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Disability insurance in The Netherlands
Insurance: Mathematics North-Holland
and Economics
101
13 (1993) 101-116
Disability insurance in The Netherlands - The policy conditions provide that disability exists if the insured directly and due solely to medical consequences of an accident and/or illness is disabled for at least 25% to perform the duties connected with his occupation as shown in the schedule, in a way reasonable for these professional activities.
F.K. Gregorius Meidoornlaan 3, Castricum, The Netherlands Received November Revised June 1993
1992
Keywords: Disability
insurance
1. Introduction 1.1. Preliminary This paper is concerned with Individual Disability Insurance, a product especially for selfemployed business-people (employees are largely covered by Social Law). Disability Insurance in the Netherlands belongs to Non-Life, though methods and technics are very similar to Life Assurance, albeit that they are much more complicated as will be shown below. Most insurers followed technical bases dated to 1982. Deteriorating results induced the organization of disability insurers (KAZO) to instruct their Actuarial Committee (AC) to advise on a revision of tariffs. The AC reported in 1990 and a year later its recommendations were implemented. The following is a summary of the AC’s study. 1.2. Specifications of individual disability insurance There
are 2 main types:
(1) The first year’s risk (A-cover). - A sick or disabled insured gets benefits, starting after a period for own risk (waiting period) and continuing ultimately up to one year after the beginning of the disablement. - The period for own risk can be chosen by the insured when he contracts his policy with possibilities: 7 or 14 days, 1, 3 or 6 months. - The benefits are based on a just as well chosen amount of an annuity. - In consequence we speak of the amount of insured A-annuity. 0167-6687/93/$06.00
0 1993 - Elsevier
Science
Publishers
(2) The after-first year’s risk (B-cover). - The benefits exist of an annuity starting after a one year waiting period from the beginning of the disablement and stopping in case of recovery or when the term of insurance expires, as a rule at an age between 55 and 65. - The policy conditions provide that disability exists if the insured directly and due solely to medical consequences of an accident and/or illness is disabled for at least 25% to perform the duties - from hereon different from Acover - which meet his abilities and capacity and which can be reasonably demanded from him in view of his education and former activities. - In establishing the measure of disableness the possibility to get an (adapted) job wil not be taken in consideration. Benefits are also granted in the case of partial disability. The insured amounts can be indexated with a fixed rate: 3, 4 or 5%. The distinction between the two types mentioned is connected with Social legislation. The so-called AAW (Algemene Arbeidsongeschiktheids Wet = General Disability Law) insures every - not employed - citizen for an annuity to an amount deducted from the legal minimum-wage. Benefits are granted, however, after a one year waiting period. Consequently, there is no cover for the first year at all and that cover can be obtained by the A-cover. After the first year a supplement to the AAW will suffice and that can be done by the B-cover. So the insured amounts can differ between the A- and B-cover.
B.V. All rights reserved
102
F.K. Gregorius
/ Disability insurance in the Netherlands
It may be clear, that the A-cover is more similar to Non-Life insurance and that especially the B-cover has many common elements with Life assurance. Policies of Disability insurance cannot be cancelled by the insurer.
2. The technique
of disability
As mentioned before the A-cover is very similar to other types of Non-Life insurance. The risk premium is based on statistics about the ratio of granted benefits and insured amounts (annuities). The premium is differentiated by the following risk factors:
- 7 and 14 days, - 1, 3 and 6 months.
(2) Age: nine groups
- 16-25 years, - 26-30 years, - 31-35 years, etc. up to 65. (3) Class of profession: Four classes according to the largeness risk of the insured’s profession.
P(i,
In the following the probability system, built around the elements i(x), r(x, s) en q(x) will be explained.
of
j, k)
=B*a(i)*b(j)*c(k). The a, b and c apply to the three (i = l-5; j = 1-9; k = l-4).
risk factors
2.2. The B-cover 2.2.1. Introduction To the B-cover actuarial methods The following bases are used: - a disability rate i(x), - A factor for class of profession A-cover),
apply.
(equal
2.2.2. The probability system This system is explained by means of the following example: Suppose, we have 1000 insureds at the beginning of the accident year t. At the end of the year, suppose that the situation is as follows: (1) 10 insureds have died. (2) 90 insureds became disabled in the period l/1-30/9 and are still disabled on 31/12. (3) 40 insureds became disabled in the period l/10-31/12 and are still disabled on 31/12. (4) 860 insured remained active. Among them are insureds, disabled during the year, but after that recovered. We can consider these observations as a realization of a probability process with the following probabilities to distinguish:
The above risk factors form a network of 5 *9*4 = 180 cells. For each of these cells statistics were available. For the structure of the premium a multiplicative model has been chosen as follows: Risk-premium
time of
insurance
2.1. The A-cover
(1) Waiting period: five classes
- a recovery rate r(x, s> with s as lapsed disability, - a mortality rate q(x), - an intrest rate.
to the
_ the probability of an x-aged to remain alive during the year: p(x). _ the conditional probability of an x-aged, if he remains alive, to become disabled in the first nine months and to remain disabled within the next three months: i*(x). - the probability of an x-aged to become disabled in the first nine months and to be still disabled at year end: i(x). _ We note that i(x) =p(x) . i*(x). The distinction between disablements in the first nine and the last three months follows from practice. In case of B-cover new disablements are only as such registered after three months have elapsed because in case of short illnesses needless clerical work should be avoided. This means that disabled of the last three months are not as such distinguished and remain registered as actives. The above group no. (3) belongs consequently to the actives!
F.lZ Gregorius
It follows further that in the above probability to be active at year end is p(x)
.(l
-i*(X)}
‘P(X)
sense
the
-i(X)
= 1 -q(x)
-i(x).
_ The probability of an (x + l)-aged who became disabled in year t at age x, to be active at year end (t + 1): T{(X) + 1)). - We note, that: Y{(X) + 1) =p(x + 1). r *((xl + 0.
This expression is applied in the recursion formulas (refer to Section 5). The probability formulas for the preceding four groups are now: 1000 q(x) for the 10 deaths; 1000 a(x) .i*(x) = 1000 i(x) for the 90 disableds; 1000 (1 -q(x) -i(x)} for the 860 actives and for the 40 disableds from the last three months. The question is now what happens with groups (2) and (3) in year (t + 1). Continuing our example we suppose at year end (t + 1) the following observations: From the 90 disabled from group (2): - 1 died, - 50 recovered, - 39 remained disabled. From the 40 disabled from group (3): - 0 died, - 18 recovered, - 22 remained disabled. Consequently we have at year end (t + 1) 39 + 22 = 61 disabled. Here we have a somewhat different probability process with the following probabilities: - the probability of an (x + l)-aged alive during the year: p(x + 1).
103
in the Netherlands
/ Disability insurance
to remain
Remark. Though there is theoretically good reason to distinguish between survival probabilities of actives and of disabled, this has not been done. For both groups population mortality has been applied. - The conditional probability of an (x + l)-aged who became disabled in year t, if he remains alive, to be active again at year end (t + 1): r *{(xl + 1). Remark. The probability of recovery depends on the elapsed time of disablement s. The general notation for an insured who became disabled at age x is: T*{(X) + s}.
The probability (t + 1) is
to be still disabled
P(X + 1). ((1 -r*{(x) =p(x+ = 1 -q(x+
1) -r{(x) 1) -r{(x)
at year end
+ 11)) + 1) + I}.
This expression, too, is applied in the recursion formulas (ref. to Section 5). This probability applies to the 90 observed disabled. The ratio, connected with it, is 61/90, though from the 61 disabled, 38 originate from the 40 ‘3 months’cases. Remarks. (1) The 18 recovered insureds originating from the disablements of the last three months of year t remain out of observation. As it were, they remain actives! But notice: they did not get any benefit as well, because of one year waiting time! [Except in case of recovery after expiration of the waiting time but before year end (t + 1); this possibility has been neglected]. (2) The given example has not taken account of the possibility of IBNR-cases. These cases are registered like those of the last three months of year t and are consequently not distinguished as IBNR-cases. We can conclude, that the probability system, together with the according observations, include IBNR-cases. This means that there is no need for a separate IBNR-provision. (3) There is even another effect: numbers of disabled namely in case of partial disability only count for the benefit percentage connected with the disability percentage. This means that also alterations in disability percentage have their influence. Year (t + 2) and following years are dealt with like year (t + 1). The r{(x) + s} are derived by comparing numbers of disabled at beginning and end of the year. Here, too, we have the effect of the balance of recoveries, changes in degrees of disability and possibly late IBNR-cases. This balance effect can cause negative r{(x) + s}-ses. This happens especially in case of high s-values (4 and 5).
104
F.K. Gregorius / Disability insurance in the Netherlands
If s is 6 or higher all these glectable and Y is put at zero.
effects
are
ne-
2.2.3. Application of the probability system The applied theory is very similar to traditional methods in Life assurance. The following four single premiums are applied: (1) Ai(x, e) = Single premium for a disability insurance (annuity) for an x-aged, payable continuously up to end-age e and with a one year waiting period. (2) ai{(x + s), e} = Single premium for a continuous annuity, immediately taking effect for an (x + s&aged disabled and payable up to end age e. The notation means: - disabled at age: x, - past duration of disablement: s. The description, too, implies, that the annuity ends in case of recovery and starts again after returned disablement. (3) aa(x, e) = Single premium for a yearly prepayable annuity, immediately taking effect for an x-aged and payable: _ in case of disablement: up to the end of the waiting period - in case of remaining active: up to one year before the end age e (the last year is premium free). In case of recovery the annuity comes into force again. This single premium aims at determining the present value of premiums to pay. (4) aia{(x) + s, e} = Single premium for a yearly prepayable conditional annuity for an (x + s)-aged disabled and payable during (the rest of) the waiting period and further only after recovery and then up to one year before end age e. This single premium aims at determining the present value of premiums to pay during (the rest of) the waiting period and after that only after possibly occuring recovery. The foregoing shows some details concerning the payment of premiums, following from the insurance conditions. The definitions apply to the unity of money. In the following for reasons of clarity the subscript e will be left out.
With the foregoing, formulas for premiums and provisions can be set up. In doing this one has to take in mind, that insureds can change from actives into disabled V.V. and also can die. There are two practical techniques: (1) Application of recursion formulas Section 5). (2) Interpretation as a Markov process Subsection 7.3).
