A Multi-state Actuarial Framework for
Smoothing and Forecasting Sick Leave
Probabilities
A prediction of the financial impact of implementing the
Generation Pact
MSc. Thesis Econometrics and Operations Research for the Actuarial Sciences
Acknowledgements:
A Multi-state Actuarial Framework for Smoothing and
Forecasting Sick Leave Probabilities
Sophie de Mol van Otterloo (s2479192)
May 2019
Abstract
The generation pact is the principle in which elderly people work part time for a relatively higher wage and maintain their pension buildup. This is an often mentioned framework by the unions and actuaries as it lessens the burden of work on older people and creates opportunities for the young. This thesis investigates the effect of part time and full time work on health using a Markov framework and generalized linear models for smoothing the resulting crude rates. The Chapman-Kolmogorov equations are used for a general solution. Finally, an example is given of the financial impact of implementing the generation pact. The smoothed rates imply that working part time does not necessarily mean a better health for the elderly. In fact, men are healthier when working full time, while women fall sick more often when working full time, but recover more often as well. Bias due to endogeneity might play a part here, as no socioeconomic factors are incorporated in the analysis and the decision to work part time might be health-related for the elderly. When comparing the future rates of a person currently aged 50 working full time and using the generation pact, both the recovery and the morbidity rates drop when starting the generation pact. An estimation of the costs associated with sick leave, using initial state ‘Healthy’ and a generation pact which entails 80% work and 90% wage, yields that costs drop by half when using the generation pact.
Contents
1 Introduction 6
1.1 The Welfare State . . . 7
1.1.1 Dutch Pension System . . . 7
1.1.2 Functional Disability Benefit . . . 9
1.2 Literature Review . . . 9
2 The Markov Framework 12 2.1 Modeling Health . . . 13 3 Data Description 15 3.1 Main Variables . . . 15 3.2 Data modification . . . 17 3.3 Descriptive Statistics . . . 18 4 Methodology 23 4.1 Graduation intensity rates . . . 24
4.2 Model selection criteria . . . 25
4.3 Out-of-sample forecast . . . 26 4.4 Chapman-Kolmogorov relationship . . . 27 5 Results 28 5.1 Model selection . . . 28 5.2 Model uncertainty . . . 33 5.3 Forecast . . . 38 5.4 General solution . . . 42
1
Introduction
One of the major challenges facing current developed economies is the aging generation of baby-boomers, born between 1945 and 1955, combined with a consistently increasing life expectancy. The significant size of this generation results in an overall aging of the population. The share of people aged 65 and older will grow from 18% in 2017 to 26% in 2040 (Stoeldraijer, van Duin & Huisman 2017). Relatively more older retirees and less younger people active on the job market puts a higher burden on the young for achieving proper redistribution of wealth. This causes fiscal imbalances in the retirement-system and threats the cost-effectiveness and sustainability of the welfare state (Bongaarts 2004). Many economies have introduced parametric pension reform such as increasing the retirement age, to ensure continuation of the welfare state. Over the last two years, three countries1, including the Netherlands, have decided to gradually increase the retire-ment age, and join several other countries2in linking it to the level of population aging afterwards (OECD 2017). This is a controversial measure, as it forces people to work longer and thus de-creases employee well-being. A solution brought up by the unions and actuaries is the generation pact. The generation pact entails that an older employee can choose to work less, while main-taining a wage and pension buildup based on the full time salary3. The vision of this framework
is to lighten the obligation to work for the elderly, while still maintaining their senior knowledge within the company and simultaneously hiring more younger people. Moreover, the generation pact increases the affordability of early partial retirement, making it a reasonable option for more people. In this thesis, the following research question is answered: What is the financial impact of using the generation pact for the employer?
In order to answer this research question, first the difference in days sick of work between part time and full time working people is estimated. The transition rates of a Markov framework are graduated using generalized linear models (Renshaw & Haberman 1995). These probabilities are then linked to average salary and sickness duration in the Netherlands to give an estimation of the difference in costs generated by the generation pact. By the notion of presenteeism, being sick, whether at home or at work, is considered to have the same financial effect as taking up a sick day. As the retirement age has been 65 for the majority of the time since the general old age pensions act (AOW) has been installed, there is a lack of data on sick days for older working people. We use the graduated data to predict sick days for people over the age of 63. The corre-sponding solution is then linked to average salary and sickness duration in The Netherlands to give an example on how these rates can be used to calculate the financial impact of the generation pact.
1Finland, Denmark and the Netherlands. 2Slovak Republic, Italy and Portugal.
The remainder of this thesis is structured as follows. The introduction will shed light on the background information concerning the set up of the welfare state in the Netherlands and gives a literature review. Section 2 explains the Markov model that is used for defining the health state of employees and section 3 gives an overview of the data and the main variables used for the anal-ysis. Section 4 then introduces the setup of the distribution and the methodology for graduating the transition rates and implementing to Kolomogorov-equations to find a solution, of which the results are given in section 5. Section 6 gives an example of the financial impact of the generation pact and section 7 concludes.
1.1
The Welfare State
The welfare state comprises two functions. The ‘Robin Hood’ function provides poverty relief, redistributes income and wealth and seeks to reduce social exclusion. The ‘piggy bank’ function provides insurance and offers a mechanism for redistribution over the life cycle (Barr 2012). The main focus of this thesis is on the pension system, and thus on the second definition of the welfare state. The chosen setting is the Dutch pensions system, as it is has a reputation of being one of the best pension plans in the world4.
1.1.1 Dutch Pension System
The success of the Dutch pension system originates from the unique combination of adequacy and sustainability, meaning that there is enough security for the whole population while not threaten-ing the sustainability of the economy5. This originates from a three-pillar system. The first pillar is the pension supplied by the government (AOW), the second the pension provided by supple-mentary collective pensions and the third individual pension savings (Anderson 2004). Adequacy is achieved by the combination of the three pillars (Alonso-Garc´ıa, Boado-Penas & Devolder 2018) and sustainability by the second and third.
The first pillar is formed by the pension benefits received from the government. The first pension plan was adopted in the Netherlands in 1913, being only for the working population. Despite the support for universal coverage in politics, there were not any real changes to this system un-til after the Second World War. In 1946, a universal insurance plan was installed to boost the economy after the war. In this plan called the Emergency Act, wage-earners provided pension for the non-wage earners over 65, creating solidarity between the two groups. After 10 more years of negotiating, the official AOW law was accepted on January 1, 1957 (Anderson 2004). Some
details have been adjusted since the implementation of the AOW, but the main principle of this pay-as-you-go pension scheme, or having the young pay for the retirement of the elderly, has not changed.
The second pillar consists of supplementary pension-income and is called occupational pension. It is collected via the employer, by paying premiums as a percentage of wage. This is a savings framework and thus, as opposed to the first pillar, each participant accumulates individual savings in a pension fund. Each company works with a certain pension fund. They can have their own fund, or join an existing industry related fund. For the self-employed and civil servants more gen-eral pension funds exist. A main activity of the pension funds is to invest the savings in order to yield a return equal to price inflation (Anderson 2004). As many people depend on the availability of supplementary pension at retirement, there are government regulations stating that each fund must have a certain solvency ratio to be assumed healthy and is obliged to cut pension benefits when the fund does not meet the minimum goal 5 years in a row. The percentages that funds have to meet depend on the size and sector of the fund6. There are two main types of contracts given out by pension funds, defined benefit (DB) and defined contribution (DC). In a DB pension scheme, the worker receives a certain percentage of (average) wage from the retirement date onward. The pay-out after retirement in a DC scheme depends on the capital that has been accumulated during employment and the corresponding investment earnings (Bodie, Marcus & Merton 1988). How-ever, since the crisis, it has been hard for companies to keep up with the promises associated with the DB plan. The more volatile discount rate increases difficulty in achieving the benefit-goal and companies were forced to make pension cuts. The most common contract now included the DC pension scheme. The weakness of the DC scheme is, however, that the contributions are too low for an adequate income after retirement. A new type of plan introduced in the Netherlands is defined ambition (DA), where the companies promise, similar to a DB plan, to pay a pension within a certain range. Now, similar to the DC plan, employers have the possibility to lower the benefit within the pre-specified range when the solvency ratios are not met (Schouten & Robinson 2012).
