Dependence between mortality and morbidity: is underwriting scoring really different for Life and Health products? Kudryavtsev Andrey St.Petersburg State University, Russia Postal address: Chaykovsky str. 62 St.Petersburg 197227 Russia Phone: +7-812-3930059 Fax/phone: +7-812-2732400 E-mail: kudr@ak1122.spb.edu Abstract: Underwriting procedure is based on the extensive systems of underwriting scores. The rules of their estimation are included into special Underwriting Manuals. The aim of the paper is to show that underwriting scores are quite close to each other for life and health insurance products. It could be problematic for the portfolio construction and modelling. The paper is based on the special investigation having hold in Russia. The medical records and reviews were used to produce the averaging underwriting scores for life and health risks. The paper shows the high enough dependence between life and health underwriting scores. The dependence could not be explained only with mortality risks in permanent health (income protection) products as it is too high. Keywords: life risks, health risks, underwriting, dependence, copula
Dependence between mortality and morbidity: is underwriting scoring really different for Life and Health products? A. Kudryavtsev, St.Petersburg State University, Russia Summary Underwriting is a powerful tool of the rating for Life and Health products. The procedure of rating is based on the extensive systems of underwriting scores which mirror the medical and other risks influenced to mortality and morbidity. The rules how to score are included into special Underwriting Manuals. The aim of the paper is to show that underwriting scores are quite close to each other for different kinds of insurance products, say for life and health insurance. It could be problematic for the portfolio construction and modelling. The paper is based on the special investigation having hold in Russia. The medical records and reviews were used to produce the averaging underwriting scores for life and health risks. The paper shows the high enough dependence between life and health underwriting scores. The dependence could not be explained only with mortality risks in permanent health (income protection) products as it is too high. The fact means that actuaries and underwriters should be careful with portfolio construction and modelling. 1. Introduction Underwriting is a powerful tool of the rating for Life and Health products. The procedure of rating is based on the extensive systems of underwriting scores which mirror the medical and other risks influenced to mortality and morbidity. The rules how to score are included into special Underwriting Manuals mostly prepared by leading reinsurance companies. Every Underwriting Manual is a complex book or file based on deep statistical investigation. The aim of the paper is to show that underwriting scores are quite close to each other for different kinds of insurance products, say for life and health insurance. It could be problematic for the portfolio construction as a result of the sales strategy because risks in the portfolio are more dependent as they are thought by management and the higher degree of risk accumulation may take a place. The paper includes nor any critique of the statistical methods used for the construction of Underwriting Manuals neither any analysis of the methods of data collection. The idea is to compare underwriting scores for life and health risks of a sample population. The population is investigated from medical point of view. It helps to understand as a by-product how accurate is usual medical underwriting procedure in life and health risks estimating. So, the paper tries to answer rather the question how to use and interpret the underwriting scores than how to estimate them. 2. Methodology The investigation underlying the paper is based on the special study with data collection for all people living in one medical district of a small town in Central Russia Lyssye Gory in Saratov Region (downstream river Volga, south-east from Moscow). It presents typical agricultural province in Russia with some industrial development (in the case, small brick factory, small regional department of oil-drilled company etc.). So, it helped to collect data