(refer
to
(refer
to
2.3. Provisions 2.3.1. The actuarial reserve In case of level premiums the applied equivalence principle implies that we have to set up an Actuarial reserve (in case of payment of yearly risk premiums which are adapted to the age reached, this is naturally not necessary). With the B-cover we can possibly be confronted with negative amounts. Some years before the end date of the insurance the risk decreases, because the influence of the quickly shortening remaining period surpasses that of still increasing disability rates. The period of premium payment should have been sufficiently shortened, but for commercial reasons this is limited to one year. As a rule a negative Actuarial reserve is put at zero. For formulas refer to Section 5. 2.3.2. The claims reserve In case of A-cover no actuarial methods are applied. The provision is an estimation of amounts to be paid, taking into account the nature of the insured’s disability. In case of B-cover, first three months of disability have to elapse before the provision is set up in order to avoid unnecessary administrative work for early recoveries. This procedure gives rise to the question whether some provision should have been set up. However, this is done indirectly. As soon as a provision is entered, it is calculated with reduced recovery rates, which are a balance of several effects. Refer to the explanation for the probability system in Subsection 2.2.2. For formulas refer to section 5.
F.K. Gregorius / Disability insurance in the Netherlands
2.3.3. The IBNR-provision As explained in Subsection 2.2.2 - refer especially to Remark (2) - the Claims provision includes IBNR. 2.3.4. Actuarial Reserve for disabled A disabled has a chance to recover and in that case an Actuarial Reserve has to be entered again. This means, that as long as there is a non-zero probability to recover - this is the case during the first five years of disablement - besides the claims reserve the Actuarial Reserve has to be partly maintained. This matter is further explained in Section 5. 2.3.5. Provisions for expenses For reasons of completeness we mention these provisions, which have to be set up for claimsand administration expenses.
Statistical data are similarly obtained by an other periodical inquiry, the so-called KAZOstatistiek. This KAZO-statistiek aims at testing the technical bases of the tariff and consequently of its structure. So this concerns invalidation, recovery and mortality and this towards the risk factors: age, class of profession and waiting period. Recovery and mortality are moreover distinguished towards past duration of disablement. 3.2. The statistics for the A-cover This cover is dealt with like other non-life classes. Incurred claims are compared with insured sums in portfolio and this for all combinations of the risk factors earlier explained. Table 1 shows an example of the inquiry of incurred claims for class of profession 3. 3.3. The statistics for the B-cover
3. Data inquiries
Observations
3.1. General In the Netherlands for some ten years a centralized office for financial and statistical data has existed: the ‘Centrum voor Verzekerings Statistiek’ (CVS). Financial data are gathered from a substantial part of the market by the ‘Comptabele Enquete’, a periodical inquiry about financial results.
Incurred Waiting 3 days
10.5
claims for class of profession
aim at:
- disability rates i(x) depending on age x, _ recovery rates r{(x) + s} depending on age x and past duration of disablement s. Mortality is observed, indeed, but because N.B. of small numbers these observations are not used. Mortality rates are taken from a recent Dutch population table (GBM 1980-1985).
Table 1 3 (disability statistics,
A-cover,
1989, benefits
period
Total 14 days
1 month
0 0 5,184 0 1,829 0 0 17,262
11,582 28,259 100,507 64,763 267,398 139,655 114,272 11,070 41,218
26,981 24,177 6,987 59,261 121,040 113,903 0 2.701
7 days
5 25 years 26-30 31-35 36-40 41-45 46-50 51-55 56-60 61-65
0
Total
0
24,275
778,724
In %
0.0
2.1
67.1
0
in Dutch guilders).
0
3 months
In %
6 months
0
0
0
0
0
0
0
0
0
0
0
0
2,755 0 0
0 0 0
11,582 55,240 124,684 76,934 326,659 262,524 230,930 11,070 61,181
355,050
2,755
0
1,160,804
30.6
0.2
0.0
1.0 4.8 10.7 6.6 28.1 22.6 19.9 1.0 5.3
100
106
F.K. Gregorius
/ Disability insurance in the Netherlands
The waiting period is standard one year and consequently not a risk factor. The class of profession is a risk factor indeed. There is good reason to assume that the structure towards x and s of disability- and recovery rates differ per class of profession. This possibility, however, has been neglected.
4. The construction
The construction of the new tariff is based on what has been explained in Subsection 2.1: ‘the technique of disability insurance’. The essential quantity is the ratio of granted benefits and insured amounts, which ratio is observed for 180 groups of combinations of the risk factors: - waiting period - age - class of profession
- 5 possibilities, - 9 groups, - 4 classes.
The risk premium ‘Pris’ for a certain combination of risk factors is determined according to the multiplicative formula: Pris = Pris( basic cell) x factor waiting
These factors, representing per factor the mutual proportions of the several possibilities, are calculated by means of a multiplicative regression model. For more details refer to Section 7.1. In Table 2 the figures used are shown. The upper part shows insured amounts of annuities, used as weighing factor. In the lower part we have percentages of becoming disabled, i.e. the earlier mentioned ratio between paid and insured amounts. The figures only show two classes of profession; four classes were introduced later on. The results for waiting period and age were (basic cells are indicated by *): 7 14 1 3 6
62 70 100 * 127 133 164 189 211 198
days days month months months
119 100* 61 23 16
For the classes of profession the ratios were determined after a special inquiry, needed because recently, concurrent with the study, four classes instead of two had been introduced. The ratios of the age groups were further modelled applying y=a.bx. This offered the opportunity to introduce a tariff per age instead of per age group. Further the ratios found for Waiting period and Class of profession were rounded off. Finally it was checked that de weighted sum of model values was equal to the sum of the observations. All this resulted in the following:
period
X factor age X factor class of profession.
period:
16-2.5 26-30 31-3.5 36-40 41-45 46-50 51-55 56-60 61-65
of the new tariff (net)
4.1. The A-couer
Waiting
Age group:
(a) ratios for Waiting period: 125 7 days: 100 14 days: 1 month: 70 3 months: 40 6 months: 30 (b) ratios for age: 32.247 times 1.036” (for x = 32 this gives 100). (c) ratios for Class of profession: class 1: 70 class 2: 85 class 3: 100 class 4: 120 (d) premium per 100 guilders annuity for the basic cell: Dfl. 2.30. In case of the Risk Tariff a solvency loading of 6% is added, because there is no Actuarial Reserve, which yields some surplus above the investment income needed. This 6% is not a result of actuarial methods.
107
F.K. Gregotius / Disability insurance in the Netherlands
technique of disability insurance’. Essential are the following bases, which will be treated in the next paragraphs: - disability rates, - recovery rates, - mortality rates,
4.2. The B-cover
4.2.1. General The construction of the new tariff is based on what has been explained in Subsection 2.2: ‘The
Table 2 Portfolio 1986 + 1987, A-cover Insured annuities Class 1 (old) Age/w
7 days
14 days
1 month
3 months
6 months
16-25 26-30 31-35 36-40 41-45 46-50 51-55 56-60 61-65
994 2734 5327 8835 9356 7366 4198 2433 712
43116 85232 118796 155982 119184 80233 48699 22665 6565
9296 28830 55725 86550 70649 47060 26509 11757 3003
1623 7934 17987 32178 28256 20819 12115 6329 1545
653 2777 6463 9237 10898 7151 4995 3129 464
3968 6745 15021 32486 50265 58236 50387 28669 4183
268916 326981 331858 347033 248759 178951 105178 44375 6367
31363 58422 71164 85879 68324 47524 29272 11751 1645
5721 10509 12456 15514 16490 12038 6982 2894 429
1949 3267 4237 7521 7655 6136 3308 1876 185
Class 2 (old) 16-25 26-30 31-35 36-40 41-45 46-50 51-55 56-60 61-65
Granted benefits as a percentage of insured annuities Class 1 (old) 16-25 26-30 31-35 36-40 41-45 46-50 51-55 56-60 61-65
1.11 2.34 1.80 3.24 2.97 5.10 5.34 3.70 2.39
1.83 1.58 2.28 2.66 2.96 3.24 3.92 4.32 4.05
0.83 0.70 1.07 1.16 1.65 1.67 2.97 2.95 5.09
0.00 0.49 0.62 0.10 0.89 0.14 0.87 2.15 0.26
0.61 0.00 0.40 0.01 0.77 0.14 0.28 1.69 1.29
3.38 2.48 4.60 5.05 4.55 5.55 6.35 6.15 7.05
1.72 2.03 2.82 3.58 3.88 4.82 5.35 6.19 5.26
0.86 1.02 1.53 2.53 1.77 2.97 3.92 5.13 3.71
0.77 0.09 0.31 0.78 0.54 0.89 2.01 4.77 0.47
0.00 0.00 0.28 0.17 0.46 0.47 1.27 2.45 0.00
Class 2 (old) 16-25 26-30 31-35 36-40 41-45 46-50 51-55 56-60 61-65
F. K. Gregorius / Disability insurance in the Netherlands
108
The rates in Table 3 are modelled applying: y = a . b”, similar to the modelling for the A-cover. The numbers of insureds have been used as weights. The factor a is adjusted such that the total weighted rate is equal to the observed total at 1.63. Here, too, an opportunity to introduce a tariff per age instead of per age-group was offered. All this resulted in the following formula:
Table 3 Age group
Disability rates
Number insureds
16-25 26-30 31-35 36-40 41-45 46-50 51-55 56-60 61-65
0.54 0.77 1.07 1.32 1.52 2.04 2.56 2.92 2.74
29137 39765 51754 73289 76188 70059 54927 30927 6424
Total
1.63
432470
of
i(x)
It should be remembered, that i(x) only refers to the first nine months of the concerning accident year (therein becoming disabled and remaining so up to the year end).
Besides these a role is played by: - The intrest rate, for which 4% was taken. In following formulas a discount factor L! could be met. This c’ is equal to l/1.04. - The factor for class of profession, for which the same factors as used for the A-cover were applied. Waiting period is not a factor, because it is standard one year. As far as it is relevant the calculations in the sequel are applicable to class of profession 3.
4.2.3. Recovery rates In Figure 1 one can see, that the recovery rates can very well be modelled by means of straight lines. The graph shows: (1) for every duration of disability s (= 1 up to 5 incl.) there are different lines and - generally spoken - the lines lie in conformity with the value of s one under the other. Only for s = 1 this does not f&y apply because of the large influence of the disabilities of the last three months of the past year (refer to Subsection 2.2.2). (2) Every line - generally spoken - descends from the left to the right, which obviously ex-
4.2.2. Disability rates Disability rates are derived from unweighted rates of becoming disabled. The observations about the period 1984-1988 that were available are listed in Table 3.