The third pillar represents private individual pension products and other private savings. People that are self-employed or do freelance work often build up their pension using this pillar. Since they are not specifically allocated to a certain pension fund in the second pillar, they are allowed to buy life annuities at fiscally attractive terms up to higher amounts than people that are employees. Other pension savings in this pillar include housing wealth or the pension that is accumulated when the working life is extended beyond the retirement age (Knoef, Been, Alessie, Caminada,
Goudswaard & Kalwij 2016).
1.1.2 Functional Disability Benefit
When an individual is not able to work, the Dutch social security system offers a benefit as part of the social security system. The original plan was called the law on functional disability (WAO). The WAO, however, resulted in almost one million people in the Netherlands receiving this benefit, yielding one of the highest functional disability percentages in the world. The government decided for change and in 2006 the law of work and income depending on working ability (WIA) was introduced. Not only were the requirements for receiving the benefit accentuated, the focus was now on trying to make the beneficiary work to his or her maximum capability (Van Sonsbeek, Kantarci & Zhang 2018). After 2 years of being functionally disabled, the government supplies a benefit depending on the measure of functional disability. The beneficiary is expected to work for the remainder of their salary. During these first two years of functional disability, the employer is obliged to maintain at least 70% of the wage pay-out7. The introduction of the WIA has resulted
in a drop of people on functional disability benefit and a higher workforce participation rate for people that are partly functionally disabled (Van Sonsbeek, Kantarci & Zhang 2018).
1.2
Literature Review
In the Netherlands, the retirement age will gradually rise from 65 to 66 in 2018 and to 67 in 2021, after which it will be adjusted based on life expectancy. Life expectancy has greatly improved in the past few decades. For instance, a 65 year old man had on average 19,6 more years to live in 20058, which increased to 20,3 years in 20189. Women had 21,3 more years to live at 65 in 2005 and 23,1 years in 2018. Taking into account this life expectancy forecast, increasing the retire-ment age seems a logical measure to solve the fiscal imbalance in the retireretire-ment system resulting from population aging. Indeed, a research performed on Austrian data confirms that increasing the retirement age results in higher employment of older workers (Staubli & Zweim¨uller 2013). Moreover, the Dutch elderly are aging relatively healthy. Declines in functional disability have been documented in the Netherlands (Lafortune & Balestat 2007), and the healthy life expectancy is increasing more rapid than normal life expectancy (van Duin & Stoeldraijer 2014). Men are expected to live 3,5 years longer in 2030 and women 2 years, while the healthy life expectancy increases with 5 and 4 years, respectively (Vogel, Ludwig & B¨orsch-Supan 2017). Furthermore, research on the effect of increasing the pension age on functional disability concludes that, if the trend continues, the negative effects are limited (Dillingh, Bolhaar, Lever, ter Rele, Swart &
van der Ven 2018) and the increase in functional disability due to the aging workforce seems to be constrained (Berendsen 2018).
Even though the increase in functional disability at older ages is limited, the most recent figures of the Dutch Central Bureau of Statistics (CBS) give higher sick leave rates for older employees. The sick leave rate was 6,4% for employees between 55 and 65 years old, and sickness has an average duration of 11,6 working days, compared to 3,9% sick leave and 6,7 days duration for employees aged 35 to 45 in 201710. The literature confirms these statistics. First, a research in which 20
countries including the Netherlands were analyzed, concludes that one of the main determinants of total sick leave days for a company is the employment of older people (Osterkamp & R¨ohn 2007). Second, the recovery rate decreases with age (Spierdijk, van Lomwel & Peppelman 2009), which confirms the most recent findings by the CBS. Lastly, a British study shows that age has a positive effect on involuntary sickness absence (Thomson, Griffiths & Davison 2000). Concluding, the growing elderly population and increasing pension age causes for a higher number of people exposed to the risk of taking up sick days (Levantesi & Menzietti 2012).
Apart from the general increase in becoming ill when increasing in age, another factor that may influence the increasing sick-leave prevalence of older workers is well-being in the workplace. A research performed by NIDI (Dutch Interdisciplinary Demographical Insititute) in 2016 discovered that the desired retirement age is around 63,2 years. This results in 31% of the people having worked 40-44 years, 49% of the people having worked 45-49 years and 67% of the people hav-ing worked 50 years and over to be (extremely) dissatisfied with the increashav-ing retirement age (Henkens, van Solinge, Damman & Dingemans 2016). Furthermore, most studies agree that hav-ing to work an involuntary amount of hours causes more sick days and a higher probability of functional disability (Joyce, Pabayo, Critchley & Bambra (2010), Costa, ˚Akerstedt, Nachreiner, Baltieri, Carvalhais, Folkard, Dresen, Gadbois, Gartner & Sukalo (2004)). A more flexible work schedule can reduce absences and coming in late (Halpern 2005). Moreover, happier employees are proven to be more productive (Oswald, Proto & Sgroi 2015).
It is noted that being sick does not necessarily imply a day off work. This notion of being sick but going to work is defined by presenteeism. Depending on the type of work and symp-toms, presenteeism lowers quantity and quality of delivered work and can lower productivity by one third or more (Hemp 2004). A research done in the U.S. analyzed the costs of 10 different health conditions, and presenteeism made up 18-60% of these costs (Goetzel, Long, Ozminkowski, Hawkins, Wang & Lynch 2004). Furthermore, it is shown that presenteeism is just as costly as
absenteeism (Chatterji & Tilley 2002).
Becoming ill is not just uncomfortable for the employee in question. The costs associated with sickness for employers are substantial. Dutch companies are obliged to continue paying at least 70% of the salary for the first two years of sick leave, with a possible exception for the first two days off work11. The increasing employee age combined with the higher number of sick days of
elderly people has a potential large cost. The total cost of sick leave was 11,5 billion in 2012 in the Netherlands12, which is 1,7% of the gross national income in that year. In 2017, the total cost
of sick leave increased to 14 billion (Van den Berg, Dirven & Souren 2017), which is 2,1% of the gross national income13.
An often proposed solution to minimize the number of sick days is to encourage part time work at older ages or to gradually enter retirement. It is a topic that has been broadly investigated. However, the effect of part time work on health is confronted with a strong causal effect (Ross & Mirowsky 1995). Evidence suggests that, when one gets sick, he or she might decide to start work-ing part time, and when one is feelwork-ing well, he or she is likely to work more hours. However, most articles argue that part time work has a positive effect on health for elderly workers compared to people that are the same age and retired (De Vaus, Wells, Kendig & Quine (2007), Forbes, Spence, Wuthrich & Rapee (2015), Dave, Rashad & Spasojevic (2006a) and Neuman (2008))14.
