrete variety of (5), i.e. the case, when Xj terms show up as a concentrated discrete distribution in nonnegative integer points. i.e. suppose that, P{Xj = m} = hm, m = 0,1,2,…

and (4) still holds. Then


1+ b / a

⎛ 1− a ⎞ ⎟⎟ , f S (0) = ⎜⎜ ⎝ 1 − a ⋅ h0 ⎠ k 1 f S (k ) = ⋅ ∑ (a + b ⋅ j / k ) ⋅ h j ⋅ f S (k − j ) , for k ≥1 1 − a ⋅ h0 j =1


Where (8) fS(k) ≡ P{S = k}. For example, it’s known that a = 0 and b = λ correspond with Puasson Distribution with the parameterλ. In this case (7) takes the following form: f S (0) = e − λ ⋅(1− h0 ) ,

f S (k ) =

λ k


⋅ ∑ j ⋅ h j ⋅ f S (k − j ) , k ≥1.


j =1

As we see, (7) and (9) equations show the pattern of a nontrivial recursion: all previoius probabilities take part in calculation of each consecutive probability. To illustrate this, the first several terms of (9) are given below:

f S (0) = e − λ ⋅(1− h0 ) f S (1) = λ ⋅ h1 ⋅ f S (0) ; ____________________________________________________________________________ 17

f S (2) = f S (3) =

f S (4) =

λ 2



λ 4

⋅ (2 ⋅ h2 ⋅ f S (0) + h1 ⋅ f S (1)) ; ⋅ (3 ⋅ h3 ⋅ f S (0) + 2 ⋅ h2 ⋅ f S (1) + h1 ⋅ f S (2)) ;

⋅ (4 ⋅ h4 ⋅ f S (0) + 3 ⋅ h3 ⋅ f S (1) + 2 ⋅ h2 ⋅ f S (2) + h1 ⋅ f S (3)) .

Note. Despite the fact that we usually model the amount of losses using a continuous distribution, we still give preference to the (5) formula over its discrete variety (7), since, in general, one can’t manage to write (5) integral in a clear analytical form and some approximate formulas are needed to carry out calculations. The second, though not less important reason is that on this occasion we didn’t try to “fit” F(x) distribution into any kind of parametric family; as we think that the distribution of cost variables won’t change seriously in the near future, the most appropriate way is to assess F(x) empirically, (i.e. to build it up using all the available data) thus, on its own, naturally creating a discrete model for the distribution of Xj terms. In particular, we took a discrete increment of 200 GEL for each of the named group and assumed that: hj = P{200⋅j < Xi ≤ 200⋅(j + 1)} ≈ # {xi : xi ∈ (200⋅j; 200⋅(j + 1)]}/ M , j = 0,1,2,…, (10) Where M is the actual (observed) number of requests in in this group and xi is the amount of i-th observed expenditure. Here, the #{A} symbol denotes the number of elements in the set A.

III. Statistical analysis of daily insurance claims As for identification of daily insurance claims (medical aid appealability) distribution, we carried it out the same way as in the previous two occasions. To illustrate this, we present 236 data items for one of the groups according to which the daily claims were distributed as follows: Table 1 Number of requests per day 0 1 2 3 4 5 6 7 8 Total

Number of days

Relative frequency

Total requests

85 34 31 17 7 4 2 3 1 184

0,4620 0,1848 0,1685 0,0924 0,0380 0,0217 0,0109 0,0163 0,0054 1

0 34 62 51 28 20 12 21 8 236

If we denote the number of requests on i-th day as N1i, then, according to the table, there were no claims in 46% of days and 1 claim in 18,5% of days ; it should be said, that due to the above mentioned reasons it’s “almost” impossible to receive more than 8 claims in Imereti region per “typical day”. Hence, this table says that possible values of the random variable N1i are 0,1,2,...,8, and the probabilities of their occurrence can be estimated by relative frequencies given in the third

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column of the table. However, it’s clear that the picture could change quite considerably in the following six-month period, giving rather different relative frequencies. Also, it is possible that in any particular occasion there will be more than eight claims per day. Therefore, in order to model random variable N1i, the relative frequencies given above should essentially be used as the estimator of the distribution “structure” rather than absolute values. In other words, only the rule to calculate a distribution (i.e. probabilities of occurrence of possible values) rather than probabilities (and even more so, relative frequencies) of the random variable N1i should be “universal”. And this can be achieved through finding (choosing) so-called parametrical models, which “quickly respond” to the above mentioned possible changes in relative frequencies during the following six-month (or even one-year) periods at the expense of changes in the parameters’ values. Thus, at this stage, the main goal of the statistical study is to find such parametrical model of the random variable N1i distribution, which will “approximate” to the actually observable relative frequencies as much as possible, i.e. the numbers given in the third column of the table. To find such parametrical model, so-called histograms or polygons are constructed and selected characteristics (means, variances, etc.) are calculated to compare the respective values of parametrical distribution with each other and to choose the model, which stands “acceptably close” to the actual picture. On our example, the polygon of the actual frequencies has the following pattern: fig. 1 0,50 0,45 0,40 0,35 0,30 0,25 0,20 0,15 0,10 0,05 0,00 0