-0,400
-I
28
23 --
duratkm dhab4ity->
1year
33
= 0.00223 times 1.0468”.
38
43
.--_)-__
Age ..... &
2 pars
3 yeam
Fig. 1. Recovery
rates 1990.
48 _ 4 years
53
58
63
_____-- - _ 5-n
e+ years
F.K. Gregorius / Disabilityinsurancein the Netherlands
presses - for each s - decreasing recovery rates with increasing ages. This has consequences for our formula, which can be of the form: r = a - b -x, but with a and b depending on s. So our five lines can be expressed as follows: r{(x)
+s)
=a(s)
-b(s).x,
109
(3) should be fulfilled for all nine x-values, but, because we have a straight line, it is sufficient to require this only for x = 22 and x = 62. Summarized, the restrictions to the regression procedure are: (1) a(s) 2 0, (2) b(s) L 0,
where a(s) =
a parameter depending on s with s = 1 up to 5 incl., b(s): ditto, x = age, but for the estimating procedure x has been taken: 22, 27, 32,. . .62 respectively (9 groups). As estimating-procedure a least squares regression method has been applied, with weighting with benefit-percentages u{(x) + s} as volume components. Consequently the sum of squares to be minimized was
(3) a(s + 1) - a(s) + 22. {b(s) - b(s + l)} 5 0 (s = 2, 3, 41, (4) a(s + 1) - a(s) + 62. {b(s) - b(s + l)} 5 0 (s = 2, 3, 4). These restrictions raise considerable complications in the calculations. The problem can be considered as a quadratic programming problem. In this way it is solved by means of a computer package. The results were: r{(x)
+ I} = 1.24111 - 0.02219x
(0.75 + -0.13467), r{(x)
+ 2} = 0.66499 - 0.01153x
(0.41-+ -0.04987), r{(x)
((a(s) -b(s) .x-r{(x)
+s)))*.
(0.16 --f - 0.05590))
the r{(x) + s} being the observed values. The foregoing, however, submits this minimizing-process to some conditions. This means: a(s)
is positive, because the line - anyway for small x - cannot lie under the x-axis. (2) b(s) is positive in connection with the beforehand chosen minus-sign in the formula of the line. (3) r{(x) + s} 2 r{(x) + s + l} with s = 2, 3, 4 leading to:
(1)
+ 3} = 0.27394 - 0.00532x
r( (x)
+ 4} = 0.23547 - 0.00470x
(0.13 + -0.05593)) r{(x)
+ 5) = 0.14166 - 0.00319x
(0.07 + -0.05612). Between brackets the course of the values from age 22 to 62 is given. Indeed, each line lies under the foregoing except the first one. But for this the restrictions did not apply. Notice the negative values for high ages, especially for s = 1.
a(S)-b(s)*x~a(s+l)-b(s+l).x, otherwise written: a(s+l)-a(s)+{b(s)-b(s+l)}.xsO. (4) r{(x) + s} 2 r{(x + 5) + s} 27 , . . . ,57 leading to a(s)
-b(s).xza(s)
with
-b(s).(x+5),
leading to b(s) L 0, which is consistent with (2).
x = 22,
4.2.4. Mortality rates Mortality rates of disabled are higher than those of actives. Further one can assume, that this rate depends on the duration of the disablement. Observations with death cases, however, is a matter of small numbers and consequently of insufficient reliability. Therefore mortality rates are taken from the population table GBM 1980-1985 for actives as well as for disabled. Generally this provides some safety margin.
110
F.K. Gregorius / Disability insurance in the Netherlands
4.2.5. The net premium for the risk tariff This is the tariff with yearly altering premiums, which only are paid for the running year and do not contain any component for future risk as is the case with level premium tariffs. The following formula holds: Pris(x)
= 1.06 * u * i(x)
* ai{
+ I}.
The single premium ai{ + 1) is determined applying the so-called ‘model without return’ (refer to Subsection 7.3.2). In this model it is assumed, that a disabled does not return to the actives, because, if that happens, he will be considered as a new active insured, paying again premiums according to his age. As the formula shows, in conformity with the A-cover, a solvency loading of 6% has been applied. 4.2.6. Level premiums (net) The net level premium is the quotient of the single premiums Ai and aa(x>, both mentioned in Subsection 2.2.3. Both single premiums are determined applying the so-called ‘model with return’. In this model it is assumed, that disabled can return to the actives again, then can become disabled again etc. etc. (refer to Subsection 7.3.2).
(a) For actives: RaAi(x)
-P.aa(x)
- R = amount of the annuity, - P = (net) policy-premium. The Ai and aa are defined in Subsection 2.2.3. For these quantities the ‘model with return’ is applied. This can be illustrated by the following recursion formulas: Ai
5.1. Introduction
In Subsection 2.4 something has already been told about provisions. Referred has been to this section for more details, especially for the formulas. In the following this will be dealt with. 5.2. The Actuarial Reserve
As mentioned in Subsection 2.4 in case of level premiums an Actuarial Reserve is set up for actives as well as disabled, for the latter as long as recovery rates are non-zero. The following prospective formulas are applied:
-i(x)}
=v*{l-q(x) +v*i(x).ai{(x)
.Ai(x+l) + I},
ai{(x)+l}=v.((l-q(x+l)-r{(x)+l))) .((OS+ai{(x) +v.r((x)
+2))) +l)
.Ai(x+2).
Notes: - The probabilities
used are explained in Subsection 2.2.2. - The ai((x> + s) for s > 1 contains a first term 0.5, representing a payment 0.5 at the beginning of the year followed by 0.5 at the end, this being an approximation of continuous payment. But in the case s = 1 the first term 0.5 is absent because of the one year waiting period, on the average falling a half year in the next insurance year. Further +v.i(x)
5. Provisions
with:
.aia{(x)
+ 1)
and, very special, aia{(x)
+ 1)
=0.5+v.((l-q(x+l)-r((x)+l))) .aia{(x)+2)+v.r{(x)+l)*aa(x+2). Note: - The term 0.5 concerns
premium to be paid over the (rest of) the waiting period, being on the average 0.5 year. For s > 1 the term 0.5 is absent.
Notice the interdependency between both pairs of formulas. The single premiums can be calculated starting with the end age and then proceeding backwards. (b) For disabled: R.Aia{(x)
+s)
-P*aia((x)
+s).
Here we have a special, up to here not introduced single premium Aia{(x) + s} representing
F.K. Gregorius / Disability insurance in the Netherlands
a conditional abled, should reads Aia((x)
insurance, namely Ai for a dishe recover. The recursion formula
=w((l-q(x+s)
-r{(x) +s+
+s}))
I} +u.r{(x) .Ai(x
+s)
An other interesting approach of the foregoing is to consider it as a Markov process. Refer to Subsection 7.3.4. 5.3. The Claims Reserve Dealing with the Claims Reserve we have to apply the ‘Model without return’. The Claims reserve only concerns the running annuity. Liabilities in case of recovery have been met by the Actuarial Reserve. Hence the formulas are = R . ai{ (x)
+ s}
with ai{
included
in
These, too, are mentioned here for reasons completeness, referring to Subsection 2.4.5.
of
+s + 1).
For the A-cover the formulas are similar albeit that _ Ai = Pris(x) + u. {l - q(x) - i(x)} .Ai(x + 1) + c’ . i(x) *Au{(x) + 1). The interpretation of this formula is, that the insurance is considered as a running annuity to the amount of the risk premium. _ The single premium ai{ + s} does not exist.
claims reserve
is, generally,
5.5. Provisions for expenses
+s}
Au{(x)
The IBNR-provision the Claims Reserve.
111
+s} =OS+v.((l-q(x+s) .((OS+ai((x)
-r{(x)
+s}))
6. Gross premiums 6.1. Introduction The gross premium - expenses, - commission, - solvency; profit.
contains
loadings
for:
To begin with the last: it is not based on calculations, except in the case of the risk tariff (6%, as mentioned before). It is provided for by safety margins in the other bases. Further, the surplus of interest (above 4%) is an important source. Commission simply is a percentage of the gross premium. The structure of the loading for expenses is of some interest. The old tariff contained a percentage loading (17%). It is obvious that this gives too high a loading in case of high premiums (and too low in case of low ones). A better structure of the loading for expenses has been studied. The following components have been distinguished:
+s+l})).
Notes: - The first term 0.5 is absent in case of s = 1. - Notice the difference with the corresponding formula in Subsection 5.2. - As mentioned in the preceding the Claims Reserve for the A-cover is not calculated by means of actuarial methods. Here, too, we refer to the Markovian application Subsection 7.3.4.
a = component for claims expenses; b = component for costs of medical examinations and reports, these for underwriting purposes; cl = component for (other) costs of underwriting, administrative costs of producing and entering the policy and acquisition costs; c2 = component for the rest of expenses as file administration, collection of premiums, bookkeeping etc.
5.4. The IBNR-provision
The structure on the following
For reasons of completeness this provision mentioned here, referring to Subsection 4.3.2.
is
of the gross premium considerations:
- Besides the insured annuity are a part of the risk. Therefore
is based
claims expenses the component a
F. K. Gregorius
112
/ Disability insurance in the Netherlands
has been related to the net premium, which is deducted from the risk premium. A loading in the shape of a fixed amount - the average costs of handling of a claim - could be considered. But realizing that high insured amounts often lead to a more intensive claims handling, one could choose for a percentage loading on the risk- or net premium. The latter has been done. - The component 6 has been related to the insured annuity as a percentage loading. Costs of medical examination - for underwriting purposes - are not part of the risk. A fixed amount per policy could have been taken into consideration. A percentage loading somewhat reflects the fact that in case of low amounts no medical examination is requested. - The components cl and c2 are expressed as in fixed loading per policy. Generally spoken they are not related to the proportions of the annuity or the premium. These considerations led to the following for the gross premium:
formula
Pgr=(l+a)~R*Pnetl+b~R+c+p~P6 with Pgr = gross premium of the policy, R = amount of the insured annuity, Pnetl = net premium per unit of insured p = fraction for commission, c = cl + c2. This leads to
amount,
Pgr=((l+a)~Przet+b~R+c}/(l-p) with Pnet = net premium
7. Mathematical
of the policy.
techniques
7.1. Regression models 7.1.1. Introduction As mentioned in Subsection 2.1, the risk factors a(i), b(j) and c(k) of the A-cover have been estimated by means of a regression technique according to the premium formula, applied in practice: P(i,
j, k) =m
* a(i)
with 5, 9 and 4 possible respectively.
