1.A. Stochastic Dependence
9.A. Various Topics
Dependence between mortality
and morbidity: is underwriting
scoring really different for Life
and Health products?
Andrey Kudryavtsev,
St.Petersburg State University,
Russia
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 companyspecific 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
up to 100
from 101 to 135
from 136 to 175
from 176 to 225
from 226 to 275
from 276 to 325
more then 326
Final score
100
125
150
200
250
300
>300
The distribution of
people investigated
Life
score
Health score
100
125
150
100
97
43
1
125
20
78
41
2
150
1
6
16
200
200
Total
250
300
2
143
2
13
156
33
6
56
118
5
12
28
45
26
27
5
5
24
26
154
520
250
1
300
>300
Total
2
118
127
58
>300
42
20
1
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 (oneto-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
standard
Total
standard
97
substandard
46
substandard
Total
21
356
377
118
402
520
143
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 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
4
3
χ2-test
8,81
1,39
p-value
0,066
0,709
Degrees of freedom
Copula
• As a first choice, the normal copula could be
used

1
1
C (u, v)   2  (u),  (v)
where  2 (,) is the bivariate Normal

distribution function with zero vector of
expected values and covariation matrix
1 
 1
Copula: conclusions
• As marginal distributions in our case are LogNormal, 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!