The declining pattern of probabilities suggests that the desired parametrical distribution might be a Puasson or negative binomial distribution. To identify which of the two is more acceptable, let’s estimate selected mean and selected variance; in this case we use the following formulas for calculations: 8

x = ∑ k ⋅ fk k =0

and 8

s2 = ∑ k 2 ⋅ fk − x 2 , k =0

where fk denotes relative frequencies. For the numerical values given in the table, we estimate:

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x = 1.2826 And s 2 = 2.7354 . As we see, x < s 2 , however, to make judgement whether or not it is significant that the selected mean is less than the selected variance, i.e. whether it is a systematic or random finding that the first numerical value is less than the other, we use so-called Gart and Boenning simple statistical criteria based on which we can “confirm” or deny the hypothesis that the population mean equals the variance. This criterium is grounded on the following statistics:

Tn =

2 ⎞ n −1 ⎛ S n ⎜ ⋅⎜ − 1⎟⎟ 2 ⎝ Xn ⎠


and the fact that if the hypothesis of equality holds, then Tn statistics is normally distributed. In our case the observed value of Tn statistics - tn equals to:

tn =

184 − 1 ⎛ 2.7354 ⎞ ⋅⎜ − 1⎟ ≈ 10.8344 , 2 ⎝ 1.2826 ⎠

and its corresponding p-value is so small that we necessarily deny the hypothesis that the population mean equals the variance and infer that the random variable N1i must be distributed only in such a way that its mean will be less than its variance. It can’t be the Puasson distribution and therefore the the random variable N1i distribution may be of a negative binomial one (or other kind of distribution characterized by the above mentioned properties). As it is known, the negative binomial distribution with the parameters m (m > 0) and q (0 < q < 1) is called an array of probabilities given by the following rule of recurrency: p0 = qm and pn = pn – 1 ⋅ (1 – q + (m – 1) ⋅ (1 – q)/ n), when n = 1,2,....


To assess the parameters, let’s use the so-called moments method, according to which the equation system is solved for the unknown parameters after equalizing the population moments with its selected characteristics:

⎧m ⋅ (1 − q ) / q = x ⎨ 2 2 ⎩m ⋅ (1 − q ) / q = s

⎧qˆ = x / s 2 . ⇒ ⎨ ⎩mˆ = x ⋅ qˆ /(1 − qˆ )

In our case,

⎧qˆ = 1.2826 / 2.7354 ≈ 0.4689 ⎨ ⎩mˆ = 1.2826 ⋅ 0.4689 /(1 − 0.4689) ≈ 1.1324 And the probabilities calculated for these parameter values (12) are given in the following table:

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Table 2 Claims per day 0 1 2 3 4 5 6 7 8 Total

Number of days 85 34 31 17 7 4 2 3 1 184

Relative frequency 0,4620 0,1848 0,1685 0,0924 0,0380 0,0217 0,0109 0,0163 0,0054 1

Probabilities 0,4242 0,2551 0,1445 0,0801 0,0440 0,0240 0,0130 0,0070 0,0081* 1

*note. The last probability is picked so that the total of probabilities is equal to 1. To compare these probabilities with the relative frequencies, it is practical to present both of them in the graphical form at the same time: fig. 2

As we see, both graphs stand quite close with each other. However, the answer to the question “whether or not this closeness is of the acceptable one” is given by so-called χ2 criterium; its statistics looks as follows:

( f k − pk ) 2 pk k =0 8

Χ 2 = n⋅∑


and, given the appropriateness (acceptability) of the model, it has has the so-called χ2 distribution with the degree of freedom d = 9 – 1 – 2 = 6. It means that we should compare the observed value of (14) statistics to the quantile of the χ2(6) distribution for the given level of values (i.e. acceptable probability of error) e.g. for α = 0.05. In our case, according to table 2, observed value of the statistics is Χ 2 = 7.94 and χ2(6;0.05) = 12.59. Therefore we got that Χ 2 <χ2(6;0.05), which is indicative of the model’s appropriateness with 95% level of confidence.