* 6(j) values
* c(k) for the i, j and k,
In the following some peculiarities, connected with the practical application of the applied model will be touched upon. Z1.2. The chosen model This is a weighted multiplicative cording to the foregoing, based on
model,
P(i,
* e(i, j, k).
j, k) =m * a(i)
* 6(j)
* c(k)
ac-
the last factor being the disturbance factor, assumed to be exponentially distributed. The a, 6 en c have been estimated by minimizing S2=SSS(i, x
j, k)g(i,
j, k)
(In m + In u(i) + In b(j) +In
c(k)
- In w(i, j, k)}*
with: SSS(i, j, k) meaning
summation
over i, j and
k, g(i, j, k) representing the weighing-factor (g of Dutch ‘gewicht’ = weight) for cell (i, j, k), being the insured amount in cell (i, j, k), w(i, j, k) representing the observation (w = Dutch ‘waarneming’ = observation) for cell (i, j, k), being the percentage of becoming disabled in cell (i, j, k). 7.1.3. Dealing with the results In Section 4 about the tariff construction we have seen that the results of the regression technique have not been followed unchanged. For Waiting period and Class of profession they were settled at values, having been rounded off, sometimes somewhat safely upwards. After settling one of the parameters, the regression procedure can be executed again for the remaining factors. The question how to deal with the data is then of some interest. Let us suppose, that the c(k) are settled. Then in the squared sum above c(k) is no more an unknown but a known quantity, that can be combined with w(i, j, k) to a new, as it were, changed observed value w*(i, j), which has to satisfy g(i,
j) .ln w*(i, = S( k)g(i,
j)
j, k) . In w(i, j, k)
113
F.K. Gregorius / Disability insurance in the Netherlands
with - g(i, j> = S(k)&, j, k), - S(k) meaning summation
7.2. Recursion formulas over k.
The recursion formulas have already been mentioned in Section 5. It has also been indicated how the calculations can be carried out. In the report of the study this subject has been worked out extensively. The reader interested in these details is referred to this report.
This implies a prescription for telescoping the g(i, j, k) and the w(i, j, k) into w*(i, j) and g(i, j), respectively. The solution of the minimizing procedure is, as is well known, executed by putting at zero the derivatives of the squared sum with respect to the parameters m, u(i), b(j) and c(k), adding some conditions to get a solution, often putting m as weighted average of the observations. Comparing the 3- and 2-dimensional formulas - the last as a result of settling the c(k) - the following conclusions can be drawn, if the above prescription is followed:
7.3. The Markovian approach 7.3.1. Introduction About insureds of a disability portfolio can be said that they pass through several possible states, which can be: - active, - disabled more, - dead,
(1) the remaining
parameters are minimally affected, (2) m* = m if the c(k) are settled so that S(k) g(i, j, k). c(k) = 0, (3) a(i) and b(j) have in both the 2- and 3-dimensional cases the same values if all g(i, j, k) are 1 and otherwise this holds approximately.
with durations
1 up to 5 incl. and 6 or
The change of group depends probabilities, which are related to:
on transition
_ to remain active (to stay in the same group also is considered as transition), - to become disabled, - to remain disabled, but with group altering indeed, because difference is made into duration of disablement, - to recover, - to die.
7.1.4. The GLIM-package The calculations, connected with the above, are executed by means of a software package called GLIM (General Linear Interactive Modelling), introduced by Nelder and Wedderburn in 1972.
Table 4 a To:
A
I(3)
I(4)
I(5)
I(6)
D
0
0
0
0
0
P(1) 0 0 0 0 0 0
0 P(2) 0 0 0 0 0
0 0 P(3) 0 0 0 0
0 0 0
0 0 0 0 P(5) 1-q 0
4 4 4 4 4 4 4 1
I(1)
I(2)
; 0 0 0 0 0 0
From: A
p(A)
I(I) I(2) I(3) I(4) I(5) I(6) D
r(I) r(2) r(3) r(4) r(5) 0 0
P(4)
0 0 0
a For reasons of clarity the subscript x has been left out. p(A) = 1- q(x)i(x); i = i(x); r(s) = r((x - s)+ s) for s = 1 up to 5 incl.; p(s) = 1 - q(x)r((x - s)+ s) voor s = 1 up to 5 incl.; q = q(x); In accordance with definitions mentioned in Section 2. From the matrix can be read, that an insured, as long as he will not be found in I(6) or D, could at any time return into a group in which he ever was. Therefore we here speak of the ‘Model with return’, which is illustrated in Figure 2.
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F.K. Gregorius / Disability insurance in the Netherlands
The several states of an insured can be seen as a Markov chain. The calculation methods, connected with Markov chains, are used to determine single premiums and provisions. Application of the matrix algebra in computer programs has a strongly compressing (short formulas) and generalizing (single premiums for all states together) effect. This will be worked out hereinafter. 7.3.2. States and transitions In the foregoing it has already been mentioned that eight states can be distinguished, namely: -A - I(1) - I(2) - I(3) - I(4) - I(5) - I(6) -D
= = = = = = = =
active, 1 year disabled, ’ 2 years disabled, 3 years disabled, 4 years disabled, 5 years disabled; 6 or more years disabled, dead.
The transitions (after 1 year as unity of time) are reflected in the transition matrix P(x) in Table 4, in which various transition probabilities, belonging to age X, are arranged. The before explained change of states has to be taken into account in case of calculation of single premiums and provisions. There are, however, two exceptions, namely: - risk premiums (refer to Subsection 4.2.5), - the Claims Reserve (refer to Subsection 5.3). For these the ‘Model without return’ has been applied. In its transition matrix the T(S) do not appear. For illustration refer to Figure 3. Closer considerations: (1) In case of the risk tariff, risk is covered for one year, which risk consists of an annuity to be paid in case of disablement up to the end-age, however without component that, after recovery, the insurance comes into force again (without premium payment). This component is not inProperly: disabled at the end of the year, in which the disablement started; this in accordance to the concepts, described in Section 2. With every next I-state the duration of the disablement is one year longer and - on an average half a year less than the subscript indicates {with adjusted interpretation for I(6)).
CIY I(6)
Fig. 2. The model with return.
chided in the risk premium. For, in case of recovery, the insured resumes premium payment in accordance with his age then reached. As it were the Risk Tariff consists of a series of one-year policies. (2) After recovery the Claims Reserve falls free; is put at zero. 7.3.3. Single premiums 7.3.3.1. Single premiums for a continuous annuity for a disabled The following general vector formula applies: A(x)
=cfpre+u.P(x).{A(x+l)
+cfpost],
where - A = general single premium symbol, - cfpre ( = cash-flow pre) indicates prepayable payments, - cfpost: ditto postpayable,
F. K. Gregorius / Disability insurance in the Netherlands
11.5
In the vectors cfpre and cfpost payments of 0.5 in case of running annuities are inserted as an approximation of continuous payment. The above does not apply to the A-cover. This will be dealt with in the following. 7.3.3.2. Single premiums for a prepayable active annuity Here, too, the general vector formula applies: A(x)=cfpre+v.P(x).{A(x+l)+cfpost). Here
A(x)’
reads
uiu{(x-1)
(uu(x),
+I),
uiu{(x-3)
+3),
uiu{(x-4)
uiu{(x-5)
+5),
O,O),
uiu{(x-2)
+2),
+4),
and cfpre’ reads { 190.5, 090,
cfpost is a zero-vector.
whereas
Fig. 3. The model without
return.
-
L’ = discountfactor, _ P(X) = the transition A(x)’ reads
matrix of the foregoing.
{Ai(x),ai{(x-1)+1},ai{(x-2)+2}, ai{(x-3)
+3},
ai((x-4)
ai{(x-5)
+5},
ai(
+4}, 0).
The element ai shows recovery rates are 0. cfpre’ reads
7.3.3.3. The single premiums for the A-cover The single premium Ai for the A-cover ’ can be considered as a single premium for an active annuity - au(x) - with the yearly risk premium Pris(x> as benefits and with an additional component Aiu{(x) + l} for a conditional A-cover insurance, namely after recovery, following on disablement. For, the A-cover insurance then comes into force again. The risk premium R-is(x) has the form a. b” (refer to Section 4). It follows from the before described, that Pris(x> gets a place in the cashflow vector cfpre, which in this case depends on x and properly should have been indicated with cfpre(x). We seem to have the same vector formula as before, but A(x)’ reads {Ai(
that
after
five years
(0, 0, 0.5, 0.5, 0.5, 0.5, 0.5, O} and ditto cfpost ‘. After working out the vector formula the accordance with the recursion formulas earlier explained, can be recognized. For Claims Reserve and Risk Tariff the Model without return applies.
0, 0, 0, 0) 7
Aiu{(x-l)+l),
Aiu{(x-3)
+3),
Aiu{(x-2)+2), Aiu{(x-4)
+4},
Aiu{ (x - 5) + 5)) 0, 0) and cfpre’ reads {pris(x), whereas
0, 0, 0, 0,
0, 0, 0) ,
cfpost is a zero-vector.
’ The notation
is the same as sub 7.3.3.1. for the B-cover. Properly the symbols should bear a subscript A or B, but, for reasons of clarity, they have been left away.
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F.K. Gregorius
/ Disability insurance in the Netherlands
7.3.4. Provisions 7.3.4.1. General The thought behind the Markovian approach also apply to the provisions. This means, that, here too, we have to think of actives and disabled combined. The provisions provide a balance between future obligations and benefits. From this follows the following general formula V(x)
=R.A(x)
-Pr.a(x)
with - I/(x> as a provision vector - R as the amount of the insured
annuity
_ A(x) as a single premium vector, in accordance with the beginning of 7.3.3.1 - Pr as the (level) premium - a(x) as active-annuity vector, as generally indicated with A(x) at the beginning of 7.3.3.2. V(x)’ {l/x(act),
reads t&($=1),
vX(s=2),
I/x(s=4),vx(s=5),vx(s>5),
vX(s=3), V(D)).