Deeg, van der Noordt, Hoogendijk, Comijs & Huisman (2018) make a plea for gradual retirement in order to create a work environment for the increasing group of people with physical disabilities, and Dave, Rashad & Spasojevic (2006b) also find that the negative effects of retirement can be mitigated by continuing to work part time.
Not all reasons for implementing the generation pact are financially oriented. Several compa-nies in the Netherlands have decided to incorporate this system in their conditions of employment. One example is Apollo Vredestein, a tire manufacturer. As the work is hard and shifts are long, they deal with a high rate of functional disability and thus have an extensive pension system. For companies such as Apollo Vredestein, where people start working at a relatively young age, the increased retirement age is not sought-after by neither the employees nor the employer. For them, the generation pact was an all-round solution to steer clear from the negative financial effects of the regulations, while also considering the well-being of the employees15. The trade union of the
11Aon: Get a grip on absence (2015).
12https://www.monitorarbeid.tno.nl/dynamics/modules/SPUB0102/view.php?pub Id=100294&att Id=4911. 13CBS.
Dutch universities have also included the idea of the generation pact in their collective labour agreement. Under the name ‘vitalitypact’, employees of all Dutch universities can now decide to start working less, 5 years before their official retirement date. The universities then maintain the 100% pension buildup and people can work 80% of their current hours for 85% of the current wage or 60% of the hours for 70% of the wage16.
Our results are of interest for several reasons. Firstly, before engaging in the generation pact, companies should perform a risk and cost assessment to see whether the wage cost compensates the increase in employee well-being. Secondly, the thesis gives an insight in the effect of part time and full time work on the health of an employee using a data set that is novel in the actuarial literature.
2
The Markov Framework
The multiple state model is a consistent manifestation in actuarial science, as they are intuitive, easy to work with and statistically sound. The formulas representing the discrete time homoge-neous multiple state model are given in equation 1 (Andersen & Keiding 2002).
For i and j in a multiple state model with set of states [0, 1, . . . , n] and multi-state process [Y (t), t = 0, 1, 2, . . . ], we have
Pij(x, t) = P [Y (t) = j | Y (0), Y (1), · · · , Y (x) = i], for x ≥ 0. (1)
In equation 1, Pij(x, t) is the probability that a life aged x being in state i, is in state j at age x + t.
The Markov framework is a special case of the multiple state model as it satisfies the Markov property that the probability of moving to a state only depends on the current state, and not on the remainder of the history of the process (Jones 1994). The mathematical formulation of the Markov property is given in equation 2.
P [Y (x + t) = i | Y (0), Y (1), . . . , Y (x)] = P [Y (x + t) = i | Y (x)] (2) The first mathematical formulas indicating a Markov framework date back to the eighteenth cen-tury (Seal 1977), but the first approach to modeling disability benefits using a similar model was published by Hamza (1900). Since then, the Markov chain has often been proposed for the ac-tuarial modeling of life contingencies and their extensions (Pitacco 1995). Many researchers have encountered unique situations in dependencies between states, time-management and size. This
resulted in numerous different breaches off the original Markov model. First of all, the hypothesis of stationarity is often not proven to be significant, especially in actuarial calculations where age is an important factor. The Markov framework has thus been used in the time-continuous and time-discrete version in actuarial modeling. See Levantesi & Menzietti (2012) for an example on discrete Markov modeling and Fong, Shao & Sherris (2015) for a continuous example. Then, the Markov assumption is often questioned, as the probability could be path-dependent. The semi-Markov model provides a solution to this dependency, by maintaining the memoryless asset at the moment where the state changes, but not in the whole process (Medhi 1994), or in other words, the future evolution does not only depend on the current state, but also on the time of entry in this state (Meira-Machado, de U˜na- ´Alvarez, Cadarso-Su´arez & Andersen 2009). Hoem (1972) discusses a number of different applications and places the semi-Markov model in a time inhomo-geneous setting. Another difficult aspect of modeling behaviour using the Markov framework is the possibility to return to a previous state. Fong, Shao & Sherris (2015) introduces a framework in which the different intensity rates, including a returning rate, are graduated, of which the tran-sition probabilities are derived, following the trantran-sition intensity approach (Haberman & Pitacco (1999), Olivieri & Pitacco (2001)). Pritchard (2006) provides another example of how recovery is modeled using interval-censored data.
2.1
Modeling Health
Figure 1: Three-state Markov process denoting the transitions between healthy, sick and function-ally disabled.
The transition from one of the states given in Figure 1 to the other can occur at any moment in time, which yields inconsistent transition intervals. Hence, to best fit the model to the data we use a continuous time Markov process (Wan, Lou, Abner & Kryscio 2016). Furthermore, as age is an important factor in calculating the effect of part time work on health for the elderly, the model is inhomogeneous. The resulting transition probabilities Pi,j from state i to j in state
space S ∈ {1, 2, 3} and transition intensities Ti,j are given in equation 3, following the Markov
assumption.
For i, j ∈ {1, 2, 3}, 0 ≤ t ≤ u we have that
Pi,j(t, u) = P {S(u) = j|S(t) = i}, (3a)
T Ii,j(t) = lim
u→t+Pi,j(t, u)/(u − t) i 6= j, (3b)
T Ii,i(t) = −
X
i6=j
T Ii,j(t). (3c)
Here, given the current age x, Pi,j(t, u) yields the probability of transitioning to state j at age
x + u while being in state i at age x + t. T Ii,j(t) denotes the transition intensity from state i to j
at age x + t.
For calculating the transition intensities of the sample data, we make a few assumptions. First, creating a completely continuous model makes calculating the transition probabilities a difficult and tedious task (Jones 1994). Hence, we use piecewise constant forces, where the transition in-tensity is calculated and assumed equal per age level17. Second, we assume that the transition
17This is a common practice in most papers performing actuarial calculations on survey data. All continuous
takes place halfway in the time interval, conform the uniform distribution18. Lastly, we assume
that each participant starts off as being healthy, or S(0) = 1.
3
Data Description
The transitions given in Figure 1 are estimated using data supplied by the Gezondheidsenquˆete (Health Survey), organized by the Dutch Bureau of Statistics (CBS). The survey is offered each year to 15.000 randomly selected Dutch people. The response rate is 60-65%, which yields data of around 9500 people each year. The survey has been distributed since 1981, but is a stand-alone survey since 2010. The survey collects data on health, visited medical specialists, lifestyle and the preventative behavior of people in the Netherlands19. The datasets ranging from 2010 until
2016 are supplied to us by the Data Archiving and Networked Services, which is part of the Royal Dutch Institute of Sciences20. Note that the random selection of participants means that we are not able to follow people and monitor what their status is each year, but that we are looking at a snapshot of a sample of the Dutch population. In other words, the data used is cross-sectional instead of longitudinal data.
3.1
Main Variables
As the dataset is cross-sectional, and the existence of state transitions is necessary for the Markov framework, the 4 years are merged into one dataset. Different variables have been used to create states within the snapshot taken in the surveys each year. From the answers given in the survey, the variables given below were created. In brackets after the variable names are the numbers of the specific survey-questions given in Appendix A that were used for that variable.
1. Status (3,4,5,6,7): whether the respondent is working part time, full time, is functionally disabled or retired. Full time work is defined as working 35 hours or more21. Any percentage of functional disability is considered to be functionally disabled.
2. General Practitioner 1 (8): whether or not the respondent has gone to the general practitioner (GP) in the past 12 months.
3. Sick (9): whether the respondent was sick in the past 2 months, specifically having had a
18This is in line with the uniform distribution of death assumption that is widely used in actuarial papers using
survey data e.g. Reuser, Bonneux & Willekens (2009).
