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 Dependence9.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 interval Final scoreup to 100 100

from 101 to 135 125from 136 to 175 150from 176 to 225 200from 226 to 275 250from 276 to 325 300more then 326 >300

The distribution of people investigated

Life score

Health score Total

100 125 150 200 250 300 >300

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

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 risks Health risks Total

standard sub-standard

standard 97 46 143

sub-standard

21 356 377

Total 118 402 520

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 witha) more conservative estimation under the

investigation than one in insurance practiceb) self-selection of potential clients with poor

healthc) full informational support in the investigation

vs. informational deficit in practice of insurance

• The latter explanation is important for insurance practice

Dependence amongsub-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 χ2-test) was achieved on

Log-Normal distribution

Marginal distributions

  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

χ2-test 8,81 1,39

p-value 0,066 0,709

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

)(),(),( 112 vuvuC

),(2

1

1

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!