The above implies, that for disabled not only the provision for the annuity, becoming effective, is generated (the Claims reserve part), but also the provision for future risk in case of recovery (the Actuarial reserve part).
and Economics
101
13 (1993) 101-116
Disability insurance in The Netherlands - The policy conditions provide that disability exists if the insured directly and due solely to medical consequences of an accident and/or illness is disabled for at least 25% to perform the duties connected with his occupation as shown in the schedule, in a way reasonable for these professional activities.
F.K. Gregorius Meidoornlaan 3, Castricum, The Netherlands Received November Revised June 1993
1992
Keywords: Disability
insurance
1. Introduction 1.1. Preliminary This paper is concerned with Individual Disability Insurance, a product especially for selfemployed business-people (employees are largely covered by Social Law). Disability Insurance in the Netherlands belongs to Non-Life, though methods and technics are very similar to Life Assurance, albeit that they are much more complicated as will be shown below. Most insurers followed technical bases dated to 1982. Deteriorating results induced the organization of disability insurers (KAZO) to instruct their Actuarial Committee (AC) to advise on a revision of tariffs. The AC reported in 1990 and a year later its recommendations were implemented. The following is a summary of the AC’s study. 1.2. Specifications of individual disability insurance There
are 2 main types:
(1) The first year’s risk (A-cover). - A sick or disabled insured gets benefits, starting after a period for own risk (waiting period) and continuing ultimately up to one year after the beginning of the disablement. - The period for own risk can be chosen by the insured when he contracts his policy with possibilities: 7 or 14 days, 1, 3 or 6 months. - The benefits are based on a just as well chosen amount of an annuity. - In consequence we speak of the amount of insured A-annuity. 0167-6687/93/$06.00
0 1993 - Elsevier
Science
Publishers
(2) The after-first year’s risk (B-cover). - The benefits exist of an annuity starting after a one year waiting period from the beginning of the disablement and stopping in case of recovery or when the term of insurance expires, as a rule at an age between 55 and 65. - The policy conditions provide that disability exists if the insured directly and due solely to medical consequences of an accident and/or illness is disabled for at least 25% to perform the duties - from hereon different from Acover - which meet his abilities and capacity and which can be reasonably demanded from him in view of his education and former activities. - In establishing the measure of disableness the possibility to get an (adapted) job wil not be taken in consideration. Benefits are also granted in the case of partial disability. The insured amounts can be indexated with a fixed rate: 3, 4 or 5%. The distinction between the two types mentioned is connected with Social legislation. The so-called AAW (Algemene Arbeidsongeschiktheids Wet = General Disability Law) insures every - not employed - citizen for an annuity to an amount deducted from the legal minimum-wage. Benefits are granted, however, after a one year waiting period. Consequently, there is no cover for the first year at all and that cover can be obtained by the A-cover. After the first year a supplement to the AAW will suffice and that can be done by the B-cover. So the insured amounts can differ between the A- and B-cover.
B.V. All rights reserved
102
F.K. Gregorius
/ Disability insurance in the Netherlands
It may be clear, that the A-cover is more similar to Non-Life insurance and that especially the B-cover has many common elements with Life assurance. Policies of Disability insurance cannot be cancelled by the insurer.
2. The technique
of disability
As mentioned before the A-cover is very similar to other types of Non-Life insurance. The risk premium is based on statistics about the ratio of granted benefits and insured amounts (annuities). The premium is differentiated by the following risk factors:
- 7 and 14 days, - 1, 3 and 6 months.
(2) Age: nine groups
- 16-25 years, - 26-30 years, - 31-35 years, etc. up to 65. (3) Class of profession: Four classes according to the largeness risk of the insured’s profession.
P(i,
In the following the probability system, built around the elements i(x), r(x, s) en q(x) will be explained.
of
j, k)
=B*a(i)*b(j)*c(k). The a, b and c apply to the three (i = l-5; j = 1-9; k = l-4).
risk factors
2.2. The B-cover 2.2.1. Introduction To the B-cover actuarial methods The following bases are used: - a disability rate i(x), - A factor for class of profession A-cover),
apply.
(equal
2.2.2. The probability system This system is explained by means of the following example: Suppose, we have 1000 insureds at the beginning of the accident year t. At the end of the year, suppose that the situation is as follows: (1) 10 insureds have died. (2) 90 insureds became disabled in the period l/1-30/9 and are still disabled on 31/12. (3) 40 insureds became disabled in the period l/10-31/12 and are still disabled on 31/12. (4) 860 insured remained active. Among them are insureds, disabled during the year, but after that recovered. We can consider these observations as a realization of a probability process with the following probabilities to distinguish:
The above risk factors form a network of 5 *9*4 = 180 cells. For each of these cells statistics were available. For the structure of the premium a multiplicative model has been chosen as follows: Risk-premium
time of
insurance
2.1. The A-cover
(1) Waiting period: five classes
- a recovery rate r(x, s> with s as lapsed disability, - a mortality rate q(x), - an intrest rate.
to the
_ the probability of an x-aged to remain alive during the year: p(x). _ the conditional probability of an x-aged, if he remains alive, to become disabled in the first nine months and to remain disabled within the next three months: i*(x). - the probability of an x-aged to become disabled in the first nine months and to be still disabled at year end: i(x). _ We note that i(x) =p(x) . i*(x). The distinction between disablements in the first nine and the last three months follows from practice. In case of B-cover new disablements are only as such registered after three months have elapsed because in case of short illnesses needless clerical work should be avoided. This means that disabled of the last three months are not as such distinguished and remain registered as actives. The above group no. (3) belongs consequently to the actives!
F.lZ Gregorius
It follows further that in the above probability to be active at year end is p(x)
.(l
-i*(X)}
‘P(X)
sense
the
-i(X)
= 1 -q(x)
-i(x).
_ The probability of an (x + l)-aged who became disabled in year t at age x, to be active at year end (t + 1): T{(X) + 1)). - We note, that: Y{(X) + 1) =p(x + 1). r *((xl + 0.
This expression is applied in the recursion formulas (refer to Section 5). The probability formulas for the preceding four groups are now: 1000 q(x) for the 10 deaths; 1000 a(x) .i*(x) = 1000 i(x) for the 90 disableds; 1000 (1 -q(x) -i(x)} for the 860 actives and for the 40 disableds from the last three months. The question is now what happens with groups (2) and (3) in year (t + 1). Continuing our example we suppose at year end (t + 1) the following observations: From the 90 disabled from group (2): - 1 died, - 50 recovered, - 39 remained disabled. From the 40 disabled from group (3): - 0 died, - 18 recovered, - 22 remained disabled. Consequently we have at year end (t + 1) 39 + 22 = 61 disabled. Here we have a somewhat different probability process with the following probabilities: - the probability of an (x + l)-aged alive during the year: p(x + 1).
103
in the Netherlands
/ Disability insurance
to remain
Remark. Though there is theoretically good reason to distinguish between survival probabilities of actives and of disabled, this has not been done. For both groups population mortality has been applied. - The conditional probability of an (x + l)-aged who became disabled in year t, if he remains alive, to be active again at year end (t + 1): r *{(xl + 1). Remark. The probability of recovery depends on the elapsed time of disablement s. The general notation for an insured who became disabled at age x is: T*{(X) + s}.
The probability (t + 1) is
to be still disabled
P(X + 1). ((1 -r*{(x) =p(x+ = 1 -q(x+
1) -r{(x) 1) -r{(x)
at year end
+ 11)) + 1) + I}.
This expression, too, is applied in the recursion formulas (ref. to Section 5). This probability applies to the 90 observed disabled. The ratio, connected with it, is 61/90, though from the 61 disabled, 38 originate from the 40 ‘3 months’cases. Remarks. (1) The 18 recovered insureds originating from the disablements of the last three months of year t remain out of observation. As it were, they remain actives! But notice: they did not get any benefit as well, because of one year waiting time! [Except in case of recovery after expiration of the waiting time but before year end (t + 1); this possibility has been neglected]. (2) The given example has not taken account of the possibility of IBNR-cases. These cases are registered like those of the last three months of year t and are consequently not distinguished as IBNR-cases. We can conclude, that the probability system, together with the according observations, include IBNR-cases. This means that there is no need for a separate IBNR-provision. (3) There is even another effect: numbers of disabled namely in case of partial disability only count for the benefit percentage connected with the disability percentage. This means that also alterations in disability percentage have their influence. Year (t + 2) and following years are dealt with like year (t + 1). The r{(x) + s} are derived by comparing numbers of disabled at beginning and end of the year. Here, too, we have the effect of the balance of recoveries, changes in degrees of disability and possibly late IBNR-cases. This balance effect can cause negative r{(x) + s}-ses. This happens especially in case of high s-values (4 and 5).
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F.K. Gregorius / Disability insurance in the Netherlands
If s is 6 or higher all these glectable and Y is put at zero.
effects
are
ne-
2.2.3. Application of the probability system The applied theory is very similar to traditional methods in Life assurance. The following four single premiums are applied: (1) Ai(x, e) = Single premium for a disability insurance (annuity) for an x-aged, payable continuously up to end-age e and with a one year waiting period. (2) ai{(x + s), e} = Single premium for a continuous annuity, immediately taking effect for an (x + s&aged disabled and payable up to end age e. The notation means: - disabled at age: x, - past duration of disablement: s. The description, too, implies, that the annuity ends in case of recovery and starts again after returned disablement. (3) aa(x, e) = Single premium for a yearly prepayable annuity, immediately taking effect for an x-aged and payable: _ in case of disablement: up to the end of the waiting period - in case of remaining active: up to one year before the end age e (the last year is premium free). In case of recovery the annuity comes into force again. This single premium aims at determining the present value of premiums to pay. (4) aia{(x) + s, e} = Single premium for a yearly prepayable conditional annuity for an (x + s)-aged disabled and payable during (the rest of) the waiting period and further only after recovery and then up to one year before end age e. This single premium aims at determining the present value of premiums to pay during (the rest of) the waiting period and after that only after possibly occuring recovery. The foregoing shows some details concerning the payment of premiums, following from the insurance conditions. The definitions apply to the unity of money. In the following for reasons of clarity the subscript e will be left out.
With the foregoing, formulas for premiums and provisions can be set up. In doing this one has to take in mind, that insureds can change from actives into disabled V.V. and also can die. There are two practical techniques: (1) Application of recursion formulas Section 5). (2) Interpretation as a Markov process Subsection 7.3).