19
https://www.cbs.nl/nl-nl/onze-diensten/methoden/onderzoeksomschrijvingen/korte-onderzoeksbeschrijvingen/gezondheidsenquete-vanaf-2010-2013.
20https://www.knaw.nl/en/homepage.
21The full time number of hours is set 35, as this is the number used by the CBS in generating their statistics on
cold, bronchitis, diarrhea or troubles with their ears, kidneys or stomach, or whether they have been vomiting.
4. General Practitioner 2 (10): whether the respondent has visited the GP following one of the diseases specified in the variable ‘Sick’.
5. Functional disability benefit (8,12,13): whether the respondent has received a social benefit for not being able to work
With these variables, we can then create two periods in the data for sickness. In brackets behind the definitions are the numbers of the variables created above, that are used for that particular definition.
Sick period 1 (1,2,4): the respondent is considered to be sick in the first period when GP 1 is true, but GP 2 is false, to ensure that the visit has occurred in the first 8 months.
Sick period 2 (1,3): sickness in period 2 occurs when the respondent was sick in the past 2 months.
Hence, the precise definition of the healthy state is not having had contact with the doctor. The definition of being sick is having been in touch with the doctor at least once, or having had a cold, bronchitis, diarrhea or troubles with their ears, kidneys or stomach, or whether having been vomiting.
In table 1, the total number of respondents is given per working status. There is some data on the number of working hours for people that are functionally disabled, and only 10 respondents of the 1026 have working hours that qualify as full time. Hence, we assume that the transition from sick to functionally disabled, or transition φ in Figure 1, can only be made by part time working people. To create two periods for functional disability, we need to analyze the interaction between functional disability and disability benefit receipt. Table 1 gives an overview of the welfare distributed among states, the questions used here are number 12 and 13 in Appendix A.
Status Total respondents On benefit % on benefit
Working part time 4925 37 0,8%
Working full time 2679 4 0,1%
Functionally disabled 1026 807 78,7% Table 1: Number of respondents having a functional disability per type of status.
to uncover the difference between respondents that have just transitioned from sick to functional disabled and the people that already were disabled, we use the fact that the functional disability benefit follows from a 2-year period of sickness. It is assumed that all respondents already receiv-ing this benefit entered the year bereceiv-ing disabled and that those who have not, are transitionreceiv-ing from sick to disabled. Summarizing, we can create the following periods for functional disability.
Functionally disabled period 1 (1,5): all respondents that are considered to be disabled and receive a disability benefit.
Functionally disabled period 2 (1,5): all respondents that are considered to be functionally disabled.
Note that this means that all people that were functionally disabled in period 1 are functionally disabled in period 2 as well. This is in line with the absorbent character of the functional disability state. Furthermore, we can add the following definition being sick in period 1.
Sick period 1 (1,5): all respondents that are considered to be functionally disabled and do not receive a disability benefit.
3.2
Data modification
The surveys are reviewed each year which may results in a slight change. Especially between 2013 and 2014, the CBS made drastic changes in the question that were asked and the answers that could be given22. As the variables that are necessary for this research result from the surveys
ranging from 2010 until 2013, only these datasets are used, which yields 58.955 observations. In order to retrieve the relevant subset and align the four datasets, a few restrictions are imposed. First, as only the working population is relevant, we impose an age restriction of at least 20, which eliminates 15.334 observations, and restrict the dataset to people that are less than 65 years old, which eliminates 8449 observations. Furthermore, all respondents that are voluntarily or invol-untarily unemployed are excluded from the analysis. The remainder of the dataset then contains 29.239 observations. Third, as we are interested in the sick leave of employees, all 3036 respondents that are self-employed are removed from the dataset as well. Some respondents failed to answer all the questions that were necessary to perform the analysis. Of the people that have indicated to be currently working, 11.476 did not answer a question related to working hours, 2 did not answer the question concerning general health, 2 more did not answer a question on sickness and 1967 failed to answer a question on a visit to the general practitioner. Of the people that have indicated to be functionally disabled, 387 did not answer all the relevant questions concerning functional
disability benefit receipt. The number of observations then is 12.369. Finally, the people that have indicated to be retired are removed from the dataset, which leaves 11.272 observations.
As mentioned before, calculating the effect of full or part time work on health is prone to en-dogeneity bias, caused by the causal relation between work and health. There are several ways to cope with this bias, such as using instrumental variables like measures that record the change in health23, which are independent of the labor-supply decision (Neuman 2008). Another possibility
is to simply ignore the bias (De Vaus, Wells, Kendig & Quine (2007), Forbes, Spence, Wuthrich & Rapee (2015)). As we are lacking the possibility of finding appropriate instrumental variables in the data set, we condition the observations related to working people to people that indicate not to be having health problems24. The number of observations is then at 11.168. Neither are the
working people included that are suffering from a long lasting illness25, as the main focus of this
research is on people that fall sick acutely26, similar to Dave, Rashad & Spasojevic (2006a). This way, it is ensured that the decision of working part time does not depend on health. The final number of observations in the sample is 8630, of which 1026 are functionally disabled, and 7604 are working.
3.3
Descriptive Statistics
Table 2 gives an overview of the respondents in the analysis. The distribution over the different age categories is approximately even, apart from the oldest category, in which more people have already stopped working. A consistent observation over the age cohorts is that a relatively large portion of the respondents are part time working women, and there is a relatively low number of female respondents that work full time. Most male respondents work full time, and the majority of men after the age of 50 works part time. The number of observations in the age interval between 60 and 65 is much lower than in other intervals in the data set. The ages 64 and 65 have the lowest number of observations, namely 88 and 11. In order to form a solid base for the out of sample forecast, these ages are excluded from the process of graduation. Hence, the number of observations on which the graduations are performed is 7505. The female numbers given in table 2 are in line with the statistics of the female working population in the Netherlands. In 2018, of the women aged 35 to 55 that are working, 74% works part time and the data given here yields a part time working percentage of 78%-81% for the age interval 35-55. For women aged below 35,
23Examples of variables that are used to measure health in Neuman (2008) are differences in self-ratings, mobility,
large muscle functions and chronic disability.
24People are denoted to be having health problems when the answer on question 11 in Appendix A is ”Bad” or
”Very Bad”.
25People are regarded to have a long lasting illness if the disease lasts at least 6 months.
the percentage given by the CBS is 65%, and 71% in this data set. For women aged above 55, the CBS yields a part time working percentage of 77% and our data gives 85%. The data set given here includes relatively more part time working men than what is registered by the CBS. The part time working percentages are around twice as high here as in the general statistics27.
Age Men Women Total
Part time Full time Part time Full time
20-30 255 516 489 368 1628 31-35 120 313 326 131 890 36-40 129 316 354 97 896 41-45 140 353 483 119 1095 46-50 161 331 423 101 1016 51-55 178 307 398 96 979 56-60 165 235 287 52 739 61-65 139 71 133 18 361 Total 1287 2442 2893 982 7604 Table 2: Number of respondents per type of work and age category.