representing an appropriate mix of both agriculture and industrial population. Some people studied in the investigation worked for the local municipal administration, in transportation (incl. railway station), in healthcare, education and in the police. The number of people studied was 769. The study took place in 2000. The basic aim of the study was mostly medical. It included two parts: deep medical investigation and survey about people s preferences in healthcare. For the purposes of the paper, only few appropriate data were used. The limitations were bound with the age interval chosen and the information which is useful for underwriting process. The age range has been limited with interval from 20 to 49 (including the latter age). The age definition used was last birthday. A result of choosing the interval was to shorten the number of people taken into account up to 520. The range were chosen for following reasons: Young people (younger than 20 year old) are presumably completely healthy (in the investigation mentioned above about 40 per cent of such people has standard life and health risks; only 10 per cent of the group have serious problems with their health). That age group really demonstrates the dependence between life and health risks as it is quite probable that there are no extra risks of both types. Old people (50+) are probably quite ill (in the investigation, about 50 per cent of them have serious problems with health and only 4,5 per cent have standard life and health risks). The dependence observed between life and health risks is basically explained with poor health (when both scores are growing with age although health score is raising faster). Only chosen age range (20 to 49) demonstrates balanced mixture of three groups (standard life and health risks, people with small problems with health, people with serious problems with health). So, the study of dependence is more interesting for them. Moreover, that age interval is more important for insurance practice. The results of medical investigations were used for the estimation of life and health risks under the usual underwriting procedure. Five basic risk factors were chosen for the procedure: 1) job/profession (including additional information about contacts with dust, chemical materials, ruining the usual sleeping time etc.); 2) height/weight index, 3) acute disease for the last 12 months before investigation and chronic disease (existing conditions); 4) addictions (tobacco smoking and alcohol drinking, there was nobody among investigated people who was addictive to narcotics); 5) heredity factors (indirectly estimated on the base of the information about the longevity and health status of close relatives). The factors were quantified with underwriting scores differently for life and health risks under usual underwriting procedures. From some different approach of the estimation of health risks, only one was chosen. The underwriting scoring was based on the Underwriting Manuals of three different companies: Skandia International Insurance Corporation, Munich Re and Cologne Re [1 4]. Those manuals have some differences in scoring procedures which could be explained with different reinsurance and underwriting experience and with the different statistical procedures used. Life and health risks were differently estimated the life risk as an extra mortality score (index) under whole life insurance contract, health risks as an extra claim score (index) for permanent health (income protection) insurance with 4 weeks of waiting periods. The latter was chosen as a middle term index of health which shows quite serious problem with health
(the term of illness more than a month), but not very rare cases like for periods of disability to work longer than 3 or 6 months. Every person investigated was estimated by the experienced underwriter with the rules taken from each manual. The estimation was quite conservative. In order to eliminate the differences in manual rules which are not explained by differences in health itself, individual score was equal to arithmetic average between company-specific scores (all three manuals for life score and Skandia and Cologne Re manuals for health score). Then, in order to eliminate the rest errors and subjectivism of estimation, the scores were classified into special boxes which represent typical standard and substandard classes (see table 1). Table 1. Rounding the individual scores Score interval Final score up to 100 100 from 101 to 135 125 from 136 to 175 150 from 176 to 225 200 from 226 to 275 250 from 276 to 325 300 more then 326 >300 Then two types of individual scores (life and health ones) were analysed and compared. Results and discussions The distribution of people investigated into life and health score boxes is shown in table 2. Although it is far from comonotonic (one-to-one functional) dependence, a kind of dependence is obvious. The coefficient of correlation is 0,6312 which is quite large (the actual t-test value is 24,6 that is much higher than the critical value that is quite close to 0). The dependence could not be explained only with mortality risks in permanent health (income protection) products as it is too high. Table 2. Frequency distribution of the rounded score for life and health risks Life Health score score 100 125 150 200 250 300 >300 Total 100 97 43 1 2 143 125 20 78 41 2 2 13 156 150 1 6 16 33 6 56 118 200 5 12 28 45 250 1 26 27 300 5 5 >300 2 24 26 Total 118 127 58 42 20 1 154 520 Special interest in the context of practical applications may be paid to standard/substandard proportions. For this purpose, table 2 should be shortened to table 3.