(refer
to
(refer
to
2.3. Provisions 2.3.1. The actuarial reserve In case of level premiums the applied equivalence principle implies that we have to set up an Actuarial reserve (in case of payment of yearly risk premiums which are adapted to the age reached, this is naturally not necessary). With the B-cover we can possibly be confronted with negative amounts. Some years before the end date of the insurance the risk decreases, because the influence of the quickly shortening remaining period surpasses that of still increasing disability rates. The period of premium payment should have been sufficiently shortened, but for commercial reasons this is limited to one year. As a rule a negative Actuarial reserve is put at zero. For formulas refer to Section 5. 2.3.2. The claims reserve In case of A-cover no actuarial methods are applied. The provision is an estimation of amounts to be paid, taking into account the nature of the insured’s disability. In case of B-cover, first three months of disability have to elapse before the provision is set up in order to avoid unnecessary administrative work for early recoveries. This procedure gives rise to the question whether some provision should have been set up. However, this is done indirectly. As soon as a provision is entered, it is calculated with reduced recovery rates, which are a balance of several effects. Refer to the explanation for the probability system in Subsection 2.2.2. For formulas refer to section 5.
F.K. Gregorius / Disability insurance in the Netherlands
2.3.3. The IBNR-provision As explained in Subsection 2.2.2 - refer especially to Remark (2) - the Claims provision includes IBNR. 2.3.4. Actuarial Reserve for disabled A disabled has a chance to recover and in that case an Actuarial Reserve has to be entered again. This means, that as long as there is a non-zero probability to recover - this is the case during the first five years of disablement - besides the claims reserve the Actuarial Reserve has to be partly maintained. This matter is further explained in Section 5. 2.3.5. Provisions for expenses For reasons of completeness we mention these provisions, which have to be set up for claimsand administration expenses.
Statistical data are similarly obtained by an other periodical inquiry, the so-called KAZOstatistiek. This KAZO-statistiek aims at testing the technical bases of the tariff and consequently of its structure. So this concerns invalidation, recovery and mortality and this towards the risk factors: age, class of profession and waiting period. Recovery and mortality are moreover distinguished towards past duration of disablement. 3.2. The statistics for the A-cover This cover is dealt with like other non-life classes. Incurred claims are compared with insured sums in portfolio and this for all combinations of the risk factors earlier explained. Table 1 shows an example of the inquiry of incurred claims for class of profession 3. 3.3. The statistics for the B-cover
3. Data inquiries
Observations
3.1. General In the Netherlands for some ten years a centralized office for financial and statistical data has existed: the ‘Centrum voor Verzekerings Statistiek’ (CVS). Financial data are gathered from a substantial part of the market by the ‘Comptabele Enquete’, a periodical inquiry about financial results.
Incurred Waiting 3 days
10.5
claims for class of profession
aim at:
- disability rates i(x) depending on age x, _ recovery rates r{(x) + s} depending on age x and past duration of disablement s. Mortality is observed, indeed, but because N.B. of small numbers these observations are not used. Mortality rates are taken from a recent Dutch population table (GBM 1980-1985).
Table 1 3 (disability statistics,
A-cover,
1989, benefits
period
Total 14 days
1 month
0 0 5,184 0 1,829 0 0 17,262
11,582 28,259 100,507 64,763 267,398 139,655 114,272 11,070 41,218
26,981 24,177 6,987 59,261 121,040 113,903 0 2.701
7 days
5 25 years 26-30 31-35 36-40 41-45 46-50 51-55 56-60 61-65
0
Total
0
24,275
778,724
In %
0.0
2.1
67.1
0
in Dutch guilders).
0
3 months
In %
6 months
0
0
0
0
0
0
0
0
0
0
0
0
2,755 0 0
0 0 0
11,582 55,240 124,684 76,934 326,659 262,524 230,930 11,070 61,181
355,050
2,755
0
1,160,804
30.6
0.2
0.0
1.0 4.8 10.7 6.6 28.1 22.6 19.9 1.0 5.3
100
106
F.K. Gregorius
/ Disability insurance in the Netherlands
The waiting period is standard one year and consequently not a risk factor. The class of profession is a risk factor indeed. There is good reason to assume that the structure towards x and s of disability- and recovery rates differ per class of profession. This possibility, however, has been neglected.
4. The construction
The construction of the new tariff is based on what has been explained in Subsection 2.1: ‘the technique of disability insurance’. The essential quantity is the ratio of granted benefits and insured amounts, which ratio is observed for 180 groups of combinations of the risk factors: - waiting period - age - class of profession
- 5 possibilities, - 9 groups, - 4 classes.
The risk premium ‘Pris’ for a certain combination of risk factors is determined according to the multiplicative formula: Pris = Pris( basic cell) x factor waiting
These factors, representing per factor the mutual proportions of the several possibilities, are calculated by means of a multiplicative regression model. For more details refer to Section 7.1. In Table 2 the figures used are shown. The upper part shows insured amounts of annuities, used as weighing factor. In the lower part we have percentages of becoming disabled, i.e. the earlier mentioned ratio between paid and insured amounts. The figures only show two classes of profession; four classes were introduced later on. The results for waiting period and age were (basic cells are indicated by *): 7 14 1 3 6
62 70 100 * 127 133 164 189 211 198
days days month months months
119 100* 61 23 16
For the classes of profession the ratios were determined after a special inquiry, needed because recently, concurrent with the study, four classes instead of two had been introduced. The ratios of the age groups were further modelled applying y=a.bx. This offered the opportunity to introduce a tariff per age instead of per age group. Further the ratios found for Waiting period and Class of profession were rounded off. Finally it was checked that de weighted sum of model values was equal to the sum of the observations. All this resulted in the following:
period
X factor age X factor class of profession.
period:
16-2.5 26-30 31-3.5 36-40 41-45 46-50 51-55 56-60 61-65
of the new tariff (net)
4.1. The A-couer
Waiting
Age group:
(a) ratios for Waiting period: 125 7 days: 100 14 days: 1 month: 70 3 months: 40 6 months: 30 (b) ratios for age: 32.247 times 1.036” (for x = 32 this gives 100). (c) ratios for Class of profession: class 1: 70 class 2: 85 class 3: 100 class 4: 120 (d) premium per 100 guilders annuity for the basic cell: Dfl. 2.30. In case of the Risk Tariff a solvency loading of 6% is added, because there is no Actuarial Reserve, which yields some surplus above the investment income needed. This 6% is not a result of actuarial methods.
107
F.K. Gregotius / Disability insurance in the Netherlands
technique of disability insurance’. Essential are the following bases, which will be treated in the next paragraphs: - disability rates, - recovery rates, - mortality rates,
4.2. The B-cover
4.2.1. General The construction of the new tariff is based on what has been explained in Subsection 2.2: ‘The
Table 2 Portfolio 1986 + 1987, A-cover Insured annuities Class 1 (old) Age/w
7 days
14 days
1 month
3 months
6 months
16-25 26-30 31-35 36-40 41-45 46-50 51-55 56-60 61-65
994 2734 5327 8835 9356 7366 4198 2433 712
43116 85232 118796 155982 119184 80233 48699 22665 6565
9296 28830 55725 86550 70649 47060 26509 11757 3003
1623 7934 17987 32178 28256 20819 12115 6329 1545
653 2777 6463 9237 10898 7151 4995 3129 464
3968 6745 15021 32486 50265 58236 50387 28669 4183
268916 326981 331858 347033 248759 178951 105178 44375 6367
31363 58422 71164 85879 68324 47524 29272 11751 1645
5721 10509 12456 15514 16490 12038 6982 2894 429
1949 3267 4237 7521 7655 6136 3308 1876 185
Class 2 (old) 16-25 26-30 31-35 36-40 41-45 46-50 51-55 56-60 61-65
Granted benefits as a percentage of insured annuities Class 1 (old) 16-25 26-30 31-35 36-40 41-45 46-50 51-55 56-60 61-65
1.11 2.34 1.80 3.24 2.97 5.10 5.34 3.70 2.39
1.83 1.58 2.28 2.66 2.96 3.24 3.92 4.32 4.05
0.83 0.70 1.07 1.16 1.65 1.67 2.97 2.95 5.09
0.00 0.49 0.62 0.10 0.89 0.14 0.87 2.15 0.26
0.61 0.00 0.40 0.01 0.77 0.14 0.28 1.69 1.29
3.38 2.48 4.60 5.05 4.55 5.55 6.35 6.15 7.05
1.72 2.03 2.82 3.58 3.88 4.82 5.35 6.19 5.26
0.86 1.02 1.53 2.53 1.77 2.97 3.92 5.13 3.71
0.77 0.09 0.31 0.78 0.54 0.89 2.01 4.77 0.47
0.00 0.00 0.28 0.17 0.46 0.47 1.27 2.45 0.00
Class 2 (old) 16-25 26-30 31-35 36-40 41-45 46-50 51-55 56-60 61-65
F. K. Gregorius / Disability insurance in the Netherlands
108
The rates in Table 3 are modelled applying: y = a . b”, similar to the modelling for the A-cover. The numbers of insureds have been used as weights. The factor a is adjusted such that the total weighted rate is equal to the observed total at 1.63. Here, too, an opportunity to introduce a tariff per age instead of per age-group was offered. All this resulted in the following formula:
Table 3 Age group
Disability rates
Number insureds
16-25 26-30 31-35 36-40 41-45 46-50 51-55 56-60 61-65
0.54 0.77 1.07 1.32 1.52 2.04 2.56 2.92 2.74
29137 39765 51754 73289 76188 70059 54927 30927 6424
Total
1.63
432470
of
i(x)
It should be remembered, that i(x) only refers to the first nine months of the concerning accident year (therein becoming disabled and remaining so up to the year end).
Besides these a role is played by: - The intrest rate, for which 4% was taken. In following formulas a discount factor L! could be met. This c’ is equal to l/1.04. - The factor for class of profession, for which the same factors as used for the A-cover were applied. Waiting period is not a factor, because it is standard one year. As far as it is relevant the calculations in the sequel are applicable to class of profession 3.
4.2.3. Recovery rates In Figure 1 one can see, that the recovery rates can very well be modelled by means of straight lines. The graph shows: (1) for every duration of disability s (= 1 up to 5 incl.) there are different lines and - generally spoken - the lines lie in conformity with the value of s one under the other. Only for s = 1 this does not f&y apply because of the large influence of the disabilities of the last three months of the past year (refer to Subsection 2.2.2). (2) Every line - generally spoken - descends from the left to the right, which obviously ex-
4.2.2. Disability rates Disability rates are derived from unweighted rates of becoming disabled. The observations about the period 1984-1988 that were available are listed in Table 3.