Let ixand exdenote the number of transitions and the exposure years of the sample data,
respec-tively. Table 3 represents the number of raw transition counts and table 4 represents the number of exposure months. Both tables are summarized by type of work (part time or full time) and gender. For clarifying purposes, age cohorts are given in the tables but data per age is used in the analysis. Men made 3839 transitions in total, and women made 4643. Around half of the transitions of women occur from healthy to sick by part time workers. In the case of men, most transitions are made by full time workers that transit from healthy to sick. For men, the exposures are twice as high for full time as they are for part time workers. For women, the exposure for part time workers is 3 times as high as it is for full time workers in the sick state. Hence, men spent relatively less time in the sick state than women.
i x , n um b er of transitions Men σ : H → S φ : S → H µ : S → F D T otal W omen σ : H → S φ : S → H µ : S → F D T otal Age PT FT PT FT PT PT FT PT FT P T 20-30 198 397 58 100 7 760 433 328 118 93 27 999 31-35 96 232 28 57 8 421 290 107 88 38 27 550 36-40 88 239 20 79 9 435 295 82 112 30 32 551 41-45 106 266 39 105 10 526 384 91 179 39 31 724 46-50 111 246 53 112 11 533 320 72 145 28 31 596 51-55 121 206 49 106 15 497 306 69 169 35 30 609 56-60 123 174 65 77 17 456 221 42 116 24 34 437 61-63 86 46 39 25 15 211 89 16 38 8 26 177 T otal 929 1806 351 661 92 3839 T otal 2338 807 965 295 238 4643
ex, exposure months Men H S Women H S Age PT FT PT FT PT FT PT FT 20-30 2098 4599 956 1569 3717 2807 2301 1,609 31-35 1000 2777 500 979 2408 1047 1546 525 36-40 1150 2696 440 1096 2847 734 1491 418 41-45 1181 2911 559 1325 3839 980 2005 448 46-50 1400 2818 574 1154 3563 844 1633 368 51-55 1554 2660 618 1012 3133 782 1685 370 56-60 1310 1961 688 847 2259 426 1245 198 61-63 823 483 455 213 891 124 519 80 Total 10.516 20.905 4790 8195 22.657 7744 12.425 4016 Total in state 31.421 12.985 30.401 16.441 Table 4: Number of exposure months summarized by gender and full time (FT) and part time (PT) work.
The transition intensities observed in the data are calculated by dividing the raw transition counts by the number of exposure months. The crude transition rates are given in Appendix B. To emphasize the linear trend, the log transformation is taken of the transition intensities for visual-ization purposes. Figure 2 gives an overview of the transition rates for the whole population. Note that dividing the raw transition counts by the exposure months yields a monthly rate, and that is constant over the course of a year. In other words, the monthly rates are time homogeneous for a year.
(a) σ: Healthy → Sick (b) φ: Sick → Healthy (c) µ: Sick → Functionally Disabled
trend for older respondents, while the transition from sick to healthy shows a small increasing trend towards higher ages. The transition from sick to functionally disabled has a lot of variabil-ity, but does show an increasing trend towards higher ages.
(a) σ: H → S, men (b) σ: H → S, women
(c) φ: S → H, men (d) φ: S → H, women
(e) µ: S → FD, men (f) µ: S → FD, women
Figure 3: Natural log of the transition rates for men, for part time and full time work separately
4
Methodology
is explained and lastly the use of the Chapman-Kolomogorov equations is elaborated on, for the purpose of transforming the monthly age dependent transition probabilities to probabilities that can be computed over future time intervals.
4.1
Graduation intensity rates
Graduation is the method of smoothing a set of crude data points, in order to provide a more suitable basis for further inferences and calculations (Haberman & Renshaw 1996). The transition intensities σ, φ and µ are graduated using generalized linear models (GLM), similar to Renshaw & Haberman (1995) and Fong, Shao & Sherris (2015). This is a flexible method of graduation as many formulas in the exponential family can be used for parametric estimation (Forfar, McCutcheon & Wilkie (1988), Renshaw (1991)). The methodology for graduating the intensity rates is illustrated using σ, but is applicable for all three types of intensities. The transition intensities are time inhomogeneous but are assumed to have a constant rate per age. Hence, the rates depend on age level x. First, we define the linear function of regressors η that represents the class of functions that are available in the GLM framework,
ηx= g(σx),
where the link function g(·) is invertible such that we have, σx= g−1(ηx).
In order to incorporate flexibility in the graduation of the intensity rates, the function η is param-eterized using unknown coefficient βj, for j ∈ [0, k], k > 0. k is thus the number of parameters
that is used in the regression. We then have the linear predictor, ηx=
X
j
fj(x)βj, (4)
which in this case is,
ηx= k
X
j=0
βjxj= β0+ β1x + β2x2+ · · · + βkxk. (5)
Note that the function η is linear in the predictors β. Recall that ixand exdenote the number of
crude transitions and exposure years at age x, respectively. As exis the central rate of exposure,
the number of transitions ix can be modeled as the response variable of a Poisson distribution
(Forfar, McCutcheon & Wilkie 1988). We thus have,
ix∼ P oi(exσx),
E(ix|ex, σx) = exσx= mx,
where mx denotes the mean of the response variables per age level x. The dispersion parameter
φ measures the variability around the mean, and is out of convention in the Poisson distribution set to unity. The log likelihood function is then given by,
log[l(i; m)] =X
x
{−mx+ ixlog(mx)} + Constant, (6)
where i is the vector of response variables, and m is the vector consisting of the mean of the response variables. The β-parameters are estimated by maximizing the log-likelihood using the relation
mx= exg−1(ηx) = exg−1
X
j
fj(x)βj.
Similar to Fong, Shao & Sherris (2015), the different versions of the model that are fitted to the data differ in the k-term of the linear predictor, given in equations 4 and 5. Here, the k-term is varied between 1 and 6. The fitted values are given by
ˆ
mx= exg−1
X
j
fj(x) ˆβj,
which yields that the optimum value of the log-likelihood is log(lc) = log(l(i; ˆm)) =
X
u
{− ˆmx+ ixlog( ˆmx)} + Constant,
where the subscript c denotes the predictor structure of the fitted model.
4.2
Model selection criteria
In order to choose the most appropriate model, a few model selection criteria are drawn or com-puted from the fitted model. The first statistic is the Bayesian Information Criterion (BIC) (Posada & Buckley 2004), which is given by
BIC = ln(n)k + 2ln(lc),
where n is the sample size, k the number of parameters that need to be estimated and lc is the
maximum value of the likelihood function that is given in equation 6.
AICc ≈ AIC + 2k
2+ 2k
n − k − 1, where AIC = 2k − 2ln(lc), and k, n and lc are defined as above.
The final statistic is the model deviance, which is defined as two times the difference between the saturated model and the optimized model in question. Note that the full, or saturated model, is the case when the fitted values, ˆmx, are equal to the actual responses, ix. This is thus the case
when
log(lf) = log(l(i, i)) =
X
x
{−ix+ ixlog(ix)} + Constant,
where lf represents the predicting structure of the full model. We then have that the deviance is
defined by D(c, f ) =X x dx= −2log(lc/lf), = −2log(lc) + 2log(lf), = 2X x {−(ix− ˆmx) + ixlog(ix/ ˆmx)}.
To indicate whether an increase in the level of k has made an significant improvement, a likelihood ratio test based on the difference in deviances is performed as well. The difference in deviances is asymptotically distributed as a chi-square distribution, in which the difference in the number of parameters is equal to the degrees of freedom.
