Dependence between mortality and morbidity: is underwriting scoring really different for Life and Health products? Andrey Kudryavtsev, St.Petersburg State University, Russia 1.A. Stochastic Dependence 9.A. Various Topics

Aim to show that underwriting scores are quite close to each other for different kinds of insurance products, say for life and health insurance If so, there are problems in portfolio construction because of –risks may be more dependent, –possible higher degree of risk accumulation

Idea to compare underwriting scores for life and health risks of a sample population Results –help to understand question how to use and interpret the underwriting scores –do NOT help to solve any questions of statistical estimation

Methodology The sample population used is investigated from medical point of view The medical records and reviews were used to produce the averaging underwriting scores for life and health risks The scores are comparing to estimate the existence and degree of correlations The idea of modelling with copula is analysed

The investigation paper is based on the special study with data collection for real group of people 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 –survey about people’s preferences in healthcare

The place of investigation Lyssye Gory – a small town in Central Russia in Saratov Region (downstream river Volga, south-east from Moscow) WHY: –typical agricultural province in Russia with some industrial development –an appropriate professional mix of population

The target group people living in one medical district additional restrictions: –age interval chosen (from 20 to 49 including the latter age) –full set of the covariates (risk factors) investigated

Reasons for age restrictions Young people (younger than 20 year old) are presumably completely healthy: probably no extra life and health risks Old people (50+) are probably quite ill: the dependence observed between life and health risks is basically explained with poor health Only chosen age range (20 to 49) demonstrates balanced mixture of risk sub-groups

The basic risk factor chosen job/profession (with additional information about working conditions) height/weight index existing conditions (current diseases) addictions (tobacco smoking and alcohol drinking) heredity factors (indirectly estimated)

The Underwriting Manuals used Insurers: –Skandia International Insurance Corporation –Munich Re –Cologne Re There are some differences in those company- specific scoring procedures Resulting 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)

Underwriting scoring Risks estimated –Life (extra mortality score under whole life insurance contract ) –Health (permanent health (income protection) insurance with 4 weeks of waiting periods) The choice of health scoring –it shows quite serious problem with health –too serious (very long) diseases are rare

Rounding the individual scores Score intervalFinal score up to from 101 to from 136 to from 176 to from 226 to from 276 to more then 326>300

The distribution of people investigated Life score Health scoreTotal > >300> Total

The distribution of people investigated there is some form of dependence the coefficient of correlation is 0,6312 quite large – the actual t-test value is 24,6 that is much higher than the critical value nevertheless, it is far from comonotonic (one- to-one functional) dependence the dependence could not be explained only with mortality risks in permanent health (income protection) products as it is too high

Standard/sub-standard proportions Life risksHealth risksTotal standardsub- standard standard sub- standard Total

Standard/sub-standard dependence: conclusions 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 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

Standard/sub-standard dependence: analysis 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 It is too small The odd of standard and sub-standard risks (1:3) is different from usual odd for life insurance portfolios (9:1)

Standard/sub-standard dependence: explanations The differencies could be explained with a)more conservative estimation under the investigation than one in insurance practice b)self-selection of potential clients with poor health c)full informational support in the investigation vs. informational deficit in practice of insurance The latter explanation is important for insurance practice

Dependence among sub-standard risks Correlation coefficient is 0,84 It is even more than for all risks The idea is to develop more formal model than simple statistical coefficient, say, copulas It helps to understand the character of dependence in more details

Marginal distributions They are conditional as the risks analysed are sub-standard The last two “boxes” (300 and ‘>300’) for health risk scores should be combined Both distributions were fitted using Maximum Likelihood method In both cases, the best goodness-of-fit (measured with χ 2 -test) was achieved on Log-Normal distribution

Marginal distributions Values for life riskshealth risks Distribution parameter μ3,7614,545 Distribution parameter σ1,0432,088 Degrees of freedom43 χ 2 -test8,811,39 p-value0,0660,709

Copula As a first choice, the normal copula could be used where is the bivariate Normal distribution function with zero vector of expected values and covariation matrix

Copula: conclusions As marginal distributions in our case are Log- Normal, the copula simply gives the bivariate Log-Normal distribution Unfortunately, the model is not well calibrated Other copulas tend to bring much more complex formulas Such models may be quite simple tools for portfolio modelling in the context of ALM or similar concepts

Thank You!