-0,400
-I
28
23 --
duratkm dhab4ity->
1year
33
= 0.00223 times 1.0468”.
38
43
.--_)-__
Age ..... &
2 pars
3 yeam
Fig. 1. Recovery
rates 1990.
48 _ 4 years
53
58
63
_____-- - _ 5-n
e+ years
F.K. Gregorius / Disabilityinsurancein the Netherlands
presses - for each s - decreasing recovery rates with increasing ages. This has consequences for our formula, which can be of the form: r = a - b -x, but with a and b depending on s. So our five lines can be expressed as follows: r{(x)
+s)
=a(s)
-b(s).x,
109
(3) should be fulfilled for all nine x-values, but, because we have a straight line, it is sufficient to require this only for x = 22 and x = 62. Summarized, the restrictions to the regression procedure are: (1) a(s) 2 0, (2) b(s) L 0,
where a(s) =
a parameter depending on s with s = 1 up to 5 incl., b(s): ditto, x = age, but for the estimating procedure x has been taken: 22, 27, 32,. . .62 respectively (9 groups). As estimating-procedure a least squares regression method has been applied, with weighting with benefit-percentages u{(x) + s} as volume components. Consequently the sum of squares to be minimized was
(3) a(s + 1) - a(s) + 22. {b(s) - b(s + l)} 5 0 (s = 2, 3, 41, (4) a(s + 1) - a(s) + 62. {b(s) - b(s + l)} 5 0 (s = 2, 3, 4). These restrictions raise considerable complications in the calculations. The problem can be considered as a quadratic programming problem. In this way it is solved by means of a computer package. The results were: r{(x)
+ I} = 1.24111 - 0.02219x
(0.75 + -0.13467), r{(x)
+ 2} = 0.66499 - 0.01153x
(0.41-+ -0.04987), r{(x)
((a(s) -b(s) .x-r{(x)
+s)))*.
(0.16 --f - 0.05590))
the r{(x) + s} being the observed values. The foregoing, however, submits this minimizing-process to some conditions. This means: a(s)
is positive, because the line - anyway for small x - cannot lie under the x-axis. (2) b(s) is positive in connection with the beforehand chosen minus-sign in the formula of the line. (3) r{(x) + s} 2 r{(x) + s + l} with s = 2, 3, 4 leading to:
(1)
+ 3} = 0.27394 - 0.00532x
r( (x)
+ 4} = 0.23547 - 0.00470x
(0.13 + -0.05593)) r{(x)
+ 5) = 0.14166 - 0.00319x
(0.07 + -0.05612). Between brackets the course of the values from age 22 to 62 is given. Indeed, each line lies under the foregoing except the first one. But for this the restrictions did not apply. Notice the negative values for high ages, especially for s = 1.
a(S)-b(s)*x~a(s+l)-b(s+l).x, otherwise written: a(s+l)-a(s)+{b(s)-b(s+l)}.xsO. (4) r{(x) + s} 2 r{(x + 5) + s} 27 , . . . ,57 leading to a(s)
-b(s).xza(s)
with
-b(s).(x+5),
leading to b(s) L 0, which is consistent with (2).
x = 22,
4.2.4. Mortality rates Mortality rates of disabled are higher than those of actives. Further one can assume, that this rate depends on the duration of the disablement. Observations with death cases, however, is a matter of small numbers and consequently of insufficient reliability. Therefore mortality rates are taken from the population table GBM 1980-1985 for actives as well as for disabled. Generally this provides some safety margin.
110
F.K. Gregorius / Disability insurance in the Netherlands
4.2.5. The net premium for the risk tariff This is the tariff with yearly altering premiums, which only are paid for the running year and do not contain any component for future risk as is the case with level premium tariffs. The following formula holds: Pris(x)
= 1.06 * u * i(x)
* ai{
+ I}.
The single premium ai{ + 1) is determined applying the so-called ‘model without return’ (refer to Subsection 7.3.2). In this model it is assumed, that a disabled does not return to the actives, because, if that happens, he will be considered as a new active insured, paying again premiums according to his age. As the formula shows, in conformity with the A-cover, a solvency loading of 6% has been applied. 4.2.6. Level premiums (net) The net level premium is the quotient of the single premiums Ai and aa(x>, both mentioned in Subsection 2.2.3. Both single premiums are determined applying the so-called ‘model with return’. In this model it is assumed, that disabled can return to the actives again, then can become disabled again etc. etc. (refer to Subsection 7.3.2).
(a) For actives: RaAi(x)
-P.aa(x)
- R = amount of the annuity, - P = (net) policy-premium. The Ai and aa are defined in Subsection 2.2.3. For these quantities the ‘model with return’ is applied. This can be illustrated by the following recursion formulas: Ai
5.1. Introduction
In Subsection 2.4 something has already been told about provisions. Referred has been to this section for more details, especially for the formulas. In the following this will be dealt with. 5.2. The Actuarial Reserve
As mentioned in Subsection 2.4 in case of level premiums an Actuarial Reserve is set up for actives as well as disabled, for the latter as long as recovery rates are non-zero. The following prospective formulas are applied:
-i(x)}
=v*{l-q(x) +v*i(x).ai{(x)
.Ai(x+l) + I},
ai{(x)+l}=v.((l-q(x+l)-r{(x)+l))) .((OS+ai{(x) +v.r((x)
+2))) +l)
.Ai(x+2).
Notes: - The probabilities
used are explained in Subsection 2.2.2. - The ai((x> + s) for s > 1 contains a first term 0.5, representing a payment 0.5 at the beginning of the year followed by 0.5 at the end, this being an approximation of continuous payment. But in the case s = 1 the first term 0.5 is absent because of the one year waiting period, on the average falling a half year in the next insurance year. Further +v.i(x)
5. Provisions
with:
.aia{(x)
+ 1)
and, very special, aia{(x)
+ 1)
=0.5+v.((l-q(x+l)-r((x)+l))) .aia{(x)+2)+v.r{(x)+l)*aa(x+2). Note: - The term 0.5 concerns
premium to be paid over the (rest of) the waiting period, being on the average 0.5 year. For s > 1 the term 0.5 is absent.
Notice the interdependency between both pairs of formulas. The single premiums can be calculated starting with the end age and then proceeding backwards. (b) For disabled: R.Aia{(x)
+s)
-P*aia((x)
+s).
Here we have a special, up to here not introduced single premium Aia{(x) + s} representing
F.K. Gregorius / Disability insurance in the Netherlands
a conditional abled, should reads Aia((x)
insurance, namely Ai for a dishe recover. The recursion formula
=w((l-q(x+s)
-r{(x) +s+
+s}))
I} +u.r{(x) .Ai(x
+s)
An other interesting approach of the foregoing is to consider it as a Markov process. Refer to Subsection 7.3.4. 5.3. The Claims Reserve Dealing with the Claims Reserve we have to apply the ‘Model without return’. The Claims reserve only concerns the running annuity. Liabilities in case of recovery have been met by the Actuarial Reserve. Hence the formulas are = R . ai{ (x)
+ s}
with ai{
included
in
These, too, are mentioned here for reasons completeness, referring to Subsection 2.4.5.
of
+s + 1).
For the A-cover the formulas are similar albeit that _ Ai = Pris(x) + u. {l - q(x) - i(x)} .Ai(x + 1) + c’ . i(x) *Au{(x) + 1). The interpretation of this formula is, that the insurance is considered as a running annuity to the amount of the risk premium. _ The single premium ai{ + s} does not exist.
claims reserve
is, generally,
5.5. Provisions for expenses
+s}
Au{(x)
The IBNR-provision the Claims Reserve.
111
+s} =OS+v.((l-q(x+s) .((OS+ai((x)
-r{(x)
+s}))
6. Gross premiums 6.1. Introduction The gross premium - expenses, - commission, - solvency; profit.
contains
loadings
for:
To begin with the last: it is not based on calculations, except in the case of the risk tariff (6%, as mentioned before). It is provided for by safety margins in the other bases. Further, the surplus of interest (above 4%) is an important source. Commission simply is a percentage of the gross premium. The structure of the loading for expenses is of some interest. The old tariff contained a percentage loading (17%). It is obvious that this gives too high a loading in case of high premiums (and too low in case of low ones). A better structure of the loading for expenses has been studied. The following components have been distinguished:
+s+l})).
Notes: - The first term 0.5 is absent in case of s = 1. - Notice the difference with the corresponding formula in Subsection 5.2. - As mentioned in the preceding the Claims Reserve for the A-cover is not calculated by means of actuarial methods. Here, too, we refer to the Markovian application Subsection 7.3.4.
a = component for claims expenses; b = component for costs of medical examinations and reports, these for underwriting purposes; cl = component for (other) costs of underwriting, administrative costs of producing and entering the policy and acquisition costs; c2 = component for the rest of expenses as file administration, collection of premiums, bookkeeping etc.
5.4. The IBNR-provision
The structure on the following
For reasons of completeness this provision mentioned here, referring to Subsection 4.3.2.
is
of the gross premium considerations:
- Besides the insured annuity are a part of the risk. Therefore
is based
claims expenses the component a
F. K. Gregorius
112
/ Disability insurance in the Netherlands
has been related to the net premium, which is deducted from the risk premium. A loading in the shape of a fixed amount - the average costs of handling of a claim - could be considered. But realizing that high insured amounts often lead to a more intensive claims handling, one could choose for a percentage loading on the risk- or net premium. The latter has been done. - The component 6 has been related to the insured annuity as a percentage loading. Costs of medical examination - for underwriting purposes - are not part of the risk. A fixed amount per policy could have been taken into consideration. A percentage loading somewhat reflects the fact that in case of low amounts no medical examination is requested. - The components cl and c2 are expressed as in fixed loading per policy. Generally spoken they are not related to the proportions of the annuity or the premium. These considerations led to the following for the gross premium:
formula
Pgr=(l+a)~R*Pnetl+b~R+c+p~P6 with Pgr = gross premium of the policy, R = amount of the insured annuity, Pnetl = net premium per unit of insured p = fraction for commission, c = cl + c2. This leads to
amount,
Pgr=((l+a)~Przet+b~R+c}/(l-p) with Pnet = net premium
7. Mathematical
of the policy.
techniques
7.1. Regression models 7.1.1. Introduction As mentioned in Subsection 2.1, the risk factors a(i), b(j) and c(k) of the A-cover have been estimated by means of a regression technique according to the premium formula, applied in practice: P(i,
j, k) =m
* a(i)
with 5, 9 and 4 possible respectively.