4.3
Out-of-sample forecast
There is little data available on morbidity and recovery rates of working people after 63 years old. As the idea of the generation pact is that people can keep on working until the retirement age, the smoothed rates are forecasted up to age 7128. As this is a short term forecast, the
rates are extrapolated and we assume that the current trend continues. Hence, the later ages (x = 64, 65, 66, 67, 68, 69, 70, 71) are filled in the fitted linear, cubic and quadratic equations using the parameters that are found using generalized linear models. The errors that are observed in-sample are not necessarily a good basis for the prediction of the errors of the forecast (Ahlburg & Land 1992). Using historical data to evaluate the quality of the forecast is recommended by Booth & Tickle (2008). Fortunately, there is some data available on the transitions of people at
28The rates are forecasted up to 71, as this is the retirement age the Dutch government is currently aiming at,
these later ages, which are used to evaluate the out of sample forecast.
It is expected that the forecast will be best for the linear models, as there is the smallest change in rates when increasing x. Forecasting out of sample is riskier for the quadratic an cubic fit, as the parameters maintain their value but the value of x2 (or x3) in the linear predictor changes more severely when x changes. Hence, it is expected that the latter two fitted models cause for more uncertainty in the forecasted values.
4.4
Chapman-Kolmogorov relationship
In order to efficiently implement the Chapman-Kolmogorov relationship, we need transition prob-abilities that are on a yearly basis. The generalized linear model as defined above and using the data given in table 3 and 4, yields monthly defined transition probabilities. The rates resulting from the GLM and prediction above can be written in matrix notation as follows,
Ax= PH,H PH,S 0 PS,H PS,S PS,F D 0 0 PF D,F D , (7)
where Pi,j denotes the probability of going from state i to j, i, j ∈ {S, H, F D}. These rates only
change for a person when he or she turns a year older. Hence, they are time homogeneous for 12 months. Furthermore, we have that PH,H(t) = 1−PH,S(t) and PS,S(t) = 1−PS,H(t)−PS,F D(t), as
all probabilities have to sum op to one. Note that for the same reason we have that PF D,F D= 1,
which implies that this is the absorbing state.
In order to calculate the effect of the generation pact on sickness probabilities, the monthly tran-sition probabilities are transformed to yearly probabilities. In order to do so, the two equations in equation 8 are used. In equation 8, p denotes a probability given in equation 7, r is the corre-sponding intensity rate and t is the time in months. The first transforms the transition probability to the transition rate, and the second uses this rate to calculate the probability for t = 12. This method is in line with the calculations used in Sonnenberg & Beck (1993), Miller & Homan (1994) and Fleurence & Hollenbeak (2007).
r = −1
tln(1 − p) (8a)
p = 1 − e−rt (8b)
B(x) = PH,H(x) PH,S(x) 0 PS,H(x) PS,S(x) PS,F D(x) 0 0 PF D,F D(x) . (9)
In order to compute the future sickness rates based on the current state, the Chapman-Kolmogorov equations are used. The Chapman-Kolmogorov relationship is an extremely useful definition for Markov models which makes efficient use of the Markov property given in equation 2. The Chapman-Kolmogorov relationship for time homogeneous models is defined as follows (Papoulis & Pillai 2002),
For i, j, r ∈ {H, S, F D}, 0 ≤ x ≤ u we have that Pi,j(x, u) =
X
r
Pi,r(x)Pr,j(x + u).
In matrix notation, for rates dependent on age x, this means that we have, B(x, u) = B(x, 0)B(x + 1, 0) · · · B(x + u, 0),
where B(x, u) denotes the transition probability matrix for person currently aged x to age u, B(x, 0) is the probability matrix of a person aged x and B(x + u, 0) the probability transition matrix of a person aged x + u.
5
Results
The results given in this thesis can be categorized in three different parts. First, the optimal number of parameters for each type of intensity rate, gender and work status is selected using the model selection statistics that are described in section 4.2. The chosen models are explained numerically and shown graphically. Also, a graphical representation of the deviance residuals and the confidence interval of the selected models and parameter uncertainty is given. Next, an extrapolation of the model to age 71 is given, including a 95% confidence interval based on parameter uncertainty. Third, the monthly transition probabilities of becoming ill are transformed to yearly probabilities.
5.1
Model selection
The model selection statistics used in this thesis are summarized per model in table 5 and 6. The AICc, BIC, deviance and significance of improvement are given for the models for k = (1, 2, 3)29,
which is related to the linear predictor as defined in equation 5. The model with the best fit is
29The models are estimated for a k ranging from 1 to 6. However, the models with the best fit fall within the
given in bold, and the asterisks indicate the significance level of the increment in the number of parameters k.
Men
Part time Full time
k AICc BIC Dc ∆Dc AICc BIC Dc ∆Dc
σ : H → S 1 249.8 -127.3 31.6 265.0 -139.3 19.7 2 247.5 -127.8 27.4 -4.2** 266.9 -135.6 19.6 -0.1 3 247.0 -126.5 24.8 -2.5 268.4 -132.3 19.1 -0.5 φ : S → H 1 199.8 -130.1 28.9 224.7 -133.3 25.6 2 201.7 -126.4 28.7 -0.1 226.7 -129.6 25.6 -0.1 3 201.7 -124.7 26.7 -2.0 227.7 -126.7 24.6 -0.9 µ : S → AO 1 135.1 -117.0 41.9 2 134.2 -116.2 39.0 -2.9* 3 130.4 -118.2 33.2 -5.8**
Table 5: AICc, BIC, model deviance and significance of increasing the number of parameter for the possible models estimated for women. ∆Dc gives the difference in deviance compared to the
Women
Part time Full time
k AICc BIC Dc ∆Dc AICc BIC Dc ∆Dc
σ : H → S 1 276.1 -140.6 18.3 235.5 -129.6 29.3 2 276.8 -138.1 17.0 -1.3 234.2 -129.2 26.0 -3.3* 3 275.6 -137.5 13.9 -3.2* 236.0 -125.6 25.8 -0.2 φ : S → H 1 261.4 -112.5 46.4 182.5 -137.8 21.1 2 251.3 -120.8 34.3 -12.1*** 184.3 -134.3 20.8 -0.3 3 250.5 -119.9 31.5 -2.8 185.8 -131.0 20.3 -0.5 µ : S → AO 1 182.4 -107.5 51.4 2 173.2 -114.9 40.3 -11.2*** 3 173.3 -112.9 38.4 -1.9
Table 6: AICc, BIC, model deviance and significance of increasing the number of parameter for the possible models estimated for women. ∆Dc gives the difference in deviance compared to the
previous model. The significance is indicated by ∗p < 0.1, ∗ ∗ p < 0.05 and ∗ ∗ ∗p < 0.01. The lines in boldface correspond to the selected model.
For the majority of men, the linear model involving only 1 parameter is considered to be the best fit. For the transition from healthy to sick (σ) for part time workers, the best fit is the quadratic model and for the transition from sick to functionally disabled (µ) the resulting model is cubic. In both cases, the increase in parameters yields a significantly better fit.
For the female part of the population, the overview of the model selection criteria is given in table 6. Here, only the transition from healthy to sick for part time working women, and the transition from sick to healthy for full time working women shows a linear fit. All others show the most resemblance to a quadratic fit.
For women, the transition from healthy to sick is higher for people that work full time at the beginning of their career. However, after age 25, women that work part time get sick more often than women that work full time. Around age 45, working full time yields a higher incidence rate, as can be seen in Figure 4b. The recovery rate for women is given in Figure 4d, and is lowest for the youngest age category that is working full time, and highest for the older age category that is working full time. For both genders, the morbidity rates in Figure 4e and 4f exhibit a small upward trend as age increases, for which male has a cubic fit.
(a) σ : H → S, men (b) σ : H → S, women
(c) φ : S → H, men (d) φ : S → H, women
(e) µ : S → F D, men (f) µ : S → F D, women
Figure 4: Smoothed transition rates per gender.