Table 3. Standard/sub-standard proportions for life and health risks Life risks Health risks standard sub-standard Total standard 97 46 143 sub-standard 21 356 377 Total 118 402 520 The actual value of χ 2 -test is 225,56 (that is much higher than the critical value that is equal to 3,84 with 1 d.f. and 5 per cent of probability). The data certainly show that there is large enough dependence between life and health scores even for age intervals where it is not highly expected from the point of view of health dynamics with age. This means that actuaries and underwriters should be more careful with assumptions about the existence of independence between different Life and Health products in context of ALM and similar concepts. The important result is that the proportion of standard risks is 27,5 per cent for life score and 22,69 per cent for health score. The odd of standard and sub-standard risks (1:3) is quite different from usual odd for life insurance portfolios (9:1). It could be explained with a) more conservative estimation under the investigation than one in insurance practice as a result of taking into account the sales strategy and underwriting strength avoiding underwriting for insurance policies with small sum assured etc., b) self-selection of potential clients with poor health, c) full informational support in the investigation vs. informational deficit in practice of insurance. Explanations (a) and (b) are not problematic as they show only the differences between insurance practice and the artificial investigation. But explanation (c), if it is correct, may be a sign of an underestimation of life and health risks by underwriters under the data shortage and long-term forecasting uncertainty. In the case, there is a huge amount of latent crosssubsidiary in life and health insurance portfolios. The next step is to study life and health dependence among sub-standard risks. Correlation coefficient is 0,84 which is even more than for all risks. The idea is to develop more formal model than simple statistical coefficient, say, copulas [5 7]. First of all, one needs to find marginal distributions (although conditional ones given risk is sub-standard). It is possible to use the last column of table 2 (without the above value of 143) for life scores frequencies and the last row of the table (without the left-hand value of 118) for health scores. The last two boxes (300 and >300 ) for health risk scores should be combined. Both distributions were fitted using Maximum Likelihood method. The best goodness-of-fit (measured with χ 2 -test) was achieved on Log-Normal distribution in both cases. The results are shown in table 4. Table 4. Frequency distribution of the rounded score for life and health risks Values for life risks health risks Distribution parameter μ 3,761 4,545 Distribution parameter σ 1,043 2,088 Degrees of freedom 4 3 χ 2 -test 8,81 1,39 p-value 0,066 0,709
1 1 As a first choice, the normal copula could be used: C( u, v) = Φ ( Φ ( u), Φ ( )), 2 v where Φ 2 (, ) is the bivariate Normal distribution function with zero vector of expected values and covariation matrix 1 α, α 1 α is the correlation coefficient (in our case 0,84), Φ 1 ( ) is the inverse function to standard Normal distribution function. As marginal distributions in our case are Log-Normal, the copula simply gives the bivariate Log-Normal distribution. Other copulas tend to bring more complex formulas. Such models may be quite simple tools for portfolio modelling in the context of ALM or similar concepts. Conclusions The example shows the high enough dependence between life and health underwriting scores. The dependence could not be explained only with mortality risks in permanent health (income protection) products as it is too high. The fact means that actuaries and underwriters should be careful with portfolio construction and modelling. Moreover, copula technique may give quite simple tool for modeling with dependence. Another important question which has been brought up with the report is whether a kind of underestimation of life and health risks exists in underwriting process as a result of informational deficit and uncertainty. The further investigation of underwriting procedures should be organized. Bibliographies 1. Life and Disability Underwriting Manual. In 2 Vol. Skandia International Insurance Corporation. Life Reassurance. 2. Life Underwriting Manual. Munich Reinsurance Company. [Münchener Rückversicherungs-Gesellschaft AG]. 3. Medical Underwriting Guidelines. In 2 Vol. [Hong Kong]: The Cologne Re, 1995. [Kölnische Rückversicherungs-Gesellschaft AG]. 4. Occupational Rating Guide. Munich Reinsurance Company. 1992. [Münchener Rückversicherungs-Gesellschaft AG]. 5. Carrière J.F. Copulas. In: Encyclopedia of Actuarial Science. Vol. 1, p. 375 379. 6. Frees G.V., Valdez E.A. Understanding Relationship Using Copulas. North American Actuarial Journal. Vol. 2, No. 1, p. 1-25. 7. Venter G. Tails of Copulas. ASTIN Colloquium. Tokyo, Japan. August 22-25, 1999.