* 6(j) values
* c(k) for the i, j and k,
In the following some peculiarities, connected with the practical application of the applied model will be touched upon. Z1.2. The chosen model This is a weighted multiplicative cording to the foregoing, based on
model,
P(i,
* e(i, j, k).
j, k) =m * a(i)
* 6(j)
* c(k)
ac-
the last factor being the disturbance factor, assumed to be exponentially distributed. The a, 6 en c have been estimated by minimizing S2=SSS(i, x
j, k)g(i,
j, k)
(In m + In u(i) + In b(j) +In
c(k)
- In w(i, j, k)}*
with: SSS(i, j, k) meaning
summation
over i, j and
k, g(i, j, k) representing the weighing-factor (g of Dutch ‘gewicht’ = weight) for cell (i, j, k), being the insured amount in cell (i, j, k), w(i, j, k) representing the observation (w = Dutch ‘waarneming’ = observation) for cell (i, j, k), being the percentage of becoming disabled in cell (i, j, k). 7.1.3. Dealing with the results In Section 4 about the tariff construction we have seen that the results of the regression technique have not been followed unchanged. For Waiting period and Class of profession they were settled at values, having been rounded off, sometimes somewhat safely upwards. After settling one of the parameters, the regression procedure can be executed again for the remaining factors. The question how to deal with the data is then of some interest. Let us suppose, that the c(k) are settled. Then in the squared sum above c(k) is no more an unknown but a known quantity, that can be combined with w(i, j, k) to a new, as it were, changed observed value w*(i, j), which has to satisfy g(i,
j) .ln w*(i, = S( k)g(i,
j)
j, k) . In w(i, j, k)
113
F.K. Gregorius / Disability insurance in the Netherlands
with - g(i, j> = S(k)&, j, k), - S(k) meaning summation
7.2. Recursion formulas over k.
The recursion formulas have already been mentioned in Section 5. It has also been indicated how the calculations can be carried out. In the report of the study this subject has been worked out extensively. The reader interested in these details is referred to this report.
This implies a prescription for telescoping the g(i, j, k) and the w(i, j, k) into w*(i, j) and g(i, j), respectively. The solution of the minimizing procedure is, as is well known, executed by putting at zero the derivatives of the squared sum with respect to the parameters m, u(i), b(j) and c(k), adding some conditions to get a solution, often putting m as weighted average of the observations. Comparing the 3- and 2-dimensional formulas - the last as a result of settling the c(k) - the following conclusions can be drawn, if the above prescription is followed:
7.3. The Markovian approach 7.3.1. Introduction About insureds of a disability portfolio can be said that they pass through several possible states, which can be: - active, - disabled more, - dead,
(1) the remaining
parameters are minimally affected, (2) m* = m if the c(k) are settled so that S(k) g(i, j, k). c(k) = 0, (3) a(i) and b(j) have in both the 2- and 3-dimensional cases the same values if all g(i, j, k) are 1 and otherwise this holds approximately.
with durations
1 up to 5 incl. and 6 or
The change of group depends probabilities, which are related to:
on transition
_ to remain active (to stay in the same group also is considered as transition), - to become disabled, - to remain disabled, but with group altering indeed, because difference is made into duration of disablement, - to recover, - to die.
7.1.4. The GLIM-package The calculations, connected with the above, are executed by means of a software package called GLIM (General Linear Interactive Modelling), introduced by Nelder and Wedderburn in 1972.
Table 4 a To:
A
I(3)
I(4)
I(5)
I(6)
D
0
0
0
0
0
P(1) 0 0 0 0 0 0
0 P(2) 0 0 0 0 0
0 0 P(3) 0 0 0 0
0 0 0
0 0 0 0 P(5) 1-q 0
4 4 4 4 4 4 4 1
I(1)
I(2)
; 0 0 0 0 0 0
From: A
p(A)
I(I) I(2) I(3) I(4) I(5) I(6) D
r(I) r(2) r(3) r(4) r(5) 0 0
P(4)
0 0 0
a For reasons of clarity the subscript x has been left out. p(A) = 1- q(x)i(x); i = i(x); r(s) = r((x - s)+ s) for s = 1 up to 5 incl.; p(s) = 1 - q(x)r((x - s)+ s) voor s = 1 up to 5 incl.; q = q(x); In accordance with definitions mentioned in Section 2. From the matrix can be read, that an insured, as long as he will not be found in I(6) or D, could at any time return into a group in which he ever was. Therefore we here speak of the ‘Model with return’, which is illustrated in Figure 2.
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F.K. Gregorius / Disability insurance in the Netherlands
The several states of an insured can be seen as a Markov chain. The calculation methods, connected with Markov chains, are used to determine single premiums and provisions. Application of the matrix algebra in computer programs has a strongly compressing (short formulas) and generalizing (single premiums for all states together) effect. This will be worked out hereinafter. 7.3.2. States and transitions In the foregoing it has already been mentioned that eight states can be distinguished, namely: -A - I(1) - I(2) - I(3) - I(4) - I(5) - I(6) -D
= = = = = = = =
active, 1 year disabled, ’ 2 years disabled, 3 years disabled, 4 years disabled, 5 years disabled; 6 or more years disabled, dead.
The transitions (after 1 year as unity of time) are reflected in the transition matrix P(x) in Table 4, in which various transition probabilities, belonging to age X, are arranged. The before explained change of states has to be taken into account in case of calculation of single premiums and provisions. There are, however, two exceptions, namely: - risk premiums (refer to Subsection 4.2.5), - the Claims Reserve (refer to Subsection 5.3). For these the ‘Model without return’ has been applied. In its transition matrix the T(S) do not appear. For illustration refer to Figure 3. Closer considerations: (1) In case of the risk tariff, risk is covered for one year, which risk consists of an annuity to be paid in case of disablement up to the end-age, however without component that, after recovery, the insurance comes into force again (without premium payment). This component is not inProperly: disabled at the end of the year, in which the disablement started; this in accordance to the concepts, described in Section 2. With every next I-state the duration of the disablement is one year longer and - on an average half a year less than the subscript indicates {with adjusted interpretation for I(6)).
CIY I(6)
Fig. 2. The model with return.
chided in the risk premium. For, in case of recovery, the insured resumes premium payment in accordance with his age then reached. As it were the Risk Tariff consists of a series of one-year policies. (2) After recovery the Claims Reserve falls free; is put at zero. 7.3.3. Single premiums 7.3.3.1. Single premiums for a continuous annuity for a disabled The following general vector formula applies: A(x)
=cfpre+u.P(x).{A(x+l)
+cfpost],
where - A = general single premium symbol, - cfpre ( = cash-flow pre) indicates prepayable payments, - cfpost: ditto postpayable,
F. K. Gregorius / Disability insurance in the Netherlands
11.5
In the vectors cfpre and cfpost payments of 0.5 in case of running annuities are inserted as an approximation of continuous payment. The above does not apply to the A-cover. This will be dealt with in the following. 7.3.3.2. Single premiums for a prepayable active annuity Here, too, the general vector formula applies: A(x)=cfpre+v.P(x).{A(x+l)+cfpost). Here
A(x)’
reads
uiu{(x-1)
(uu(x),
+I),
uiu{(x-3)
+3),
uiu{(x-4)
uiu{(x-5)
+5),
O,O),
uiu{(x-2)
+2),
+4),
and cfpre’ reads { 190.5, 090,
cfpost is a zero-vector.
whereas
Fig. 3. The model without
return.
-
L’ = discountfactor, _ P(X) = the transition A(x)’ reads
matrix of the foregoing.
{Ai(x),ai{(x-1)+1},ai{(x-2)+2}, ai{(x-3)
+3},
ai((x-4)
ai{(x-5)
+5},
ai(
+4}, 0).
The element ai shows recovery rates are 0. cfpre’ reads
7.3.3.3. The single premiums for the A-cover The single premium Ai for the A-cover ’ can be considered as a single premium for an active annuity - au(x) - with the yearly risk premium Pris(x> as benefits and with an additional component Aiu{(x) + l} for a conditional A-cover insurance, namely after recovery, following on disablement. For, the A-cover insurance then comes into force again. The risk premium R-is(x) has the form a. b” (refer to Section 4). It follows from the before described, that Pris(x> gets a place in the cashflow vector cfpre, which in this case depends on x and properly should have been indicated with cfpre(x). We seem to have the same vector formula as before, but A(x)’ reads {Ai(
that
after
five years
(0, 0, 0.5, 0.5, 0.5, 0.5, 0.5, O} and ditto cfpost ‘. After working out the vector formula the accordance with the recursion formulas earlier explained, can be recognized. For Claims Reserve and Risk Tariff the Model without return applies.
0, 0, 0, 0) 7
Aiu{(x-l)+l),
Aiu{(x-3)
+3),
Aiu{(x-2)+2), Aiu{(x-4)
+4},
Aiu{ (x - 5) + 5)) 0, 0) and cfpre’ reads {pris(x), whereas
0, 0, 0, 0,
0, 0, 0) ,
cfpost is a zero-vector.
’ The notation
is the same as sub 7.3.3.1. for the B-cover. Properly the symbols should bear a subscript A or B, but, for reasons of clarity, they have been left away.
116
F.K. Gregorius
/ Disability insurance in the Netherlands
7.3.4. Provisions 7.3.4.1. General The thought behind the Markovian approach also apply to the provisions. This means, that, here too, we have to think of actives and disabled combined. The provisions provide a balance between future obligations and benefits. From this follows the following general formula V(x)
=R.A(x)
-Pr.a(x)
with - I/(x> as a provision vector - R as the amount of the insured
annuity
_ A(x) as a single premium vector, in accordance with the beginning of 7.3.3.1 - Pr as the (level) premium - a(x) as active-annuity vector, as generally indicated with A(x) at the beginning of 7.3.3.2. V(x)’ {l/x(act),
reads t&($=1),
vX(s=2),
I/x(s=4),vx(s=5),vx(s>5),
vX(s=3), V(D)).
The above implies, that for disabled not only the provision for the annuity, becoming effective, is generated (the Claims reserve part), but also the provision for future risk in case of recovery (the Actuarial reserve part).