Part time Full time Men σ : H → S φ : S → H µ : S → F D σ : H → S φ : S → H β0 -1.6715*** -3.2726*** -22.6778*** -2.4388*** -3.2740*** β1 -0.0395** 0.0145*** 1.3023** -0.0003 0.0176*** β2 0.0005** - -0.0293** - -β3∗ 10−2 - - 0.0213** - -Women σ : H → S φ : S → H µ : S → F D σ : H → S φ : S → H β0 -1.9944*** -4.8707*** -2.1046** -1.1516*** -3.3102*** β1 -0.0066*** 0.0956*** -0.1489*** -0.0524** 0.0182*** β2 - -0.0009*** 0.0020*** 0.0006* -β3∗ 10−2 - - - -
-Table 7: Coefficients for the fitted models corresponding to the response variable ix. The
signifi-cance of the paramters is indicated using ∗p < 0.1, ∗ ∗ p < 0.05 and ∗ ∗ ∗p < 0.01.
5.2
Model uncertainty
(a) σ: H → S, men (b) σ: H → S, women
(c) φ: S → H, men (d) φ: S → H, women
(e) φ: S → FD, men (f) φ: S → FD, women
Figure 5: Deviance residuals of the estimated transition rates.
The confidence interval of the graduated rates is calculated using the delta method30, and the
parameter uncertainty is computed by performing a bootstrap. The result is given in Figure 6 and
30The 95% standard error of the nested model is calculated using SE = ˆλ ± 1.96 ∗
q
ˆ λ
n, where ˆλ denotes the
(a) σ: H → S, Part time (b) σ: H → S, Full time
(c) φ: S → H, Part time (d) φ: S → H, Full time
(e) φ: S→ FD, Part time
(a) σ: H → S, Part time (b) σ: H → S, Full time
(c) φ: S → H, Part time (d) φ: S → H, Full time
(e) φ: S → FD, Part time
5.3
Forecast
In order to forecast what the effect of part time or full time work is on sick days for people past 63, the fitted models are extrapolated to ages up to 71. The extrapolated rates are graphically exhibited in Figure 8. The solid lines represent the smoothed crude rates, and the striped lines the 95% confidence interval concerning parameter uncertainty31. The blue lines correspond to
part time workers and the green lines to people that work full time. In all transitions that do not include functional disability, the perceptions are the same as in to non-extrapolated case, but they are magnified. The rate of falling ill is higher for part time working men than it is for full time working men, and the opposite is true for women. Full time working men and women recover more often than part time working men. Furthermore, the higher variability in the rates concerning functional disability become clear here now, as the extrapolation shows a large confidence interval, especially in the male case. Given that the perceptions are the same, but magnified, the conclusion drawn from the fitted rates is applicable here as well. For elderly men, these forecasts imply that working full time is beneficial for health. Several theories may explain this. There may still be an endogeneity bias causing this relationship, or men feel better when kept busy and feeling useful. For women, working full time results in a higher morbidity rate, but a higher recovery rate as well. This may imply that women have more less severe illnesses for which they seek help.
31The 95% confidence interval corresponding to process uncertainty can not be computed using SE = ˆλ ± 1.96 ∗
q
ˆ λ
(a) σ: H → S, men (b) σ: H → S, women
(c) φ: S → H, men (d) φ: S → H, women
(e) µ: S → FD, men (f) µ: S → FD, women
Figure 8: Forecast to age 71 for the transition rates. The solid and striped line represent the smoothed model with 95% confidence interval of parameter uncertainty. The dotted line represents the forecast and the stripe-dottedline the corresponding 95% confidence interval of parameter uncertainty of the forecast.
(a) σ: H → S, Part time (b) σ: H → S, Full time
(c) φ: S → H, Part time (d) φ: S → H, Full time
(e) φ: S → FD, Part time
(a) σ: H → S, Part time (b) σ: H → S, Full time
(c) φ: S → H, Part time (d) φ: S → H, Full time
(e) φ: S → FD, Part time
Figure 10: Forecasts up to age 71, including 95% parameter uncertainty for men. Crude rates and corresponding smoothed lines are given in blue.
5.4
General solution
(a) σ: H → S, Men (b) σ: H → S, Women
(c) φ: S → H, Men (d) φ: S → H, Women
(e) φ: S → FD, Men (f) φ: Sick → FD, Women
Figure 11: Fitted and predicted transition probabilities over a 1-year period.
6
Financial impact generation pact
conclusions are given regarding the financial impact.
6.1
Transition probabilities
(a) σ: H → S, Men (b) σ: H → S, Women
(c) φ: S → H, Men (d) φ: S → H, Women
(e) φ: S → FD, Men (f) φ: S → FD, Women
Figure 12: Future transition probabilities over 20 years for a person currently aged 50, when implementing the generation pact (blue line) and when working full time (green line).
healthy to sick as for sick to healthy, for men and women. Hence, part time work decreases the future probability of falling ill, but it also decreases the probability of recovery. The probability for functional disability increases in both cases, which is in line with the expectations. As the transition from sick to functionally disabled can only be made from a part time perspective, no comparison can be made with respect to the generation pact. Furthermore, the forecasts are too high to match the literature or seem realistic, which is why this transition is omitted in calculating the effect of the generation pact.
6.2
Financial impact
To estimate the financial impact of the generation pact of this one person, a few assumptions are made. It is not possible to estimate the costs for a person that is sick in the initial state. The probability of moving from state sick to sick is defined by 1 − PS,H− PS,F D, while the transition
from sick to functionally disabled is not used. We thus assume that the initial state of this person is healthy. Furthermore, the generation pact is designed such that the work percentage is 80%, for 90% wage. The pension buildup is not incorporated in this comparison, as it is the same in both cases, namely 100%. A few variables are drawn from the CBS. First, for the generation of wages, the average full time wage32 is registered for 6 age intervals given in table 8. For people that
work part time, this full time wage is multiplied with the part time percentage to yield the part time wage and the number of working days per year is set at 260. Lastly, the duration of sickness depends on gender and age, of which the numbers are given in table 933. Then, the financial
impact is calculated using
F Ix(t) =
5
7PH,S(x, t)dx+tsx+tP Tx+t,
where dx+t is the duration dependent on age as given in table 9, sx+t is the average wage as given
in table 8. PH,S(x, t) gives the probability of being in state ‘Sick’ at time x + t, given that this
person is in state ‘Healthy’ at x. The variable P Tx+t is set at 1, and changes to 0.8 when turning
6034in the case of the generation pact. Note that the probability of being sick is multiplied with5 7
in both cases as the sick days are uniformly distributed over the week. In the case of the generation pact, this probability is multiplied with 0.8 as well, as this is the percentage of working days. The resulting costs are given in table 10. For both men and women, the generation pact yields costs that are half of the full time scenario. Especially at later ages, the costs are significantly lower for the generation pact, which results from the lower probability of becoming ill at later ages, but from the lower duration of sickness as well.
32CBS, Werkzame beroepsbevolking; gemiddeld inkomen.