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Thus, the negative binomial model (12) with the above mentioned parameters may serve as the probability distribution of daily claims with high level of confidence (95%)

IV. Calculating the Distribution of the Aggregate Yearly Expenditure As we mentioned above, when a random variable falls into to the Panjer’s class, the distribution of the random variable S N =


∑X j =1


satisfies the (7) equation. Panjer’s class probabilities are those

ones which satisfy the recurrent ratio (4). If we draw comparisons between the equations (4) and (12) in the premius section, we’ll find that the negative binomial distribution is truly one of the Panjer’s classes, with the parameters: a1 = 1 – q and b1 = (m – 1)⋅(1 – q). (15) As we saw, in the previous section, the number of daily claims is the random variable N1i with the negative binomial distribution and the parameters m (m > 0) and q (0 < q < 1). It’s known, that the total yearly claims N, which amount to the sum of 365 separate N1i – i.e. N =


∑N i =1


, will have the

same distribution but with the parameters 365⋅m and q. Hence, the random variable N is that of the Panjer’s classes with the parameters: a = 1 – q and b = (365⋅m – 1)⋅(1 – q). (16) The last step in estimating the aggregate yearly expenditure distribution is to recalculate consecutevely fS(k) probabilities using the recurrent ratio (7) 1+bˆ / aˆ

k ⎛ 1 − aˆ ⎞ 1 ⎟⎟ f S (0) = ⎜⎜ , f S (k ) = ⋅ ∑ (aˆ + bˆ ⋅ j / k ) ⋅ h j ⋅ f S (k − j ) , for k ≥ 1 1 − aˆ ⋅ h0 j =1 ⎝ 1 − aˆ ⋅ h0 ⎠ ˆ into the equation (16); relative Where aˆ and bˆ are the numbers resulting from putting qˆ and m

frequencies, calculated from the equation (10) are used as the hj probabilities: hj = # {xi : xi ∈ (200⋅j; 200⋅(j + 1)]}/M , j = 0,1,2,…. Lastly, the distribution function of the random variable SN, i.e. FS(x)=P{SN≤x} probabilities are calculated as follows: FS(x) = P{SN ≤ x} = f S (k ) . (17)

k ≤ x / 200

To illustrate, we present the results of calculating the “planned” essential characteristics and the distribution function of the group.

ˆ = 1.60007, and according to the equation (16): For this group it turned out that qˆ = 0.03777 and m

aˆ = 1 – 0.03777 = 0.96223 and bˆ = (365⋅1.60007 – 1)⋅ 0.96223 = 561.00595. hj probabilities for this group are given in the following table:

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Table 3 j 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

200⋅j 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 3600 3800

hj 0.34806 0.30573 0.02614 0.05447 0.08111 0.05261 0.03423 0.01906 0.01636 0.00877 0.00152 0.00337 0.00152 0.00000 0.00219 0.00101 0.00101 0.00084 0.00000 0.00135

j 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

200⋅j 4000 4200 4400 4600 4800 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000 7200 7400 7600 7800

hj 0.00000 0.00000 0.00000 0.00000 0.00051 0.00067 0.00051 0.00202 0.00219 0.00067 0.00017 0.00202 0.00017 0.00017 0.00084 0.00759 0.00051 0.00236 0.00000 0.00287

j 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59

200⋅j 8000 8200 8400 8600 8800 9000 9200 9400 9600 9800 10000 10200 10400 10600 10800 11000 11200 11400 11600 11800

hj 0.00101 0.01551 0.00000 0.00000 0.00017 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00017 0.00000 0.00000 0.00017 0.00000 0.00000 0.00034

As for the probabilities f S (x) and FS(x), they are given in the following graphs: fig. 3 f S (x ) 1.60000E-04 1.40000E-04 1.20000E-04 1.00000E-04 8.00000E-05 6.00000E-05 4.00000E-05 2.00000E-05 0.00000E+00 8000000







____________________________________________________________________________ 23

fig. 4

F S (x ) 1.000000 0.950000 0.900000 0.850000 0.800000 0.750000 0.700000 0.650000 0.600000 0.550000 0.500000 0.450000 0.400000 0.350000 0.300000 0.250000 0.200000 0.150000 0.100000 0.050000 0.000000 8000000







From the respective tables of the last figures (which due to their detailed nature are available only in the excel format) we can readily find the quantiles of FS(x) distribution. For example, FS(12225900) = 0.95 and FS(12595100) = 0.99, which indicates that with 95% of confidence the yearly expenditure on the planned group doesnâ&#x20AC;&#x2122;t exceed 12 225 900 GEL and with 99% of confidence it doesnâ&#x20AC;&#x2122;t exceed 12 595 100 GEL. The values of the same characteristics and parameters for the rest of the groups are given in the following tables:


Table 4 Type





Company 1

Company 2

Company 3

Company 4





































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Table 5 Planned




Company 1

Company 2

Company 3

Company 4





















0.9 0.95 0.96 0.97

12,031,900 12,225,900 12,282,800 12,353,000

1,112,200 1,120,800 1,123,300 1,126,400

6,743,800 6,828,800 6,853,700 6,884,400

4,996,000 5,083,900 5,109,800 5,141,700

2,729,000 2,767,300 2,778,500 2,792,500

1,665,100 1,701,500 1,712,200 1,725,500

5,043,900 5,113,600 5,133,900 5,159,100

2,077,700 2,108,000 2,116,800 2,127,700










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