33CBS, Ziekteverzuim volgens werknemers; geslacht en leeftijd (2017). 34Hence, P T
Concluding, the generation pact has as positive effect on the costs associated with sickness in this example. Adding to this, an elderly person working less also implies that a new younger em-ployee can be hired. The financial impact is influenced by the difference in salary of the younger employee as well. In table 8 can be seen that the average salaries of older employees are twice as high as that of younger people, which is probably because people are the height of their careers when approaching retirement. Hence, the financial aspect of the generation pact is also influenced by the percentages of part time work and the wage that is maintained, and the wages of younger people at the company. When implementing the generation pact, it is important to keep this fact in mind. Furthermore, the rates generated here are based on the entire Dutch population and it is advised to draw a parallel with company-specific data. Using this data, an estimation on the probability of staying in the ‘Sick’ state can be used to generate the same calculations as above for a person with initial state ‘Sick’.
Age interval Full time wage Daily wage Daily PT wage
20-25 23.500 90 81 25-35 40.800 157 141 35-45 55.400 213 128 45-55 62.100 239 215 55-65 60.900 234 211 65-75 64.600 248 223
Table 8: Average wage over 6 different age intervals. For the daily wage, 260 working days per year are assumed. For the daily part time wage, a percentage of 90% is assumed.
Age interval Men Women 20-25 1,8 2,5 25-35 4,4 7 35-45 5,5 8 45-55 7,3 9,2 55-65 12 11,1 65-71 4,4 3,7
t Man FT Man GP Woman FT Woman GP 0 823 474 1072 617 1 541 312 683 393 2 634 365 820 472 3 597 343 765 441 4 606 349 782 451 5 599 345 772 445 6 962 554 913 526 7 956 551 910 524 8 952 548 908 523 9 947 545 907 522 10 942 543 906 522 11 938 506 905 416 12 934 454 904 381 13 929 409 904 317 14 925 359 904 273 15 921 312 904 224 16 356 102 320 65 17 355 83 320 52 18 353 65 320 41 19 352 49 321 31 Total 14.621 7268 15.238 7235
Table 10: Estimated costs (ine ) over 20 years for taking a day off sick, for person that is now aged 50 and healthy. In case of the generation pact the part time percentage is set at 80% and the wage percentage is 90%.
7
Conclusion
results imply that working part time is beneficial for the health of women, which is in line with the literature. For men, the rates imply that working full time is beneficial for their health, which does not correspond with the majority of the literature.
When estimating the future rates of a person currently aged 50, the generation pact does de-crease the morbidity rates, but also dede-creases the recovery rate. This is in line with the literature stating that the duration of disease for the elderly is longer. Furthermore, estimating the costs associated with sick leave for 20 years for a healthy person now aged 50, yields that the generation pact cuts these costs by half. This does not only result from lower morbidity probabilities, but also from lower costs associated with part time work when not going to work. Furthermore, the balance between wage percentage and the hiring of new younger employees is an important factor as well.
7.1
Discussion
There are some limitations in the study with respect to the assumptions used in the Markov model, the content of the data and the implementation of the generation pact.
First, regarding the Markov model, using a cross-sectional database instead of a longitudinal database can have some shortcomings. First and foremost, the data is aggregated and no distinc-tion has been made in time period. Hence, any development in the health of people which may influence the rates is not incorporated in this research. However, given the short time period of 4 years that is studied, this should not be a major issue. Also, over the two created periods, there are several definitions for being sick. One could be sick for going to the general practitioner or one could be sick by going to the general practitioner following a number of illnesses. Furthermore, no distinction has been made in the reasons for a visit or between physical and cognitive health. For the computation of the number of exposure months and transitions it is assumed that everyone starts in the healthy state. However, in 2017, 44% of the whole working population had called in sick35. Also, it could be possible for people to work full time and suddenly become
function-ally disabled. This scenario is not taken into consideration in this model. These are thus both simplifying assumptions that are not necessarily true. Retirement is a slow process and having longitudinal data would add extra value in capturing this transition. Other papers in the literature have used the Health and Retirement Study and exploited their longitudinal nature to provide estimates of long-term care usage. However, this dataset does not provide the same level of work and health related information as the database used. Also, the policy recommendations are solely relevant for the U.S. population whereas in our study we are interested in the Dutch context and
the generation pact.
Some limitations regarding the content of the data are present as well. First, a lot of people are eliminated for having a long term illness or not feeling well to avoid the endogeneity bias. This could affect the outcome of the results, as the morbidity rates are higher when including people that generally do not feel well, and thus results in an underestimation of risk. Furthermore, it is not possible to empirically check the absorbent feature of the functional disability state given the limited amount of states available. In general, the transition to functional disability shows some limitations. There are very little observations and the extrapolation to later ages regarding this transition is thus not trustworthy. Furthermore, the data used to create the crude transition rates is very indirect, and can thus easily be influenced. Also, the percentage of men that works part time is much higher in this study than the statistics given by the CBS imply. However, as these per-centages do not influence the smoothed transition rates, the effect on the results should be limited.
Other important side notes regard the implementation of the generation pact. In the example given here, data provided by the CBS is used, which may be subject to endogeneity bias as well. In general, forcing people to work part time may not beneficial for their health, and thus it is important to maintain the freedom of choice. Furthermore, good communication concerning the different options on transitioning from work to retirement is important.
7.2
Further research
Further research is necessary for optimal use of the generation pact, so that the people that gain the most benefit are offered to work part time. An example is to do more research on the differ-ences in the effect of part time work between white, blue and pink collar workers, or to distinct between people that work part time, full time and extra full time, which means that people work overtime. Another option is to dive deeper in the socioeconomic factors that have an effect on the health of full time working elderly people. Also, in order to get a complete picture of the effect of the generation pact, it is useful to incorporate the increase in utility of the employee resulting from having more free hours.
8
Appendix
A
Questions overview
Question Possible answers
1 What is your date of birth? Date of birth
2 What is your gender? Man/Woman
3 How many hours do you work on average per week? (not including extra or unpaid hours)
[0..95]
4 How many hours of low intensity work do you have per week? (work seated or standing with regular walks, like work behind a desk or standing work with light burden)
number of hours or not applicable
5 How many hours of high intensity work do you have per week? (work seated or standing involving regular heavy lifting)
number of hours or not applicable
6 Which description fits you best? 1. working with payment, 2. homemaker, 3. student, 4. volunteer, 5. something else 7 Position in the workforce Independent, without work, functionally
dis-abled, working, retired. 8 Have you been in contact with your general practitioner
in the past 12 months?
Yes/No
9 Below are listed a number of acute disease and symp-toms. Please supple per disease or symptom whether you have had it in the past 2 months?
For a cold/flu/throat infection/frontal si-nus infection, acute bronchitis/lung infec-tion, ear infection, infection of the kid-neys/bladder/urinary canal, diarrhea, vomit-ing, gastric ulcer: Yes/No
10 Please supply per disease whether you have gone to the general practitioner concerning this illness.
idem
11 How is your health in general? 1.Very good 2.Good 3.Normal 4.Bad 5.Very bad 12 Do you have any extra welfare, not including study
benefits?
Yes/No
13 Is that wholly or partially a functional disability ben-efit?
Yes/No
14 Do you have one or more long term illnesses? Yes/No
B
Crude transition intensities
(a) σ: Healthy → Sick (b) φ: Sick → Healthy (c) µ: Sick → Functionally Disabled
Figure 13: Crude transition rates of the full working population.
(a) σ: Healthy → Sick (b) φ: Sick → Healthy (c) µ: Sick → Functionally Disabled
Figure 14: Crude transition rats of the full working population, summarized by full time and part time work.
(a) σ: Healthy → Sick (b) φ: Sick → Healthy (c) µ: Sick → Functionally Disabled
(a) σ: Healthy → Sick (b) φ: Sick → Healthy (c) µ: Sick → Functionally Disabled
